research article linguistic intuitionistic fuzzy sets and...

Research ArticleLinguistic Intuitionistic Fuzzy Sets and Application in MAGDM

Huimin Zhang

School of Management Henan University of Technology Zhengzhou 450001 China

Correspondence should be addressed to Huimin Zhang zhm76126com

Received 19 January 2014 Revised 5 April 2014 Accepted 8 April 2014 Published 29 April 2014

Academic Editor Reinaldo Martinez Palhares

Copyright copy 2014 Huimin Zhang 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

To better deal with imprecise and uncertain information in decision making the definition of linguistic intuitionistic fuzzysets (LIFSs) is introduced which is characterized by a linguistic membership degree and a linguistic nonmembership degreerespectively To compare any two linguistic intuitionistic fuzzy values (LIFVs) the score function and accuracy function aredefined Then based on 119905-norm and 119905-conorm several aggregation operators are proposed to aggregate linguistic intuitionisticfuzzy information which avoid the limitations in exiting linguistic operation In addition the desired properties of these linguisticintuitionistic fuzzy aggregation operators are discussed Finally a numerical example is provided to illustrate the efficiency of theproposed method in multiple attribute group decision making (MAGDM)

1 Introduction

Intuitionistic fuzzy set (IFS) [1] which is characterized bya degree of membership and a degree of nonmembershipis a very powerful tool to process vague information Afterthe pioneering study of Atanassov [1] the IFS has capturedmuch attention from researchers in various fields and manyachievements have been made such as entropy measure ofIFS [2ndash7] distance or similarity measure between IFSs [8ndash13] and aggregation operators of IFS [14ndash21] In additionrelated to IFS some authors proposed several other toolsto handle vague and imprecise information whereby twoor more sources of vagueness appear simultaneously [22]Atanassov andGargov [23] introduced the notion of interval-valued intuitionistic fuzzy set (IVIFS) which is characterizedby a membership function and a nonmembership functionwith interval values Torra [24] and Torra and Narukawa [25]gave a definition of hesitant fuzzy set (HFS) which can betterdeal with the situations where several values are possible todetermine the membership of an element Zhu et al [26]defined dual hesitant fuzzy set in terms of two functions thatreturn two sets of membership values and nonmembershipvalues respectively for each element in the domain

Although the foregoing fuzzy tools are suitable fordealing with problems that are defined as quantitative sit-uations [22] uncertainty is often because of the vaguenessof meanings that is used by experts in problems whose

nature is rather qualitative For example for reason of theincreasing complexity of the decision making environmenttime pressure and the lack of data or knowledge aboutthe problem domain in the process of decision makingunder intuitionistic fuzzy environment a decision makermay have difficulty in expressing the degree of membershipand nonmembership as exact values whereas he or she maythink the use of linguistic values is more straightforwardand suitable to express the degree of membership and non-membership Similar to IFS linguistic intuitionistic fuzzy set(LIFS) is characterized by a linguisticmembership degree anda linguistic nonmembership degree respectively By usingthe LIFS decision makers are able to consider a linguistichesitancy degree in the belongingness of an element to a setwhere they cannot easily express their subjective judgmentwith a single linguistic term

The outline of the paper is organized as follows Thefollowing section presents a brief introduction to the basicknowledge that will be used in the definition of LIFSSection 3 gives the concept of LIFS and constructs thescore function and accuracy function for LIFS Section 4develops several aggregation operators for LIFS Section 5proposes a MAGDM method with linguistic intuitionisticfuzzy information In Section 6 an application of the newapproach is presented Finally conclusions are provided inSection 7

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 432092 11 pageshttpdxdoiorg1011552014432092

2 Journal of Applied Mathematics

2 Preliminaries

In the following some basic concepts and knowledge relatedto IFS and linguistic approach are briefly described

Definition 1 (see [1]) Let119883 be a universal set An IFS 119860 in119883

is given as

119860 = (119909 120583119860 (119909) V119860 (119909)) | 119909 isin 119883 (1)

where the functions 120583119860(119909) 119883 rarr [0 1] V

119860(119909) 119883 rarr [0 1]

stand for the degree of membership and nonmembershipof the element 119909 to 119860 respectively Any 119909 isin 119883 meets thecondition 0 le 120583

119860(119909) + V

119860(119909) le 1

120587119860(119909) is called intuitionistic index or degree of indeter-

minacy of 119909 to 119860 120587119860(119909) = 1 minus 120583

119860(119909) minus V

119860(119909) Obviously if

120587119860(119909) = 0 IFS 119860 is reduced to a fuzzy setSome basic definitions and operations on IFS are pre-

sented as follows

Definition 2 (see [14 15]) If 119860 and 119861 are two IFSs of the set119883 then

(1) 119860 = 119861 if and only if forall119909 isin 119883 120583119860(119909) = 120583

119861(119909) and

V119860(119909) = V

119861(119909)

(2) 119860119888 = (119909 V119860(119909) 120583119860(119909)) | 119909 isin 119883 where 119860

119888 is thecomplement of 119860

(3) 119860 and 119861 = (min(120583119860(119909) 120583119861(119909))max(V

119860(119909) V119861(119909)))

(4) 119860 or 119861 = (max(120583119860(119909) 120583119861(119909))min(V

119860(119909) V119861(119909)))

(5) 119860 oplus 119861 = (120583119860(119909) + 120583

119861(119909) minus 120583

119860(119909)120583119861(119909) V119860(119909)V119861(119909))

(6) 119860 otimes 119861 = (120583119860(119909)120583119861(119909) V119860(119909) + V

119861(119909) minus V

119860(119909)V119861(119909))

(7) 120582119860 = (1 minus (1 minus 120583119860(119909))120582 V119860(119909)120582) 120582 gt 0

(8) 119860120582 = (120583119860(119909)120582 1 minus (1 minus V

119860(119909))120582) 120582 gt 0

In real world many decision making problems presentqualitative aspects that are complex to assess by means ofnumerical values In such cases it may be more suitable toconsider them as linguistic variables

Let 119878 = 119904119894| 119894 = 0 1 119905 be a finite linguistic term set

with odd cardinality where 119904119894represents a possible linguistic

term for a linguistic variable For example a set of seven terms119878 can be expressed as follows

119878 = 1199040= N (none) 1199041 = VL (very low) 119904

2= L (low)

1199043= M (medium) 1199044 = H (high)

1199045= VH (very high) 119904

6= P (perfect)

(2)

It is required that the linguistic term set should satisfy thefollowing characteristics [27ndash30]

(1) The set is ordered 119904119894gt 119904119895 if and only if 119894 gt 119895

(2) There is a negation operator Neg(119904119894) = 119904119895such that

119895 = 119905 minus 119894(3) Max operator max(119904

119894 119904119895) = 119904119894 if and only if 119894 ge 119895

(4) Min operator min(119904119894 119904119895) = 119904119894 if and only if 119894 le 119895

To preserve all the given information Xu [31] extendedthe discrete term set 119878 to a continuous linguistic term set 119878 =

119904120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] where if 119904

120572isin 119878 then 119904

120572is called

the original linguistic term Otherwise 119904120572is called the virtual

linguistic term

Definition 3 (see [31 32]) Consider any two linguistic terms119904120572 119904120573

isin 119878 and 120583 1205831 1205832

isin [0 1] the add and multiplyoperations of linguistic variable are defined as follows

119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904120572+120573

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904120572120573

(3)

119905-norm and 119905-conorm have been widely used to constructoperations for fuzzy sets and IFSs

Definition 4 (see [33 34]) A 119905-norm is a mapping 119879 [0 1] times

[0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]

(1) 119879(119909 1) = 119909

(2) 119879(119909 119910) = 119879(119910 119909)

(3) 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)

(4) 119879(119909 119910) le 119879(119909 119911) whenever 119910 le 119911

The four basic 119905-norms 119879119872 119879119875 119879119871 and 119879

119863are given as

follows

119879119872(119909 119910) = min(119909 119910) (lattice operation)

119879119875(119909 119910) = 119909 sdot 119910 (algebraic operation)

119879119871(119909 119910) = max(119909+119910minus1 0) (Lukasiewicz operation)

119879119863(119909 119910) =

0 if (119909119910)isin[01)2min(119909119910) otherwise (drastic operation)

Definition 5 (see [33 34]) A t-conorm is a mapping 119878

[0 1] times [0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]

(1) 119878(119909 0) = 119909

(2) 119878(119909 119910) = 119878(119910 119909)

(3) 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)

(4) 119878(119909 119910) le 119878(119909 119911) whenever 119910 le 119911

The four basic 119905-conorms 119878119872 119878119875 119878119871 and 119878

119863are given as

follows

119878119872(119909 119910) = max(119909 119910) (lattice operation)

119878119875(119909 119910) = 119909 + 119910 minus 119909 sdot 119910 (algebraic operation)

119878119871(119909 119910) = min(119909 + 119910 1) (Lukasiewicz operation)

119878119863(119909 119910) =

max(119909119910) if (119909119910)isin(01]21 otherwise

(drastic operation)

3 Linguistic Intuitionistic Fuzzy Set

The concept of linguistic intuitionistic fuzzy set (LIFS) isgiven as follows

Journal of Applied Mathematics 3

Definition 6 Let119883 be a finite universal set and 119878 = 119904120572| 1199040lt

119904120572le 119904119905 120572 isin [0 119905] a continuous linguistic term set A LIFS 119860

in119883 is given as

119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)

where 119904120579(119909) 119904120590(119909) isin 119878 stand for the linguistic membership

degree and linguistic nonmembership of the element 119909 to 119860respectively

For any 119909 isin 119883 the condition 0 le 120579 + 120590 le 119905 is alwayssatisfied 120587(119909) is called linguistic indeterminacy degree of 119909to 119860 120587(119909) = 119904

119905minus120579minus120590 Obviously if 120579 + 120590 = 119905 then LIFS

119860 has the minimum linguistic indeterminacy degree thatis 120587(119909) = 119904

0 which means the membership degree of 119909 to

119860 can be precisely expressed with a single linguistic termand LIFS 119860 is reduced to a linguistic variable Oppositelyif 120579 = 120590 = 0 then LIFS 119860(119909) has the maximum linguisticindeterminacy degree that is 120587(119909) = 119904

119905 Similar to IFS

the LIFS 119860(119909) can be transformed into an interval linguisticvariable [119904

