research article some generalized dependent aggregation ...the power generalized interval-valued...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 705159 24 pageshttpdxdoiorg1011552013705159

Research ArticleSome Generalized Dependent Aggregation Operators withInterval-Valued Intuitionistic Fuzzy Information and TheirApplication to Exploitation Investment Evaluation

Xiao-wen Qi12 Chang-yong Liang12 and Junling Zhang3

1 School of Management Hefei University of Technology Hefei 230009 China2 Key Laboratory of Process Optimization and Intelligent Decision-Making Ministry of Education Hefei 230009 China3 School of Economics and Management Zhejiang Normal University Jinhua 321004 China

Correspondence should be addressed to Junling Zhang springoasis zhang126com

Received 28 January 2013 Accepted 21 April 2013

Academic Editor Francisco Chiclana

Copyright copy 2013 Xiao-wen Qi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate multiple attribute group decision making (MAGDM) problems with arguments taking the form of interval-valuedintuitionistic fuzzy numbers In order to relieve influence of unfair arguments a Gaussian distribution-based argument-dependentweighting method and a hybrid support-function-based argument-dependent weighting method are devised by respectivelymeasuring support degrees of arguments indirectly and directly based on which the Gaussian generalized interval-valuedintuitionistic fuzzy ordered weighted averaging operator (Gaussian-GIIFOWA) and geometric operator (Gaussian-GIIFOWG)the power generalized interval-valued intuitionistic fuzzy ordered weighted averaging (P-GIIFOWA) operator and geometric (P-GIIFOWA) operator are proposed to generalize a wide range of aggregation operators for decision makers to flexibly choose indecisionmodelling And some desirable properties of the proposed operators are also analyzed Further application of an approachintegrating proposed operators to exploitation investment evaluation of tourist spots has shown the effectiveness and practicalityof developed methods experimental results also verify the properties of proposed operators

1 Introduction

Multiple attribute group decision making (MAGDM) is animportant part of decision theories and the purpose ofMAGDM is to find a desirable solution from finite alterna-tives by a group of experts assessing on multiple attributeswith different types of decision information such as crispnumbers [1ndash5] interval values [6ndash8] linguistic scales [9ndash11]and fuzzy numbers [12ndash17] In order to better handle thefuzziness and uncertainty in decision process intuitionisticfuzzy set (IFS) [18] and interval-valued intuitionistic fuzzy set(IVIFS) [19] have been introduced and increasing approaches[20ndash31] for MAGDM with intuitionistic fuzzy informationcan be found in related research literatures Among theprocedures of those MAGDM approaches a very commoninformation aggregation technique is the OWA [32] operatorwhich can provide a parameterized family of aggregationoperators including the maximum the minimum and the

average criteria Since its appearance the OWA operator hasbeen developed and used in a wide range of applications indecision making and expert systems [8 10 13 21ndash24 33ndash40]

The important and fundamental step of OWA operatorand its extended versions is to determine the associatedweights Many researches have been carried out on thisissue and useful methods have been developed under differ-ent decision environments such as crisp numbers intervalnumbers and linguistic scales which can be mainly clas-sified into two categories [41] (1) argument-independentapproaches [42ndash48] (2) argument-dependent approaches [45 9 11 23 41 42 44 45 49ndash52] The weights derived bythe argument-independent approaches are associated withparticular ordered positions of the aggregated argumentsand have no connection with the aggregated argumentswhile the argument-dependent approaches determine theweights based on input arguments As for the argument-dependent approaches the prominent characteristic is that

2 Journal of Applied Mathematics

they can relieve the influence of unfair arguments on theaggregated results by assigning low weights to those ldquofalserdquoand ldquobiasedrdquo ones In viewing of this merit researches onargument-dependent approaches under intuitionistic fuzzyenvironments and interval-valued intuitionistic fuzzy envi-ronments have started to accumulate recently such as thelinear programming-based aggregation operators with par-tial weight information [53] the aggregation operators [24]based on power method [54] the induced aggregation oper-ators [55] based on Choquet integral and Dempster-Shafertheory the power average operators [27] with trapezoidalintuitionistic fuzzy information the generalized intuitionisticfuzzy power averaging operators [23] and the generalizeddependent aggregation operators [40] for MAGDM withintuitionistic linguistic information

Another practical and interesting research issue of apply-ing OWA operator to MAGDM is the generalized exten-sions utilizing generalizedmeans and quasiarithmeticmeanswhich are respectively known as the generalized OWA(GOWA) operators [35 56] and the Quasi-OWA operators[57] And the main advantages of generalized operators arethat they can generalize a wide range of aggregation operatorsincluding the average the OWA and the OWG operatorsand that they can flexibly reflect the interests and actualneeds of decision makers such as the generalized extensions[58] of induced OWA (IOWA) operator [59] the generalizedweighted exponential proportional aggregation operators [4]for group decisionmaking with crisp numbers the expandedgeneralized hybrid averaging (GHA) operator [37 60] forfuzzy multiattribute decision making environments theinduced linguistic generalized OWA (ILGOWA) operators[61] and the generalized power aggregation operators forlinguistic environment [9]

As for the decision making situations with intuitionisticfuzzy information several researches have been conductedto address suitable generalized operators recently Zhao et al[62] investigated extensions ofGOWAoperator to present thegeneralized intuitionistic fuzzy weighted averaging operatorgeneralized intuitionistic fuzzy ordered weighted averagingoperator generalized intuitionistic fuzzy hybrid averagingoperator and Li [36] presented another extensions of GOWAoperator to accommodate intuitionistic fuzzy informationHowever all the above operators are unsuitable for aggre-gating individual preference relations into group preferencerelation when some experts prefer to aggregate the vari-ables with an inducing order So Xu and Xia [55] studiedthe induced generalized intuitionistic fuzzy Choquet inte-gral operators and induced generalized intuitionistic fuzzyDempster-Shafer operators Xu and Wang [22] developedthe induced generalized intuitionistic fuzzy orderedweightedaveraging (IGIFOWA) operator based on the GIFOWA [62]and the I-IFOWA [22 63] operators And based on theIGOWA operator introduced by Merigo and Gil-Lafuente[58] Su et al [21] presented another induced intuitionisticgeneralized fuzzy ordered weighted averaging (IG-IFOWA)operator In addition Zhou et al [23] proposed a generalizedoperator based on the power aggregation operator and gen-eralized mean but the same as most researches [9 11 24 3864] that focused on extended power aggregation operators

they did not discuss construction methodology of supportfunction in their presented operators Comparatively currentresearch on generalized operators for decision making situa-tions with interval-valued intuitionistic fuzzy information isstill in its infancy only few papers can be found in the liter-ature Representatively Zhao et al [62] further extended theGOWA operators to present some basic generalized aggrega-tion operators for dealing with interval-valued intuitionisticfuzzy information including the generalized interval-valuedintuitionistic fuzzy weighted averaging operator generalizedinterval-valued intuitionistic fuzzy ordered weighted averag-ing operator and generalized interval-valued intuitionisticfuzzy hybrid average operator Based on the Choquet integralmethod andDempster-Shafer theory Xu andXia [55] investi-gated patulous induced generalized operators for aggregationof interval-valued intuitionistic fuzzy information Andmostrecently Xu and Wang [22] also studied the induced versionof generalized OWA operators for interval-valued intuition-istic fuzzy group decision making

The aim of this paper is to develop some generalizedargument-dependent aggregation operatorsmore suitable fortackling with uncertainty inmultiple attribute group decisionmaking with interval-valued intuitionistic fuzzy informationInspired by the Gaussian distribution method we presentthe Gaussian generalized interval-valued intuitionistic fuzzyordered weighted averaging (Gaussian-GIIFOWA) operatorand Gaussian generalized interval-valued intuitionistic fuzzyorderedweighted geometric (Gaussian-GIIFOWG) operatorand a hybrid method is developed for construction of sup-port degree function based on which we further presentthe power generalized interval-valued intuitionistic fuzzyordered weighted averaging (P-GIIFOWA) operator andthe power generalized interval-valued intuitionistic fuzzyordered weighted geometric (P-GIIFOWG) operator Themain advantages of these operators are that they depend oninput arguments neatly and allow arguments being aggre-gated to support each other so that they can relieve the influ-ence of unfair assessments on decision results by assigninglowweights to those ldquofalserdquo and ldquobiasedrdquo ones and simultane-ously they can include a wide range of aggregation operatorsas particular cases such as interval-valued intuitionistic fuzzyaveraging (IIFA) operator interval-valued intuitionistic fuzzygeometric (IIFG) operator Gaussian interval-valued intu-itionistic fuzzy ordered weighted geometric (Gaussian-IIFOWG) operator and averaging (Gaussian-IIFOWA) oper-ator power interval-valued intuitionistic fuzzy orderedweighted geometric (P-IIFOWG) operator and averaging(Gaussian-IIFOWA) operator and generalized IIFA (GIIFA)operator and generalized IIFG (GIIFG) operator Further-more an approach based on the proposed operators is devel-oped and applied to solve a practical MAGDM problem con-cerning exploitation investment evaluation of tourist spotsThis approach can give a more completely view of decisionproblems with decision information aggregation dependingon input arguments and can also be suitable for solving othergroup decisionmaking problems including supplier selectiondecision making strategic management decision makinghuman resourcemanagement and emergency solutions eval-uation

Journal of Applied Mathematics 3

The remainder of this paper is organized as follows InSection 2 we give a concise review of fundamental conceptsrelated to intuitionistic fuzzy sets and interval-valued intu-itionistic fuzzy sets In Section 3 we first introduce somerelated basic aggregation operators and then we present twomethods to obtain argument-dependent attribute weights byGaussian distribution method and by support degree func-tion respectively based on which the Gaussian-GIIFOWAoperator Gaussian-GIIFOWG operator P-GIIFOWA oper-ator and P-GIIFOWG operator are presented In additionsome desirable properties of these operators are analyzedIn Section 4 an approach for multiple attribute groupdecision making under interval-valued intuitionistic fuzzyenvironments is constructed based on the four generalizeddependent aggregation operators In Section 5 applicationstudy on exploitation investment evaluation of tourist spots isconducted to verify the validity and practicality of developedmethods Finally conclusions are given in Section 6

2 Preliminaries

In this section we briefly review some basic concepts tofacilitate future discussions

Atanassov [18] generalized the concept of fuzzy set anddefined the concept of intuitionistic fuzzy set as shown in thefollowing Definition 1

Definition 1 (see [18]) An intuitionistic fuzzy set (IFS) 119860 is ageneralized fuzzy set and can be defined as

119860 = ⟨119909 120583119860 (119909) ]119860 (119909)⟩ | 119909 isin 119883 (1)

in which 120583119860means a membership function and ]

119860means a

nonmembership function with the condition 0 le 120583119860(119909) +

]119860(119909) le 1 120583

119860(119909) ]119860(119909) isin [0 1] for all 119909 isin 119883 Particularly

119860 = 120583119860(119909) = ]

119860(119909) the given IFS 119860 is degraded to an

ordinary fuzzy setIn reality itmay not be easy to identify the exact values for

the membership and nonmembership degrees of an elementa set In this case a range of values should be a moreappropriatemeasurement to accommodate the vagueness SoAtanassov and Gargov [19] introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS)

Definition 2 (see [19]) An interval-valued intuitionistic fuzzyset (IVIFS) 119860 in119883 can be defined as

119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883

= ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] []

119871

119860(119909) ]

119880

119860(119909)]⟩ | 119909 isin 119883

(2)

where 0 le 120583119871

119860(119909) le 120583

119880

119860(119909) le 1 0 le ]119871

119860(119909) le ]119880

119860(119909) le 1

0 le 120583119880

119860(119909) + ]119880

119860(119909) le 1 for all 119909 isin 119883

Similarly the intervals 120583119860(119909) and ]

119860(119909) denote the mem-

bership and non-membership of an element a setIf each of the intervals 120583

119860(119909) and ]

119860(119909) contains only one

value for each 119909 isin 119883 we have

120583119860(119909) = 120583

119871

119860(119909) = 120583

119880

119860(119909) ]

119860(119909) = ]

119871

119860(119909) = ]

119880

119860(119909)

(3)

Then the given IVIFS 119860 is degraded to an ordinaryIFS

In order to aggregate interval-valued intuitionistic fuzzyinformation Xu [65] defined the following relations and basicoperations

Definition 3 (see [65]) Let = ([119886 119887] [119888 119889]) 1= ([1198861 1198871]

[1198881 1198891]) 2= ([1198862 1198872] [1198882 1198892]) be interval-valued intuition-

istic fuzzy numbers (IVIFNs) then

(1) 1oplus 2= ([1198861+ 1198862minus 11988611198862 1198871+ 1198872minus 11988711198872] [11988811198882 11988911198892])

(2) 1otimes 2= ([11988611198862 11988711198872] [1198881+ 1198882minus 11988811198882 1198891+ 1198892minus 11988911198892])

(3) 120582 = ([1 minus (1 minus 119886)120582 1 minus (1 minus 119887)

120582] [119888120582 119889120582])

(4) 120582 = ([119886120582 119887120582] [1 minus (1 minus 119888)

120582 1 minus (1 minus 119889)

120582])

Usually the following normalized distance measure for-mulae listed inDefinition 4 can be introduced to calculate thedistance of IVIFSs

Definition 4 Suppose that two interval-valued intuitionisticfuzzy sets (IVIFSs) 119860 and 119861 in119883 can be defined as

119860 = ⟨119909119894 120583119860(119909119894) ]119860(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119860(119909119894) 120583119880

119860(119909119894)] []119871

119860(119909119894) ]119880

119860(119909119894)]⟩ | 119909

119894isin 119883

119861 = ⟨119909119894 120583119861(119909119894) ]119861(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119861(119909119894) 120583119880

