interval-valuedpythagoreanhesitantfuzzysetandits...

Research ArticleInterval-Valued Pythagorean Hesitant Fuzzy Set and ItsApplication to Multiattribute Group Decision-Making

Maoyin Zhang1 Tingting Zheng 1 Wanrong Zheng1 and Ligang Zhou12

1School of Mathematical Sciences Anhui University Hefei 230601 China2China Institute of Manufacturing Development Nanjing University of Information Science and TechnologyNanjing 210044 China

Correspondence should be addressed to Tingting Zheng tt-zheng163com

Received 4 May 2019 Revised 22 July 2019 Accepted 12 August 2019 Published 13 February 2020

Academic Editor Lingzhong Guo

Copyright copy 2020Maoyin Zhang et alis 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

Pythagorean hesitant fuzzy sets are widely watched because of their excellent ability to deal with uncertainty imprecise and vagueinformation is paper extends Pythagorean hesitant fuzzy environments to interval-valued Pythagorean hesitant fuzzy en-vironments and proposes the concept of interval-valued Pythagorean hesitant fuzzy set (IVPHFS) which allows the membershipof each object to be a set of several pairs of possible interval-valued Pythagorean fuzzy elements Furthermore we develop a seriesof aggregation operators for interval-valued Pythagorean hesitant fuzzy information and apply them to multiattribute groupdecision-making (MAGDM) problems en some desired operational laws and properties of IVPHFSs are studied Especiallyconsidering an interval-valued Pythagorean fuzzy element (IVPHFE) is formed by several pairs of interval values this paperproposes the concepts of score function and accuracy function in the form of two interval numbers which can retain interval-valued Pythagorean fuzzy information as much as possible en the relationship among these operators is discussed bycomparing the interval numbers Eventually an illustrative example fully shows the feasibility practicality and effectiveness of theproposed approach

1 Introduction

ere are numerous uncertain imprecise and incompleteproblems in the real world Zadehrsquos fuzzy set theory [1] is asuccessful and effective tool to solve many similar problemse theory and its primary extensions such as intuitionisticfuzzy sets [2] interval-valued intuitionistic fuzzy sets [3]type-2 fuzzy sets [4] interval type-2 fuzzy sets [5] and q-rung orthopair fuzzy sets [6] have been successfully appliedto pattern recognition [7] medical diagnosis [8] fuzzy logic[9] decision-making [10ndash18] and so on

In many practical cases it is difficult to define amembership function of a universe because people wouldface several possible membership degrees of one object to bechosen and hesitate about which one would be themost rightone Hesitant fuzzy set theory introduced by Torra andNarukawa [19 20] has provided successful results dealingwith hesitant situations which are not well managed by the

previous tools [21] It constructs possible membership de-grees of an object as a set and keeps more information in realenvironment Many scholars have focused on hesitant fuzzysets and proposed diverse corresponding extensions such asdual hesitant fuzzy sets [22] interval-valued dual fuzzy sets[23 24] intuitionistic hesitant fuzzy sets [25] interval-valued hesitant fuzzy sets [26ndash28] and interval-valuedintuitionistic fuzzy sets [29] ese theories have been ap-plied to decision-making [30 31] clustering analysis [32]and so on

As another extension of fuzzy set theory Yager [33 34]proposed another class of nonstandard fuzzy sets calledPythagorean fuzzy sets e sets are represented by pairs oftwo values langμP(x) ]P(x)rang which satisfies μ2P(x) +]2P(x)le1Obviously their application range is broader than that ofintuitionistic fuzzy sets ese situations are more commonin different real-world problems So Pythagorean fuzzy setshave been paid attention in a short period of time Yager

HindawiComplexityVolume 2020 Article ID 1724943 26 pageshttpsdoiorg10115520201724943

[35 36] Zhang and Xu [37] Ren et al [38] Liu et al [39]and Teng et al [39] have studied several kinds of Pythag-orean fuzzy aggregation operators and applied them todecision-making problems Furthermore Peng and Yang[40] proposed the definition of interval-valued Pythagoreanfuzzy set and Rahamn et al [41 42] developed group de-cision-making with interval-valued Pythagorean fuzzy en-vironments Yi et al [43] applied it to multicriteria decision-making problems Also Liang et al [44] introduced theinterval-valued Pythagorean fuzzy extended Bonferronimean operators

Recently some scholars have tried to combine Py-thagorean fuzzy sets and hesitant fuzzy sets and all calledthem Pythagorean hesitant fuzzy sets (PHFSs) but theirconstruction methods are controversial LetA langx ΓA(x)ΨA(x)rang | x isin U1113864 1113865 be a Pythagorean fuzzy setin U where ΓA(x) μA(x) | μA(x) isin [0 1]1113864 1113865 andΨA(x) ]A(x) | ]A(x) isin [0 1]1113864 1113865 Liu and He [45] definedthat for any x any μA(x) isin ΓA(x) and any ]A(x) isin ΨA(x)(μA(x))2 + (]A(x))2 le 1 holds Khan et al [46] defined thatfor any x and for any μA(x) isin ΓA(x) there is ]A(x) isin ΨA(x)such that (μA(x))2 + (]A(x))2 le 1 Also for any]A(x) isin ΨA(x) there is μA(x) isin ΓA(x) such that(μA(x))2 + (]A(x))2 le 1 Wei et al[47] also defined anotherPythagorean fuzzy set P langx hP(x)rang | x isin U1113864 1113865 wherehP(x) is a set of some Pythagorean fuzzy elements in U ethree definitions have their own merits Considering thepairing of possible membership degrees and possible non-membership degrees it is more conducive to the aggregationof hesitant fuzzy numbers In this paper we choose Weirsquosdefinition as the definition of PHFS

From the above analysis we can see that PHFSs are moreconvenient to deal with fuzzy information than hesitantfuzzy sets or Pythagorean fuzzy sets However for realmultiattribute group decision-making (MAGDM) prob-lems it is difficult for decision makers to provide some exactand crisp fuzzy values to depict uncertain or insufficientalternatives because of the increasing complexity of socialand economic life e aim of this paper is to extend PHFSsto interval-valued Pythagorean hesitant fuzzy sets(IVPHFSs) and develop MAGDM approaches to interval-valued Pythagorean hesitant fuzzy environments based onnewly constructed aggregation operators In particular sincethere is no one-to-one correspondence between intervalnumbers and real numbers this paper directly uses intervalnumbers to define score functions and accuracy functionswhich can preserve interval-valued Pythagorean fuzzy in-formation as much as possible

e rest of this paper is organized as follows In Section2 we review some basic concepts and results of hesitantfuzzy sets interval-valued hesitant fuzzy sets and interval-valued Pythagorean fuzzy sets Section 3 proposes the def-inition of IVPHFS and discusses some basic operationallaws In particular we propose the concepts of the scorefunction and accuracy function which are both appeared asinterval values Some operators for aggregating interval-valued Pythagorean hesitant fuzzy information are studiedand developed in section 4 Section 5 shows the applicationto MAGDMs in interval-valued Pythagorean hesitant fuzzy

environments and illustrates the feasibility and applicabilityof the proposed method Concluding remarks are made inSection 6

2 Preliminaries

is section will briefly review the basic notations and resultsof hesitant fuzzy sets interval-valued Pythagorean fuzzy sets

Hereinafter without explanation let U be a nonemptyfinite set called the universe of discourse and D[x y] be theset of all closed subintervals of the interval [x y]

21 Hesitant Fuzzy Set (HFS) HFS was introduced by Torraand Narukawa [19] and Torra [20] which permits themembership being a set of possible values It is very suitablefor describing problems that are difficult to determine withonly one membership

Definition 1 (see [20]) A hesitant fuzzy set (HFS) E on Uis described as E langx hE(x)rang | x isin U1113864 1113865 where hE(x)

μE(x) | μE(x) isin [0 1]1113864 1113865 represents the set of possiblemembership degrees of E at x

For convenience Xia and Xu [30] proposed the conceptof a hesitant fuzzy element (HFE) denoted by h hE(x)where h c | c isin [0 1]1113864 1113865 ey gave some operations onHFEs