120579(119909) 119904119905minus120590

(119909)] which indicates that the minimumandmaximum linguisticmembership degrees of the elements119909 to 119860 are 119904

120579and 119904119905minus120590

respectivelyFor notational simplicity we suppose both LIFS 119860 and

119861 contain only one element which stand for linguisticintuitionistic fuzzy values (LIFVs) that is the pairs 119860 =

(119904120579 119904120590) and 119861 = (119904

120583 119904])

To compare any two LIFVs the score function andaccuracy function are defined as follows

Definition 7 Let 119860 = (119904120579 119904120590) and 119861 = (119904

120583 119904]) be two LIFVs

with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] The score

function of 119860 is defined as

119878 (119860) = 119904(119905+120579minus120590)2

(5)

and the accuracy function is defined as

119867(119860) = 119904120579+120590

(6)

Thus 119860 and 119861 can be ranked by the following procedure

(1) if 119878(119860) gt 119878(119861) then 119860 gt 119861(2) if 119878(119860) = 119878(119861) and

(a) 119867(119860) = 119867(119861) then 119860 = 119861(b) 119867(119860) gt 119867(119861) then 119860 gt 119861(c) 119867(119860) lt 119867(119861) then 119860 lt 119861

It is easy to see that 0 le (119905+120579minus120590)2 le 119905 and 0 le 120579+120590 le 119905which means 119904

(119905+120579minus120590)2 119904120579+120590

isin 119878

Example 8 Let 119860 = (1199041 1199043) 119861 = (119904

0 1199044) and 119862 = (119904

1 1199045) be

LIFVs which are derived from 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin

[0 6]Applying formulas (5) and (6) we have

119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904

1

119867 (119861) = 1199044lt 119867 (119862) = 119904

6

(7)

Thus we obtain 119860 gt 119862 gt 119861

4 Aggregation Operators for LinguisticIntuitionistic Fuzzy Sets

Since the definition of LIFS is given it is necessary tointroduce the operations and computations between them


120583 119904]) be two LIFVs

then

(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]

(2) 119860119888 = (119904120590 119904120579) where 119860119888 is the complement of 119860

(3) the intersection of 119860 and 119861 119860 and 119861 =

(min(119904120579 119904120583)max(119904

120590 119904]))

(4) the union of 119860 and 119861 119860 or 119861 =

(max(119904120579 119904120583)min(119904

120590 119904]))

Motivated by 119905-norm and 119905-conorm we propose thefollowing operation laws for linguistic variables

Definition 10 Considering any two linguistic terms 119904120572 119904120573isin

119878 = 119904120574| 1199040lt 119904120574le 119904119905 120574 isin [0 119905] the add and multiply

operations of linguistic variable are defined as follows

119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)

(8)

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)

(9)

where 119878(120572119905 120573119905) and 119879(120572119905 120573119905) are 119905-conorm and 119905-normrespectively

Since 119878(120572119905 120573119905) 119879(120572119905 120573119905) isin [0 1] we have 119905119878(120572119905

120573119905) 119905119879(120572119905 120573119905) isin [0 119905] which indicate the operationresults match the original linguistic term set 119878 that is119904119905119878(120572119905120573119905)

119904119905119879(120572119905120573119905)

isin 119878 In addition it is worth noting thatbecause of the monotonicity of 119905-conorm and 119905-norm thevalues of function 119905119878(120572119905 120573119905) and 119905119879(120572119905 120573119905) aremonoton-ically increasing with the increasing of 120572 and 120573 which meansthe operation results obtained by (8) and (9) are in accordwith our intuition

If we take the well-known 119878119875(120572119905 120573119905) and 119879

119875(120572119905 120573119905)

into (8) and (9) respectively then they can be rewritten asfollows

119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905(120572119905+120573119905minus120572120573119905

2)= 119904120572+120573minus120572120573119905

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905

2)= 119904120572120573119905

(10)

Example 11 Let 119878 = 119904120572| 1199040lt 119904120572le 1199046 120572 isin [0 6] Applying

(10) we have 1199043oplus 1199044= 1199043+4minus3times46

= 1199045 1199044oplus 1199045= 1199044+5minus4times56

=

1199045667

1199043otimes 1199044= 1199043times46

= 1199042 and 119904

4otimes 1199045= 1199044times56

= 1199043333

Thus we obtain 119904

3oplus 1199044lt 1199044oplus 1199045and 1199043otimes 1199044lt 1199044otimes 1199045 Such

results seem to be intuitive and can be easily accepted

Alternatively if we take the operation laws ofDefinition 3we have 119904

3oplus 1199044= 1199047notin 119878 119904

4otimes 1199045= 11990420

notin 119878 where thesubscripts of 119904

7and 119904

20are bigger than the cardinality of

linguistic term set 119878 In addition if we extend the discreteterm set 119878 to a continuous term set 119878

1015840

= 119904120572

| 120572 isin

[0 119902] [35] where 119902 (119902 gt 119905) is a sufficiently large positive


integer there is an unavoidable question on how to definethe semantics for 119904

7and 11990420 Obviously 119904

7or 11990420has different

semantics in different linguistic term set 119878 with differentcardinalities As a result it is unrealistic to assign semanticsto a given linguistic value 119904

120572derived from linguistic term set

with variable cardinality If we follow the method of 119904120572oplus 119904120573=

min119904120572+120573

119904119905 and 119904

120572otimes 119904120573= min119904

120572120573 119904119905 [36] for 119904

120572119904120573isin 119878 =

119904120574

| 1199040

lt 119904120574

le 119904119905 120574 isin [0 119905] then we have 119904

3oplus 1199044

=

1199044oplus 1199045= 1199046and 1199043otimes 1199044= 1199044otimes 1199045= 1199046 Such results seem to be

counter-intuitive and may not be easily accepted Applying(10) we can overcome the limitations that the subscripts ofthe linguistic variable are bigger than the cardinality of thecorresponding linguistic term set 119878 and obtain results agreedwith our intuition

Based on (10) we can get the following operation laws forLIFVs


120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 gt 0 then

119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905

2) 119904119905(120590]1199052)) = (119904

120579+120583minus120579120583119905 119904120590]119905) (11)

119860 otimes 119861 = (119904119905(120579120583119905

2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904

120579120583119905 119904120590+]minus120590]119905) (12)

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582 ) (13)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) (14)

Some special cases of 120582119860 and 119860120582 are obtained as follows

If 119860 = (119904120579 119904120590) = (119904119905 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus119905119905)

120582) 119904119905(0119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(119905119905)120582 119904119905(1minus(1minus0119905)

120582)) = (119904

119905 1199040)

(15)

If 119860 = (119904120579 119904120590) = (1199040 119904119905) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus0119905)

120582) 119904119905(119905119905)120582) = (119904

0 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(0119905)120582 119904119905(1minus(1minus119905119905)

120582)) = (119904

0 119904119905)

(16)

If 119860 = (119904120579 119904120590) = (1199040 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582)=(119904119905(1minus(1minus0119905)

120582) 119904119905(0119905)

120582)=(1199040 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582))=(119904119905(0119905)

120582 119904119905(1minus(1minus0119905)

120582))=(1199040 1199040)

(17)

If 120582 rarr 0 then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (1199040 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905 1199040)

(18)

If 120582 rarr +infin then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

0 119904119905)

(19)

Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904

120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 1205821 1205822gt 0 Then one has

(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861

(2) 1205821119860 oplus 120582

2119860 = (120582

1+ 1205822)119860

(3) 119860120582 otimes 119861120582= (119860 otimes 119861)

120582

(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822

Proof (1) By (11) we have 119860 oplus 119861 = (119904120579+120583minus120579120583119905

119904120590]119905) Thus

based on (13) we have

120582 (119860 oplus 119861) = (119904119905(1minus(1minus(120579+120583minus120579120583119905)119905)

120582) 119904119905((120590]119905)119905)120582)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(20)

Similarly since 120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) and 120582119861 =

(119904119905(1minus(1minus120583119905)

120582) 119904119905(]119905)120582) then

120582119860 oplus 120582119861

= (119904119905(1minus(1minus120579119905)

120582)+119905(1minus(1minus120583119905)

120582)minus119905(1minus(1minus120579119905)

120582)(1minus(1minus120583119905)

120582)

119904119905(120590119905)

120582119905(]119905)120582119905)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(21)

Hence we obtain 120582(119860 oplus 119861) = 120582119860 oplus 120582119861(2) By (13) we have 120582

1119860 = (119904

119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)

1205821 ) and1205822119860 = (119904

119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)

1205822 ) thus we obtain

1205821119860 oplus 120582

2119860

= (119904119905(1minus(1minus120579119905)

1205821 )+119905(1minus(1minus120579119905)1205822 )minus119905(1minus(1minus120579119905)

1205821 )(1minus(1minus120579119905)1205822 )

119904119905(120590119905)

1205821 (120590119905)1205822 )

= (119904119905(1minus120579119905)

1205821 (1minus120579119905)1205822 119904119905(120590119905)

1205821+1205822 )

= (119904119905(1minus120579119905)

1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582

1+ 1205822) 119860

(22)

(3) By (14) we get 119860120582 = (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) and 119861

120582=

(119904119905(120583119905)

120582 119904119905(1minus(1minus]119905)120582)) thus based on (12) we have

119860120582otimes 119861120582

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582)+119905(1minus(1minus]119905)120582)minus119905(1minus(1minus120590119905)120582)(1minus(1minus]119905)120582))

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(23)


Since 119860 otimes 119861 = (119904120579120583119905

119904120590+]minus120590]119905) then by (14) we have

(119860 otimes 119861)120582

= (119904119905(120579120583119905119905)

120582 119904119905(1minus(1minus(120590+]minus120590]119905)119905)120582))

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905minus]119905minus120590]1199052)120582))

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(24)

Hence we obtain 119860120582otimes 119861120582= (119860 otimes 119861)

120582(4) By (14) we have 119860

1205821 = (119904

119905(120579119905)1205821 119904119905(1minus(1minus120590119905)

1205821 )) and

1198601205822 = (119904119905(120579119905)

1205822 119904119905(1minus(1minus120590119905)1205822 )) thus we have

1198601205821otimes 1198601205822

= (119904119905(120579119905)

1205821 (120579119905)1205822

119904119905(1minus(1minus120590119905)


1205821 )(1minus(1minus120590119905)1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821 )(1minus120590119905)

1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )

) = 1198601205821+1205822

(25)

which completes the proof of Theorem 13

Motivated by the intuitionistic fuzzy aggregation oper-ators [14 15] in what follows we define some aggregationoperators for LIFVs

Definition 14 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedaveraging (LIFWA) operator is defined as

LIFWA (1198601 1198602 119860

119899) = 11990811198601oplus 11990821198602oplus sdot sdot sdot oplus 119908

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 )

(26)

where 119908 = (1199081 1199082 119908

119899)119879 is the weight vector of 119860

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1

In Particular if 119908 = (1119899 1119899 1119899)119879 then the

LIFWA operator is reduced to a linguistic intuitionistic fuzzyaveraging (LIFA) operator that is

LIFWA (1198601 1198602 119860

119899) =

1

1198991198601oplus1

1198991198602oplus sdot sdot sdot oplus

1

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(27)

Based on Definition 14 we get some properties of theLIFWA operator

Theorem 15 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a set ofLIFVs and 119908 = (119908

1 1199082 119908

119899)119879 the weight vector of 119860

119894(119894 =

0 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has

the following

(1) Idempotency If all 119860119894(119894 = 1 2 119899) are equal that

is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (28)

(2) Monotonicity Let 1198601015840119894= (1199041205791015840

119894

1199041205901015840

119894

) (119894 = 1 2 119899) be aset of LIFVs If 119904

1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119860(119860

1 1198602 119860

119899)

(29)

for any 119908(3) Boundary Consider

(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(30)

Proof (1) Since 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

LIFWA (1198601 1198602 119860

119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579119905)

119908119894 ) 119904119905prod119899

119894=1(120590119905)119908119894 )

= (119904119905(1minus(1minus120579119905))

119904119905(120590119905)

) = (119904120579 119904120590)

(31)

(2) If 1199041205791015840

119894

ge 119904120579119894

that is 1205791015840119894ge 120579119894 for any 119894 then we have

1205791015840

119894ge 120579119894997904rArr 0 le 1 minus

1205791015840

119894

119905le 1 minus

120579119894

119905le 1

997904rArr (1 minus1205791015840

119894

119905)

119908119894

le (1 minus120579119894

119905)

119908119894

997904rArr

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

le

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 1 minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

ge 1 minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 119905(1minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

) ge 119905(1minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

)

(32)

Similarly when 1199041205901015840

119894

le 119904120590119894

for any 119894 we can get119905prod119899

119894=1(1205901015840

119894119905)1119899

le 119905prod119899

119894=1(120590119894119905)1119899

According to Definition 7 we obtain

(119904119905(1minusprod

119899

119894=1(1minus1205791015840

119894119905)1119899) 119904119905prod119899

119894=1(1205901015840

119894119905)1119899)

ge (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(33)

that is

LIFWA (1198601015840

1 1198601015840

2 119860

1015840

119899) ge LIFWA (119860

1 1198602 119860

119899)

(34)


(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894

) for any 119894 then based on the monotonicityof Theorem 15 we derive

(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted averaging (LIFOWA) operator is defined as

LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)

oplus sdot sdot sdot oplus 119908119899119860(119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)

) is the 119894th largest of 1198601 1198602 119860

119899

and 119908 = (1199081 1199082 119908

119899)119879 is the associated weight vector of

119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1

Similar to Theorem 15 we have some properties of theLIFOWA operator

Theorem 17 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a set oflinguistic intuitionistic fuzzy values and119908 = (119908

1 1199082 119908

119899)119879

the associated weight vector of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1

119908119894= 1 then one has the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894

) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values If 119904

1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)

(4) Commutativity Let 1198601015840119894= (1199041205791015840

119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)

be a set of linguistic intuitionistic fuzzy values then forany 119908

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840

119899) is any permutation of

(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy weightedgeometric (LIFWG) operator is defined as

LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082

2otimes sdot sdot sdot otimes 119860

119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908

119899)119879is the weight vector of 119860

119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1

The LIFWG operator has the following properties

Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894

) (119894 = 1 2 119899) be a setof LIFVs Then the linguistic intuitionistic fuzzy orderedweighted geometric (LIFOWG) operator is defined as

LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082

(2)otimes sdot sdot sdot otimes 119860

119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1

Similar to Theorem 15 we have some properties of theLIFOWG operator


Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879 the associated weight vector

of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then

one has the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)

(4) Commutativity Let1198601015840119894= (1199041205791015840

119894

1199041205901015840

119894

) (119894 = 1 2 119899) be aset of linguistic intuitionistic fuzzy values then for any119908

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)

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

119899

Based on Lemma 22 we have the following theorem

Theorem 23 Let 119860119894= (119904120579119894

119904120590119894

)(119894 = 1 2 119899) be a set ofLIFVs then one has

119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908

119899)119879 be the weight vector of

119860119894(119894 = 1 2 119899) with 119908

119894isin [0 1] and sum

119899

119894=1119908119894= 1 then

by Lemma 22 we have 1 minusprod119899

119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894

with equality if and only if 1205791

= 1205792

= sdot sdot sdot = 120579119899 that is

119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899

119894=1(120579119894119905)119908119894 with equality if and

only if 1205791= 1205792= sdot sdot sdot = 120579

119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if

and only if 1205901= 1205902= sdot sdot sdot = 120590

119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590

119894119905)119908119894) with equality if and only if 120590

1= 1205902=

sdot sdot sdot = 120590119899

Consequently by Definition 7 we obtain(119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 )) with equality if and only if 120579

1= 1205792

=

sdot sdot sdot = 120579119899and 120590

1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899

Similarly we can also prove LIFOWA(1198601 1198602 119860

119899) ge

LIFOWG(1198601 1198602 119860

119899) with equality if and only if 119860

1=

1198602= sdot sdot sdot = 119860

119899

Besides the above properties we can derive the followingdesirable results of the LIFOWAandLIFOWGoperators

Theorem 24 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908119899)119879 the associated weight vector

of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)

) is the 119894th largest of1198601 1198602 119860

119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)

and 1198604= (1199044 1199041) be LIFVs which are derived from 119878 = 119904

120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)

119879 be theweight vector of 119860

119894(119894 = 1 2 3 4)


Applying (26) and (41) we obtain the aggregated LIFVsas follows

LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)

It is easy to see that

LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)

By Definition 7 we calculate the following values of scorefunction and accuracy function

119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)

Assume that 119908 = (0155 0345 0345 0155)119879 which are

determined by the normal distribution based method [39] isthe associated weight vector of 119860

(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)

= (1199046(1minusprod

4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)

5 MAGDM Method with LinguisticIntuitionistic Fuzzy Information

In the following we present a handling method forMAGDMproblems where the weight vector of attributes is known andthe attribute performance values take the form of LIFVs

Let 119860 = (1198601 1198602 119860

119898) be the set of alternatives and

119862 = (1198621 1198622 119862

119899) the set of attributes whose weight vector

is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863

119897) be the set of decision makers Suppose

119877119896= (119903119896

119894119895)119898times119899

is the decision matrix where 119903119896

119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)

denotes the preference value and takes the form of LIFVwhich is given by decision makers 119863

119896for alternative 119860

119894with

respect to attribute 119862119895and 119904119896

120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905

The proposed method is described as follows

Step 1 Construct the linguistic intuitionistic fuzzy decisionmatrix 119877

119896= (119903119896

119894119895)119898times119899

Step 2 Utilize the LIFOWA or LIFOWG operator to derivethe aggregated decision matrix 119877 = (119903

119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)

where the LIFOWA and LIFOWGweights are determined bythe normal distribution based method [39]

Step 3 Aggregate 119903119894119895(119895 = 1 2 119899) to yield the collective

overall preference values 119903119894for each alternative 119860

119894(119894 =

1 2 119898) based on the LIFWA or LIFWG operator

Step 4 Rank the alternatives in accordance with 119903119894 according

to Definition 7

6 Numerical Example

In this section we consider an example adapted fromHerrera and Herrera-Viedma [40] Suppose an investmentcompany which wants to invest a sum of money in thebest option There is a panel with four possible alternativesof where to invest the money 119860

1is a car industry 119860

2

is a food company 1198603is a computer company 119860

4is

an arms industry The investment company must make adecision according to four criteria 119862

1is the risk analysis

1198622is the growth analysis 119862

3is the social-political impact

analysis 1198624is the environmental impact analysis The weight

vector of attributes is 119908 = (03 01 02 04) Three expertsare invited to provide their preferences for each alterna-tive on each attribute with the linguistic term set 119878 =

1199040

= extremely poor 1199041

= very poor 1199042

= poor 1199043

=

slightly poor 1199044= fair 119904

5= slightly good 119904

6= good 119904

7=

very good 1199048= extremely good


Table 1 The decision matrix 1198771given by119863

1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)

Step 1 The decision makers provide their evaluation val-ues and construct the linguistic intuitionistic fuzzy deci-sion matrix 119877

119896= (119903

119896

119894119895)119898times119899

(119896 = 1 2 3) as shown inTables 1 2 and 3 respectively

Step 2 By the normal distribution based method [39]the associated weight vector is determined as 120596 =

(0243 0514 0243)119879 Then utilize the LIFOWA operator

to derive the aggregated decision matrix 119877 = (119903119894119895)4times4

which is shown in Table 4 Alternatively if the LIFOWGoperator instead of LIFOWAoperator is applied in Step 2 theaggregated decision matrix 119877 = (119903

119894119895)119898times119899

can be derived asshown in Table 5

Step 3 Aggregate 119903119894119895

(119895 = 1 2 4) to yield the collectiveoverall preference values 119903

119894for each alternative 119860

119894(119894 =

1 2 4) based on the LIFWA or LIFWG operator with theweight vector 119908 = (03 01 02 04) as shown in Table 6

Step 4 By ranking 119903119894(119894 = 1 2 3 4) based on Definition 7

the priorities of the alternatives can be obtained which arelisted in Table 6 Obviously both the methods come to thesame conclusion that the best alternative is 119860

4

In the above linguistic intuitionistic fuzzy aggregationoperators the weights of the arguments are supposed tobe crisp numbers However for reason of complexity oruncertainty of decision making decision makers may havedifficulty in assigning weights of the attributes with crispnumbers and may be inclined to appoint the weights ofthe attributes as linguistic values interval linguistic valuesor LIFVs The existing linguistic aggregation operators can-not deal with such cases According to the operation laws

Table 4 The aggregated decision matrix 119877 by LIFOWA operator

1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)