119861(119909119894)] []119871

119861(119909119894) ]119880

119861(119909119894)]⟩ | 119909

119894isin 119883

(4)

then we can have

(1) the normalized Euclidean distance measure

1198631(119860 119861)

= (1

6119899

119899

sum

119894=1

[(120583119871

119860(119909119894) minus 120583119871

119861(119909119894))2

+ (120583119880

119860(119909119894) minus 120583119880

119861(119909119894))2

+ (]119871

119860(119909119894) minus ]119871

119861(119909119894))2

+ (]119880

119860(119909119894) minus ]119880

119861(119909119894))2

+ (120587119871

119860(119909119894) minus 120587119871

119861(119909119894))2

+ (120587119880

119860(119909119894) minus 120587119880

119861(119909119894))2

])

12

(5)


(2) the normalized Hamming distance measure

1198632(119860 119861)

=1

6119899

119899

sum

119894=1

10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(6)

(3) the normalized Hausdorff distance measure

1198633(119860 119861)

=1

119899

119899

sum

119894=1

max 10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583

119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(7)

In order to rank alternatives it is necessary to considerhow to compare two interval-valued intuitionistic fuzzynumbers so Xu [66] devised an approach to compare twoIVIFNs based on the concepts of score function and accuracyfunction

Definition 5 (see [66]) For any three IVIFNs = ([120583119871 120583119880]

[]119871 ]119880]) 1

= ([120583119871

1 120583119880

1] []1198711 ]1198801]) and

2= ([120583

119871

2 120583119880

2] []1198712

]1198802]) score function can be defined as 119904() = (12)(120583

119871+

120583119880minus ]119871 minus ]119880) accuracy function can be defined as ℎ() =

(12)(120583119871+ 120583119880+ ]119871 + ]119880) and

if 119904(1) lt 119904(

2) then

1is smaller than

2 1lt 2

if 119904(1) gt 119904(

2) then

1is greater than

2 1gt 2

if 119904(1) = 119904(

2) then

if ℎ(1) lt ℎ(

2) then

1is smaller than

2 1lt

2

if ℎ(1) gt ℎ(

2) then

1is greater than

2 1gt

2

if ℎ(1) = ℎ(

2) then

1and

2represent the

same information denoted by 1= 2

3 Proposed Generalized DependentInterval-Valued Intuitionistic FuzzyOrdered Weighted Aggregation Operators

31 Basic Operators Up to now some useful operators havebeen proposed for aggregating the interval-valued intuition-istic fuzzy informationThemost commonly used two opera-tors for aggregating interval-valued intuitionistic fuzzy argu-ments are the interval-valued intuitionistic fuzzy weightedaveraging (IIFWA)operator and geometric (IIFWG)operatoras defined by Xu [65] in the following definitions

Definition 6 (see [65]) An interval-valued intuitionisticfuzzy weighted averaging (IIFWA) operator of dimension 119899

is a mapping IIFWA Ω119899 rarr Ω which has an argumentassociated vector 120596 = (120596

1 1205962 120596

119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWA120596(1 2

119899) = 12059611oplus 12059622oplus sdot sdot sdot oplus 120596

119899119899 (8)

Let 119886119894= ([119886119894 119887119894] [119888119894 119889119894]) (119894 = 1 2 119899) be a collection

of interval-valued intuitionistic fuzzy numbers then theiraggregated value by using the IIFWA operator can be shownas

IIFWA120596(1 2

119899)

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119887119895)120596119895]

]

[

[

119899

prod

119895=1

119888120596119895

119895

119899

prod

119895=1

119889120596119895

119895]

]

)

(9)

Particularly when 120596 = (1119899 1119899 1119899)119879 the IIFWA

operator reduces to the interval-valued intuitionistic fuzzyaveraging (IIFA) operator that is

IIFWA120596(1 2

119899)

=1

1198991oplus

1

1198992oplus sdot sdot sdot oplus

1

119899119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

]

]

[

[

119899

prod

119895=1

1198881119899

119895

119899

prod

119895=1

1198891119899

119895]

]

)

= IIFA (1 2

119899)

(10)

Definition 7 (see [65]) An interval-valued intuitionisticfuzzy weighted geometric (IIFWG) operator of dimension119899 is a mapping IIFWG Ω119899 rarr Ω which has an argument


associated vector 120596 = (1205961 1205962 120596

119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWG120596(1 2

119899)

= 1205961

1otimes 1205962

2otimes sdot sdot sdot otimes

120596119899

119899

= ([

[

119899

prod

119895=1

119886120596119895

119895

119899

prod

119895=1

119887120596119895

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119889119895)120596119895]

]

)

(11)

Particularly when 120596 = (1119899 1119899 1119899)119879 the IIFWG

operator reduces to the interval-valued intuitionistic fuzzygeometric (IIFG) operator that is

IIFWG120596(1 2

119899)

= 1119899

1otimes 1119899


1119899

119899

= ([

[

119899

prod

119895=1

1198861119899

119895

119899

prod

119895=1

1198871119899

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119889119895)1119899

]

]

)

= IIFG (1 2

119899)

(12)

Considering ordered positions of interval-valued intu-itionistic fuzzy arguments rather than weighting the interval-valued intuitionistic fuzzy arguments themselves Xu andChen [67] proposed an interval-valued intuitionistic fuzzyordered weighted averaging (IIFOWA) operator and aninterval-valued intuitionistic fuzzy ordered weighted geo-metric (IIFOWG) operator as shown in the following defi-nitions

Definition 8 (see [67]) Let (1 2

119899) be a collec-

tion of interval-valued intuitionistic fuzzy arguments and119895= ([119886119895 119887119895] [119888119895 119889119895])The interval-valued intuitionistic fuzzy

ordered weighted averaging (IIFOWA) operator of dimen-sion 119899 is a mapping IIFOWA 119877119899 rarr 119877 which has an asso-ciated weight vector 119908 = (119908

1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and

119908119895isin [0 1] then

IIFOWA119908(1 2


119899120573119899(13)

where (1205731 1205732 120573

119899) is a permutation of (

1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895

Particularly when 119908 = (1119899 1119899 1119899)119879 the IIFOWA

operator reduces to the IIFA operator that is

IIFOWA119908(1 2

119899)

=1

1198991205731oplus

1


1

119899120573119899

= IIFA119908(1 2

119899)

(14)


119899) be a collection

of interval-valued intuitionistic fuzzy arguments and 119895

=

([119886119895 119887119895] [119888119895 119889119895]) The IIFOWG operator of dimension 119899 is a

mapping IIFOWG 119877119899 rarr 119877 which has an associated weightvector 119908 = (119908

1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin [0 1]

then

IIFOWG119908(1 2

119899) = 1205731199081

1otimes 1205731199082

2otimes sdot sdot sdot otimes 120573

119908119899

119899 (15)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895

Particularly when 119908 = (1119899 1119899 1119899)119879 the IIFOWG

operator reduces to the IIFG operator that is

IIFOWG119908(1 2

119899)

= 1205731119899

1otimes 1205731119899


1119899

119899

= IIFG119908(1 2

119899)

(16)

From another important and practical aspect Yager [56]defined a generalized version of OWA operators as thegeneralized ordered weighted averaging (GOWA) operatorthen Zhao et al [62] extended it to the situations whereinput arguments are IVIFNs and presented a generalizedinterval-valued intuitionistic fuzzy ordered weighted averag-ing (GIIFOWA) operator and geometric (GIIFOWG) opera-tor as defined in Definitions 10 and 11




=

([119886119895 119887119895] [119888119895 119889119895]) The GIIFOWA operator of dimension 119899 is

a mapping GIIFOWA 119877119899 rarr 119877 which has an associatedweight vector 119908 = (119908

1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895

If 119908 = (1119899 1119899 1119899)119879 then the GIIFOWA operator

reduces to the GIIFA operator that is

GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=

([119886119895 119887119895] [119888119895 119889119895]) The GIIFOWG operator of dimension 119899 is

a mapping GIIFOWG 119877119899 rarr 119877 which has an associatedweight vector 119908 = (119908

1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895

If 119908 = (1119899 1119899 1119899)119879 then the GIIFOWG operator

reduces to the GIIFG operator that is

GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)

From Definition 8 to Definition 11 it can be seen thatone important and basic step of interval-valued intuitionis-tic fuzzy ordered weighted aggregation operators and gen-eralized versions is to determine the associated weightsIn the following subsections we will focus on investigat-ing argument-dependent operators in which the associatedweights can be determined objectively only depending on theinterval-valued intuitionistic fuzzy input arguments

32 Proposed Gaussian Generalized Interval-Valued Intuition-istic Fuzzy Aggregation Operators According to the basicoperational rules listed in Definition 3 and IIFWA operatorin Definition 6 for aggregating IVIFNs here we can naturallydefinemean value of a set of IVIFNs as shown in the followingdefinition Obviously the mean value 120583 is still an IVIFN

Definition 12 Let (1 2

119899) be a collection of inter-

val-valued intuitionistic fuzzy arguments where 119895= ([119886119895

119887119895] [119888119895 119889119895]) Let 120583 be the mean value of (

1 2

119899) and

120583 = ([119886120583 119887120583] [119888120583 119889120583]) then 120583 can be obtained by IIFWA ope-

rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886

120583 119887120583] [119888120583 119889120583]) denotes mean value

of (1 2

119899) then the variance of

1 2

119899can be

computed according to

120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)

In real world a collection of 119899 aggregated arguments(1205721 1205722 120572

119899) usually takes the form of a collection of 119899

preference values provided by 119899 different decision makersSome decisionmakers may assign unduly high or unduly lowpreference values to their preferred or repugnant objects Insuch case very lowweights should be assigned to these ldquofalserdquoor ldquobiasedrdquo opinions that is to say the closer a preferencevalue argument is to the mid one(s) the more the weightconversely the further a preference value is apart from themid one(s) the less the weight So Xu [44] and Xu [49]developed Gaussian (normal) distribution-based method todetermine OWA weights by utilizing orderings of arguments


assessed with crisp numbers and interval numbers respec-tively Inspired by these ideas by using predefinedmean value120583 of IVIFNs we extended the Gaussian distribution methodto obtain the dependentweights here calledGaussianweight-ing vector according to interval-valued intuitionistic fuzzyinput arguments

Definition 14 Let 120583 be the mean value of given interval-valued intuitionistic fuzzy arguments 120590 the variance ofgiven interval-valued intuitionistic fuzzy arguments then theGaussian weighting vector 120596 = (120596

1 1205962 120596

119899)119879 can be

defined as

120596119895=

1

radic2120587120590119890minus1198892(120573119895minus120583)2120590

2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Consider that 120596119895isin [0 1] and sum

119899

119895=1120596119895= 1 are commonly

required in aggregation operators then we can normalize theGaussian weighting vector according to

120596119895=

(1radic2120587120590) 119890minus1198892(120573119895minus120583)2120590

2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)

Then by (17) we can define a Gaussian generalized inter-val-valued intuitionistic fuzzy ordered weighted averaging(Gaussian-GIIFOWA) operator as shown in the followingdefinition

Definition 15 A Gaussian-GIIFOWA operator of dimension119899 is a mapping Gaussian-GIIFOWA Ω119899 rarr Ω which has an

associated Gaussian weighting vector 120596 = (1205961 1205962 120596

119899)119879

with 120596119894isin [0 1] and sum

119899

119894=1120596119894= 1 then

Gaussian-GIIFOWA (1 2

119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582

120590(2)oplus sdot sdot sdot oplus 120596

120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2

oplus sdot sdot sdot oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2

oplus sdot sdot sdot oplus1


2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Similarly we can define the Gaussian generalized inter-val-valued intuitionistic fuzzy ordered weighted geometric(Gaussian-GIIFOWG) operator

Definition 16 A Gaussian-GIIFOWG operator of dimension119899 is a mapping Gaussian-GIIFOWGΩ119899 rarr Ω which has anassociated Gaussian weighting vector 120596 = (120596

1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then

Gaussian-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)

otimes sdot sdot sdot otimes (120582120573119899)120596120573(119899)

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes sdot sdot sdot otimes (120582120573119899)(1radic2120587120590)119890

minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)

]) then by Definition 3 Gaussian-GIIFOWA oper-ator and Gaussian-GIIFOWG operator can be transformedinto the following forms

Gaussian-GIIFOWA (1 2 119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

(1radic2120587120590)119890minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)

Gaussian-GIIFOWG (1 2 119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2

1205731205822 oplus sdot sdot sdot oplus

1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)


then we can rewrite as


119899)

= (120596(1)

120582

1oplus 120596(2)

120582

2oplus sdot sdot sdot oplus 120596

(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)

Obviously the aggregated results of Gaussian-GIIFOWAoperator and Gaussian-GIIFOWG operator are indepen-dent of orderings thus Gaussian-GIIFOWA and Gaussian-GIIFOWG are neat and dependent operators

Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be

a collection of interval-valued intuitionistic fuzzy argumentsand the 120596 = (120596

1 1205962 120596

119899)119879 be the Gaussian weighting

vector related to Gaussian-GIIFOWA operator and Gaussian-GIIFOWG operator with 120596

119895isin [0 1] and sum

119899

119895=1120596119895= 1 then

Gaussian-GIIFOWA operator and Gaussian-GIIFOWG ope-rator have the following properties

(1) Commutativity let (lowast1 lowast

2

lowast

119899) be any a permuta-

tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)

(2) Idempotency let 119895= for all 119895 = 1 2 119899 then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)

(3) Boundedness the Gaussian-GIIFOWA operator andthe Gaussian-GIIFOWG operator lie between the maxand min operators

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)

Theorem 18 Let 119895

= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)

]) be two collections of interval-valuedintuitionistic fuzzy arguments (120573

1 1205732 120573

119899) a permutation

of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 and

120596 = (1205961 1205962 120596

119899)119879 the Gaussian weighting vector related

to the Gaussian-GIIFOWA operator and Gaussian-GIIFOWGoperator with 120596


119899

119895=1120596119895= 1 then

(1) if 120582 = 1 then the Gaussian-GIIFOWA operator andGaussian-GIIFOWG operator reduce to the followingGaussian-IIFOWA operator and Gaussian-IIFOWGoperator