Definition 2 (see [19 20 30]) Let h h1 and h2 be three HFEsand λgt 0 then

(1) hC 1 minus c | c isin h1113864 1113865

(2) hλ cλ | c isin h1113864 1113865

(3) λh 1 minus (1 minus c)λ | c isin h1113966 1113967

(4) h1 oplus h2 c1 + c2 minus c1c2 | c1 isin h1 c2 isin h21113864 1113865

(5) h1 otimes h2 c1c2 | c1 isin h1 c2 isin h21113864 1113865

(6) h1 cup h2 max c1 c21113864 1113865 | c1 isin h1 c2 isin h21113864 1113865

(7) h1 cap h2 min c1 c21113864 1113865 | c1 isin h1 c2 isin h21113864 1113865

Furthermore to compare the HFEs the followingcomparison laws are given

Definition 3 (see [30]) For an HFE h S(h) 1|h|1113936c isin hc iscalled the score function of h with |middot| denoting the cardi-nality here and below For two HFEs h1 and h2 ifS(h1)gt S(h2) then h1 is superior to h2 denoted by h1 ≻ h2 ifS(h1) S(h2) then h1 is equivalent to h2 denoted byh1 h2

22 Interval-Valued Hesitant Fuzzy Set (IVHFS) Chen et al[26 27] generalized HFSs to interval-valued hesitant fuzzysets (IVPHFSs) in which the membership degree of eachobject of the universe is denoted by several possible intervalvalues

Definition 4 (see [26]) An interval-valued hesitant fuzzy set(IVHFS) 1113957E on U is described as 1113957E langx 1113957h1113957E(x)rang | x isin U1113966 1113967

2 Complexity

where 1113957h1113957E(x) 1113957μ1113957E(x) | 1113957μ1113957E(x)≜ [μminus

1113957E(x) μ+

1113957E(x)] isin D[0 1]1113966 1113967

represents the set of possible membership intervals of 1113957E at xFor convenience 1113957h1113957E(x) is called an interval-valued

hesitant fuzzy element (IVHFE) denoted by 1113957h

1113957μ | 1113957μ≜ [μminus μ+]1113864 1113865 Especially if μminus μ+ then 1113957h degeneratesto an HFS

Definition 5 (see [26 27]) Let 1113957h 1113957h1 and 1113957h2 be three IVHFEsand λgt 0 then

(1) 1113957hC

[1 minus μ+ 1 minus μminus ] | 1113957μ isin 1113957h1113966 1113967

(2) 1113957hλ

[(μminus )λ (μ+)λ] | 1113957μ isin 1113957h1113966 1113967

(3) λ1113957h [1 minus (1 minus μminus )λ 1 minus (1 minus μ+)λ] | 1113957μ isin 1113957h1113966 1113967

(4) 1113957h1 oplus 1113957h2 [μminus1 + μminus

2 minus μminus1 middot μminus

2 μ+1 + μ+

2 minus μ+1 middot μ+

2 ]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2

(5) 1113957h1 otimes 1113957h2 [μminus1 middot μminus

2 μ+1 middot μ+

2 ] | 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h21113966 1113967

(6) 1113957h1 cup 1113957h2 [max μminus1 μminus

21113864 1113865max μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2

(7) 1113957h1 cap 1113957h2 [min μminus1 μminus

21113864 1113865min μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2

Definition 6 (see [48]) Let a [aminus a+] and b [bminus b+] betwo interval numbers λgt 0 e interval arithmetic is de-fined as

(1) a + b [aminus + bminus a+ + b+]

(2) a minus b [aminus minus b+ a+ minus bminus ]

(3) λa [λaminus λa+]

(4) ab [min aminus bminus aminus b+ a+bminus a+b+ max aminus bminus

aminus b+ a+bminus a+b+]

Definition 7 (see [49]) Let a [aminus a+] and b [bminus b+] betwo interval numbers and l(a) a+ minus aminus and l(b) b+ minus bminus then the possibility degree of agt b is defined as follows

P(age b) max 1 minus maxb+ minus aminus

l(a) + l(b) 01113896 1113897 01113896 1113897 (1)

e equation of the possibility degree is used to comparetwo interval numbers If P(age b)gt 05 then a is superior tob denoted by agt b if P(age b) 05 then a is equivalent tob denoted by a b

Since an IVHFE is formed by several interval numbersthe comparative analysis of IVHFEs is different from that ofHFEs Based on the possibility degree of interval numbers in[49] Chen et al [26] gave the following comparison laws

Definition 8 (see [26]) For an IVHFE 1113957h S(1113957h) 1|1113957h|11139361113957μ isin1113957h1113957μis called the score function of 1113957h with | middot | denoting thecardinality and S(1113957h) is an interval value belonging to [0 1]For two IVHFEs 1113957h1 and 1113957h2 if P(S(1113957h1)ge S(1113957h2))gt 05 then 1113957h1is superior to 1113957h2 denoted by 1113957h1 ≻ 1113957h2

23 Interval-Valued Pythagorean Fuzzy Set (IVPFS) A Py-thagorean fuzzy set (PFS) is introduced by Yager [33] which

is characterized by a membership function and a non-membership function where the sum of the square of themembership degree and the nonmembership degree of x isless than or equal to 1 while an intuitionistic fuzzy set is alsocharacterized by them where the sum is less than or equal to1 Obviously PFSs are more general than intuitionistic fuzzysets A PFS has emerged as an effective tool to solve mul-tiattribute decision-making problems [37]

Definition 9 (see [33 35]) A Pythagorean fuzzy set (PFS) Pon U is described as

P x μP(x) ]P(x)1113866 1113867 | μ2P(x) + ]2P(x)le 1 μP(x) ]P(x)1113966

isin 0 1] x isin U[

(2)

where μP(x) and ]P(x) represent the Pythagorean mem-bership degree and the Pythagorean nonmembership degreeof P at x respectively

Since people often find it difficult to exactly quantifytheir opinions facing with incomplete fuzzy decision-making problems interval-valued fuzzy elements can pro-vide a better solving way Peng et al [40] focused on interval-valued Pythagorean fuzzy sets (IVPFSs) whose ideas aresimilar to interval-valued intuitionistic fuzzy sets

Definition 10 (see [40]) An interval-valued Pythagoreanfuzzy set (IVPFS) 1113957P on U is described as

1113957P x μminus

1113957P(x) μ+

1113957P(x)1113960 1113961 ]minus

1113957P(x) ]+

1113957P(x)1113960 11139611113924 1113925 | μminus

1113957P(x) μ+

1113957P(x)1113960 11139611113882

middot ]minus

1113957P(x) ]+

1113957P(x)1113960 1113961 isin D[0 1] μ+

1113957P(x)1113872 11138732

+ ]+

1113957P(x)1113872 11138732le 1 x isin U1113883

(3)

where [μminus

1113957P(x) μ+

1113957P(x)] and []minus

1113957P(x) ]+

1113957P(x)] are the Pythago-rean membership intervals and the Pythagorean non-membership intervals of 1113957P at x respectively

Each pair lang[μminus

1113957P(x) μ+

1113957P(x)] []minus

1113957P(x) ]+

1113957P(x)]rang is called aninterval-valued Pythagorean fuzzy element (IVPFE)denoted by 1113957P lang[μminus μ+] []minus ]+]rang Obviously for anyIVPFE 1113957P on U [μminus μ+] and []minus ]+] are both singletons theIVPFS degenerates into a PFS

Definition 11 (see [40]) Let 1113957P lang[μminus μ+] []minus ]+]rang1113957P1 lang[μminus

1 μ+1 ] []minus

1 ]+1 ]rang and 1113957P2 lang[μminus

2 μ+2 ] []minus

2 ]+2 ]rang be

three IVPFEs and λgt 0 e operational laws of IVPFEs aredefined as follows

(1) 1113957PC

lang[]minus ]+] [μminus μ+]rang

(2) 1113957Pλ

lang[(μminus )λ (μ+)λ] [

1 minus (1 minus (]minus )2)λ1113969

1 minus (1 minus (]+)2)λ1113969

]rang

(3) λ1113957P lang[

1 minus (1 minus (μminus )2)λ1113969

1 minus (1 minus (μ+)2)λ1113969

] [(]minus )λ (]+)λ]rang

(4) 1113957P1 oplus 1113957P2 lang[

(μminus1 )2 + (μminus

2 )2 minus (μminus1 )2(μminus

2 )21113969

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

] []minus1]minus

2 ]+1]+

2 ]rang

Complexity 3

(5) 1113957P1 otimes 1113957P2 lang[μminus1μ

minus2 μ+

1μ+2 ] [

(]minus1 )2 + (]minus

2 )2 minus (]minus1 )2

1113969

(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969

]rang

(6) 1113957P1 cup 1113957P2 lang[max μminus1 μminus

21113864 1113865max μ+1 μ+

21113864 1113865]