Table 5 The aggregated decision matrix 119877 by LIFOWG operator

1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)

described by (11) and (12) we can also solve such decisionmaking problems For example suppose the decision makerstake the weights of attributes as linguistic values in thisexample that is 119908

1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and

1199084= [1199044 1199047] which can be transformed to LIFVs that is119908

1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by

(11) and (12) the collective overall preference values 119903119894for each

alternative 119860119894(119894 = 1 2 4) with linguistic weights can be

obtained by the following

119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)

Without loss of generalization on the basis of Table 4 wehave the following results

1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)

by which we can obtain the rankings of alternatives that is1198604gt 1198603gt 1198601gt 1198602


Table 6 The collective overall preference values and the rankings of alternatives

1198601

1198602

1198603

1198604

RankingLIFOWALIFWA (119904

3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601

Such results do not exceed the cardinality of the cor-responding linguistic term set and can be easily acceptedBy contrast if we transform the above LIFVs into intervallinguistic values and take the operation laws defined byDefinition 3 we may have similar problems discussed inExample 11

7 Conclusions

In this paper we introduce the concept of LIFS whichcan be seen as a generalization of IFS and is suitable todeal with imprecise and uncertain information in decisionmaking We further define the score function and accuracyfunction to compare the LIFVsMoreover we propose severalaggregation operators for LIFSs such as the LIFWA operatorLIFOWA operator LIFWG operator and LIFOWG operatortogether with their desired properties Comparing with theexisting linguistic operation laws we can obtain more intu-itive and acceptable results by these aggregation operatorsproposed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to thank the anonymous referees fortheir valuable comments and suggestions which helped toimprove this paper This work was supported by the NationalNatural Science Foundation of China under Grant noU1304701 the High-Level Talent Project of Henan Universityof Technology under Grant no 2013BS015 and the Plan ofNature Science Fundamental Research in Henan Universityof Technology under Grant no 2013JCYJ14

References

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

[2] P Burillo and H Bustince ldquoEntropy on intuitionistic fuzzy setsand on interval-valued fuzzy setsrdquo Fuzzy Sets and Systems vol78 no 3 pp 305ndash316 1996

[3] E Szmidt and J Kacprzyk ldquoEntropy for intuitionistic fuzzy setsrdquoFuzzy Sets and Systems vol 118 no 3 pp 467ndash477 2001

[4] H M Zhang ldquoEntropy for intuitionistic fuzzy set basedon distance and intuitionistic indexrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 21 pp139ndash155 2013

[5] E Szmidt J Kacprzyk and P Bujnowski ldquoHow to measure theamount of knowledge conveyed by Atanassovrsquos intuitionisticfuzzy setsrdquo Information Sciences vol 257 pp 276ndash285 2013

[6] C-P Wei P Wang and Y-Z Zhang ldquoEntropy similaritymeasure of interval-valued intuitionistic fuzzy sets and theirapplicationsrdquo Information Sciences vol 181 no 19 pp 4273ndash4286 2011

[7] J Li G Deng H Li and W Zeng ldquoThe relationship betweensimilarity measure and entropy of intuitionistic fuzzy setsrdquoInformation Sciences vol 188 pp 314ndash321 2012

[8] K Atanassov Intuitionistic Fuzzy Sets Theory and ApplicationsPhysica Wyrzburg Germany 1999

[9] W-L Hung and M-S Yang ldquoOn similarity measures betweenintuitionistic fuzzy setsrdquo International Journal of IntelligentSystems vol 23 no 3 pp 364ndash383 2008

[10] Z Yue ldquoDeriving decision makerrsquos weights based on distancemeasure for interval-valued intuitionistic fuzzy group decisionmakingrdquo Expert Systems with Applications vol 38 no 9 pp11665ndash11670 2011

[11] Z Liang and P Shi ldquoSimilarity measures on intuitionistic fuzzysetsrdquo Pattern Recognition Letters vol 24 no 15 pp 2687ndash26932003

[12] E Szmidt and J Kacprzyk ldquoDistances between intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 114 no 3 pp 505ndash5182000

[13] M Xia and Z S Xu ldquoSome new similarity measures for intu-itionistic fuzzy values and their application in group decisionmakingrdquo Journal of Systems Science and Systems Engineeringvol 19 no 4 pp 430ndash452 2010

[14] Z S Xu andR R Yager ldquoSome geometric aggregation operatorsbased on intuitionistic fuzzy setsrdquo International Journal ofGeneral Systems vol 35 no 4 pp 417ndash433 2006

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

[16] R R Yager ldquoOWA aggregation of intuitionistic fuzzy setsrdquoInternational Journal of General Systems vol 38 no 6 pp 617ndash641 2009

[17] J Ye ldquoMulticriteria fuzzy decision-making method based ona novel accuracy function under interval-valued intuitionisticfuzzy environmentrdquo Expert Systems with Applications vol 36no 3 pp 6899ndash6902 2009

[18] Z S Xu ldquoA method based on distance measure for interval-valued intuitionistic fuzzy group decisionmakingrdquo InformationSciences vol 180 no 1 pp 181ndash190 2010

[19] C Tan and X Chen ldquoIntuitionistic fuzzy Choquet integraloperator for multi-criteria decision makingrdquo Expert Systemswith Applications vol 37 no 1 pp 149ndash157 2010

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

[21] X Yu and Z S Xu ldquoPrioritized intuitionistic fuzzy aggregationoperatorsrdquo Information Fusion vol 14 pp 108ndash116 2013


[22] R M Rodrıguez L Martinez and F Herrera ldquoHesitant fuzzylinguistic term sets for decision makingrdquo IEEE Transactions onFuzzy Systems vol 20 no 1 pp 109ndash119 2012

[23] K Atanassov and G Gargov ldquoInterval valued intuitionisticfuzzy setsrdquo Fuzzy Sets and Systems vol 31 no 3 pp 343ndash3491989

[24] V Torra ldquoHesitant fuzzy setsrdquo International Journal of IntelligentSystems vol 25 no 6 pp 529ndash539 2010

[25] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the 2009 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Republic of KoreaAugust 2009

[26] B Zhu Z S Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 879629 13 pages2012

[27] F Herrera and LMartınez ldquoA 2-tuple fuzzy linguistic represen-tation model for computing with wordsrdquo IEEE Transactions onFuzzy Systems vol 8 no 6 pp 746ndash752 2000

[28] Y Dong Y Xu H Li and B Feng ldquoThe OWA-based consensusoperator under linguistic representation models using positionindexesrdquo European Journal of Operational Research vol 203 no2 pp 455ndash463 2010

[29] F Herrera and L Martınez ldquoA model based on linguistic 2-tuples for dealing with multigranular hierarchical linguisticcontexts in multi-expert decision-makingrdquo IEEE Transactionson Systems Man and Cybernetics B Cybernetics vol 31 no 2pp 227ndash234 2001

[30] F Herrera L Martınez and P J Sanchez ldquoManaging non-homogeneous information in group decision makingrdquo Euro-pean Journal of Operational Research vol 166 no 1 pp 115ndash1322005

[31] Z S Xu ldquoA method based on linguistic aggregation operatorsfor group decision making with linguistic preference relationsrdquoInformation Sciences vol 166 no 1ndash4 pp 19ndash30 2004

[32] Y-J Xu and Q-L Da ldquoStandard and mean deviation methodsfor linguistic group decision making and their applicationsrdquoExpert Systems with Applications vol 37 no 8 pp 5905ndash59122010

[33] G Klir and B Yuan Fuzzy Sets and Fuzzy Logic Theory andApplications Prentice Hall Upper Saddle River NJ USA 1995

[34] E P Klement R Mesiar and E Pap Triangular Norms KluwerAcademic Publishers Dordrecht The Netherlands 2000

[35] Z S Xu ldquoDeviation measures of linguistic preference relationsin group decision makingrdquo Omega vol 33 no 3 pp 249ndash2542005

[36] Z S Xu ldquoInduced uncertain linguistic OWA operators appliedto group decision makingrdquo Information Fusion vol 7 no 2 pp231ndash238 2006

[37] O Holder ldquoUber einen Mittelwertsatzrdquo in GottingenNachrichten pp 38ndash47 1889

[38] J L W V Jensen ldquoSur les fonctions convexes et les inegalitesentre les valeurs moyennesrdquo Acta Mathematica vol 30 no 1pp 175ndash193 1906

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

[40] FHerrera andEHerrera-Viedma ldquoLinguistic decision analysissteps for solving decision problems under linguistic informa-tionrdquo Fuzzy Sets and Systems vol 115 no 1 pp 67ndash82 2000

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Preliminaries

In the following some basic concepts and knowledge relatedto IFS and linguistic approach are briefly described

Definition 1 (see [1]) Let119883 be a universal set An IFS 119860 in119883

is given as

119860 = (119909 120583119860 (119909) V119860 (119909)) | 119909 isin 119883 (1)

where the functions 120583119860(119909) 119883 rarr [0 1] V

119860(119909) 119883 rarr [0 1]

stand for the degree of membership and nonmembershipof the element 119909 to 119860 respectively Any 119909 isin 119883 meets thecondition 0 le 120583

119860(119909) + V

119860(119909) le 1

120587119860(119909) is called intuitionistic index or degree of indeter-

minacy of 119909 to 119860 120587119860(119909) = 1 minus 120583

119860(119909) minus V

119860(119909) Obviously if

120587119860(119909) = 0 IFS 119860 is reduced to a fuzzy setSome basic definitions and operations on IFS are pre-

sented as follows

Definition 2 (see [14 15]) If 119860 and 119861 are two IFSs of the set119883 then

(1) 119860 = 119861 if and only if forall119909 isin 119883 120583119860(119909) = 120583

119861(119909) and

V119860(119909) = V

119861(119909)

(2) 119860119888 = (119909 V119860(119909) 120583119860(119909)) | 119909 isin 119883 where 119860

119888 is thecomplement of 119860

(3) 119860 and 119861 = (min(120583119860(119909) 120583119861(119909))max(V

119860(119909) V119861(119909)))

(4) 119860 or 119861 = (max(120583119860(119909) 120583119861(119909))min(V

119860(119909) V119861(119909)))

(5) 119860 oplus 119861 = (120583119860(119909) + 120583

119861(119909) minus 120583

119860(119909)120583119861(119909) V119860(119909)V119861(119909))

(6) 119860 otimes 119861 = (120583119860(119909)120583119861(119909) V119860(119909) + V