119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2

otimes sdot sdot sdot otimes 120573(1radic2120587120590)119890

minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)

(2) if 120582 rarr 0 then the Gaussian-GIIFOWA operator re-duces to the Gaussian-IIFOWG operator

(3) if 120596 = (1119899 1119899 1119899)119879 then the Gaussian-

GIIFOWA operator and Gaussian-GIIFOWG


operator reduce to the GIIFA operator and GIIFGoperator

(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then

the Gaussian-GIIFOWA operator and Gaussian-GIIFOWG operator reduce to the IIFA operator andIIFG operator

(5) if120596 = (1119899 1119899 1119899)119879 and 120582 rarr 0 then the Gaus-

sian-GIIFOWA operator reduces to the IIFG operator

Lemma 19 Assume that 119909119895gt 0 120582

119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)

with equality if and only if 1199091= 1199092= sdot sdot sdot = 119909

119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573

119899) is a permuta-

tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596

119899)119879 be the Gaussian weighting vector

related to the Gaussian-GIIFOWA operator and Gaussian-GIIFOWG operator with 120596


119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

Proof Based on Lemma 19 we can have

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus (1 minus 119886

120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)

Obviously the above inequations (a) (b) and (c) are alsovalid for 119887

120573(119895) 119888120573(119895)

and 119889120573(119895)

Then by Lemma 19 we can have

119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)

and thus complete the proof of Theorem 20

Example 21 For a group decision making problem supposethat there are six decision makers 119889

119895(119895 = 1 2 6)

these decision makers provide their individual preferenceswith interval-valued intuitionistic fuzzy numbers Then thepreference arguments are collected as follows

1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)

Utilizing (21) and (22) the mean value and variancevalue 120590 can be obtained

= ([04273 0664] [0 03238]) 120590 = 01271 (41)

Then by (23) and (24) we can compute the Gaussianweighting vector

120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675

Given 120582 = 5 according to (27) and (28) it follows that


119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)

33 Proposed Power Generalized Interval-Valued IntuitionisticFuzzy Aggregation Operators The above-presented Gaussiandistribution-based methods can obtain argument-dependentweights according to the indirectly calculated support degreeof arguments by considering the distances between argu-ments and the mid one (mean value) On the other hand todirectly consider the support degree of each argument Yager[54] developed the power average (PA) operator and a powerordered weighted average (POWA) operator which allow thearguments being aggregated to support each other Then Xuand Yager [39] developed power geometric average (PGA)operator and power ordered weighted average (POWA) ope-rator Most recently Zhou and Chen [9] further studiedextensions of power operator to linguistic decision environ-ment Motivated by these ideas here we first devise a hybridsupport function for interval-valued intuitionistic fuzzy inputarguments to not only consider the support degrees of eachargument by other arguments but also consider the sup-port degrees between argument values and mid one (meanvalue)Then a power generalized interval-valued intuitionis-tic fuzzy ordered weighted averaging (P-GIIFOWA) operatorand a power generalized interval-valued intuitionistic fuzzyordered weighted geometric (P-GIIFOWG) operator aredefined in which associated weights are obtained by thedevised hybrid support function



119899) be a collection of interval-

valued intuitionistic fuzzy arguments and let 120583 denote themean value then the hybrid support function can be definedas

Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)

Then we can use Sup(119894 119895) to denote the support degree

between 119886119894and 119895and Sup(

119894 120583) to denote the support degree

between 119894and 120583

Obviously Sup(119894 119895) and Sup(

119894 120583) satisfy the following

properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where

119889 is a certain distance measure for interval-valuedintuitionistic fuzzy numbers

Then utilizing hybrid support function in Definition 22we can manage to obtain the associated argument weightscalled power weighting vector according to

120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)

that is to say the closer a preference argument is to otherarguments or the closer a preference argument is tomid valuethe more the argument weighs

And let (1205731 1205732 120573

119899) be a permutation of (

1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 then we can have the

power weighting vector derived according to

120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)

Further we can define the P-GIIFOWA operator and P-GIIFOWG operator as follows

Definition 23 A P-GIIFOWA operator of dimension 119899 is amapping P-GIIFOWA Ω119899 rarr Ω 120596 = (120596

1 1205962 120596

119899)119879 is

associated power weighting vector 120596119894isin [0 1] and sum

119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582

2oplus sdot sdot sdot oplus Sup (120573

119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Definition 24 A P-GIIFOWG operator of dimension 119899 is amapping P-GIIFOWG Ω119899 rarr Ω 120596 = (120596

1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)

otimes sdot sdot sdot otimes (120582120573119899)Sup(120573119899)sum

119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)

otimes sdot sdot sdot otimes (120582120573119899)Sup(120573119899)

)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)

]) then P-GIIFOWA operator and P-GIIFOWGoperator can be transformed into the following forms

P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))sum

119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

1sum119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)

Obviously P-GIIFOWA and P-GIIFOWG are also neatand dependent operators

Theorem 25 Let (1 2


valued intuitionistic fuzzy arguments and (1205731 1205732 120573

119899) is

a permutation of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be

a collection of interval-valued intuitionistic fuzzy argumentsand 120596 = (120596

1 1205962 120596

119899)119879 the weighting vector derived by

hybrid supportmethod related to the P-GIIFOWAoperator andP-GIIFOWG operator with 120596


119899

119895=1120596119895= 1 then


the P-GIIFOWA operator and the P-GIIFOWG operator havethe following properties


2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)

(3) Boundedness the P-GIIFOWA operator and the P-GIIFOWG operator lie between the max and minoperators

minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596

119899)119879 the weighting vector derived by hybrid

support method related to the P-GIIFOWA operator and P-GIIFOWG operator 120596

119895isin [0 1] sum119899

119895=1120596119895= 1 then

(1) if 120582 = 1 then the P-GIIFOWA operator and P-GIIFOWG operator reduce to the following P-IIFOWAoperator and P-IIFOWG operator

119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2

otimes sdot sdot sdot otimes 120573Sup(120573119899)119899

)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)

(2) if 120582 rarr 0 then the P-GIIFOWA operator reduces to theP-IIFOWG operator

(3) if120596 = (1119899 1119899 1119899)119879 then the P-GIIFOWA oper-

ator and P-GIIFOWG operator reduce to the GIIFAoperator and GIIFG operator

(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-

GIIFOWA operator and P-GIIFOWG operator reduceto the IIFA operator and IIFG operator

(5) if 120596 = (1119899 1119899 1119899)119879 and 120582 rarr 0 then the P-

GIIFOWA operator reduces to the IIFG operator


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596

119899)119879 the weight vector derived by hybrid support

method related to the P-GIIFOWA operator and P-GIIFOWGoperator with 120596


119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)


Proof Similar to the proof of Theorem 20 Theorem 28 canbe proved by mathematical induction method so proof stepsare omitted here


119895(119895 = 1 2 6) to pro-

vide their individual preferences with interval-valued intui-tionistic fuzzy numbers Then the preference arguments canbe collected as follows

1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)

According to (44) and (45) we can have the powerweighting vector

120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625

Suppose 120582 = 5 then according to (51) and (52) it followsthat

P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)

]) be two collections of interval-valuedintuitionistic fuzzy arguments and let 120574 be the interval-valuedintuitionistic fuzzy number obtained by applying 119866119868119868119865119874119882119860

120582

or 119866119868119868119865119874119882119866120582on 119895and 120573

119895 then one can have

(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0

Proof For the proposition (1-a) if 119888120573(119895)

= 0 then we can have

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)

so the proposition (1-a) is right Correspondingly proposition(1-b) and proposition (1-c) can be proved in the same way

For the proposition (2-a) if 119886120573(119895)

= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)

so the proposition (2-a) is right and proposition (2-b) andproposition (2-c) can also be proved similarly

Thus according to Theorem 30 for the situation that119888120573(119895)

= 0 or 119889120573(119895)

= 0 GIIFOWG120582operators should be


better choices than GIIFOWA120582operators to consider more

completely the preference information indicated by nonzeroarguments while for the situation 119886

120573(119895)= 0 or 119887

120573(119895)= 0

GIIFOWA120582operators can use preference information more

completely than GIIFOW119866120582operators

4 An Approach forMultiple Attribute Group DecisionMaking with Interval-Valued IntuitionisticFuzzy Information

For the multiple attribute group decision making problemsin which both the attribute weights and the expert weightstake the form of real numbers and the attribute argumentstake the form of interval-valued intuitionistic fuzzy num-bers we develop a decision making approach based onthe above-presented dependent interval-valued intuitionisticfuzzy aggregation operators

Let 119883 = 1199091 1199092 119909

119899 be a set of alternatives 119866 =

1198921 1198922 119892

119898 a set of attributes 120596 = 120596

1 1205962 120596

119898119879 the

weighting vector of attributes where 120596119895isin [0 1] sum119899

119895=1120596119895=

1 119863 = 1198891 1198892 119889

119905 a set of decision makers and 120582 =

(120582(1)

120582(2)

120582(119905)) the weighting vector of decision makers

The proposed approach involves the following steps

Step 1 Construct individual interval-valued intuitionisticfuzzy evaluation matrices

(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)

119894119895] indicates the degree to which the alternative 119909

119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895

] indicates the degree to which thealternative 119909

119894(119894 = 1 2 119899) does not satisfies the attribute

119892119895(119895 = 1 2 119898)

Step 2 Calculate argument weighting vector 120596(119896)

= (120596(119896)

1

120596(119896)

2 120596

(119896)

119899)119879 associated with the interval-valued intuition-

istic fuzzy value 119903(119896)

119894119895in 119896th individual matrix

(119896) accordingto (24) or (46)

Step 3 Utilize Gaussian-GIIFOWA operator P-GIIFOWAoperator Gaussian-GIIFOWG operator or P-GIIFOWGoperator to aggregate the arguments in 119894th row of 119896th decisionmakerrsquos assessmentmatrix (119896) as the corresponding interval-valued intuitionistic fuzzy value 119903

119894119896in the group decision

matrix for each 119909119894

Step 4 Utilize IIFWA operator or IIFWG operator to derivethe overall group interval-valued intuitionistic fuzzy decisionvector 119903 for all the alternatives by aggregating the values ineach row of

Step 5 Calculate score values 119904(119903119894) (119894 = 1 2 119899) and

accuracy values ℎ(119903119894) (119894 = 1 2 119899) of alternative 119909

119894and

then rank all the alternatives to select the optimal one(s)according to Definition 5

Step 6 End

5 Application to Exploitation InvestmentEvaluation of Tourist Spots

51 Application Study Suppose that a tourism developmentand investment company is about to choose the mostdesirable project(s) to invest from several candidate touristspots which are filtered out through initial screening andadvance to an investment expert committee for detailed com-prehensive due diligence such as evaluation of exploitationfeasibility and evaluation of sustainable management strate-gies [69] Given that five filtered alternative tourist spots119909119894(119894 = 1 2 3 4 5) advance to be reviewed for acceptance the

corresponding investment criteria about exploitation feasibi-lity of tourist spots could be constructed according to [69]from the following five aspects variety (119892

1) orientability

(1198922) monopoly (119892

3) destructibility (119892

4) and novelty (119892

5)

And three domain experts are organized as decision makersDM 119889

119896(119896 = 1 2 3) in the investment expert committee

to assess alternative tourist spots 119909119894by interval-valued intu-

itionistic fuzzy numbers with respect to each investmentcriterion 119892

119895 Suppose the decision makersrsquo weighting vector

120582 = (03 03 04)119879 According to Section 4 the procedure

for solving this practical MAGDM problem contains thefollowing steps

Step 1 According to the opinions of decision makers theinterval-valued intuitionistic fuzzy decision matrix

(119896)=

(119903(119896)

119894119895)119899times119898

(119896 = 1 2 3) can be firstly constructed and theassessments are listed in Tables 1 2 and 3

Step 2 Respectively calculate Gaussian weighting vectoraccording to (24) and power weighting vector according to(46)

Gaussian weighting vector

120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)

power weighting vector

120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)

Step 3 Then respectively utilize the Gaussian-GIIFOWAoperator P-GIIFOWA operator Gaussian-GIIFOWG oper-ator or P-GIIFOWG operator to aggregate each interval-valued intuitionistic fuzzy arguments in 119894th row of 119896th deci-sion makerrsquos assessment matrix

(119896) and get the group deci-sionmatrix for each 119909

119894 Here suppose 120582 = 1 and the results

are shown in Tables 4 5 6 and 7

Step 4 Aggregate each row in using IIFWA operator orIIFWG operator to derive the interval-valued intuitionistic


Table 1 Decision matrix (1) by 119889

1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])

Table 4 Group decision matrix obtained by utilizing Gaussian-GIIFOWA operator

1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])

Table 5 Group decision matrix obtained by utilizing P-GIIFOWA operator

1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])

Table 6 Group decision matrix obtained by utilizing Gaussian-GIIFOWG operator

1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])


Table 7 Group decision matrix obtained by utilizing P-GIIFOWG operator

1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)

Table 8 Overall group decision assessment values for all alternatives

Combination ofoperators 119909

11199092

1199093

1199094

1199095

Gaussian-GIIFOWAand IIFWA

([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])

Gaussian-GIIFOWGand IIFWG

([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])

Table 9 Orderings of the alternatives obtained by using differentoperators

Different combination of operators OrderingGaussian-GIIFOWA and IIFWA 119909

2≺ 1199093≺ 1199095≺ 1199091≺ 1199094

P-GIIFOWA and IIFWA 1199092≺ 1199093≺ 1199095≺ 1199091≺ 1199094

Gaussian-GIIFOWG and IIFWG 1199092≺ 1199093≺ 1199091≺ 1199095≺ 1199094

P-GIIFOWG and IIFWG 1199092≺ 1199093≺ 1199091≺ 1199095≺ 1199094

fuzzy overall group decision assessment values for all alter-natives The results are shown in Table 8