[min ]minus1 ]minus

21113864 1113865 min ]+1 ]+

21113864 1113865]rang

(7) 1113957P1 cap 1113957P2 lang[min μminus1 μminus

21113864 1113865 min μ+1 μ+

21113864 1113865] [max ]minus1 ]minus

21113864 1113865

max ]+1 ]+

21113864 1113865]rang

Definition 12 (see [40]) For an IVPFE 1113957P lang[μminus μ+]

[]minus ]+]rang the score function of 1113957P is defined as follows

S(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

minus ]minus( )

2minus ]+

( 11138572

1113960 1113961 (4)

e accuracy function of 1113957P is defined as follows

H(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

+ ]minus( )

2+ ]+

( 11138572

1113960 1113961 (5)

For two IVPFEs 1113957P1 and 1113957P2

(1) If S(1113957P1)gt S(1113957P2) then 1113957P1 ≻ 1113957P2

(2) If S(1113957P1) S(1113957P2) then

(21) If H(1113957P1)gtH(1113957P2) then 1113957P1 ≻ 1113957P2(22) If H(1113957P1) H(1113957P2) then 1113957P1 1113957P2

3 Interval-Valued Pythagorean Hesitant FuzzySets (IVPHFSs)

31 Interval-Valued Pythagorean Hesitant Fuzzy ElementsAs mentioned earlier in many practical problems it isdifficult for decision makers to determine precise mem-bership degrees or nonmembership degrees and the eval-uation with relatively reasonable interval values often existsin decision-making In order to better avoid the informationloss and enhance the flexibility and applicability of thedecision-making models in dealing with qualitative infor-mation we propose the concept of interval-valued Py-thagorean hesitant fuzzy set (IVPHFS)

Definition 13 An interval-valued Pythagorean hesitantfuzzy set (IVPHFS) P on U is described as

P x hP(x)1113866 1113867 | x isin U1113864 1113865 (6)

wherehP(x) 1113957μP(x) 1113957]P(x)1113866 1113867 | 1113957μP(x) μminus

P(x) μ+P(x)1113858 1113859 isin D[0 1]1113864

1113957]P(x) ]minusP(x) ]+

P(x)1113858 1113859 isin D[0 1] μ+P(x)( 1113857

2+ ]+

P(x)( 11138572 le 11113967

(7)

where 1113957μP(x) and 1113957]P(x) are the possible Pythagoreanmembership intervals and the possible Pythagorean non-membership intervals of P at x respectively e set of allIVPHFEs on U is denoted by Ω

Obviously for each x isin U if hP(x) includes only onepair of intervals the IVPHFS degenerates into an IVPFS ifboth 1113957μP(x) and 1113957]P(x) degenerate one singleton theIVPHFS can be seen as a PHFS if 1113957]P(x) [0 0] theIVPHFS can be seen as an IVHFS if μ+

P(x) + ]+P(x)le 1 the

IVPHFS can be seen as an interval-valued intuitionistichesitant fuzzy set [29]

For convenience we call each pair 1113957P hP(x) as aninterval-valued Pythagorean hesitant fuzzy element(IVPHFE) where 1113957P lang1113957μ 1113957]rang | 1113957μ [μminus μ+] 1113957] []minus ]+]1113864 1113865

Based on the operators of IVHFEs [27] and IVPFEs [40]the operational laws of IVPHFEs are defined as follows

Definition 14 Let 1113957P lang1113957μ1113957]rang| 1113957μ[μminus μ+]1113957][]minus ]+]1113864 1113865 1113957P1

lang1113957μ11113957]1rang| 1113957μ1 [μminus1 μ+

1 ]1113957]1 []minus1 ]+

1 ]1113864 1113865 and 1113957P2 lang1113957μ21113957]2rang| 1113957μ2 1113864

[μminus2 μ+

2 ]1113957]2 []minus2 ]+

2 ] be three IVPHFEs and λgt0 e op-erational laws of IVPHFEs are defined as follows

(1) 1113957PC

lang[]minus ]+] [μminus μ+]rang | lang1113957μ 1113957]rang isin 1113957P1113966 1113967

(2) 1113957Pλ



1113882

1 minus (1 minus (]+)2)λ1113969

]rang | lang1113957μ 1113957]rang isin 1113957P

(3) λ 1113957P [


1 minus (1 minus (μ+)2)λ1113969

]1113882

lang[(]minus )λ (]+)λ]rang | lang1113957μ 1113957]rang isin 1113957P

(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

1113882

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

]

[]minus1]minus

2 ]+1]+

2 ]rang | lang1113957μi 1113957]irang isin 1113957Pi i 1 2

(5) 1113957P1 otimes 1113957P2 lang[μminus11113864 μminus

2 μ+1μ

+2 ][

(]minus

1 )1113968 2 + (]minus


(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969

]rang | lang1113957μi 1113957]irang isin1113957Pi i 1 2


21113864 1113865max μ+1 μ+

21113864 1113865]1113864

[min ]minus1 ]minus

21113864 1113865min ]+1 ]+

21113864 1113865]rang | lang1113957μi 1113957]irang isin 1113957Pi i 1 2


21113864 1113865min μ+1 μ+

21113864 1113865]1113864

[max ]minus1 ]minus

21113864 1113865max ]+1 ]+


Proposition 1 Let 1113957P 1113957P1 and 1113957P2 be three IVPHFEs andλgt 0 then 1113957P

C 1113957Pλ λ 1113957P 1113957P1 oplus 1113957P2 1113957P1 otimes 1113957P2 1113957P1 cup 1113957P2 and

1113957P1 cap 1113957P2 are all IVPHFESs

Proof Obviously 1113957PC is an IVPHFE

For any lang1113957μ 1113957]rang isin 1113957P since (μ+)2 + (]+)2 le 1

μ+( 1113857

λ1113874 1113875

2+

1 minus 1 minus ]+( )2( 1113857λ

1113969

1113874 11138752

μ+( 1113857

2λ+ 1 minus 1 minus ]+

( 11138572

1113872 1113873λ

le μ+( 1113857

2λ+ 1 minus μ+

( 11138572λ

1

(8)

So 1113957Pλ is an IVPHFE Similarly λ 1113957P is also an IVPHFE

As for 1113957P1 oplus 1113957P2

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ ]+

1( 11138572 ]+

2( 11138572

le μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

1113872 1113873 1 minus ]+2( 1113857

21113872 1113873

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

minus μ+2( 1113857

2

+ μ+1( 1113857

2 μ+2( 1113857

2 1

(9)

4 Complexity

So 1113957P1 oplus 1113957P2 is an IVPHFE Similarly 1113957P1 otimes 1113957P2 is also anIVPHFE

At last we prove the last two claims Assume μ+1 lt μ+

2 en (max μ+

1 μ+21113864 1113865)2 (μ+

2 )2 le 1 minus (]+2 )2 le 1 minus (min ]+

1 1113864

]+2 )2 So 1113957P1 cup 1113957P2 is an IVPHFE Similarly 1113957P1 cap 1113957P2 is alsoan IVPHFE All the claims are proved

Proposition 2 Let 1113957P 1113957P11113957P2 and 1113957P3 be four IVPHFEs

and λ λ1 λ2 gt 0 then

(1) 1113957P1 oplus 1113957P2 1113957P2 oplus 1113957P1 and 1113957P1 otimes 1113957P2 1113957P2 otimes 1113957P1

(2) λ 1113957P1oplusλ 1113957P2 λ( 1113957P1 oplus 1113957P2) and ( 1113957P1)λ otimes ( 1113957P2)

λ ( 1113957P1otimes 1113957P2)

λ

(3) λ1 1113957Poplus λ2 1113957P (λ1 + λ2) 1113957P and 1113957Pλ1 otimes 1113957P

λ2 1113957P

(λ1+λ2)

(4) ( 1113957PC

)λ (λ 1113957P)C and λ( 1113957PC

) ( 1113957Pλ)C

(5) ( 1113957P1)Coplus( 1113957P2)

C ( 1113957P1 otimes 1113957P2)C and ( 1113957P1)

C otimes ( 1113957P2)C

( 1113957P1 oplus 1113957P2)C

(6) ( 1113957P1 oplus 1113957P2)oplus 1113957P3 1113957P1 oplus ( 1113957P2 oplus 1113957P3) and ( 1113957P1 otimes 1113957P2)

otimes 1113957P3 1113957P1 otimes ( 1113957P2 otimes 1113957P3)

Proof (1) Based on Definition 14 claim (1) is obvious sohere the proof process is overleaped