119861(119909) minus V

119860(119909)V119861(119909))

(7) 120582119860 = (1 minus (1 minus 120583119860(119909))120582 V119860(119909)120582) 120582 gt 0

(8) 119860120582 = (120583119860(119909)120582 1 minus (1 minus V

119860(119909))120582) 120582 gt 0

In real world many decision making problems presentqualitative aspects that are complex to assess by means ofnumerical values In such cases it may be more suitable toconsider them as linguistic variables

Let 119878 = 119904119894| 119894 = 0 1 119905 be a finite linguistic term set

with odd cardinality where 119904119894represents a possible linguistic

term for a linguistic variable For example a set of seven terms119878 can be expressed as follows

119878 = 1199040= N (none) 1199041 = VL (very low) 119904

2= L (low)

1199043= M (medium) 1199044 = H (high)

1199045= VH (very high) 119904

6= P (perfect)

(2)

It is required that the linguistic term set should satisfy thefollowing characteristics [27ndash30]

(1) The set is ordered 119904119894gt 119904119895 if and only if 119894 gt 119895

(2) There is a negation operator Neg(119904119894) = 119904119895such that

119895 = 119905 minus 119894(3) Max operator max(119904

119894 119904119895) = 119904119894 if and only if 119894 ge 119895

(4) Min operator min(119904119894 119904119895) = 119904119894 if and only if 119894 le 119895

To preserve all the given information Xu [31] extendedthe discrete term set 119878 to a continuous linguistic term set 119878 =

119904120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] where if 119904

120572isin 119878 then 119904

120572is called

the original linguistic term Otherwise 119904120572is called the virtual

linguistic term

Definition 3 (see [31 32]) Consider any two linguistic terms119904120572 119904120573

isin 119878 and 120583 1205831 1205832

isin [0 1] the add and multiplyoperations of linguistic variable are defined as follows

119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904120572+120573

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904120572120573

(3)

119905-norm and 119905-conorm have been widely used to constructoperations for fuzzy sets and IFSs

Definition 4 (see [33 34]) A 119905-norm is a mapping 119879 [0 1] times

[0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]

(1) 119879(119909 1) = 119909

(2) 119879(119909 119910) = 119879(119910 119909)

(3) 119879(119909 119879(119910 119911)) = 119879(119879(119909 119910) 119911)

(4) 119879(119909 119910) le 119879(119909 119911) whenever 119910 le 119911

The four basic 119905-norms 119879119872 119879119875 119879119871 and 119879

119863are given as

follows

119879119872(119909 119910) = min(119909 119910) (lattice operation)

119879119875(119909 119910) = 119909 sdot 119910 (algebraic operation)

119879119871(119909 119910) = max(119909+119910minus1 0) (Lukasiewicz operation)

119879119863(119909 119910) =

0 if (119909119910)isin[01)2min(119909119910) otherwise (drastic operation)

Definition 5 (see [33 34]) A t-conorm is a mapping 119878

[0 1] times [0 1] rarr [0 1] satisfying for all 119909 119910 119911 isin [0 1]

(1) 119878(119909 0) = 119909

(2) 119878(119909 119910) = 119878(119910 119909)

(3) 119878(119909 119878(119910 119911)) = 119878(119878(119909 119910) 119911)

(4) 119878(119909 119910) le 119878(119909 119911) whenever 119910 le 119911

The four basic 119905-conorms 119878119872 119878119875 119878119871 and 119878

119863are given as

follows

119878119872(119909 119910) = max(119909 119910) (lattice operation)

119878119875(119909 119910) = 119909 + 119910 minus 119909 sdot 119910 (algebraic operation)

119878119871(119909 119910) = min(119909 + 119910 1) (Lukasiewicz operation)

119878119863(119909 119910) =

max(119909119910) if (119909119910)isin(01]21 otherwise

(drastic operation)

3 Linguistic Intuitionistic Fuzzy Set

The concept of linguistic intuitionistic fuzzy set (LIFS) isgiven as follows





119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)










120579(119909) 119904119905minus120590


120579and 119904119905minus120590



(119904120579 119904120590) and 119861 = (119904

120583 119904])



120583 119904]) be two LIFVs

with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904



119878 (119860) = 119904(119905+120579minus120590)2

(5)


119867(119860) = 119904120579+120590

(6)





(119905+120579minus120590)2 119904120579+120590

isin 119878

Example 8 Let 119860 = (1199041 1199043) 119861 = (119904

0 1199044) and 119862 = (119904

1 1199045) be



119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904

1

119867 (119861) = 1199044lt 119867 (119862) = 119904

6

(7)





120583 119904]) be two LIFVs

then

(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]



(min(119904120579 119904120583)max(119904

120590 119904]))


(max(119904120579 119904120583)min(119904

120590 119904]))





119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)

(8)

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)

(9)




119904119905119879(120572119905120573119905)



119875(120572119905 120573119905)



2)= 119904120572+120573minus120572120573119905

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905

2)= 119904120572120573119905

(10)




=

1199045667

1199043otimes 1199044= 1199043times46

= 1199042 and 119904

4otimes 1199045= 1199044times56

= 1199043333





3oplus 1199044= 1199047notin 119878 119904

4otimes 1199045= 11990420


7and 119904



1015840

= 119904120572

| 120572 isin









min119904120572+120573

119904119905 and 119904

120572otimes 119904120573= min119904

120572120573 119904119905 [36] for 119904

120572119904120573isin 119878 =

119904120574

| 1199040

lt 119904120574


3oplus 1199044

=





120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 gt 0 then

119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905

2) 119904119905(120590]1199052)) = (119904

120579+120583minus120579120583119905 119904120590]119905) (11)

119860 otimes 119861 = (119904119905(120579120583119905

2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904

120579120583119905 119904120590+]minus120590]119905) (12)

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582 ) (13)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) (14)


If 119860 = (119904120579 119904120590) = (119904119905 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus119905119905)

120582) 119904119905(0119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(119905119905)120582 119904119905(1minus(1minus0119905)

120582)) = (119904

119905 1199040)

(15)

If 119860 = (119904120579 119904120590) = (1199040 119904119905) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus0119905)

120582) 119904119905(119905119905)120582) = (119904

0 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(0119905)120582 119904119905(1minus(1minus119905119905)

120582)) = (119904

0 119904119905)

(16)

If 119860 = (119904120579 119904120590) = (1199040 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582)=(119904119905(1minus(1minus0119905)

120582) 119904119905(0119905)

120582)=(1199040 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582))=(119904119905(0119905)

120582 119904119905(1minus(1minus0119905)

120582))=(1199040 1199040)

(17)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (1199040 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905 1199040)

(18)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

0 119904119905)

(19)

Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904

120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 1205821 1205822gt 0 Then one has

(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861

(2) 1205821119860 oplus 120582

2119860 = (120582

1+ 1205822)119860

(3) 119860120582 otimes 119861120582= (119860 otimes 119861)

120582

(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822


119904120590]119905) Thus



120582) 119904119905((120590]119905)119905)120582)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(20)


120582) 119904119905(120590119905)

120582) and 120582119861 =

(119904119905(1minus(1minus120583119905)

120582) 119904119905(]119905)120582) then

120582119860 oplus 120582119861

= (119904119905(1minus(1minus120579119905)

120582)+119905(1minus(1minus120583119905)


120582)(1minus(1minus120583119905)

120582)

119904119905(120590119905)

120582119905(]119905)120582119905)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(21)


1119860 = (119904

119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)

1205821 ) and1205822119860 = (119904

119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)


1205821119860 oplus 120582

2119860

= (119904119905(1minus(1minus120579119905)


1205821 )(1minus(1minus120579119905)1205822 )

119904119905(120590119905)

1205821 (120590119905)1205822 )

= (119904119905(1minus120579119905)

1205821 (1minus120579119905)1205822 119904119905(120590119905)

1205821+1205822 )

= (119904119905(1minus120579119905)

1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582

1+ 1205822) 119860

(22)

(3) By (14) we get 119860120582 = (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) and 119861

120582=

(119904119905(120583119905)


119860120582otimes 119861120582

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(23)


Since 119860 otimes 119861 = (119904120579120583119905


(119860 otimes 119861)120582

= (119904119905(120579120583119905119905)


= (119904119905(120579119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(24)


120582(4) By (14) we have 119860

1205821 = (119904

119905(120579119905)1205821 119904119905(1minus(1minus120590119905)

1205821 )) and

1198601205822 = (119904119905(120579119905)


1198601205821otimes 1198601205822

= (119904119905(120579119905)

1205821 (120579119905)1205822

119904119905(1minus(1minus120590119905)


1205821 )(1minus(1minus120590119905)1205822 ))

= (119904119905(120579119905)


1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )

) = 1198601205821+1205822

(25)



Definition 14 Let 119860119894= (119904120579119894

119904120590119894


LIFWA (1198601 1198602 119860


119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 )

(26)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



LIFWA (1198601 1198602 119860

119899) =

1

1198991198601oplus1


1

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(27)


Theorem 15 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

0 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has

the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (28)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119860(119860

1 1198602 119860

119899)

(29)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(30)

Proof (1) Since 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

LIFWA (1198601 1198602 119860

119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579119905)

119908119894 ) 119904119905prod119899

119894=1(120590119905)119908119894 )

= (119904119905(1minus(1minus120579119905))

119904119905(120590119905)

) = (119904120579 119904120590)

(31)

(2) If 1199041205791015840

119894

ge 119904120579119894


1205791015840

119894ge 120579119894997904rArr 0 le 1 minus

1205791015840

119894

119905le 1 minus

120579119894

119905le 1

997904rArr (1 minus1205791015840

119894

119905)

119908119894

le (1 minus120579119894

119905)

119908119894

997904rArr

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

le

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 1 minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

ge 1 minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 119905(1minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

) ge 119905(1minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

)

(32)

Similarly when 1199041205901015840

119894

le 119904120590119894


119894=1(1205901015840

119894119905)1119899

le 119905prod119899

119894=1(120590119894119905)1119899


(119904119905(1minusprod

119899

119894=1(1minus1205791015840

119894119905)1119899) 119904119905prod119899

119894=1(1205901015840

119894119905)1119899)


119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(33)

that is

LIFWA (1198601015840

1 1198601015840

2 119860

1015840

119899) ge LIFWA (119860

1 1198602 119860

119899)

(34)


(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894


LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)


= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 17 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879


(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)


119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)


119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840


(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894


LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082


119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894


LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082


119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119860 = (119909 119904120579 (119909) 119904120590 (119909)) | 119909 isin 119883 (4)










120579(119909) 119904119905minus120590


120579and 119904119905minus120590



(119904120579 119904120590) and 119861 = (119904

120583 119904])



120583 119904]) be two LIFVs

with 119904120579 119904120590 119904120583 119904] isin 119878 = 119904



119878 (119860) = 119904(119905+120579minus120590)2

(5)


119867(119860) = 119904120579+120590

(6)





(119905+120579minus120590)2 119904120579+120590

isin 119878

Example 8 Let 119860 = (1199041 1199043) 119861 = (119904

0 1199044) and 119862 = (119904

1 1199045) be



119878 (119860) = 1199042gt 119878 (119861) = 119878 (119862) = 119904

1

119867 (119861) = 1199044lt 119867 (119862) = 119904

6

(7)





120583 119904]) be two LIFVs

then

(1) 119860 = 119861 if and only if 119904120579= 119904120583and 119904120590= 119904]



(min(119904120579 119904120583)max(119904

120590 119904]))


(max(119904120579 119904120583)min(119904

120590 119904]))





119904120572oplus 119904120573= 119904120573oplus 119904120572= 119904119905119878(120572119905120573119905)

(8)

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905119879(120572119905120573119905)

(9)




119904119905119879(120572119905120573119905)



119875(120572119905 120573119905)



2)= 119904120572+120573minus120572120573119905

119904120572otimes 119904120573= 119904120573otimes 119904120572= 119904119905(120572120573119905

2)= 119904120572120573119905

(10)




=

1199045667

1199043otimes 1199044= 1199043times46

= 1199042 and 119904

4otimes 1199045= 1199044times56

= 1199043333





3oplus 1199044= 1199047notin 119878 119904

4otimes 1199045= 11990420


7and 119904



1015840

= 119904120572

| 120572 isin









min119904120572+120573

119904119905 and 119904

120572otimes 119904120573= min119904

120572120573 119904119905 [36] for 119904

120572119904120573isin 119878 =

119904120574

| 1199040

lt 119904120574


3oplus 1199044

=





120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 gt 0 then

119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905

2) 119904119905(120590]1199052)) = (119904

120579+120583minus120579120583119905 119904120590]119905) (11)

119860 otimes 119861 = (119904119905(120579120583119905

2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904

120579120583119905 119904120590+]minus120590]119905) (12)

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582 ) (13)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) (14)


If 119860 = (119904120579 119904120590) = (119904119905 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus119905119905)

120582) 119904119905(0119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(119905119905)120582 119904119905(1minus(1minus0119905)

120582)) = (119904

119905 1199040)

(15)

If 119860 = (119904120579 119904120590) = (1199040 119904119905) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus0119905)

120582) 119904119905(119905119905)120582) = (119904

0 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(0119905)120582 119904119905(1minus(1minus119905119905)

120582)) = (119904

0 119904119905)

(16)

If 119860 = (119904120579 119904120590) = (1199040 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582)=(119904119905(1minus(1minus0119905)

120582) 119904119905(0119905)

120582)=(1199040 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582))=(119904119905(0119905)

120582 119904119905(1minus(1minus0119905)

120582))=(1199040 1199040)

(17)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (1199040 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905 1199040)

(18)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

0 119904119905)

(19)

Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904

120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 1205821 1205822gt 0 Then one has

(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861

(2) 1205821119860 oplus 120582

2119860 = (120582

1+ 1205822)119860

(3) 119860120582 otimes 119861120582= (119860 otimes 119861)

120582

(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822


119904120590]119905) Thus



120582) 119904119905((120590]119905)119905)120582)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(20)


120582) 119904119905(120590119905)

120582) and 120582119861 =

(119904119905(1minus(1minus120583119905)

120582) 119904119905(]119905)120582) then

120582119860 oplus 120582119861

= (119904119905(1minus(1minus120579119905)

120582)+119905(1minus(1minus120583119905)


120582)(1minus(1minus120583119905)

120582)

119904119905(120590119905)

120582119905(]119905)120582119905)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(21)


1119860 = (119904

119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)

1205821 ) and1205822119860 = (119904

119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)


1205821119860 oplus 120582

2119860

= (119904119905(1minus(1minus120579119905)


1205821 )(1minus(1minus120579119905)1205822 )

119904119905(120590119905)

1205821 (120590119905)1205822 )

= (119904119905(1minus120579119905)

1205821 (1minus120579119905)1205822 119904119905(120590119905)

1205821+1205822 )

= (119904119905(1minus120579119905)

1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582

1+ 1205822) 119860

(22)

(3) By (14) we get 119860120582 = (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) and 119861

120582=

(119904119905(120583119905)


119860120582otimes 119861120582

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(23)


Since 119860 otimes 119861 = (119904120579120583119905


(119860 otimes 119861)120582

= (119904119905(120579120583119905119905)


= (119904119905(120579119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(24)


120582(4) By (14) we have 119860

1205821 = (119904

119905(120579119905)1205821 119904119905(1minus(1minus120590119905)

1205821 )) and

1198601205822 = (119904119905(120579119905)


1198601205821otimes 1198601205822

= (119904119905(120579119905)

1205821 (120579119905)1205822

119904119905(1minus(1minus120590119905)


1205821 )(1minus(1minus120590119905)1205822 ))

= (119904119905(120579119905)


1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )

) = 1198601205821+1205822

(25)



Definition 14 Let 119860119894= (119904120579119894

119904120590119894


LIFWA (1198601 1198602 119860


119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 )

(26)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



LIFWA (1198601 1198602 119860

119899) =

1

1198991198601oplus1


1

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(27)


Theorem 15 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

0 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has

the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (28)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119860(119860

1 1198602 119860

119899)

(29)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(30)

Proof (1) Since 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

LIFWA (1198601 1198602 119860

119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579119905)

119908119894 ) 119904119905prod119899

119894=1(120590119905)119908119894 )

= (119904119905(1minus(1minus120579119905))

119904119905(120590119905)

) = (119904120579 119904120590)

(31)

(2) If 1199041205791015840

119894

ge 119904120579119894


1205791015840

119894ge 120579119894997904rArr 0 le 1 minus

1205791015840

119894

119905le 1 minus

120579119894

119905le 1

997904rArr (1 minus1205791015840

119894

119905)

119908119894

le (1 minus120579119894

119905)

119908119894

997904rArr

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

le

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 1 minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

ge 1 minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 119905(1minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

) ge 119905(1minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

)

(32)

Similarly when 1199041205901015840

119894

le 119904120590119894


119894=1(1205901015840

119894119905)1119899

le 119905prod119899

119894=1(120590119894119905)1119899


(119904119905(1minusprod

119899

119894=1(1minus1205791015840

119894119905)1119899) 119904119905prod119899

119894=1(1205901015840

119894119905)1119899)


119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(33)

that is

LIFWA (1198601015840

1 1198601015840

2 119860

1015840

119899) ge LIFWA (119860

1 1198602 119860

119899)

(34)


(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894


LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)


= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 17 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879


(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)


119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)


119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840


(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894


LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082


119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894


LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082


119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















min119904120572+120573

119904119905 and 119904

120572otimes 119904120573= min119904

120572120573 119904119905 [36] for 119904

120572119904120573isin 119878 =

119904120574

| 1199040

lt 119904120574


3oplus 1199044

=





120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 gt 0 then

119860 oplus 119861 = (119904119905(120579119905+120583119905minus120579120583119905

2) 119904119905(120590]1199052)) = (119904

120579+120583minus120579120583119905 119904120590]119905) (11)

119860 otimes 119861 = (119904119905(120579120583119905

2) 119904119905(120590119905+]119905minus120590]1199052)) = (119904

120579120583119905 119904120590+]minus120590]119905) (12)

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582 ) (13)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) (14)


If 119860 = (119904120579 119904120590) = (119904119905 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus119905119905)

120582) 119904119905(0119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(119905119905)120582 119904119905(1minus(1minus0119905)

120582)) = (119904

119905 1199040)

(15)

If 119860 = (119904120579 119904120590) = (1199040 119904119905) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905(1minus(1minus0119905)

120582) 119904119905(119905119905)120582) = (119904

0 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905(0119905)120582 119904119905(1minus(1minus119905119905)

120582)) = (119904

0 119904119905)

(16)

If 119860 = (119904120579 119904120590) = (1199040 1199040) then

120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582)=(119904119905(1minus(1minus0119905)

120582) 119904119905(0119905)

120582)=(1199040 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582))=(119904119905(0119905)

120582 119904119905(1minus(1minus0119905)

120582))=(1199040 1199040)

(17)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (1199040 119904119905)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

119905 1199040)

(18)


120582119860 = (119904119905(1minus(1minus120579119905)

120582) 119904119905(120590119905)

120582) = (119904119905 1199040)

119860120582= (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) = (119904

0 119904119905)

(19)

Theorem 13 Let 119860 = (119904120579 119904120590) and 119861 = (119904

120583 119904]) be two LIFVs

where 119904120579 119904120590 119904120583 119904] isin 119878 = 119904

120572| 1199040lt 119904120572le 119904119905 120572 isin [0 119905] with

120582 1205821 1205822gt 0 Then one has

(1) 120582(119860 oplus 119861) = 120582119860 oplus 120582119861

(2) 1205821119860 oplus 120582

2119860 = (120582

1+ 1205822)119860

(3) 119860120582 otimes 119861120582= (119860 otimes 119861)

120582

(4) 1198601205821 otimes 1198601205822 = 1198601205821+1205822


119904120590]119905) Thus



120582) 119904119905((120590]119905)119905)120582)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(20)


120582) 119904119905(120590119905)

120582) and 120582119861 =

(119904119905(1minus(1minus120583119905)

120582) 119904119905(]119905)120582) then

120582119860 oplus 120582119861

= (119904119905(1minus(1minus120579119905)

120582)+119905(1minus(1minus120583119905)


120582)(1minus(1minus120583119905)

120582)

119904119905(120590119905)

120582119905(]119905)120582119905)

= (119904119905(1minus(1minus120579119905)

120582(1minus120583119905)

120582) 119904119905(120590119905)

120582(]119905)120582)

(21)


1119860 = (119904

119905(1minus(1minus120579119905)1205821 ) 119904119905(120590119905)