Step 5 Calculate the scores 119878(119903119894) (119894 = 1 2 3 4 5) of the

group overall intuitionistic fuzzy assessment values and rankall alternatives in accordance with scores 119878(119903

119894) the obtained

ordering results are listed in Table 9

As can be seen from Table 9 for all four combinations ofoperators alternative 119909

4is consistently distinguished as the

best one and alternative 1199092and 119909

3are consistently distin-

guished as the worst ones The ordering of 1199091and 119909

5shows

difference with IIFWA or IIFWG adopted The first twocombinations of averaging operators yield the same rankingresult as 119909

2≺ 1199093≺ 1199095≺ 1199091≺ 1199094and the last two combina-

tions of geometric operators also generate the same rankingresult as 119909

2≺ 1199093≺ 1199091≺ 1199095≺ 1199094 which show that the pro-

posed Gaussian distribution-based operators and powermethod-based operators can help to effectively differentiatethe most desirable one(s) Generally from the aspect of dif-ferent support degree measurement methods adopted theGaussian-GIIFOWA operator and Gaussian-GIIFOWG ope-rator appear to be more straight and concise than the P-GIIFOWA operator and P-GIIFOWG operator while the

latter two operators can utilize preference more completelyby considering not only support degree of each argumentby other arguments but also the support degree between theaggregated argument and the mean value So for differentpractical decision making problems decision makers maychoose different operators according to their preference andthe related facts

52 Further Discussion In order to further verify proper-ties of the proposed four generalized argument-dependentaggregation operators experiments are conducted in thissubsection with attitudinal parameter 120582 varying in a crisprange 15 14 13 12 1 2 3 4 and 5 For clarity the proposedGaussian-GIIFOWA operator Gaussian-GIIFOWG opera-tor P-GIIFOWA operator and P-GIIFOWG operator arerespectively applied on assessment matrix given by decisionmaker119889

1(as shown inTable 4) and corresponding results are

listed in Table 10 to Table 13From comparison with the last columns of Table 10 to

Table 13 it is can be seen that the best and worst alternativesare totally consistent and only the orderings of 119909

2and 119909

5

exhibit some difference which shows that all the proposedfour aggregation operators can effectively distinguish themost desirable alternatives And from the view of resultsobtained by Gaussian-GIIFOWA and Gaussian-GIIFOWGwith ranging120582 it is can be observed that all the score values inTable 11 are smaller than the score values in Table 10 with 120582 =

1 (Gaussian-GIIFOWA is reduced to be Gaussian-IIFOWA)and that all the score values in Table 10 are bigger than thescore values in Table 11 with 120582 = 1 (Gaussian-GIIFOWGreduces to Gaussian-IIFOWG) These observed facts justverify the validness of the inequations given in Theorem 20And similarly the same facts verifying the validness ofTheo-rem 28 can also be observed by comparing the score valueslisted in Tables 12 and 13


Table 10 Ranking results obtained by applying Gaussian-GIIFOWA on assessments given by 1198891

Aggregation operator 120582 Score values Ranking

15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Table 11 Ranking results obtained by applying Gaussian-GIIFOWG on assessments given by 1198891

Aggregation operator 120582 Score value Ranking

15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

Table 12 Ranking results obtained by applying P-GIIFOWA on assessments given by 1198891


15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


Table 13 Ranking results obtained by applying P-GIIFOWG on assessments given by 1198891


15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

In summary the proposed four generalized argument-dependent operators can effectively and objectively distin-guish the most desirable alternative(s) The Gaussian weight-ing vector directly measures support degree of argumentswith less computation complexity by only considering thedistance between arguments with mean value while thepower weighting vector obtained by hybrid support functioncan consider objective preference information by measuringboth the support degrees between arguments and the supportdegrees between arguments andmid values In practical deci-sion modelling process decision makers can select suitablypresented operators and proper parameter 120582 in accordancewith attitudinal preference or problem characteristics to helpeffectively solve multiple attribute group decision makingproblems under interval-valued intuitionistic fuzzy environ-ments

6 Conclusions

We have investigated some generalized argument-dependentaggregation operators for MAGDM under interval-valuedintuitionistic fuzzy environments First a Gaussian distribu-tion-based method for deriving weighting vector by measur-ing distances between arguments and mean value has beenpresented based on which we have proposed the Gaussian-GIIFOWA operator and the Gaussian-GIIFOWG operatorThen a hybrid support degree function has been devisedfor measuring both the support degrees between argumentsand the support degrees between arguments and mean valuebased onwhich we have also proposed the P-GIIFOWAoper-ator and the P-GIIFOWG operator And some desirable pro-perties of the proposed dependent operators have been ana-lyzed The main advantages of developed operators are thatthey can relieve the influence of unfair assessments on theresults by assigning lowweights to those false andbiased onesand they can include a wide range of other aggregation ope-rators for decisionmakers to flexibly choose in practical deci-sion modelling An approach has been formed based ondeveloped operators and applied to solve the MAGDM

problem concerning exploitation investment evaluation oftourist spots for verifying effectiveness and practicality ofproposed methods Furthermore the results of comparativeexperiments have also verified the properties of developedoperators In the future we will continue working on theextension and application of these operators to other fieldsunder different environments such as the information fusiondata mining and hybrid decision making indices with hesi-tant fuzzy preference

Acknowledgments

This work was supported in part by the National ScienceFoundation of China (no 71201145 no 71271072) the Re-search Fund for the Doctoral Program of Higher Educationof China (no 20110111110006) the Social Science Foundationof Ministry of Education of China (no 11YJC630283) andZhejiang Provincial Natural Science Foundation of China(no Y6110345)

References

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[2] Y C Dong G Q Zhang W C Hong and Y F Xu ldquoConsensusmodels for AHP group decision making under row geometricmean prioritization methodrdquo Decision Support Systems vol 49no 3 pp 281ndash289 2010

[3] LA Yu andKK Lai ldquoAdistance-based groupdecision-makingmethodology for multi-person multi-criteria emergency deci-sion supportrdquo Decision Support Systems vol 51 no 2 pp 307ndash315 2011

[4] L G Zhou H Y Chen and J P Liu ldquoGeneralized weightedexponential proportional aggregation operators and their appli-cations to group decision makingrdquo Applied Mathematical Mod-elling vol 36 no 9 pp 4365ndash4384 2012

[5] L G Zhou H Y Chen and J P Liu ldquoGeneralized logarithmicproportional averaging operators and their applications to


group decision makingrdquo Knowledge-Based Systems vol 36 pp268ndash279 2012

[6] Q-WCao and JWu ldquoThe extendedCOWGoperators and theirapplication tomultiple attributive group decisionmaking prob-lems with interval numbersrdquo Applied Mathematical Modellingvol 35 no 5 pp 2075ndash2086 2011

[7] L G Zhou H Y Chen J M Merigo and A M Gil-LafuenteldquoUncertain generalized aggregation operatorsrdquo Expert Systemswith Applications vol 39 pp 1105ndash1117 2012

[8] L-G Zhou and H-Y Chen ldquoContinuous generalized OWAoperator and its application to decisionmakingrdquo Fuzzy Sets andSystems vol 168 pp 18ndash34 2011

[9] LG Zhou andHYChen ldquoA generalization of the power aggre-gation operators for linguistic environment and its applicationin group decision makingrdquo Knowledge-Based Systems vol 26pp 216ndash224 2012

[10] J M Merigo A M Gil-Lafuente L G Zhou and H Y ChenldquoInduced and linguistic generalized aggregation operators andtheir application in linguistic group decision makingrdquo GroupDecision and Negotiation vol 21 pp 531ndash549 2010

[11] GWWei and X F Zhao ldquoSome dependent aggregation opera-tors with 2-tuple linguistic information and their application tomultiple attribute group decision makingrdquo Expert Systems withApplications vol 39 pp 5881ndash5886 2012

[12] G W Wei ldquoFIOWHM operator and its application to multipleattribute group decision makingrdquo Expert Systems with Applica-tions vol 38 no 4 pp 2984ndash2989 2011

[13] J M Merigo and A M Gil-Lafuente ldquoFuzzy induced general-ized aggregation operators and its application in multi-persondecision makingrdquo Expert Systems with Applications vol 38 no8 pp 9761ndash9772 2011

[14] P D Liu ldquoA weighted aggregation operators multi-attributegroup decision-makingmethod based on interval-valued trape-zoidal fuzzy numbersrdquo Expert Systems with Applications vol 38no 1 pp 1053ndash1060 2011

[15] F Zandi andMTavana ldquoA fuzzy groupmulti-criteria enterprisearchitecture framework selection modelrdquo Expert Systems withApplications vol 39 pp 1165ndash1173 2012

[16] K Khalili-Damghani and S Sadi-Nezhad ldquoA hybrid fuzzy mul-tiple criteria group decision making approach for sustainableproject selectionrdquo Applied Soft Computing vol 13 pp 339ndash3522013

[17] B Vahdani SMMousavi HHashemiMMousakhani and RTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 pp 779ndash788 2013

[18] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986

[19] K Atanassov and G Gargov ldquoInterval valued intuitionistic fuz-zy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash349 1989

[20] D-F Li G-H Chen and Z-G Huang ldquoLinear programmingmethod formultiattribute group decisionmaking using IF setsrdquoInformation Sciences vol 180 no 9 pp 1591ndash1609 2010

[21] Z X Su G P Xia M Y Chen and L Wang ldquoInduced gen-eralized intuitionistic fuzzy OWA operator for multi-attributegroup decision makingrdquo Expert Systems with Applications vol39 pp 1902ndash1910 2012

[22] Y J Xu and H MWang ldquoThe induced generalized aggregationoperators for intuitionistic fuzzy sets and their application ingroup decision makingrdquo Applied Soft Computing vol 12 pp1168ndash1179 2012

[23] L G Zhou H Y Chen and J P Liu ldquoGeneralized power aggre-gation operators and their applications in group decision mak-ingrdquo Computers amp Industrial Engineering vol 62 pp 989ndash9992012

[24] Z S Xu ldquoApproaches to multiple attribute group decision mak-ing based on intuitionistic fuzzy power aggregation operatorsrdquoKnowledge-Based Systems vol 24 no 6 pp 749ndash760 2011

[25] Z P Chen and W Yang ldquoA new multiple attribute group deci-sion making method in intuitionistic fuzzy settingrdquo AppliedMathematical Modelling vol 35 no 9 pp 4424ndash4437 2011

[26] Z X Su M Y Chen G P Xia and L Wang ldquoAn interactivemethod for dynamic intuitionistic fuzzy multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 38 pp15286ndash15295 2011

[27] S PWan ldquoPower average operators of trapezoidal intuitionisticfuzzy numbers and application to multi-attribute group deci-sionmakingrdquoAppliedMathematicalModelling vol 37 pp 4112ndash4126 2013

[28] Z L Yue and Y Y Jia ldquoAn application of soft computing tech-nique in group decision making under interval-valued intu-itionistic fuzzy environmentrdquo Applied Soft Computing vol 13pp 2490ndash2503 2013

[29] D J Yu Y YWu and T Lu ldquoInterval-valued intuitionistic fuzzyprioritized operators and their application in group decisionmakingrdquo Knowledge-Based Systems vol 30 pp 57ndash66 2012

[30] M M Xia and Z S Xu ldquoEntropycross entropy-based groupdecisionmaking under intuitionistic fuzzy environmentrdquo Infor-mation Fusion vol 13 pp 31ndash47 2012

[31] Y Jiang Z S Xu and X H Yu ldquoCompatibility measures andconsensusmodels for group decisionmaking with intuitionisticmultiplicative preference relationsrdquo Applied Soft Computingvol 13 pp 2075ndash2086 2013

[32] R R Yager ldquoOn ordered weighted averaging aggregation ope-rators in multicriteria decisionmakingrdquo IEEE Transactions onSystems Man and Cybernetics vol 18 no 1 pp 183ndash190 1988

[33] J M Merigo and M Casanovas ldquoDecision-making with dis-tance measures and induced aggregation operatorsrdquo Computersand Industrial Engineering vol 60 no 1 pp 66ndash76 2011

[34] G W Wei ldquoA method for multiple attribute group decisionmaking based on the ET-WG and ET-OWG operators with 2-tuple linguistic informationrdquo Expert Systems with Applicationsvol 37 no 12 pp 7895ndash7900 2010

[35] N B Karayiannis ldquoSoft learning vector quantization and clus-tering algorithms based on ordered weighted aggregation oper-atorsrdquo IEEE Transactions on Neural Networks vol 11 no 5 pp1093ndash1105 2000

[36] D F Li ldquoMultiattribute decision making method based on gen-eralized OWA operators with intuitionistic fuzzy setsrdquo ExpertSystems with Applications vol 37 no 12 pp 8673ndash8678 2010

[37] J M Merigo and M Casanovas ldquoFuzzy generalized hybrid ag-gregation operators and its application in fuzzy decision mak-ingrdquo International Journal of Fuzzy Systems vol 12 no 1 pp15ndash24 2010

[38] Y J Xu J M Merigo and H M Wang ldquoLinguistic poweraggregation operators and their application tomultiple attributegroup decision makingrdquo Applied Mathematical Modelling vol36 no 11 pp 5427ndash5444 2012

[39] Z S Xu and R R Yager ldquoPower-geometric operators and theiruse in group decisionmakingrdquo IEEE Transactions on Fuzzy Sys-tems vol 18 no 1 pp 94ndash105 2010


[40] P D Liu ldquoSome generalized dependent aggregation operatorswith intuitionistic linguistic numbers and their application togroup decision makingrdquo Journal of Computer and System Sci-ences vol 79 no 1 pp 131ndash143 2013