λ 1113957P1 oplus λ 1113957P2

1 minus 1 minus μminus1( 1113857

21113872 1113873

λ1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus1( 1113857

λ ]+

1( 1113857λ

1113876 11138771113898 1113899 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113896 1113897

oplus


21113872 1113873

λ1113970

1 minus 1 minus μ+2( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus2( 1113857

λ ]+

2( 1113857λ

1113876 11138771113898 1113899 1113957μ2 1113957]21113866 1113867 isin 1113957P211138681113868111386811138681113896 1113897


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

λ 1113957P1 oplus 1113957P21113872 1113873

λ

μminus1( 1113857

2+ μminus

2( 11138572

minus μminus1( 1113857

2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

λ


21113872 1113873 1 minus μminus

2( 11138572

1113872 1113873

1113969

1 minus 1 minus μ+1( 1113857

21113872 1113873 1 minus μ+

2( 11138572

1113872 1113873

1113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

(10)

So λ 1113957P1 oplus λ 1113957P2 λ( 1113957P1 oplus 1113957P2) holds Similarly we have( 1113957P1)

λ otimes ( 1113957P2)λ ( 1113957P1 otimes 1113957P2)

λ

λ1 1113957Poplus λ2 1113957P

1 minus 1 minus μminus( 11138572

1113872 1113873λ1

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1

1113970

1113890 1113891 ]minus( )

λ1 ]+( 1113857

λ11113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

oplus


1113872 1113873λ2

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ2

1113970

1113890 1113891 ]minus( )

λ2 ]+( 1113857

λ21113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ1+λ2( )

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1+λ2( )

1113970

1113890 1113891 ]minus( )

λ1+λ2( ) ]+( 1113857

λ1+λ2( )1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

λ1 + λ2( 1113857 1113957P

(11)

Similarly we have 1113957Pλ1 otimes 1113957P

λ2 1113957P(λ1+λ2)

Complexity 5

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)

Similarly we have λ( 1113957PC

) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+

11113858 11138591113866 1113867 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113966 1113967oplus ]minus

2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572

minus ]minus1( 1113857

2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)

Similarly we have ( 1113957P1 otimes 1113957P2)otimes 1113957P3 1113957P1 otimes ( 1113957P2 otimes 1113957P3)All the claims of the proposition are proved


and λgt 0 then

(1) 1113957P1 cup 1113957P2 1113957P2 cup 1113957P1 and 1113957P1 cap 1113957P2 1113957P2 cap 1113957P1

(2) λ 1113957P1cupλ 1113957P2 λ( 1113957P1 cup 1113957P2) and ( 1113957P1)λ cap ( 1113957P2)

λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C

(4) ( 1113957P1 cup 1113957P2)cup 1113957P3 1113957P1 cup ( 1113957P2 cap 1113957P3) and ( 1113957P1 cap 1113957P2)

cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)

(5) ( 1113957P1 cup 1113957P2)cap 1113957P2 1113957P2 and ( 1113957P1 cap 1113957P2)cup 1113957P2 1113957P2

Proof ey are trial We omit them

6 Complexity

32 Score Function of IVPHFEs To determine the prioritiesof the alternatives of an interval-valued Pythagorean hes-itant fuzzy group decision-making problem we need theconcept of score functions for IVPHFEs Since an IVPHFEincludes several pairs formed by possible Pythagoreanmembership intervals and possible Pythagorean non-membership intervals if we use a method similar toDefinition 12 the intervals are represented by the averagevalues of the intervals which must lose some informationbecause there is no one-to-one correspondence between aninterval number and a value

To facilitate comparison of IVPHFEs we shall give thefollowing comparison laws

Definition 15 Let 1113957P lang1113957μ 1113957]rang | 1113957μ [μminus μ+] 1113957] []minus ]+]1113864 1113865

be an IVPHFE e score function S( 1113957P) is described as

S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)

e accuracy function H( 1113957P) is described as

H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)

e idea of the above concepts originated from Defi-nition 8 and used the definition of interval arithmetic inDefinition 6e score function of an IVPHFE is themean ofthe difference between possible Pythagorean membershipintervals and possible Pythagorean nonmembership inter-vals also the accuracy function reflects the overall accuracydegree of an IVPHFE For keeping fuzzy information asmuch as possible both the two functions are represented byinterval values For an IVPHFE 1113957P the score functionS( 1113957P) isin [minus 1 1] and the accuracy function H( 1113957P) isin [0 1]

hold obviously Based on Definition 7 we have the followingdefinition

Definition 16 Let 1113957P1 and 1113957P2 be two IVPHFEs

(1) If P(S( 1113957P1)gt S( 1113957P2))lt 05 then we say 1113957P1 ≺ 1113957P2

(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then

(21) If P(H( 1113957P1)gtH( 1113957P2))lt 05 we say 1113957P1 ≺ 1113957P2(22) If P(H( 1113957P1)gtH( 1113957P2)) 05 we say 1113957P1 1113957P2

Proposition 4 Let 1113957P1 lang1113957μ1j 1113957]1jrang | 1113957μ1j [μminus1j μ+

1j] 1113957]1j 1113966

[]minus1j ]+

1j] j 1 2 middot middot middot m and 1113957P2 lang1113957μ2j 1113957]2jrang | 1113957μ2j [μminus2j1113966

μ+2j] 1113957]2j []minus

2j ]+2j] j 1 2 middot middot middot m be two IVPHFEs If for

any j 1 middot middot middot m μminus1j le μminus

2j μ+1j le μ+

2j ]minus1j ge ]minus

2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2

Proof Based on Definition 15 we have

S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732

minus μminus1j1113872 1113873

21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)

Hence we obtain P(S( 1113957P1)gt S( 1113957P2)) max1 minus max L 0 0 by Definition 7

For any j 1 middot middot middot m μminus1j le μminus

2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)

2 ge 0 then 05leLle 1 So 0le 1 minus max L 0

1 minus Lle 05 Hence P(S( 1113957P1)gt S( 1113957P2)) 1 minus Lle 05 atmeans 1113957P1 ≼ 1113957P2 holds

4 Aggregation Operators for Interval-ValuedPythagorean Hesitant Fuzzy Information

In multiattribute decision-making problems the selectionof aggregation operators is a basis problem which is alsoimportant in the interval-valued Pythagorean hesitant

fuzzy environment Considering an IVPHFE is regardedas the extension of an IVHFE IVPFE or PHFE wepropose a series of aggregation operators for interval-valued Pythagorean hesitant fuzzy information inthis section based on the discussion of aggregation op-erators in [27 45 50 51] and deduce some desirableproperties

41 lte IVPHFWA IVPHFWG GIVPHFWA andGIVPHFWG Operators

Definition 17 Let 1113957Pi (i 1 2 middot middot middot n) be a collection of someIVPHFEs and ω (ω1ω2 middot middot middot ωn)T be the weight vector of1113957Pi (i 1 2 middot middot middot n) with ωi isin [0 1] 1113936

ni1ωi 1 and λgt 0

Complexity 7

(1) An interval-valued Pythagorean hesitant fuzzyweighted averaging (IVPHFWA) operator can beseen a map IVPHFWA Ωn⟶Ω such thatIVPHFWA 1113957P1 middot middot middot 1113957Pn1113872 1113873 ω1

1113957P1 oplus middot middot middot oplusωn1113957Pn

1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899

| 1113957μi 1113957]1113866 1113867i isin 1113957Pi i 1 middot middot middot n⎫⎬

⎭

(19)

(2) An interval-valued Pythagorean hesitant fuzzyweighted geometric (IVPHFWG) operator can beseen a map IVPHFWG Ωn⟶Ω such that

IVPHFWG 1113957P1 middot middot middot 1113957Pn1113872 1113873 1113957Pω11 otimes middot middot middot otimes 1113957P

ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867

isin 1113957Pi i 1 middot middot middot n⎫⎬

⎭

(20)

(3) A generalized interval-valued Pythagorean hesitantfuzzy weighted averaging (GIVPHFWA) operatorcan be seen a map GIVPHFWA Ωn⟶Ω whichsatisfies

GIVPHFWAλ1113957P1 middot middot middot 1113957Pn1113872 1113873 ω1

1113957Pλ1 oplus middot middot middot oplusωn

1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1 minus 1113945n

i11 minus 1 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972


i11 minus 1 minus ]+

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867

isin 1113957Pi i 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0

(4) A generalized interval-valued Pythagorean hesitantfuzzy weighted geometric (GIVPHFWG) operator

can be seen a map GIVPHFWG Ωn⟶Ω whichsatisfies

GIVPHFWGλ1113957P1 middot middot middot 1113957Pn1113872 1113873

1λ

λ 1113957P11113872 1113873ω1 otimes middot middot middot otimes λ 1113957Pn1113872 1113873