1205821 ) and1205822119860 = (119904

119905(1minus(1minus120579119905)1205822 ) 119904119905(120590119905)


1205821119860 oplus 120582

2119860

= (119904119905(1minus(1minus120579119905)


1205821 )(1minus(1minus120579119905)1205822 )

119904119905(120590119905)

1205821 (120590119905)1205822 )

= (119904119905(1minus120579119905)

1205821 (1minus120579119905)1205822 119904119905(120590119905)

1205821+1205822 )

= (119904119905(1minus120579119905)

1205821+1205822 119904119905(120590119905)1205821+1205822 ) = (120582

1+ 1205822) 119860

(22)

(3) By (14) we get 119860120582 = (119904119905(120579119905)

120582 119904119905(1minus(1minus120590119905)

120582)) and 119861

120582=

(119904119905(120583119905)


119860120582otimes 119861120582

= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(23)


Since 119860 otimes 119861 = (119904120579120583119905


(119860 otimes 119861)120582

= (119904119905(120579120583119905119905)


= (119904119905(120579119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(24)


120582(4) By (14) we have 119860

1205821 = (119904

119905(120579119905)1205821 119904119905(1minus(1minus120590119905)

1205821 )) and

1198601205822 = (119904119905(120579119905)


1198601205821otimes 1198601205822

= (119904119905(120579119905)

1205821 (120579119905)1205822

119904119905(1minus(1minus120590119905)


1205821 )(1minus(1minus120590119905)1205822 ))

= (119904119905(120579119905)


1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )

) = 1198601205821+1205822

(25)



Definition 14 Let 119860119894= (119904120579119894

119904120590119894


LIFWA (1198601 1198602 119860


119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 )

(26)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



LIFWA (1198601 1198602 119860

119899) =

1

1198991198601oplus1


1

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(27)


Theorem 15 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

0 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has

the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (28)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119860(119860

1 1198602 119860

119899)

(29)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(30)

Proof (1) Since 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

LIFWA (1198601 1198602 119860

119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579119905)

119908119894 ) 119904119905prod119899

119894=1(120590119905)119908119894 )

= (119904119905(1minus(1minus120579119905))

119904119905(120590119905)

) = (119904120579 119904120590)

(31)

(2) If 1199041205791015840

119894

ge 119904120579119894


1205791015840

119894ge 120579119894997904rArr 0 le 1 minus

1205791015840

119894

119905le 1 minus

120579119894

119905le 1

997904rArr (1 minus1205791015840

119894

119905)

119908119894

le (1 minus120579119894

119905)

119908119894

997904rArr

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

le

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 1 minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

ge 1 minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 119905(1minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

) ge 119905(1minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

)

(32)

Similarly when 1199041205901015840

119894

le 119904120590119894


119894=1(1205901015840

119894119905)1119899

le 119905prod119899

119894=1(120590119894119905)1119899


(119904119905(1minusprod

119899

119894=1(1minus1205791015840

119894119905)1119899) 119904119905prod119899

119894=1(1205901015840

119894119905)1119899)


119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(33)

that is

LIFWA (1198601015840

1 1198601015840

2 119860

1015840

119899) ge LIFWA (119860

1 1198602 119860

119899)

(34)


(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894


LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)


= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 17 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879


(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)


119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)


119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840


(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894


LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082


119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894


LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082


119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 119860 otimes 119861 = (119904120579120583119905


(119860 otimes 119861)120582

= (119904119905(120579120583119905119905)


= (119904119905(120579119905)


= (119904119905(120579119905)

120582(120583119905)120582 119904119905(1minus(1minus120590119905)

120582(1minus]119905)120582))

(24)


120582(4) By (14) we have 119860

1205821 = (119904

119905(120579119905)1205821 119904119905(1minus(1minus120590119905)

1205821 )) and

1198601205822 = (119904119905(120579119905)


1198601205821otimes 1198601205822

= (119904119905(120579119905)

1205821 (120579119905)1205822

119904119905(1minus(1minus120590119905)


1205821 )(1minus(1minus120590119905)1205822 ))

= (119904119905(120579119905)


1205822 ))

= (119904119905(120579119905)

1205821+1205822 119904119905(1minus(1minus120590119905)1205821+1205822 )

) = 1198601205821+1205822

(25)



Definition 14 Let 119860119894= (119904120579119894

119904120590119894


LIFWA (1198601 1198602 119860


119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 )

(26)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



LIFWA (1198601 1198602 119860

119899) =

1

1198991198601oplus1


1

119899119860119899

= (119904119905(1minusprod

119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(27)


Theorem 15 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

0 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has

the following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (28)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119860(119860

1 1198602 119860

119899)

(29)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(30)

Proof (1) Since 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

LIFWA (1198601 1198602 119860

119899)

= (119904119905(1minusprod

119899

119894=1(1minus120579119905)

119908119894 ) 119904119905prod119899

119894=1(120590119905)119908119894 )

= (119904119905(1minus(1minus120579119905))

119904119905(120590119905)

) = (119904120579 119904120590)

(31)

(2) If 1199041205791015840

119894

ge 119904120579119894


1205791015840

119894ge 120579119894997904rArr 0 le 1 minus

1205791015840

119894

119905le 1 minus

120579119894

119905le 1

997904rArr (1 minus1205791015840

119894

119905)

119908119894

le (1 minus120579119894

119905)

119908119894

997904rArr

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

le

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 1 minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

ge 1 minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

997904rArr 119905(1minus

119899

prod

119894=1

(1 minus1205791015840

119894

119905)

119908119894

) ge 119905(1minus

119899

prod

119894=1

(1 minus120579119894

119905)

119908119894

)

(32)

Similarly when 1199041205901015840

119894

le 119904120590119894


119894=1(1205901015840

119894119905)1119899

le 119905prod119899

119894=1(120590119894119905)1119899


(119904119905(1minusprod

119899

119894=1(1minus1205791015840

119894119905)1119899) 119904119905prod119899

119894=1(1205901015840

119894119905)1119899)


119899

119894=1(1minus120579119894119905)1119899) 119904119905prod119899

119894=1(120590119894119905)1119899)

(33)

that is

LIFWA (1198601015840

1 1198601015840

2 119860

1015840

119899) ge LIFWA (119860

1 1198602 119860

119899)

(34)


(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894


LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)


= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 17 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879


(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)


119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)


119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840


(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894


LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082


119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894


LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082


119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(3) Since min119894(119904120579119894

) le 119904120579119894

le max119894(119904120579119894

) and max119894(119904120590119894

) le

119904120590119894

le min119894(119904120590119894


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le LIFWA (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(35)

Definition 16 Let 119860119894= (119904120579119894

119904120590119894


LIFOWA (1198601 1198602 119860

119899)

= 1199081119860(1)

oplus 1199082119860(2)


= (119904119905(1minusprod

119899

119894=1(1minus120579(119894)119905)119908119894 ) 119904119905prod119899

119894=1(120590(119894)119905)119908119894 )

(36)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 17 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908

119899)119879


(119894 = 1 2 119899) with 119908119894isin

[0 1] and sum119899119894=1



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = (119904

120579 119904120590) (37)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(38)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119860(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(39)


119894

1199041205901015840

119894

) (119894 = 0 1 2 119899)


119871119868119865119874119882119860(1198601015840

1 1198601015840

2 119860

1015840

119899) = 119871119868119865119874119882119860(119860

1 1198602 119860

119899)

(40)

where (1198601015840

1 1198601015840

2 119860

1015840


(1198601 1198602 119860

119899)

Definition 18 Let 119860119894= (119904120579119894

119904120590119894


LIFWG (1198601 1198602 119860

119899) = 1198601199081

1otimes 1198601199082


119908119899

119899

= (119904119905prod119899

119894=1(120579119894119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590119894119905)119908119894 ))

(41)

where 119908 = (1199081 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1


Theorem 19 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


119894(119894 =

1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then one has the

following


is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119882119866 (1198601 1198602 119860

119899) = (119904

120579 119904120590) (42)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119882119866(1198601015840

1 1198601015840

2 119860

1015840

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

(43)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119882119866 (1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(44)

Definition 20 Let 119860119894= (119904120579119894

119904120590119894


LIFOWG(1198601 1198602 119860

119899)

= 1198601199081

(1)otimes 1198601199082


119908119899

(119899)

= (119904119905prod119899

119894=1(120579(119894)119905)119908119894 119904119905(1minusprod

119899

119894=1(1minus120590(119894)119905)119908119894 ))

(45)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

and 119908 = (1199081 1199082 119908


119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1



Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Theorem 21 Let 119860119894= (119904120579119894

119904120590119894


1 1199082 119908


of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] and sum

119899

119894=1119908119894= 1 then



is 119860119894= (119904120579119894

119904120590119894

) = (119904120579 119904120590) for any 119894 then

119871119868119865119874119882119866(1198601 1198602 119860

119899) = (119904

120579 119904120590) (46)


119894

1199041205901015840

119894


1205791015840

119894

ge 119904120579119894

and 1199041205901015840

119894

le 119904120590119894

for any 119894 then

119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(47)


(min119894

(119904120579119894

) max119894

(119904120590119894

)) le 119871119868119865119874119882119866(1198601 1198602 119860

119899)

le (max119894

(119904120579119894

) min119894

(119904120590119894

))

(48)


119894

1199041205901015840

119894


119871119868119865119874119882119866(1198601015840

1 1198601015840

2 1198601015840

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(49)

where (1198601015840

1 1198601015840

2 1198601015840


(1198601 1198602 119860

119899)

Lemma 22 (see [37 38]) Let 119909119894gt 0 120582

119894gt 0 119894 = 1 2 119899

and sum119899119894=1

120582119894= 1 then

119899

prod

119894=1

119909120582119894

119894le

119899

sum

119894=1

120582119894119909119894 (50)


119899


Theorem 23 Let 119860119894= (119904120579119894

119904120590119894


119871119868119865119882119860(1198601 1198602 119860

119899) ge 119871119868119865119882119866 (119860

1 1198602 119860

119899)

119871119868119865119874119882119860(1198601 1198602 119860

119899) ge 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

(51)