[41] Z S Xu ldquoDependent OWA operatorsrdquo Lecture Notes in Com-puter Science vol 3885 pp 172ndash178 2006

[42] D P Filev and R R Yager ldquoOn the issue of obtaining OWAoperator weightsrdquo Fuzzy Sets and Systems vol 94 no 2 pp 157ndash169 1998

[43] R Fuller and P Majlender ldquoAn analytic approach for obtainingmaximal entropy OWA operator weightsrdquo Fuzzy Sets andSystems vol 124 no 1 pp 53ndash57 2001

[44] Z S Xu ldquoAn overview of methods for determining OWAweightsrdquo International Journal of Intelligent Systems vol 20 no8 pp 843ndash865 2005

[45] R R Yager ldquoFamilies of OWA operatorsrdquo Fuzzy Sets and Sys-tems vol 59 no 2 pp 125ndash148 1993

[46] R R Yager and D P Filev ldquoParametrized and-like and or-likeownoperatorsrdquo International Journal of General Systems vol 22no 3 pp 297ndash316 1994

[47] P D Liu and F Jin ldquoMethods for aggregating intuitionisticuncertain linguistic variables and their application to groupdecision makingrdquo Information Sciences vol 205 pp 58ndash712012

[48] P D Liu ldquoSome geometric aggregation operators based oninterval intuitionistic uncertain linguistic variables and theirapplication to group decision makingrdquo Applied MathematicalModelling vol 37 no 4 pp 2430ndash2444 2013

[49] Z S Xu ldquoDependent uncertain ordered weighted aggregationoperatorsrdquo Information Fusion vol 9 no 2 pp 310ndash316 2008

[50] Z S Xu and Q L Da ldquoThe uncertain OWA operatorrdquo Interna-tional Journal of Intelligent Systems vol 17 no 6 pp 569ndash5752002

[51] G W Wei ldquoMethod of uncertain linguistic multiple attributegroup decision making based on dependent aggregation opera-torsrdquo Systems Engineering and Electronics vol 32 no 4 pp 764ndash769 2010

[52] J Wu C Y Liang and Y Q Huang ldquoAn argument-dependentapproach to determining OWA operator weights based on therule of maximum entropyrdquo International Journal of IntelligentSystems vol 22 no 2 pp 209ndash221 2007

[53] Z J Wang K W Li and W Z Wang ldquoAn approach to multi-attribute decision making with interval-valued intuitionisticfuzzy assessments and incomplete weightsrdquo Information Sci-ences vol 179 no 17 pp 3026ndash3040 2009

[54] R R Yager ldquoThe power average operatorrdquo IEEE Transactions onSystems Man and Cybernetics Part A vol 31 no 6 pp 724ndash7312001

[55] Z S Xu and M M Xia ldquoInduced generalized intuitionisticfuzzy operatorsrdquo Knowledge-Based Systems vol 24 no 2 pp197ndash209 2011

[56] R R Yager ldquoGeneralized OWA aggregation operatorsrdquo FuzzyOptimization and Decision Making vol 3 no 1 pp 93ndash1072004

[57] J Fofor J L Marichal and M Roubens ldquoCharacterization ofthe ordered weighted averaging operatorsrdquo IEEE Transactionson Fuzzy Systems vol 3 pp 236ndash240 1995

[58] J M Merigo and A M Gil-Lafuente ldquoThe induced generalizedOWA operatorrdquo Information Sciences vol 179 no 6 pp 729ndash741 2009

[59] R R Yager andD P Filev ldquoInduced orderedweighted averagingoperatorsrdquo IEEE Transactions on Systems Man and CyberneticsPart B vol 29 no 2 pp 141ndash150 1999

[60] J M Merigo and M Casanovas ldquoThe generalized hybrid ave-raging operator and its application in decision makingrdquo Journalof Quantitative Methods For Economics and Business Adminis-tration vol 9 pp 69ndash84 2010

[61] J M Merigo and A M Gil-Lafuente ldquoThe induced linguisticgeneralized OWA operatorrdquo in Proceedings of the FLINS Inter-national Conference pp 325ndash330 Madrid Spain 2008

[62] H Zhao Z S XuM F Ni and S Liu ldquoGeneralized aggregationoperators for intuitionistic fuzzy setsrdquo International Journal ofIntelligent Systems vol 25 no 1 pp 1ndash30 2010

[63] G W Wei ldquoSome induced geometric aggregation operatorswith intuitionistic fuzzy information and their application togroup decision makingrdquo Applied Soft Computing vol 10 no 2pp 423ndash431 2010

[64] Y J Xu and H M Wang ldquoApproaches based on 2-tuple lin-guistic power aggregation operators formultiple attribute groupdecision making under linguistic environmentrdquo Applied SoftComputing vol 11 no 5 pp 3988ndash3997 2011

[65] Z S Xu ldquoMethods for aggregating interval-valued intuitionisticfuzzy information and their application to decision makingrdquoControl and Decision vol 22 no 2 pp 215ndash219 2007

[66] Z S Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEETran-sactions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

[67] Z S Xu and J Chen ldquoAn approach to group decision makingbased on interval-valued intuitionistic judgment matricesrdquo Sys-tem Engineering Theory and Practice vol 27 no 4 pp 126ndash1332007

[68] Z S Xu and R R Yager ldquoIntuitionistic and interval-valuedintuitionistic fuzzy preference relations and their measures ofsimilarity for the evaluation of agreementwithin a grouprdquo FuzzyOptimization and Decision Making vol 8 no 2 pp 123ndash1392009

[69] S H Ren and K Q Cai Research on Evaluation of ExploitationFeasibility and Sustainable Development Strategies for TourismResources in ZhouShan Sea Islands of China Ocean PressBeijing China 2010

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Stochastic AnalysisInternational Journal of


they can relieve the influence of unfair arguments on theaggregated results by assigning low weights to those ldquofalserdquoand ldquobiasedrdquo ones In viewing of this merit researches onargument-dependent approaches under intuitionistic fuzzyenvironments and interval-valued intuitionistic fuzzy envi-ronments have started to accumulate recently such as thelinear programming-based aggregation operators with par-tial weight information [53] the aggregation operators [24]based on power method [54] the induced aggregation oper-ators [55] based on Choquet integral and Dempster-Shafertheory the power average operators [27] with trapezoidalintuitionistic fuzzy information the generalized intuitionisticfuzzy power averaging operators [23] and the generalizeddependent aggregation operators [40] for MAGDM withintuitionistic linguistic information

Another practical and interesting research issue of apply-ing OWA operator to MAGDM is the generalized exten-sions utilizing generalizedmeans and quasiarithmeticmeanswhich are respectively known as the generalized OWA(GOWA) operators [35 56] and the Quasi-OWA operators[57] And the main advantages of generalized operators arethat they can generalize a wide range of aggregation operatorsincluding the average the OWA and the OWG operatorsand that they can flexibly reflect the interests and actualneeds of decision makers such as the generalized extensions[58] of induced OWA (IOWA) operator [59] the generalizedweighted exponential proportional aggregation operators [4]for group decisionmaking with crisp numbers the expandedgeneralized hybrid averaging (GHA) operator [37 60] forfuzzy multiattribute decision making environments theinduced linguistic generalized OWA (ILGOWA) operators[61] and the generalized power aggregation operators forlinguistic environment [9]

As for the decision making situations with intuitionisticfuzzy information several researches have been conductedto address suitable generalized operators recently Zhao et al[62] investigated extensions ofGOWAoperator to present thegeneralized intuitionistic fuzzy weighted averaging operatorgeneralized intuitionistic fuzzy ordered weighted averagingoperator generalized intuitionistic fuzzy hybrid averagingoperator and Li [36] presented another extensions of GOWAoperator to accommodate intuitionistic fuzzy informationHowever all the above operators are unsuitable for aggre-gating individual preference relations into group preferencerelation when some experts prefer to aggregate the vari-ables with an inducing order So Xu and Xia [55] studiedthe induced generalized intuitionistic fuzzy Choquet inte-gral operators and induced generalized intuitionistic fuzzyDempster-Shafer operators Xu and Wang [22] developedthe induced generalized intuitionistic fuzzy orderedweightedaveraging (IGIFOWA) operator based on the GIFOWA [62]and the I-IFOWA [22 63] operators And based on theIGOWA operator introduced by Merigo and Gil-Lafuente[58] Su et al [21] presented another induced intuitionisticgeneralized fuzzy ordered weighted averaging (IG-IFOWA)operator In addition Zhou et al [23] proposed a generalizedoperator based on the power aggregation operator and gen-eralized mean but the same as most researches [9 11 24 3864] that focused on extended power aggregation operators

they did not discuss construction methodology of supportfunction in their presented operators Comparatively currentresearch on generalized operators for decision making situa-tions with interval-valued intuitionistic fuzzy information isstill in its infancy only few papers can be found in the liter-ature Representatively Zhao et al [62] further extended theGOWA operators to present some basic generalized aggrega-tion operators for dealing with interval-valued intuitionisticfuzzy information including the generalized interval-valuedintuitionistic fuzzy weighted averaging operator generalizedinterval-valued intuitionistic fuzzy ordered weighted averag-ing operator and generalized interval-valued intuitionisticfuzzy hybrid average operator Based on the Choquet integralmethod andDempster-Shafer theory Xu andXia [55] investi-gated patulous induced generalized operators for aggregationof interval-valued intuitionistic fuzzy information Andmostrecently Xu and Wang [22] also studied the induced versionof generalized OWA operators for interval-valued intuition-istic fuzzy group decision making

The aim of this paper is to develop some generalizedargument-dependent aggregation operatorsmore suitable fortackling with uncertainty inmultiple attribute group decisionmaking with interval-valued intuitionistic fuzzy informationInspired by the Gaussian distribution method we presentthe Gaussian generalized interval-valued intuitionistic fuzzyordered weighted averaging (Gaussian-GIIFOWA) operatorand Gaussian generalized interval-valued intuitionistic fuzzyorderedweighted geometric (Gaussian-GIIFOWG) operatorand a hybrid method is developed for construction of sup-port degree function based on which we further presentthe power generalized interval-valued intuitionistic fuzzyordered weighted averaging (P-GIIFOWA) operator andthe power generalized interval-valued intuitionistic fuzzyordered weighted geometric (P-GIIFOWG) operator Themain advantages of these operators are that they depend oninput arguments neatly and allow arguments being aggre-gated to support each other so that they can relieve the influ-ence of unfair assessments on decision results by assigninglowweights to those ldquofalserdquo and ldquobiasedrdquo ones and simultane-ously they can include a wide range of aggregation operatorsas particular cases such as interval-valued intuitionistic fuzzyaveraging (IIFA) operator interval-valued intuitionistic fuzzygeometric (IIFG) operator Gaussian interval-valued intu-itionistic fuzzy ordered weighted geometric (Gaussian-IIFOWG) operator and averaging (Gaussian-IIFOWA) oper-ator power interval-valued intuitionistic fuzzy orderedweighted geometric (P-IIFOWG) operator and averaging(Gaussian-IIFOWA) operator and generalized IIFA (GIIFA)operator and generalized IIFG (GIIFG) operator Further-more an approach based on the proposed operators is devel-oped and applied to solve a practical MAGDM problem con-cerning exploitation investment evaluation of tourist spotsThis approach can give a more completely view of decisionproblems with decision information aggregation dependingon input arguments and can also be suitable for solving othergroup decisionmaking problems including supplier selectiondecision making strategic management decision makinghuman resourcemanagement and emergency solutions eval-uation



2 Preliminaries




119860 = ⟨119909 120583119860 (119909) ]119860 (119909)⟩ | 119909 isin 119883 (1)


119860means a


]119860(119909) le 1 120583


119860 = 120583119860(119909) = ]





119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883

= ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] []

119871

119860(119909) ]

119880

119860(119909)]⟩ | 119909 isin 119883

(2)

where 0 le 120583119871

119860(119909) le 120583

119880

119860(119909) le 1 0 le ]119871

119860(119909) le ]119880

119860(119909) le 1

0 le 120583119880

119860(119909) + ]119880

119860(119909) le 1 for all 119909 isin 119883




119860(119909) and ]



120583119860(119909) = 120583

119871

119860(119909) = 120583

119880

119860(119909) ]

119860(119909) = ]

119871

119860(119909) = ]

119880

119860(119909)

(3)









120582] [119888120582 119889120582])

(4) 120582 = ([119886120582 119887120582] [1 minus (1 minus 119888)

120582 1 minus (1 minus 119889)

120582])



119860 = ⟨119909119894 120583119860(119909119894) ]119860(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119860(119909119894) 120583119880

119860(119909119894)] []119871

119860(119909119894) ]119880

119860(119909119894)]⟩ | 119909

119894isin 119883

119861 = ⟨119909119894 120583119861(119909119894) ]119861(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119861(119909119894) 120583119880

119861(119909119894)] []119871

119861(119909119894) ]119880

119861(119909119894)]⟩ | 119909

119894isin 119883

(4)

then we can have


1198631(119860 119861)

= (1

6119899

119899

sum

119894=1

[(120583119871

119860(119909119894) minus 120583119871

119861(119909119894))2

+ (120583119880

119860(119909119894) minus 120583119880

119861(119909119894))2

+ (]119871

119860(119909119894) minus ]119871

119861(119909119894))2

+ (]119880

119860(119909119894) minus ]119880

119861(119909119894))2

+ (120587119871

119860(119909119894) minus 120587119871

119861(119909119894))2

+ (120587119880

119860(119909119894) minus 120587119880

119861(119909119894))2

])

12

(5)



1198632(119860 119861)

=1

6119899

119899

sum

119894=1

10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(6)


1198633(119860 119861)

=1

119899

119899

sum

119894=1

max 10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583

119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(7)