ωn1113872 1113873


i11 minus 1 minus μminus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972


i11 minus 1 minus μ+

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

(22)

where λgt 0 If λ 1 then the GIVPHFWA andGIVPHFWG operators reduce to the IVPHFWA andIVPHFWG operators respectively

Example 1 Suppose 1113957P1 lang[03 05] [02 04]rang

8 Complexity

1113957P2 [01 04] [03 06]lang rang [01 04] [05 07]lang rang

1113957P3 [03 04] [02 06]lang rang [01 03] [02 06]lang rang [04 07] [02 06]lang rang (23)

are three IVPHFEs and their weight vector isω (015 030 055)T

en by Definition 17 we have

IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭

[0258304173] [0225905646]lang rang [0149803701] [0225905646]lang rang

middot [0327006102] [0225905646]lang rang [0258304173] [0263305913]lang rang


IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P

ω23 otimes 1113957P

ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

middot | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎬

⎭


middot [0252805627] [0235105774]lang rang [0215804137] [0329406142]lang rang [0117903531]lang

middot 0329406142][ rang [0252805627] [0329406142]lang rang

GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1 minus 11139453


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭

[0282704243] [0225405589]lang rang [0219003982] [0225405589]lang rang [0365006418]lang

middot 0225405589][ rang [0282704243] [0258205805]lang rang [0219003982] [0258205805]lang rang [0365006418]lang

middot 0258205805][ rang

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭

[0213504129] [0253205858]lang rang [0117403508] [0253205858]lang rang [0247105437] [0253205858]lang rang

middot [0213504129] [0410106298]lang rang [0117403508] [0410106298]lang rang [0247105437] [0410106298]lang rang⎫⎬

⎭

(24)

Complexity 9

Based on Proposition 1 we know the above four ag-gregation operators are all IVPHFEs Also we can obtainthat the score values are

S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]

S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]

S GIVPHFWG31113957P1

1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]

So P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840

P SGIVPHFWG31113957P1

1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876

Hence IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)

In fact we have the following

Lemma 1 (see [52]) Let xi gt 0 ωi gt 0 (i 1 2 middot middot middot n) and1113936

ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)

with equality if and only if x1 x2 middot middot middot xn

Proposition 5 Let 1113957Pi (i 1 2 middot middot middot n) be a collection ofIVPHFEs ωi gt 0 and 1113936

ni1ωi 1 then

IVPHFWG 1113957P11113957P2 middot middot middot

1113957Pn1113872 1113873≼IVPHFWA 1113957P11113957P2 middot middot middot

1113957Pn1113872 1113873

(27)

with equality if and only if 1113957P1 1113957P2 middot middot middot 1113957Pn

Proof Suppose lang[μminusi μ+

i ] []minusi ]+

i ]rang isin 1113957Pi for any i Based onLemma 1 we have

1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n

i1ωi 1 minus μminus

i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le

1 minus 1113937ni1(1 minus (μminus

i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)

By Definition 17 and Proposition 4P(S( 1113957P

ω11 otimes middot middot middot otimes 1113957P

ωn

n )ge S(ω11113957P1 oplus middot middot middot oplusωn

1113957Pn))le 05holds which means



1113957Pn1113872 1113873

(30)

Proposition 6 Let 1113957Pi (i 1 2 middot middot middot n) be a collection ofIVPHFEs λgt 0 ωi gt 0 and 1113936

ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1

1113957P2 middot middot middot 1113957Pn1113872 1113873

GIVPHFWGλ1113957P1

1113957P2 middot middot middot 1113957Pn1113872 1113873≼IVPHFWA 1113957P1


(31)


i ] []minusi ]+

i ]rang isin 1113957Pi for any i Since1113937

ni1((μminus

i )2)ωi (1113937ni1((μminus

i )2λ)ωi )1λ le (1113936ni1ωi(μminus

i )2λ)1λ

(1 minus 1113936ni1ωi(1 minus (μminus

i )2λ))1λ le 1 minus (1113937ni1(1 minus (μminus

i )2λ)ωi )1λwe have 1113937

ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le

(1 minus 1113937ni1(1 minus (μminus

i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le

(1 minus 1113937ni1(1 minus (μ+

i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le


i1ωi 1 minus 1 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n

i1ωi 1 minus ]minus

i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly

1 minus (1 minus 1113937ni1(1 minus (1 minus (]+

i )2)λ)ωi )1λ1113969

le

1 minus 1113937ni1(1 minus (]+

i )2)ωi

1113969

holdsAll imply that IVPHFWG( 1113957P1 middot middot middot 1113957Pn)≼

GIVPHFWAλ(1113957P1 middot middot middot 1113957Pn)

In the same way we have GIVPHFWGλ(1113957P1 middot middot middot 1113957Pn)≼

IVPHFWA( 1113957P1 middot middot middot 1113957Pn) and complete the proof of Prop-osition 6

Proposition 42 shows no matter how the parameterλ (λgt 0) changes the values obtained by IVPHFWG op-erators are not bigger than the ones obtained byGIVPHFWA operators and the values obtained byGIVPHFWG operators are not bigger than the ones ob-tained by IVPHFWA operators

Proposition 7 Let 1113957Pi (i 1 2 middot middot middot n) be a collection ofIVPHFEs λ2 ge λ1 gt 0 ωi gt 0 and 1113936

ni1ωi 1 then

GIVPHFWAλ11113957P1

1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873

GIVPHFWGλ11113957P1

1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)

Proof According to eorem 38 in [31] we have

1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)

Also we can conclude that



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11

erefore by Proposition 4 we have


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)

42 lte IVPHFOWA IVPHFOWG GIVPHFOWA andGIVPHFOWG Operators

Definition 18 Let 1113957Pi (i 1 2 middot middot middot n) be a collection of someIVPHFEs 1113957Pσ(i) be the ith largest of them and

κ (κ1 κ2 middot middot middot κn)T be the associated vector such thatκi isin [0 1] 1113936

ni1κi 1 and λgt 0 en

(1) An interval-valued Pythagorean hesitant fuzzy or-dered weighted averaging (IVPHFOWA) operatorcan be seen a map IVPHOFWA Ωn⟶Ω such that

IVPHFOWA 1113957P1 middot middot middot 1113957Pn1113872 1113873 κ1 1113957Pσ(1) oplus middot middot middot oplus κn1113957Pσ(n)

1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

middot | 1113957μσ(i) 1113957]σ(i)1113924 1113925 isin 1113957Pσ(i) i 1 middot middot middot n1113897

(38)

(2) An interval-valued Pythagorean hesitant fuzzy or-dered weighted geometric (IVPHFOWG) operator

can be seen a map IVPHFOWG Ωn⟶Ω suchthat

IVPHFOWG 1113957P1 middot middot middot 1113957Pn1113872 1113873 1113957Pκ1σ(1) otimes middot middot middot otimes 1113957P

κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)

(3) A generalized interval-valued Pythagorean hesitantfuzzy ordered weighted averaging (GIVPHFOWA)

operator can be seen a map GIVPHFOWA

Ωn⟶Ω which satisfies

GIVPHFOWAλ1113957P1 middot middot middot 1113957Pn1113872 1113873 κ1

_1113957Pσ(1)λ oplus middot middot middot oplus κn_1113957Pσ(n)1113874 1113875

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n

i11 minus 1 minus _]minus

σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


i11 minus 1 minus _]+

σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

middot | _1113957μσ(i)_1113957]σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0

(4) A generalized interval-valued Pythagorean hesitantfuzzy ordered weighted geometric (GIVPHFOWG)

operator can be seen a mapGIVPHFOWG Ωn⟶Ω which satisfies

GIVPHFOWGλ1113957P1 middot middot middot 1113957Pn1113872 1113873

1λ

λ 1113957Peuro

σ(1)1113874 1113875κ1otimes middot middot middot otimes λ 1113957P

euroσ(n)1113874 1113875

κn

1113874 1113875


n

i11 minus 1 minus euroμminus

σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n

i11 minus 1 minus euroμ+

σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n

i11 minus euro]minus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

middot | euro1113957μσ(i)euro1113957]σ(i)1113924 1113925 isin 1113957P

euroσ(i) i 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

(41)

where λgt 0 If λ 1 then the GIVPHFOWA andGIVPHFOWG operators reduce to the IVPHFOWA andIVPHFOWG operators respectively