119899

Proof Let 119908 = (1199081 1199082 119908


119860119894(119894 = 1 2 119899) with 119908


119899

119894=1119908119894= 1 then


119894=1(1 minus 120579

119894119905)119908119894 ge 1 minus sum

119899

119894=1119908119894(1 minus

120579119894119905) = 1minussum

119899

119894=1119908119894+sum119899

119894=1119908119894120579119894119905 = sum

119899

119894=1119908119894120579119894119905 ge prod

119899

119894=1(120579119894119905)119908119894


= 1205792


119905(1 minus prod119899

119894=1(1 minus 120579

119894119905)119908119894) ge 119905prod

119899



119899 and prod

119899

119894=1(120590119894119905)119908119894 le sum

119899

119894=1119908119894120590119894119905 =

1 minus sum119899

119894=1119908119894(1 minus 120590

119894119905) le 1 minus prod

119899

119894=1(1 minus 120590

119894119905)119908119894 with equality if


119899 that is 119905(prod119899

119894=1(120590119894119905)119908119894) le

119905(1 minus prod119899

119894=1(1 minus 120590


1= 1205902=



119899

119894=1(1minus120579119894119905)119908119894 ) 119904119905prod119899

119894=1(120590119894119905)119908119894 ) ge (119904

119905prod119899

119894=1(120579119894119905)119908119894

119904119905(1minusprod

119899


1= 1205792

=


1= 1205902= sdot sdot sdot = 120590

119899 that is

LIFWA (1198601 1198602 119860

119899) ge LIFWG (119860

1 1198602 119860

119899) (52)


119899


119899) ge

LIFOWG(1198601 1198602 119860


1=

1198602= sdot sdot sdot = 119860

119899


Theorem 24 Let 119860119894= (119904120579119894

119904120590119894



of 119860(119894)

(119894 = 1 2 119899) with 119908119894isin [0 1] andsum119899

119894=1119908119894= 1 Then


(1) If 119908 = (1 0 0)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= max119894

(119860119894)

(53)

(2) If 119908 = (0 0 1)119879 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= min119894

(119860119894)

(54)

(3) If 119908119894= 1 and 119908

119895= 0 for 119895 = 119894 then

119871119868119865119874119882119860(1198601 1198602 119860

119899) = 119871119868119865119874119882119866(119860

1 1198602 119860

119899)

= 119860(119894)

(55)

where 119860(119894)

= (119904120579(119894)

119904120590(119894)


119899

Example 25 Let 1198601= (1199041 1199042) 1198602= (1199041 1199043) 1198603= (1199042 1199044)


120572|

1199040lt 119904120572le 1199046 120572 isin [0 6] and let 119908 = (02 03 04 01)


119894(119894 = 1 2 3 4)



LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











LIFWA (1198601 1198602 119860

4) = (119904

6(1minusprod4

119894=1(1minus1205791198946)119908119894 ) 1199046prod4

119894=1(1205901198946)119908119894 )

= (1199041827

1199042781

)

LIFWG (1198601 1198602 119860

4) = (119904

6prod4

119894=1(1205791198946)119908119894 1199046(1minusprod

4

119894=1(1minus1205901198946)119908119894 ))

= (1199041516

1199043157

)

(56)


LIFWA (1198601 1198602 119860

4) = (119904

1827 1199042781

)

gt LIFWG (1198601 1198602 119860

4)

= (1199041516

1199043157

)

(57)


119878 (1198601) = 119904(6+1minus2)2

= 11990425 119878 (119860

2) = 119904(6+1minus3)2

= 1199042

119878 (1198603) = 119904(6+2minus4)2

= 1199042 119878 (119860

4) = 119904(6+4minus1)2

= 11990445

119867 (1198602) = 1199041+3

= 1199044 119867 (119860

3) = 1199042+4

= 1199046

(58)

Since 119878(1198604) gt 119878(119860

1) gt 119878(119860

2) = 119878(119860

3) and 119867(119860

3) gt

119867(1198602) then

119860(1)

= 1198604= (1199044 1199041) 119860

(2)= 1198601= (1199041 1199042)

119860(3)

= 1198603= (1199042 1199044) 119860

(4)= 1198602= (1199041 1199043)

(59)



(119894) Then by (36) and (45)

we have

LIFOWA (1198601 1198602 119860

119899)


4

119894=1(1minus120579(119894)6)119908119894 ) 1199046prod119899

119894=1(120590(119894)6)119908119894)

= (1199041983

1199042430

)

LIFOWG (1198601 1198602 119860

119899)

= (1199046prod119899

119894=1(120579(119894)6)119908119894 1199046(1minusprod

119899

119894=1(1minus120590(119894)6)119908119894 ))

= (1199041575

1199042882

)

(60)


LIFOWA (1198601 1198602 119860

4) = (119904

1983 1199042430

)

gt LIFOWG (1198601 1198602 119860

4)

= (1199041575

1199042882

)

(61)



Let 119860 = (1198601 1198602 119860


119862 = (1198621 1198622 119862


is 119908 = (1199081 1199082 119908

119899) with 119908


119899

119894=1119908119894= 1 Let

119863 = (1198631 1198632 119863


119877119896= (119903119896

119894119895)119898times119899


119894119895= (119904119896

120579119894119895

119904119896

120590119894119895

)



119894with


120579119894119895

119904119896

120590119894119895

isin 119878 = 119904119894| 119894 = 0 1 119905



119896= (119903119896

119894119895)119898times119899


119894119895)119898times119899

119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWA (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

or 119903119894119895= (119904120579119894119895

119904120590119894119895

) = LIFOWG (1199031

119894119895 1199032

119894119895 119903

119897

119894119895)

(62)




119894(119894 =



to Definition 7

6 Numerical Example



2


4is







1199040


= very poor 1199042

= poor 1199043

=



6= good 119904

7=




1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

1198621

1198622

1198623

1198624

1198601

(1199046 1199041) (119904

3 1199041) (119904

3 1199043) (119904

1 1199046)

1198602

(1199043 1199044) (119904

3 1199044) (119904

2 1199045) (119904

2 1199044)

1198603

(1199041 1199043) (119904

2 1199043) (119904

3 1199042) (119904

6 1199041)

1198604

(1199046 1199042) (119904

4 1199043) (119904

5 1199041) (119904

7 1199041)


2

1198621

1198622

1198623

1198624

1198601

(1199043 1199042) (119904

4 1199041) (119904

3 1199044) (119904

2 1199043)

1198602

(1199045 1199042) (119904

2 1199041) (119904

3 1199044) (119904

2 1199045)

1198603

(1199042 1199043) (119904

3 1199043) (119904

1 1199042) (119904

3 1199043)

1198604

(1199045 1199042) (119904

3 1199043) (119904

5 1199042) (119904

4 1199041)


3

1198621

1198622

1198623

1198624

1198601

(1199043 1199043) (119904

3 1199045) (119904

6 1199041) (119904

2 1199046)

1198602

(1199043 1199042) (119904

2 1199044) (119904

2 1199041) (119904

3 1199044)

1198603

(1199046 1199041) (119904

2 1199045) (119904

3 1199044) (119904

1 1199043)

1198604

(1199045 1199041) (119904

4 1199044) (119904

6 1199042) (119904

5 1199042)


119896= (119903

119896

119894119895)119898times119899






119894119895)119898times119899


Step 3 Aggregate 119903119894119895



119894(119894 =




4



1198621

1198622

1198623

1198624

1198601

(1199043998

1199041865

) (1199043264

1199041479

) (1199043998

1199042463

) (1199041771

1199045070

)

1198602

(1199043584

1199042367

) (1199042537

1199042856

) (1199042537

1199043051

) (1199042260

1199044223

)

1198603

(1199043230

1199042297

) (1199042260

1199043397

) (1199042574

1199042856

) (1199043657

1199042297

)

1198604

(1199045282

1199041401

) (1199043777

1199043478

) (1199045282

1199041401

) (1199044720

1199041184

)


1198621

1198622

1198623

1198624

1198601

(1199043550

1199042041

) (1199043217

1199042303

) (1199043550

1199042860

) (1199041690

1199045501

)

1198602

(1199043397

1199042563

) (1199042463

1199043417

) (1199042462

1199043727

) (1199042207

1199044270

)

1198603

(1199042207

1199042574

) (1199042207

1199043584

) (1199042719

1199043129

) (1199042719

1199042774

)

1198604

(1199045227

1199041505

) (1199043730

1199043542

) (1199045227

1199041505

) (1199044838

1199041257

)


1= (1199043 1199042) 1199082= 1199041 1199083= [1199042 1199044] and


1=

(1199043 1199042) 1199082= (1199041 1199047) 1199083= (1199042 1199044) and 119908

4= (1199044 1199041) Thus by




119903119894= (1199081otimes 1199031198941) oplus (119908

2otimes 1199031198942) oplus (119908

3otimes 1199031198943)

oplus (1199084otimes 1199031198944)

(63)


1199031= ((1199043 1199042) otimes (1199043998

1199041865

)) oplus ((1199041 1199047) otimes (1199043264

1199041479

))

oplus ((1199042 1199044) otimes (1199043998

1199042463

)) oplus ((1199044 1199041) otimes (1199041771

1199045070

))

= (1199043200

1199041245

)

1199032= ((1199043 1199042) otimes (1199043584

1199042367

)) oplus ((1199041 1199047) otimes (1199042537

1199042856

))

oplus ((1199042 1199044) otimes (1199042537

1199043051

)) oplus ((1199044 1199041) otimes (1199042260

1199044223

))

= (1199042931

1199041221

)

1199033= ((1199043 1199042) otimes (1199043230

1199042297

)) oplus ((1199041 1199047) otimes (1199042260

1199043397

))

oplus ((1199042 1199044) otimes (1199042574

1199042856

)) oplus ((1199044 1199041) otimes (1199043657

1199042297

))

= (1199043355

1199040882

)

1199034= ((1199043 1199042) otimes (1199045282

1199041401

)) oplus ((1199041 1199047) otimes (1199043777

1199043478

))

oplus ((1199042 1199044) otimes (1199045282

1199041401

)) oplus ((1199044 1199041) otimes (1199044720

1199041184

))

= (1199045109

1199040424

)

(64)




1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198601

1198602

1198603

1198604


3142 1199042874

) (1199042772

1199043199

) (1199043198

1199042495

) (1199044938

1199041435

) 1198604gt 1198603gt 1198601gt 1198602

LIFOWGLIFWG (119904

2612 1199043932

) (1199042596

1199043619

) (1199042501

1199042876

) (1199044990

1199041650

) 1198604gt 1198603gt 1198602gt 1198601


7 Conclusions




Acknowledgments


References

















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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