[]119871 ]119880]) 1

= ([120583119871

1 120583119880

1] []1198711 ]1198801]) and

2= ([120583

119871

2 120583119880

2] []1198712


119871+


(12)(120583119871+ 120583119880+ ]119871 + ]119880) and

if 119904(1) lt 119904(

2) then

1is smaller than

2 1lt 2

if 119904(1) gt 119904(

2) then

1is greater than

2 1gt 2

if 119904(1) = 119904(

2) then

if ℎ(1) lt ℎ(

2) then

1is smaller than

2 1lt

2

if ℎ(1) gt ℎ(

2) then

1is greater than

2 1gt

2

if ℎ(1) = ℎ(

2) then

1and

2represent the






1 1205962 120596

119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWA120596(1 2


119899119899 (8)



IIFWA120596(1 2

119899)

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119887119895)120596119895]

]

[

[

119899

prod

119895=1

119888120596119895

119895

119899

prod

119895=1

119889120596119895

119895]

]

)

(9)



IIFWA120596(1 2

119899)

=1

1198991oplus

1


1

119899119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

]

]

[

[

119899

prod

119895=1

1198881119899

119895

119899

prod

119895=1

1198891119899

119895]

]

)

= IIFA (1 2

119899)

(10)




119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWG120596(1 2

119899)

= 1205961

1otimes 1205962


120596119899

119899

= ([

[

119899

prod

119895=1

119886120596119895

119895

119899

prod

119895=1

119887120596119895

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119889119895)120596119895]

]

)

(11)



IIFWG120596(1 2

119899)

= 1119899

1otimes 1119899


1119899

119899

= ([

[

119899

prod

119895=1

1198861119899

119895

119899

prod

119895=1

1198871119899

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119889119895)1119899

]

]

)

= IIFG (1 2

119899)

(12)






1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and

119908119895isin [0 1] then

IIFOWA119908(1 2


119899120573119899(13)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWA119908(1 2

119899)

=1

1198991205731oplus

1


1

119899120573119899

= IIFA119908(1 2

119899)

(14)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin [0 1]

then

IIFOWG119908(1 2

119899) = 1205731199081

1otimes 1205731199082


119908119899

119899 (15)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWG119908(1 2

119899)

= 1205731119899

1otimes 1205731119899


1119899

119899

= IIFG119908(1 2

119899)

(16)





=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)







1 2

119899) and


rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886


of (1 2


1 2

119899can be


120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)







1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 Preliminaries




119860 = ⟨119909 120583119860 (119909) ]119860 (119909)⟩ | 119909 isin 119883 (1)


119860means a


]119860(119909) le 1 120583


119860 = 120583119860(119909) = ]





119860 = ⟨119909 120583119860(119909) ]

119860(119909)⟩ | 119909 isin 119883

= ⟨119909 [120583119871

119860(119909) 120583

119880

119860(119909)] []

119871

119860(119909) ]

119880

119860(119909)]⟩ | 119909 isin 119883

(2)

where 0 le 120583119871

119860(119909) le 120583

119880

119860(119909) le 1 0 le ]119871

119860(119909) le ]119880

119860(119909) le 1

0 le 120583119880

119860(119909) + ]119880

119860(119909) le 1 for all 119909 isin 119883




119860(119909) and ]



120583119860(119909) = 120583

119871

119860(119909) = 120583

119880

119860(119909) ]

119860(119909) = ]

119871

119860(119909) = ]

119880

119860(119909)

(3)









120582] [119888120582 119889120582])

(4) 120582 = ([119886120582 119887120582] [1 minus (1 minus 119888)

120582 1 minus (1 minus 119889)

120582])



119860 = ⟨119909119894 120583119860(119909119894) ]119860(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119860(119909119894) 120583119880

119860(119909119894)] []119871

119860(119909119894) ]119880

119860(119909119894)]⟩ | 119909

119894isin 119883

119861 = ⟨119909119894 120583119861(119909119894) ]119861(119909119894)⟩ | 119909119894isin 119883

= ⟨119909119894 [120583119871

119861(119909119894) 120583119880

119861(119909119894)] []119871

119861(119909119894) ]119880

119861(119909119894)]⟩ | 119909

119894isin 119883

(4)

then we can have


1198631(119860 119861)

= (1

6119899

119899

sum

119894=1

[(120583119871

119860(119909119894) minus 120583119871

119861(119909119894))2

+ (120583119880

119860(119909119894) minus 120583119880

119861(119909119894))2

+ (]119871

119860(119909119894) minus ]119871

119861(119909119894))2

+ (]119880

119860(119909119894) minus ]119880

119861(119909119894))2

+ (120587119871

119860(119909119894) minus 120587119871

119861(119909119894))2

+ (120587119880

119860(119909119894) minus 120587119880

119861(119909119894))2

])

12

(5)



1198632(119860 119861)

=1

6119899

119899

sum

119894=1

10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(6)


1198633(119860 119861)

=1

119899

119899

sum

119894=1

max 10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583

119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(7)



[]119871 ]119880]) 1

= ([120583119871

1 120583119880

1] []1198711 ]1198801]) and

2= ([120583

119871

2 120583119880

2] []1198712


119871+


(12)(120583119871+ 120583119880+ ]119871 + ]119880) and

if 119904(1) lt 119904(

2) then

1is smaller than

2 1lt 2

if 119904(1) gt 119904(

2) then

1is greater than

2 1gt 2

if 119904(1) = 119904(

2) then

if ℎ(1) lt ℎ(

2) then

1is smaller than

2 1lt

2

if ℎ(1) gt ℎ(

2) then

1is greater than

2 1gt

2

if ℎ(1) = ℎ(

2) then

1and

2represent the






1 1205962 120596

119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWA120596(1 2


119899119899 (8)



IIFWA120596(1 2

119899)

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119887119895)120596119895]

]

[

[

119899

prod

119895=1

119888120596119895

119895

119899

prod

119895=1

119889120596119895

119895]

]

)

(9)



IIFWA120596(1 2

119899)

=1

1198991oplus

1


1

119899119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

]

]

[

[

119899

prod

119895=1

1198881119899

119895

119899

prod

119895=1

1198891119899

119895]

]

)

= IIFA (1 2

119899)

(10)




119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWG120596(1 2

119899)

= 1205961

1otimes 1205962


120596119899

119899

= ([

[

119899

prod

119895=1

119886120596119895

119895

119899

prod

119895=1

119887120596119895

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119889119895)120596119895]

]

)

(11)



IIFWG120596(1 2

119899)

= 1119899

1otimes 1119899


1119899

119899

= ([

[

119899

prod

119895=1

1198861119899

119895

119899

prod

119895=1

1198871119899

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119889119895)1119899

]

]

)

= IIFG (1 2

119899)

(12)






1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and

119908119895isin [0 1] then

IIFOWA119908(1 2


119899120573119899(13)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWA119908(1 2

119899)

=1

1198991205731oplus

1


1

119899120573119899

= IIFA119908(1 2

119899)

(14)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin [0 1]

then

IIFOWG119908(1 2

119899) = 1205731199081

1otimes 1205731199082


119908119899

119899 (15)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWG119908(1 2

119899)

= 1205731119899

1otimes 1205731119899


1119899

119899

= IIFG119908(1 2

119899)

(16)





=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)







1 2

119899) and


rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886


of (1 2


1 2

119899can be


120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)







1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198632(119860 119861)

=1

6119899

119899

sum

119894=1

10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

+10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(6)


1198633(119860 119861)

=1

119899

119899

sum

119894=1

max 10038161003816100381610038161003816120583119871

119860(119909119894) minus 120583119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120583119880

119860(119909119894) minus 120583

119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119871

119860(119909119894) minus ]119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816]119880

119860(119909119894) minus ]119880

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119871

119860(119909119894) minus 120587119871

119861(119909119894)10038161003816100381610038161003816

10038161003816100381610038161003816120587119880

119860(119909119894) minus 120587119880

119861(119909119894)10038161003816100381610038161003816

(7)



[]119871 ]119880]) 1

= ([120583119871

1 120583119880

1] []1198711 ]1198801]) and

2= ([120583

119871

2 120583119880

2] []1198712


119871+


(12)(120583119871+ 120583119880+ ]119871 + ]119880) and

if 119904(1) lt 119904(

2) then

1is smaller than

2 1lt 2

if 119904(1) gt 119904(

2) then

1is greater than

2 1gt 2

if 119904(1) = 119904(

2) then

if ℎ(1) lt ℎ(

2) then

1is smaller than

2 1lt

2

if ℎ(1) gt ℎ(

2) then

1is greater than

2 1gt

2

if ℎ(1) = ℎ(

2) then

1and

2represent the






1 1205962 120596

119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWA120596(1 2


119899119899 (8)



IIFWA120596(1 2

119899)

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119887119895)120596119895]

]

[

[

119899

prod

119895=1

119888120596119895

119895

119899

prod

119895=1

119889120596119895

119895]

]

)

(9)



IIFWA120596(1 2

119899)

=1

1198991oplus

1


1

119899119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

]

]

[

[

119899

prod

119895=1

1198881119899

119895

119899

prod

119895=1

1198891119899

119895]

]

)

= IIFA (1 2

119899)

(10)




119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWG120596(1 2

119899)

= 1205961

1otimes 1205962


120596119899

119899

= ([

[

119899

prod

119895=1

119886120596119895

119895

119899

prod

119895=1

119887120596119895

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119889119895)120596119895]

]

)

(11)



IIFWG120596(1 2

119899)

= 1119899

1otimes 1119899


1119899

119899

= ([

[

119899

prod

119895=1

1198861119899

119895

119899

prod

119895=1

1198871119899

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119889119895)1119899

]

]

)

= IIFG (1 2

119899)

(12)






1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and

119908119895isin [0 1] then

IIFOWA119908(1 2


119899120573119899(13)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWA119908(1 2

119899)

=1

1198991205731oplus

1


1

119899120573119899

= IIFA119908(1 2

119899)

(14)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin [0 1]

then

IIFOWG119908(1 2

119899) = 1205731199081

1otimes 1205731199082


119908119899

119899 (15)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWG119908(1 2

119899)

= 1205731119899

1otimes 1205731119899


1119899

119899

= IIFG119908(1 2

119899)

(16)





=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)







1 2

119899) and


rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886


of (1 2


1 2

119899can be


120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)







1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119899)119879 with 120596

119895isin [0 1] and

sum119899

119895=1120596119895= 1 such that

IIFWG120596(1 2

119899)

= 1205961

1otimes 1205962


120596119899

119899

= ([

[

119899

prod

119895=1

119886120596119895

119895

119899

prod

119895=1

119887120596119895

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)120596119895 1 minus

119899

prod

119895=1

(1 minus 119889119895)120596119895]

]

)

(11)



IIFWG120596(1 2

119899)

= 1119899

1otimes 1119899


1119899

119899

= ([

[

119899

prod

119895=1

1198861119899

119895

119899

prod

119895=1

1198871119899

119895]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888119895)1119899

1 minus

119899

prod

119895=1

(1 minus 119889119895)1119899

]

]

)

= IIFG (1 2

119899)

(12)






1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and

119908119895isin [0 1] then

IIFOWA119908(1 2


119899120573119899(13)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWA119908(1 2

119899)

=1

1198991205731oplus

1


1

119899120573119899

= IIFA119908(1 2

119899)

(14)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin [0 1]

then

IIFOWG119908(1 2

119899) = 1205731199081

1otimes 1205731199082


119908119899

119899 (15)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



IIFOWG119908(1 2

119899)

= 1205731119899

1otimes 1205731119899


1119899

119899

= IIFG119908(1 2

119899)

(16)





=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)







1 2

119899) and


rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886


of (1 2


1 2

119899can be


120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)







1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

(17)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWA120582(1 2

119899) = (

119899

oplus119895=1

(1

119899120573120582

119895))

1120582

= GIIFA120582(1 2

119899)

(18)




=



1 1199082 119908

119899)119879 sum119899119895=1

119908119895= 1 and 119908

119895isin

[0 1] 120582 gt 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

(19)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895



GIIFOWG120582(1 2

119899) =

1

120582(119899

otimes119895=1

(120582120573119895)1119899

)

= GIIFG120582(1 2

119899)

(20)







1 2

119899) and


rator with 120596 = (1119899 1119899 1119899)119879 where

119886120583= 1 minus

119899

prod

119895=1

(1 minus 119886119895)1119899

119887120583= 1 minus

119899

prod

119895=1

(1 minus 119887119895)1119899

119888120583=

119899

prod

119895=1

1198881119899

119895 119889

120583=

119899

prod

119895=1

1198891119899

119895

(21)




=

([119886119895 119887119895] [119888119895 119889119895]) 120583 = ([119886


of (1 2


1 2

119899can be


120590 = radic1

119899

119899

sum

119895=1

(119889 (119895 120583))2

(22)







1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 1205962 120596

119899)119879 can be

defined as

120596119895=

1


2

119895 = 1 2 119899 (23)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


119899

119895=1120596119895= 1 are commonly


120596119895=


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2 119895 = 1 2 119899 (24)




119899)119879


119899

119894=1120596119894= 1 then


119899)

= (120596120590(1)

120582

120590(1)oplus 120596120590(2)

120582


120590(119899)120582

120590(119899))1120582

= (


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

1

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

2



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573120582

119899)

1120582

= (1


2

120573120582

1oplus

1


2

120573120582

2



2

120573120582

119899)

1120582

times ((

119899

sum

119895=1

1


2

)

1120582

)

minus1

(25)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899



1 1205962 120596

119899)119879


119899

119894=1120596119894= 1 then


119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

=1

120582((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(26)


where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where (1205731 1205732 120573


1 2

119899)with

120573119895minus1

ge 120573119895for all 119895 = 2 119899

Let 119894

= ([119886(119894)

119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894

= ([119886120573(119894)

119887120573(119894)

]

[119888120573(119894)

119889120573(119894)



[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(27)


[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

]

]

)

(28)

1


2

1205731205821 oplus

1


2


1


2

120573120582119899

=1


2

1205821 oplus

1


2

1205822



2

120582119899

(

119899

sum

119895=1

1


2

)