By comparing Definition 17 with Definition 18 wecan find that the IVPHFOWA IVPHFOWG GIVPH-FOWA and GIVPHFOWG operators weight the orderedpositions of the IVPHFEs instead of weighting theIVPHFEs themselves Also their characteristics lie inreordering original IVPHFEs according to a decreasingorder and in aggregating through the associated weightsof positions where the weight κi is associated with the ithposition in the collection of the IVPHFEs during ag-gregation processes

Example 2 Suppose1113957P1 [03 05] [02 04]lang rang



[04 07] [02 06]lang rang

(42)

are three IVPHFEs and the aggregation-associated vector isκ (02 045 035)T

en by Definition 15 we have the score values of1113957P1

1113957P2 and 1113957P3 as follows

S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877

[minus 02075 minus 0005]

S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)

Based on Definition 16 we can obtain thatP(S( 1113957P1)gt S( 1113957P3)) 05678 andP(S( 1113957P3)gt S( 1113957P2))

07024 and then 1113957Pσ(1) 1113957P1 1113957Pσ(2) 1113957P3 and 1113957Pσ(3) 1113957P2

According to Definition 18 we have

Complexity 13

IVPHFOWA 1113957P11113957P2

1113957P31113872 1113873 κ1 1113957Pσ(1) oplus κ1 1113957Pσ(2) oplus κn1113957Pσ(3)

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

middot | 1113957μσ(i) 1113957]σ(i)1113924 1113925 isin 1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭

[02505 04229] [02305 05533]lang rang [02505 04229] [02756 05839]lang rang [01629 03856]lang

[02305 05533]rang [01629 03856] [02756 05839]lang rang [03097 05865] [02305 05930]lang rang

[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P

κ2σ(2) otimes 1113957P

κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

middot | 1113957μσ(i) 1113957]σ(i)1113924 1113925 isin 1113957Pσ(i) i 1 2 31113967

[02793 04310] [02299 05466]lang rang [02793 04310] [02697 05706]lang rang



GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

middot | 1113957μσ(i) 1113957]σ(i)1113924 1113925 isin 1113957Pσ(i) i 1 2 3⎫⎬

⎭




(44)

14 Complexity

Based on Definition 14 we can obtain that

S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]

S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]

S GIVPHFOWG31113957P1

1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038

P S GIVPHFOWG31113957P1

1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)

SoIVPHFOWG 1113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)

Similar to Proposition 5ndash7 we easily obtain the followingproperties of ordered aggregation operators of IVPHFEs

Proposition 8 Let 1113957Pi (i 1 2 middot middot middot n) be a collection ofIVPHFEs λgt 0 and λ2 gt λ1 gt 0 then

IVPHFOWG 1113957P11113957P2 middot middot middot 1113957Pn1113872 1113873≼ IVPHFOWA 1113957P1


IVPHFOWG 1113957P11113957P2 middot middot middot 1113957Pn1113872 1113873≼GIVPHFOWAλ

1113957P11113957P2 middot middot middot 1113957Pn1113872 1113873

GIVPHFOWGλ1113957P1

1113957P2 middot middot middot 1113957Pn1113872 1113873≼ IVPHFOWA 1113957P11113957P2 middot middot middot 1113957Pn1113872 1113873

GIVPHFOWAλ11113957P1

1113957P2 middot middot middot 1113957Pn1113872 1113873≼GIVPHFOWAλ21113957P1



1113957P2 middot middot middot 1113957Pn1113872 1113873≽GIVPHFOWGλ21113957P1


(48)

From Definition 17 and Definition 18 we know that theIVPHFWA IVPHFWG GIVPHFWA and GIVPHFWGoperators weight the interval-valued hesitant fuzzy argumentitself and there is no consideration of the impact of the order ofparameters while the IVPHFOWA IVPHFOWG GIVPH-FOWA and GIVPHFOWG operators only weight the orderedposition of each given IVPHFE and there is no considerationof the impact of the argument itself Hence we propose a newhybrid aggregation operator which can weight both all thegiven arguments and their ordered positions

43 lte IVPHFHA IVPHFHG GIVPHFHA andGIVPHFHG Operators

Definition 19 For a collection of some IVPHFEs1113957Pi (i 1 2 middot middot middot n) ω (ω1ω2 middot middot middot ωn)T is the weightvector of them with ωi isin [0 1] and 1113936

ni1ωi 1 Suppose κ

(κ1 κ2 middot middot middot κn)T is the associated vector such that κi isin [0 1]1113936


(1) An interval-valued Pythagorean hesitant fuzzy hy-brid averaging (IVPHFHA) operator can be seen amap IVPHFHA Ωn⟶Ω such that

IVPHFHA 1113957P1 middot middot middot 1113957Pn1113872 1113873 κ1_1113957Pσ(1) oplus middot middot middot oplus κn

_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)

_1113957]σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 middot middot middot n⎫⎬

⎭ (49)

Complexity 15

where _1113957Pσ(i) is the largest ith of _1113957Pk nωk

1113957Pk (k 1 2 middot middot middot n)

(2) An interval-valued Pythagorean hesitant fuzzy hy-brid geometric (IVPHFHG) operator can be seen amap IVPHFHG Ωn⟶Ω such that

IVPHFHG 1113957P1 middot middot middot 1113957Pn1113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes middot middot middot otimes σ(n)1113872 1113873

κn

1113945n

i1euroμminusσ(i)1113872 1113873

κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩

| euro1113957μσ(i)euro1113957]σ(i)1113924 1113925 isin euro1113957Pσ(i) i 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

(50)

where euro1113957Pσ(i) is the largest ith of euro1113957Pk ( 1113957Pk)nωk

(k 1 2 middot middot middot n)

(3) A generalized interval-valued Pythagorean hesitantfuzzy hybrid averaging (GIVPHFHA) operator can

be seen a map GIVPHFHA Ωn⟶Ω whichsatisfies

GIVPHFHAλ1113957P1 middot middot middot 1113957Pn1113872 1113873 κ1

_1113957Pσ(1)1113874 1113875λoplus middot middot middot oplus κn

_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| _1113957μσ(i)_1113957]σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 middot middot middot n

(51)



(4) A generalized interval-valued Pythagorean hesitantfuzzy hybrid geometric (GIVPHFHG) operator can

be seen a map GIVPHFHG Ωn⟶Ω whichsatisfies

GIVPHFHGλ1113957P1 middot middot middot 1113957Pn1113872 1113873

1λ

λ euro1113957Pσ(1)1113874 1113875κ1otimes middot middot middot otimes λ euro1113957Pσ(n)1113874 1113875

κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μσ(i)_1113957]σ(i)1113924 1113925 isin euro1113957Pσ(i) i 1 middot middot middot n

⎫⎬

⎭

(52)

where euro1113957Pσ(i) is the largest ith of _1113957Pk ( 1113957Pk)nωk (k 12 middot middot middot n)

16 Complexity

If λ 1 then the GIVPHFHA and GIVPHFHG operatorsreduce to the IVPHFHAand IVPHFHGoperators respectively




are three IVPHFEs the weight vector of the attributes isω (015 030 055)T and the aggregation-associated vec-tor is κ (02 045 035)T en we can obtain

_1113957P1 (3 times 015)otimes 1113957P1 [02038 03485] [04847 06621]lang rang

_1113957P2 (3 times 03)otimes 1113957P2 [00949 03811] [03384 06314]lang rang [00949 03811] [05359 07254]lang rang


[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)

By Definition 16 we have P(S(_1113957P1)leS(

_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))

05509 and P(S(euro1113957P2)leS(

euro1113957P1)) 1 So we can get

_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17

According to Definition 19 we can obtain

IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩

| euro1113957μσ(i)euro1113957]σ(i)1113924 1113925 isin euro1113957Pσ(i) i 1 2 3

⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13

3 euro1113957Pσ(1)1113874 1113875κ1otimes 3 euro1113957Pσ(2)1113874 1113875

κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity

5 The Application of the AggregationOperators of IVPHFEs for MAGDM

In multiattribute group decision-making (MAGDM)problems because the different decision makers usuallycome from different research directions and backgroundsthey usually have diverging opinionsus how to constructan optimal model that can be obtained with the maximumdegree of consensus or agreement of these experts for thegiven alternatives is an important and interesting researchcontent in MAGDM In this section we propose a newMAGDM method based on the interval-valued Pythagorashesitant fuzzy integration operators Furthermore theranking of integrated interval-valued Pythagorean hesitantfuzzy numbers is accomplished by adopting the scorefunction e specific details of the MAGDM method isshown in following