1120582

= (

119899

sum

119895=1

1


2

)

1120582

((1205821205731)(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

otimes (1205821205732)(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2


minus1198892(120573119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ((1205821)(1radic2120587120590)119890

minus1198892(1minus120583)2120590

2

otimes (1205822)(1radic2120587120590)119890

minus1198892(2minus120583)2120590

2


minus1198892(119899minus120583)2120590

2

)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(119895minus120583)2120590

2

(29)




119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

(30)


119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


) (31)


Theorem 17 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then



2

lowast


tion of (1 2

119899) then

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899)

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(32)


119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860120596120582

(1 2

119899) =

119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120596120582

(1 2

119899) = 120572

(33)


minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(34)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(35)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then


119866119886119906119904119904119894119886119899-119868119868119865119874119882119860(1 2

119899) =


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205731

oplus


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

21205732



2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2120573119899

=1


2

1205731oplus

1


2

1205732



2

120573119899

times (

119899

sum

119895=1

1


2

)

minus1


= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

[119888(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119889(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)])

(36)

119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2 119899) = 120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1

otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

119899

= (120573(1radic2120587120590)119890

minus1198892(1205731minus120583)2120590

2

1otimes 120573(1radic2120587120590)119890

minus1198892(1205732minus120583)2120590

2

2


minus1198892(120573119899minus120583)2120590

2

119899)

1sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

= ([

[

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

119899

prod

119895=1

119887(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

]

]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

1 minus

119899

prod

119895=1

(1 minus 119889120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

]

]

)

(37)






(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then





119895gt 0 119895 = 1 2 119899 and

sum119899

119895=1120582119895= 1 then

119899

prod

119895=1

119909120582119895

119895le

119899

sum

119895=1

120582119895119909119895 (38)


119899


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


tion of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899

and let 120596 = (1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)

(2) 119866119886119906119904119904119894119886119899-119868119868119865119874119882119866(1 2

119899) le 119866119886119906119904119904119894119886119899-

119866119868119868119865119874119882119860120582(1 2

119899)

(3) 119866119886119906119904119904119894119886119899-119866119868119868119865119874119882119866120582(1 2

119899) le 119866119886119906119904119904119894119886119899-

119868119868119865119874119882119860(1 2

119899)


119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

le

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120573(119895)

= 1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))

le 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(a)

119899

prod

119895=1

119886(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

120573(119895)

= (

119899

prod

119895=1

(119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2119886120582

120573(119895))

1120582

= (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120582

120573(119895)))

1120582

le (1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

(b)


1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)


2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

le 1 minus (1 minus

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590


120573(119895))120582

))

1120582

= 1 minus (

119899

sum

119895=1


2

sum119899

119895=1(1radic2120587120590) 119890

minus1198892(120573119895minus120583)2120590

2(1 minus 119886

120573(119895))120582

)

1120582

le 1 minus (

119899

prod

119895=1

(1 minus 119886120573(119895)

)120582(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

)

1120582

= 1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

sum119899119895=1(1radic2120587120590)119890

minus1198892(120573119895minus120583)2120590

2

(c)


120573(119895) 119888120573(119895)

and 119889120573(119895)


119899

otimes119895=1

(120573120596119895

119895) le119899

oplus119895=1

(120596119895120573119895)

119899

otimes119895=1

(120573120596119895

119895) le (

119899

oplus119895=1

(120596119895120573120582

119895))

1120582

1

120582(119899

otimes119895=1

(120582120573119895)119908119895) le119899

oplus119895=1

(120596119895120573119895)

(39)



119895(119895 = 1 2 6)


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(40)


= ([04273 0664] [0 03238]) 120590 = 01271 (41)


120596 = (1205961 1205962 120596

6) (42)

where 1205961= 01391 120596

2= 0128 120596

3= 01867 120596

4= 0192

1205965= 01867 and 120596

6= 01675



119899)

= ([04676 06846] [00 03083])


119899)

= ([0381 06038] [02166 03554])

(43)






Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Sup (119895) =

1

119899 minus 1

119899

sum

119896=1119895 = 119896

(1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583))

=1

119899 minus 1

119899

sum

119896=1119895 = 119896

Sup (119895 119896) + Sup (

119895 120583)

(44)







properties

(1) Sup(119894 119895) isin [0 1] Sup(

119894 120583) isin [0 1]

(2) Sup(119894 119895) = Sup(

119895 119894)

(3) Sup(119894 119895) ge Sup(

119904 119901) if 119889(

119894 119895) lt 119889(

119904 119901) and

Sup(119894 120583) ge Sup(

119895 120583) if 119889(

119894 120583) lt 119889(

119895 120583) where



120596119895=

Sup (119895)

sum119899

119895=1Sup (

119895)

119895 = 1 2 119899 (45)


And let (1205731 1205732 120573


1 2

119899) with 120573



120596120573(119895)

=

Sup (120573119895)

sum119899

119895=1Sup (120573

119895)

119895 = 1 2 119899 (46)



1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWA (1 2

119899)

= (

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

120573120582

1oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

120573120582

2


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573120582

119899)

1120582

= (

Sup (1205731) 120573120582

1oplus Sup (120573

2) 120573120582


119899) 120573120582

119899

sum119899

119895=1Sup (120573

119895)

)

1120582

(47)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899


1 1205962 120596

119899)119879 is


119899

119894=1120596119894=

1 then

P-GIIFOWG (1 2

119899)

=1

120582((1205821205731)Sup(1205731)sum

119899119895=1 Sup(120573119895)

otimes (1205821205732)Sup(1205732)sum

119899119895=1 Sup(120573119895)


119899119895=1 Sup(120573119895)

)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

(48)

where (1205731 1205732 120573


1 2

119899)

with 120573119895minus1

ge 120573119895for all 119895 = 2 119899

Given 119894= ([119886

(119894) 119887(119894)

] [119888(119894)

119889(119894)

]) 120573119894= ([119886

120573(119894) 119887120573(119894)

]

[119888120573(119894)

119889120573(119894)


P-GIIFOWA (1 2

119899) = ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

120596120573(119895)

)

1120582

]

]


[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

)

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(49)

P-GIIFOWG (1 2

119899) = ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

120596120573(119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

120596120573(119895)

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

120596120573(119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

120596120573(119895)

)

1120582

]

]

)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582)Sup(120573119895)sum

119899119895=1 Sup(120573119895))

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

)

1120582

]

]

)

(50)


By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










By (45) we can have

P-GIIFOWA (1 2

119899) = (120596

120573(1)120573120582

1oplus 120596120573(2)

120573120582


120573(119899)120573120582

119899)1120582

= (

sum119899

119895=1Sup(120573

119895)120573120582

119895

sum119899

119895=1Sup(120573

119895)

)

1120582

= (

sum119899

119895=1(sum119899

119896=1119895 = 119896((1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))) 120573

120582

119895

sum119899

119895=1sum119899

119896=1119895 = 119896(1 minus 119889 (120573

119895 120573119896)) + (1 minus 119889 (120573

119895 120583))

)

1120582

(51)

P-GIIFOWG (1 2

119899) =

1

120582((1205821205731)120596120573(1)

otimes (1205821205732)120596120573(2)


)

=1

120582((1205821205731)Sup(1205731)

otimes (1205821205732)Sup(1205732)


)

1sum119899119895=1 Sup(120573119895)

=1

120582(

119899

prod

119895=1


119899119895=1 sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)

=1

120582(

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

)


(52)

Since

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583))) 120573

119895

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

119895

119899

prod

119895=1

(120582120573119895)sum119899119896=1119895 = 119896(1minus119889(120573119895 120573119896))+(1minus119889(120573119895 120583))

=

119899

prod

119895=1

(120582119895)sum119899119896=1119895 = 119896(1minus119889(119895 119896))+(1minus119889(119895 120583))

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (120573119895 120573119896)) + (1 minus 119889 (120573

119895 120583)))

=

119899

sum

119895=1

119899

sum

119896=1119895 = 119896

((1 minus 119889 (119895 119896)) + (1 minus 119889 (

119895 120583)))

(53)

then we can have

P-GIIFOWA (1 2

119899)

= (120596(1)

120582

1oplus 120596(2)

120582


(119899)120582

119899)1120582

P-GIIFOWG (1 2

119899)

=1

120582((1205821)120596(1)

otimes (1205822)120596(2)


)

(54)


Theorem 25 Let (1 2



119899) is


119899) with 120573

119895minus1ge 120573119895for all 119895 =

2 119899 If Sup(120573119894) ge Sup(120573

119895) then 120596

120573(119894)ge 120596120573(119895)

Theorem 26 Let 119895= ([119886119895 119887119895] [119888119895 119889119895]) (119895 = 1 2 119899) be


1 1205962 120596




119899

119895=1120596119895= 1 then




2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2

lowast


tion of (1 2

119899) then

119875-119866119868119868119865119874119882119860120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119860120596120582

(1 2 119899)

119875-119866119868119868119865119874119882119866120596120582

(lowast

1 lowast

2

lowast

119899)

= 119875-119866119868119868119865119874119882119866120596120582

(1 2

119899)

(55)


119875-119866119868119868119865119874119882119860120596120582

(1 2

119899) = 120572

119875-119866119868119868119865119874119882119866120596120582

(1 2

119899) =

(56)


minusle 119875-119866119868119868119865119874119882119860

120596120582(1 2

119899) le +

minusle 119875-119866119868119868119865119874119882119866

120596120582(1 2

119899) le +

(57)

where

minus= ([min

119895

(119886119895) min119895

(119887119895)] [max

119895

(119888119895) max119895

(119889119895)])

+= ([max

119895

(119886119895) max119895

(119887119895)] [min

119895

(119888119895) min119895

(119889119895)])

(58)


= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573


120596 = (1205961 1205962 120596



119895isin [0 1] sum119899

119895=1120596119895= 1 then


119875-119868119868119865119874119882119860(1 2

119899)

=

Sup (1205731)

sum119899

119895=1Sup (120573

119895)

1205731oplus

Sup (1205732)

sum119899

119895=1Sup (120573

119895)

1205732


Sup (120573119899)

sum119899

119895=1Sup (120573

119895)

120573119899

= ([

[

1 minus

119899

prod

119895=1

(1 minus 119886120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119887120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)]

]

[119888Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119889

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)])

(59)

119875-119868119868119865119874119882119866(1 2

119899)

= (120573Sup(1205731)1

otimes 120573Sup(1205732)2


)

1sum119899119895=1 Sup(120573119895)

= ([119886Sup(120573119895)sum

119899119895=1 Sup(120573119895)

120573(119895) 119887

Sup(120573119895)sum119899119895=1 Sup(120573119895)

120573(119895)]

[

[

1 minus

119899

prod

119895=1

(1 minus 119888120573(119895)

)Sup(120573119895)sum

119899119895=1 Sup(120573119895)

1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

Sup(120573119895)sum119899119895=1 Sup(120573119895)

]

]

)

(60)




(4) if 120596 = (1119899 1119899 1119899)119879 and 120582 = 1 then the P-





= ([119886(119895)

119887(119895)

] [119888(119895)

119889(119895)

]) 120573119895

=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


1 1205732 120573


of (1 2

119899) with 120573

119895minus1ge 120573119895for all 119895 = 2 119899 120596 =

(1205961 1205962 120596




119899

119895=1120596119895= 1 then

(1) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119868119868119865119874119882119860(

1 2

119899)

(2) 119875-119868119868119865119874119882119866(1 2

119899) le 119875-119866119868119868119865119874119882119860

120582(1 2

119899)

(3) 119875-119866119868119868119865119874119882119866120582(1 2

119899) le 119875-119868119865119874119882119860(

1 2

119899)




119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119895(119895 = 1 2 6) to pro-


1= ([05 06] [02 04])

2= ([03 06] [03 04])

3= ([04 07] [02 03])

4= ([03 05] [01 04])

5= ([04 07] [02 03])

6= ([06 08] [00 02])

(61)


120596 = (1205961 1205962 1205963 1205964 1205965 1205966) (62)

where 1205961= 01653 120596

2= 0164 120596

3= 01715 120596

4= 01651

1205965= 01715 and 120596

6= 01625


P-GIIFOWA (1 2

119899)

= ([04691 06828] [00 0299])

P-GIIFOWG (1 2

119899)

= ([03808 06049] [02225 03422])

(63)

Theorem 30 Let 119895= ([119886

(119895) 119887(119895)

] [119888(119895)

119889(119895)

]) and 120573119895=

([119886120573(119895)

119887120573(119895)

] [119888120573(119895)

119889120573(119895)


120582

or 119866119868119868119865119874119882119866120582on 119895and 120573


(1-a) if 119888120573(119895)

= 0 120574 = 119866119868119868119865119874119882119860120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119888 = 0(1-b) if 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119889 = 0(1-c) if 119888

120573(119895)= 0 and 119889

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119888 = 119889 = 0(2-a) if 119886

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119886 = 0(2-b) if 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) = ([119886 119887] [119888 119889])

then 119887 = 0(2-c) if 119886

120573(119895)= 0 and 119887

120573(119895)= 0 120574 = 119866119868119868119865119874119882119860

120582(119895 120573119895) =

([119886 119887] [119888 119889]) then 119886 = 119887 = 0



GIIFOWA120582(1 2

119899)

= (119899

oplus119895=1

(119908119895120573120582

119895))

1120582

= ([

[

(1 minus

119899

prod

119895=1

(1 minus 119886120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119887120582

120573(119895))

119908119895

)

1120582

]

]

[

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119888120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119889120573(119895)

)120582

)

119908119895

)

1120582

]

]

)

= ([119886 119887] [0 119889])

(64)



= 0 then

GIIFOWG120582(1 2

119899)

=1

120582(119899

otimes119895=1

(120582120573119895)119908119895)

= ([

[

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119886120573(119895)

)120582

)

119908119895

)

1120582

1 minus (1 minus

119899

prod

119895=1

(1 minus (1 minus 119887120573(119895)

)120582

)

119908119895

)

1120582

]

]

[

[

(1 minus

119899

prod

119895=1

(1 minus 119888120582

120573(119895))

119908119895

)

1120582

(1 minus

119899

prod

119895=1

(1 minus 119889120582

120573(119895))

119908119895

)

1120582

]

]

)

= ([0 119887] [119888 119889])

(65)



= 0 or 119889120573(119895)





120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120573(119895)= 0 or 119887

120573(119895)= 0





Let 119883 = 1199091 1199092 119909


1198921 1198922 119892


1 1205962 120596

119898119879 the


119895=1120596119895=

1 119863 = 1198891 1198892 119889


(120582(1)

120582(2)




(119896) (119896)

= (119903(119896)

119894119895)119899times119898

=

(120583(119896)

119894119895 ](119896)119894119895

)119899times119898

= ([120583119871(119896)

119894119895 120583119880(119896)

119894119895] []119871(119896)119894119895

]119880(119896)119894119895

])119899times119898

where [120583119871(119896)119894119895

120583119880(119896)


119894satisfies

the attribute 119892119895 []119871(119896)119894119895

]119880(119896)119894119895



119892119895(119895 = 1 2 119898)


= (120596(119896)

1

120596(119896)

2 120596

(119896)




(119896) accordingto (24) or (46)







119894and


Step 6 End




1) orientability

(1198922) monopoly (119892



5)









(119896)=

(119903(119896)

119894119895)119899times119898




120596(1)

= (02443 0159 02682 01661 01623)119879

120596(2)

= (01719 02185 03227 01169 017)119879

120596(3)

= (01613 02245 02058 02721 01363)119879

(66)


120596(1)

= (02022 0197 02046 01976 01985)119879

120596(2)

= (01982 02030 02072 01901 02015)119879

120596(3)

= (01972 02041 02029 02069 01889)119879

(67)








1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

1198921

1198922

1198923

1198924

1198925

1199091

([04 05] [03 04]) ([05 06] [01 02]) ([06 07] [02 03]) ([07 08] [01 02]) ([07 08] [00 02])

1199092

([06 08] [01 02]) ([05 06] [03 04]) ([04 05] [03 04]) ([04 06] [03 04]) ([04 07] [01 03])

1199093

([05 06] [03 04]) ([05 07] [01 02]) ([05 06] [03 04]) ([03 04] [02 05]) ([06 07] [02 03])

1199094

([05 06] [03 04]) ([07 08] [00 01]) ([04 05] [02 04]) ([05 07] [01 02]) ([05 07] [02 03])

1199095

([04 07] [02 03]) ([05 06] [02 04]) ([03 06] [03 04]) ([06 08] [01 02]) ([04 05] [02 03])


2

1198921

1198922

1198923

1198924

1198925

1199091

([04 06] [03 04]) ([05 07] [00 02]) ([05 06] [02 04]) ([06 08] [01 02]) ([04 07] [02 03])

1199092

([05 08] [01 02]) ([03 05] [02 03]) ([03 06] [02 04]) ([04 05] [02 04]) ([03 06] [02 03])

1199093

([05 06] [00 01]) ([05 08] [01 02]) ([04 07] [02 03]) ([02 04] [02 03]) ([05 08] [00 02])

1199094

([05 07] [01 03]) ([04 06] [00 01]) ([03 05] [02 04]) ([07 09] [00 01]) ([03 05] [02 02])

1199095

([07 08] [00 01]) ([04 06] [00 02]) ([04 07] [02 03]) ([03 05] [01 03]) ([06 07] [01 02])


3

1198921

1198922

1198923

1198924

1198925

1199091

([03 04] [04 05]) ([08 09] [01 01]) ([07 08] [01 02]) ([04 05] [03 05]) ([02 04] [03 06])

1199092

([05 07] [01 03]) ([04 07] [02 03]) ([04 05] [02 02]) ([06 08] [01 02]) ([02 03] [00 01])

1199093

([02 04] [01 02]) ([04 05] [02 04]) ([05 08] [00 01]) ([04 06] [02 03]) ([05 06] [02 03])

1199094

([07 08] [00 02]) ([05 07] [01 02]) ([06 07] [01 03]) ([04 05] [01 02]) ([07 08] [01 02])

1199095

([05 06] [02 04]) ([05 08] [00 02]) ([04 07] [02 03]) ([03 06] [02 03]) ([07 08] [00 01])


1198891

1198892

1198893

1199091

([05836 06885] [00 02642]) ([04815 06701] [00 03019]) ([05666 06954] [01959 02958])

1199092

([04721 06578] [01919 03223]) ([03511 06173] [01775 03175]) ([04574 06650] [00 02128])

1199093

([04900 06099] [02205 03549]) ([04397 07080] [00 02122]) ([04095 06107] [00 02391])

1199094

([05159 06539] [00 02730]) ([04215 06386] [00 02126]) ([05689 06945] [00 02174])

1199095

([04321 06554] [01988 03172]) ([04938 06837] [00 02122]) ([04694 07064] [00 02470])


1198891

1198892

1198893

1199091

([05951 07002] [00 02500]) ([04845 06879] [00 02874]) ([05457 06792] [02024 03094])

1199092

([04667 06562] [01932 03284]) ([03641 06194] [01743 03104]) ([04322 06338] [00 02047])

1199093

([04887 06132] [02058 03445]) ([04322 06925] [00 02048]) ([04104 06071] [00 02337])

1199094

([05307 06741] [00 02507]) ([04598 06820] [00 01905]) ([05970 07189] [00 02175])

1199095

([04486 06560] [01895 03109]) ([05037 06766] [00 02048]) ([05006 07153] [00 02344])


1198891

1198892

1198893

1199091

([05553 06574] [01658 02805]) ([04733 06588] [01677 03217]) ([04555 05881] [02392 03904])

1199092

([04576 06285] [02247 03400]) ([03387 05930] [01836 03307]) ([04213 06035] [01321 02279])

1199093

([04732 05894] [02388 03752]) ([04180 06725] [01141 02302]) ([03861 05724] [01463 02724])

1199094

([04969 06292] [01818 03117]) ([03851 05905] [01202 02588]) ([05400 06647] [00846 02217])

1199095

([04104 06345] [02129 03299]) ([04562 06658] [00972 02302]) ([04351 06871] [01329 02719])



1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198891

1198892

1198893

1199091

([05669 06689] [01473 02655]) ([04735 06745] [01663 03070]) ([04247 05680] [02473 04063])

1199092

([04537 06317] [02258 03443]) ([03506 05913] [01811 03239]) ([03927 05645] [01240 02235])

1199093

([04687 05886] [02245 03684]) ([04011 06443] [01042 02234]) ([03819 05663] [01424 02661])

1199094

([05105 06503] [01671 02907]) ([04134 06202] [01060 02312]) ([05693 06926] [00810 02218])

1199095

([04270 06319] [02032 03244]) ([04592 06535] [00838 02234]) ([04636 06959] [01244 02662)



11199092

1199093

1199094

1199095


([05481 06859][00 02877])

([04322 06491][00 02718])

([04437 06427][00 02597])

([05125 06664][00 02312])

([04661 06850][00 02544])

P-GIIFOWA andIIFWA

([05442 06882][00 02839])

([04235 06365][00 02673])

([04414 06367][00 02524])

([05394 06951][00 02181])

([04865 06869][00 02450])


([04890 06292][01965 03385])

([04045 06077][01762 02943])

([04203 06061][01660 02930])

([04759 06310][01254 02608])

([04337 06646][01475 02778])

P-GIIFOWG andIIFWG

([04785 06281][01943 03371])

([03964 05921][01728 02919])

([04121 05955][01570 02864])

([05005 06575][01151 02459])

([04510 06634][01372 02719])



2≺ 1199093≺ 1199095≺ 1199091≺ 1199094














5shows











2and 119909

5






15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












15 119904(1199031) = 09965 119904(119903

2) = 06022 119904(119903

3) = 05121 119904(119903

4) = 08839 119904(119903

5) = 05594 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWA

14 119904(1199031) = 09972 119904(119903

2) = 06030 119904(119903

3) = 05128 119904(119903

4) = 08874 119904(119903

5) = 05601 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 09984 119904(119903

2) = 06044 119904(119903

3) = 05141 119904(119903

4) = 08860 119904(119903

5) = 05613 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 1008 119904(119903

2) = 06071 119904(119903

3) = 05167 119904(119903

4) = 08886 119904(119903

5) = 05638 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 1008 119904(119903

2) = 06156 119904(119903

3) = 05245 119904(119903

4) = 08968 119904(119903

5) = 05716 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 10231 119904(119903

2) = 06341 119904(119903

3) = 0541 119904(119903

4) = 09147 119904(119903

5) = 05887 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 10386 119904(119903

2) = 06542 119904(119903

3) = 0558 119904(119903

4) = 09343 119904(119903

5) = 06075 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 1054 119904(119903

2) = 06755 119904(119903

3) = 0575 119904(119903

4) = 09552 119904(119903

5) = 06272 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

5 119904(1199031) = 10688 119904(119903

2) = 06972 119904(119903

3) = 05917 119904(119903

4) = 09767 119904(119903

5) = 06471 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093



15 119904(1199031) = 0841 119904(119903

2) = 05523 119904(119903

3) = 04714 119904(119903

4) = 07077 119904(119903

5) = 05225 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

Gaussian-GIIFOWG

14 119904(1199031) = 08327 119904(119903

2) = 05505 119904(119903

3) = 04701 119904(119903

4) = 0699 119904(119903

5) = 05213 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

13 119904(1199031) = 08215 119904(119903

2) = 05475 119904(119903

3) = 04678 119904(119903

4) = 06873 119904(119903

5) = 05193 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

12 119904(1199031) = 08041 119904(119903

2) = 05413 119904(119903

3) = 04632 119904(119903

4) = 06694 119904(119903

5) = 05152 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

1 119904(1199031) = 07664 119904(119903

2) = 05215 119904(119903

3) = 04486 119904(119903

4) = 06325 119904(119903

5) = 05022 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

2 119904(1199031) = 07059 119904(119903

2) = 04811 119904(119903

3) = 04168 119904(119903

4) = 05795 119904(119903

5) = 04741 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06513 119904(119903

2) = 04449 119904(119903

3) = 03838 119904(119903

4) = 05371 119904(119903

5) = 04459 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

4 119904(1199031) = 06024 119904(119903

2) = 04149 119904(119903

3) = 03513 119904(119903

4) = 05019 119904(119903

5) = 04194 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 056 119904(119903

2) = 03905 119904(119903

3) = 03199 119904(119903

4) = 04725 119904(119903

5) = 03952 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093



15 119904(1199031) = 10344 119904(119903

2) = 05892 119904(119903

3) = 05375 119904(119903

4) = 09417 119904(119903

5) = 05918 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWA

14 119904(1199031) = 1035 119904(119903

2) = 05899 119904(119903

3) = 05383 119904(119903

4) = 09424 119904(119903

5) = 05925 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 10361 119904(119903

2) = 05911 119904(119903

3) = 05398 119904(119903

4) = 09437 119904(119903

5) = 05938 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 10384 119904(119903

2) = 05936 119904(119903

3) = 05427 119904(119903

4) = 09462 119904(119903

5) = 05963 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 10453 119904(119903

2) = 06013 119904(119903

3) = 05517 119904(119903

4) = 0954 119904(119903

5) = 06042 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 10595 119904(119903

2) = 06182 119904(119903

3) = 05704 119904(119903

4) = 09708 119904(119903

5) = 06214 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

3 119904(1199031) = 10741 119904(119903

2) = 06367 119904(119903

3) = 05895 119904(119903

4) = 0989 119904(119903

5) = 064 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 10884 119904(119903

2) = 06564 119904(119903

3) = 06083 119904(119903

4) = 10081 119904(119903

5) = 06594 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 11021 119904(119903

2) = 06767 119904(119903

3) = 06263 119904(119903

4) = 10275 119904(119903

5) = 06788 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093




15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












15 119904(1199031) = 09026 119904(119903

2) = 05437 119904(119903

3) = 04907 119904(119903

4) = 07869 119904(119903

5) = 05531 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

P-GIIFOWG

14 119904(1199031) = 08938 119904(119903

2) = 0542 119904(119903

3) = 04892 119904(119903

4) = 07773 119904(119903

5) = 05518 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

13 119904(1199031) = 08818 119904(119903

2) = 05392 119904(119903

3) = 04867 119904(119903

4) = 07642 119904(119903

5) = 05497 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

12 119904(1199031) = 08631 119904(119903

2) = 05335 119904(119903

3) = 04814 119904(119903

4) = 07442 119904(119903

5) = 05453 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

1 119904(1199031) = 08229 119904(119903

2) = 05154 119904(119903

3) = 04644 119904(119903

4) = 07029 119904(119903

5) = 05313 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

2 119904(1199031) = 07581 119904(119903

2) = 04783 119904(119903

3) = 04271 119904(119903

4) = 06435 119904(119903

5) = 05012 119909

1≻ 1199094≻ 1199092≻ 1199095≻ 1199093

3 119904(1199031) = 06987 119904(119903

2) = 04451 119904(119903

3) = 03884 119904(119903

4) = 05956 119904(119903

5) = 04709 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

4 119904(1199031) = 06453 119904(119903

2) = 04176 119904(119903

3) = 03505 119904(119903

4) = 05555 119904(119903

5) = 04424 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093

5 119904(1199031) = 05987 119904(119903

2) = 0395 119904(119903

3) = 03146 119904(119903

4) = 05217 119904(119903

5) = 04164 119909

1≻ 1199094≻ 1199095≻ 1199092≻ 1199093


6 Conclusions



Acknowledgments


References
















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article some generalized dependent aggregation ...the power generalized interval-valued...

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