Let Y Yi | i 1 2 middot middot middot m1113864 1113865 be a finite set of alternativesC Cj | j 1 2 middot middot middot n1113966 1113967 be a set of attributes andD Dk | k 1 2 middot middot middot l1113864 1113865 be a set of l decision makers en

let M(k) ( 1113957P(k)

ij )mtimesn is an interval-valued Pythagoreanhesitant fuzzy decision matrix (IVPHFDM) of the kth de-cision maker where 1113957P

(k)

ij lang1113957μ(k)ij 1113957](k)

ij rang1113966 1113967 (i 1 2 middot middot middot m

j 1 2 middot middot middot n) is an IVPHFE given by the decision makerDK in which 1113957μ(k)

ij indicates the possible membership in-tervals that the alternative Yi satisfies the attribute Cj and1113957](k)

ij indicates the possible nonmembership intervals that thealternative Yi nonmembership intervals not satisfy the at-tribute Cj

In the actual MAGDM problem we will encounter someattributes are benefit (ie the bigger the attribute values thebetter) and the other attributes are cost (ie the smaller theattribute values the better) In such cases transform the costattribute values into the benefit attribute values and normalizethe IVPHFDM M(k) ( 1113957P

(k)

ij )mtimesn into the correspondingIVPHFDM N(k) (1113957P

(k)

ij )mtimesn by the method in [53] where

1113957P(k)

ij

1113957P(k)

ij for bene fit attributeCj

1113957P(k)

ij1113874 1113875C

for cost attributeCj

⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)

ij )C is the complement of 1113957P(k)

ij Based on the above discussion and analysis we develop

an approach for multiattribute decision-making in interval-valued Pythagorean hesitant fuzzy environments e al-gorithm involves the following steps

Step 1 construct the IVPHFDM M(k) ( 1113957P(k)

ij )mtimesn andtransform M(k) into the corresponding normalizedmatrix N(k) ( 1113957P

(k)

ij )mtimesnStep 2 utilize the GIVPHFWA operator (or theGIVPHFWG operator) to aggregate all the IVPHFDMsN(k) (1113957P

(k)

ij )mtimesn (k 1 2 middot middot middot l) into the collectiveIVPHFDM N (1113957Pij)mtimesn where ξ (ξ1 ξ2 middot middot middot ξl)

T isthe weight vector of the decision makersDk (k 1 2 middot middot middot l) and the specific integration oper-ation is as follows

1113957Pij GIVPHFWAλ1113957P

(1)

ij middot middot middot 1113957P(l)

ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l

k11 minus 1 minus ]minus

1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

1 minus 1 minus 1113945l

k11 minus 1 minus ]+

1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)

ij i 1 middot middot middot m j 1 middot middot middot n⎫⎬

⎭

(58)

or

1113957Pij GIVPHFWGλ1113957P

(1)


ij1113874 1113875


l

k11 minus 1 minus μminus

1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l

k11 minus 1 minus μ+

1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)

ij i 1 middot middot middot m j 1 middot middot middot n

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19

Step 3 utilize the GIVPHFHA operator (or theGIVPHFHG operator) to aggregate all the preferencevalues 1113957Pi again where κ (κ1 κ2 middot middot middot κn)T is the

associated weight vector e specific integration op-eration is as follows

1113957Pi GIVPHFHAλ1113957Pi1 middot middot middot 1113957Pin( 1113857

1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩


j11 minus 1 minus _]minus

1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n

j11 minus 1 minus _]+

1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)

1113884 1113885 isin _1113957Piσ(j) i 1 middot middot middot m j 1 middot middot middot n⎫⎬

⎭

(60)

or 1113957Pi GIVPHFHGλ(1113957Pi1 middot middot middot 1113957Pin)


j11 minus 1 minus euroμminus

1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972


j11 minus 1 minus euroμ+

1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n

j11 minus euro]minus

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)

1113884 1113885 isin euro1113957Piσ(j) i 1 middot middot middot m j 1 middot middot middot n⎫⎬

⎭

(61)

Here let ω (ω1ω2 middot middot middot ωn)T be the weight vector ofthe attributes Cj (j 1 2 middot middot middot n) where ωj isin [0 1]

(j 1 2 middot middot middot n) and 1113936nj1ωj 1 So we can get _1113957Piσ(j)

(or euro1113957Piσ(j)) defined by

_1113957Pij n times ωj1113872 1113873otimes 1113957Pij

1 minus 1 minus μminus

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij

1113884 1113885 isin 1113957Pij i 1 middot middot middot m j 1 middot middot middot n

⎫⎪⎪⎬

⎪⎪⎭

euro1113957Pij 1113957Pij1113872 1113873ntimesωj( 1113857

μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦

1 minus 1 minus ]minus

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity

Step 4 compute the score values S(1113957Pi) and the accuracyvalues H(1113957Pi) of 1113957Pi (i 1 2 middot middot middot m) by Definition 15Step 5 get the priority of the alternatives Yi by rankingS(1113957Pi) (i 1 2 middot middot middot m) based on Definition 16 End

Example 4 A company needs to select a new site forbuildingsree alternatives Yi (i 1 2 3) are available andthe three decision makers Dk (k 1 2 3) consider threeattributes to decide which project to choose (1) C1 (price)(2) C2 (environment) and (3) C3 (location) Among theconsidered attributes C1 is of cost type and C2 andC3 are ofbenefit type e weight vector of the decision makersDk (k 1 2 3) is ξ (01 06 03)T e weight vector ofthe attributes Cj (j 1 2 3) is ω (05 02 03)T Supposethat the decision makers provide their own IVPHFDMM(k) ( 1113957P

(k)

ij )3times3 as listed in Tables 1ndash3 respectively where1113957P

(k)

ij is an IVPHFE given by the decision maker Dk

(Tables 4ndash7)

Step 1 transform the decision matrix M(k) into thecorresponding normalized matrix N(k) (1113957P

(k)

ij )3times3(k 1 2 3) (in Tables 4ndash6)

Step 2 suppose λ 5 and utilize the GIVPHFWAoperator to aggregate the three IVPHFDMsN(k) (1113957P

(k)

ij )3times3 (k 1 2 3) into the collectiveIVPHFDM N (1113957Pij)3times3 (in Table 7)Step 3 aggregate all the preference values1113957Pij (j 1 2 3) in the ith line of N based on theGIVPHFHA operators whose associated weightingvector is κ (025 065 01)TStep 4 compute the three score values as follows

S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)

Step 5 get the priority of the alternatives by ranking thescore functions We can get the ranking order of allalternatives Y1 ≻Y3 ≻Y2 So the optimal scheme is Y1

Furthermore we study the change of the GIVPHFHAand GIVPHFHG operators with the parameter λ

In Table 8 we can find that the score values obtained by theGIVPHFHA operators become bigger as the parameter λ

Table 1 IVPHFDM M(1)

C1 C2 C3

Y1 lang[04 05] [03 05]rang lang[02 03] [06 08]rang

lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang

lang[01 04] [06 07]ranglang[08 09] [01 01]rang lang[03 05] [04 04]rang


C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang

lang[05 07] [02 03]ranglang[09 09] [01 02]rang

Y2 lang[07 08] [01 02]rang lang[08 09] [01 03]rang lang[05 08] [02 03]rang

lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21

increases for the same aggregation arguments and the decisionmakers can choose the values of λ according to their prefer-ences see that as the parameter λ changes we have differentresults Suppose λ 01 02 middot middot middot 50 thenwe analyze the impactof the role of λ on the aggregation results and see the trend inFigure 1 Because of space limitations we gave the image withinthe range of λ isin [07 119] the same hereinafter

From Figure 1 we can see that all of S(1113957Pi)(i 1 2 3)

increase with the increase of λ When λ 07 the ranking

order of the three alternatives is Y2 ≻Y3 ≻Y1 and the bestchoice is Y2 when λ isin [08 15] the ranking order of thethree alternatives is Y3 ≻Y2 ≻Y1 and the best choice is Y3when λ isin [16 114] the ranking order of the three alter-natives is Y1 ≻Y3 ≻Y2 and the best choice is Y1 whenλ isin [115 50] the ranking order of the three alternatives isY2 ≻Y1 ≻Y3 and the best choice is Y2

In Table 9 we use the GIVPHFHG operators to ag-gregate the values of the alternatives We find the score

Table 4 IVPHFDM N(1)

C1 C2 C3

Y1 lang[03 05] [04 05]rang lang[02 03] [06 08]rang lang[06 08] [01 03]rang lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang

lang[04 05] [05 05]rang lang[06 07] [01 02]rang


Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3

Y1lang[01 02] [07 09]rang lang[01 02] [07 08]rang

lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity

Table 9 Score values based on the GIVPHFHG operators

λ 01 λ 10 λ 20Y1 [0392 04493] [minus 03225 minus 02264] [minus 03633 minus 02567]

Y2 [minus 00184 00047] [minus 02448 minus 01454] [minus 02774 minus 01621]


Ranking Y1 ≻Y2 ≻Y3 Y2 ≻Y1 ≻Y3 Y2 ≻Y1 ≻Y3

λ 30 λ 40 λ 50Y1 [minus 03793 minus 02681] [minus 03877 minus 02344] [minus 03928 minus 02011]

Y2 [minus 02899 minus 00968] [minus 02217 minus 00990] [minus 02244 minus 01006]



Table 8 Score values based on the GIVPHFHA operators

λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ

λ = 16 P(S(Y1) ge S(Y3)) gt 05 P(S(Y3) ge S(Y2))05 so Y1 gt Y3 gt Y2

λ = 08 P(S(Y3) ge S(Y2)) gt 05 P(S(Y2) ge S(Y1))05 so Y3 gt Y2 gt Y1λ = 07 P(S(Y2) ge S(Y3)) gt 05 P(S(Y3) ge S(Y1))05 so Y2 gt Y3 gt Y1


70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3

Figure 1 Variation of the score functions of the GIVPHFHA operators by the parameter λ



λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3

Figure 2 Variation of the score functions of the GIVPHFHG operators by the parameter λ

Complexity 23

values obtained by the GIVPHFHG operators becomesmaller as the parameter λ increases for the same aggregationarguments and the decision makers can choose the values ofλ according to their preferences See the details in Figure 2


decrease with the increase of λ When λ isin [07 12] theranking order of the three alternatives is Y1 ≻Y2 ≻Y3 andthe best choice is Y1 when λ isin [13 50] the ranking order ofthe three alternatives is Y2 ≻Y1 ≻Y3 and the best choice isY2

6 Conclusions

is paper extends IVHFs and IVPFs to IVPHFSs Firstlythe new score functions and accuracy functions of IVPHFSsare introduced to compare the size of IVPHFEs based onthe comparison of interval numbers en we focus on anew series of interval-valued Pythagorean hesitant fuzzyaggregation operators including the IVPHFWAIVPHFWG GIVPHFWA GIVPHFWG IVPHFOWAIVPHFOWG GIVPHFOWA GIVPHFOWG IVPHFHAIVPHFHG GIVPHFHA and GIVPHFHG operators erelations of these operators are developed Furthermore anew approach has been developed based on the proposedoperators to solve theMAGDMproblems under the IVPHFenvironment Finally a numerical example is used to il-lustrate the effectiveness and feasibility of our proposedmethod

e following work is to enhance the study of the ag-gregation operators of weighted IVPHFSs we shall developthe new potential application of the proposed operators inanother field such as clustering analysis image processingpattern recognition and so on We hope that these willenrich and provide more new ideas and new methods forthese fields under the interval-valued Pythagorean hesitantfuzzy environment

Abbreviations

GIVPHFHA Generalized interval-valued Pythagoreanhesitant fuzzy hybrid averaging

GIVPHFHG Generalized interval-valued Pythagoreanhesitant fuzzy hybrid geometric

GIVPHFOWA Generalized interval-valued Pythagoreanhesitant fuzzy ordered weighted averaging

GIVPHFOWG Generalized interval-valued Pythagoreanhesitant fuzzy ordered weighted geometric

GIVPHFWA Generalized interval-valued Pythagoreanhesitant fuzzy weighted averaging

GIVPHFWG Generalized interval-valued Pythagoreanhesitant fuzzy weighted geometric

HFE Hesitant fuzzy elementHFS Hesitant fuzzy setIVHFS Interval-valued hesitant fuzzy setIVHFE Interval-valued hesitant fuzzy elementIVPHFDM Interval-valued Pythagorean hesitant fuzzy

decision matrixIVPHFE Interval-valued Pythagorean hesitant fuzzy

element

IVPHFHA Interval-valued Pythagorean hesitant fuzzyhybrid averaging

IVPHFHG Interval-valued Pythagorean hesitant fuzzyhybrid geometric

IVPHFOWA Interval-valued Pythagorean hesitant fuzzyordered weighted averaging

IVPHFOWG Interval-valued Pythagorean hesitant fuzzyordered weighted geometric

IVPHFWA Interval-valued Pythagorean hesitant fuzzyweighted averaging

IVPHFWG Interval-valued Pythagorean hesitant fuzzyweighted geometric

IVPHFS Interval-valued Pythagorean hesitant fuzzyset

IVPFE Interval-valued Pythagorean fuzzy elementIVPFS Interval-valued Pythagorean fuzzy setMAGDM Multiattribute group decision-makingPHFE Pythagorean hesitant fuzzy elementPHFS Pythagorean hesitant fuzzy set

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e work was supported by the National Natural ScienceFoundation of China (nos 61806001 71771001 71701001and 71871001) Natural Science Foundation of AnhuiProvince (no 1708085MF163) Natural Science Foundationfor Distinguished Young Scholars of Anhui Province (no1908085J03) Research Funding Project of Academic andtechnical leaders and reserve candidates in Anhui Province(no 2018H179) and Provincial Natural Science ResearchProject of Anhui Colleges (no KJ2017A026)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8no 3 pp 338ndash353 1965

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

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

[4] L A Zadeh ldquoe concept of a linguistic variable and itsapplication to approximate reasoning-Irdquo Information Sci-ences vol 8 no 3 pp 199ndash249 1975

[5] J M Mendel R I John and F Liu ldquoInterval type-2 fuzzylogic systems made simplerdquo IEEE Transactions on FuzzySystems vol 14 no 6 pp 808ndash821 2006

[6] R R Yager ldquoGeneralized orthopair fuzzy setsrdquo IEEETransactions on Fuzzy Systems vol 26 no 5 pp 1222ndash12302017

24 Complexity

[7] W Pedrycz ldquoFuzzy sets in pattern recognition methodologyand methodsrdquo Pattern Recognition vol 23 no 1-2pp 121ndash146 1990

[8] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquo Mathematical and Computer Mod-elling vol 53 no 1-2 pp 91ndash97 2011

[9] F Liu ldquoAn efficient centroid type-reduction strategy forgeneral type-2 fuzzy logic systemrdquo Information Sciencesvol 178 no 9 pp 2224ndash2236 2008

[10] P Liu and P Wang ldquoSome q-Rung orthopair fuzzy aggre-gation operators and their applications to multiple-attributedecision makingrdquo International Journal of Intelligent Systemsvol 33 no 2 pp 259ndash280 2018

[11] P Liu and J Liu ldquoSome q-rung orthopai fuzzy Bonferronimean operators and their application to multi-attribute groupdecision makingrdquo International Journal of Intelligent Systemsvol 33 no 2 pp 315ndash347 2018

[12] P Liu and P Wang ldquoMultiple-attribute decision-makingbased on archimedean Bonferroni operators of q-Rungorthopair fuzzy numbersrdquo IEEE Transactions on Fuzzy Sys-tems vol 27 no 5 pp 834ndash848 2019

[13] P Liu S-M Chen and P Wang ldquoMultiple-attribute groupdecision-making based on q-rung orthopair fuzzy powerMaclaurin symmetric mean operatorsrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash16 2019

[14] T Wu X Liu and F Liu ldquoAn interval type-2 fuzzy TOPSISmodel for large scale group decision making problems withsocial network informationrdquo Information Sciences vol 432pp 392ndash410 2018

[15] T Wu X Liu and J Qin ldquoA linguistic solution for doublelarge-scale group decision-making in E-commercerdquo Com-puters amp Industrial Engineering vol 116 pp 97ndash112 2018

[16] Q Wu P Wu L Zhou H Chen and X Guan ldquoSome newHamacher aggregation operators under single-valued neu-trosophic 2-tuple linguistic environment and their applica-tions to multi-attribute group decision makingrdquo Computers ampIndustrial Engineering vol 116 pp 144ndash162 2018

[17] Q Wu W Lin L Zhou Y Chen and H Chen ldquoEnhancingmultiple attribute group decision making flexibility based oninformation fusion technique and hesitant Pythagorean fuzzysetsrdquo Computers amp Industrial Engineering vol 127pp 954ndash970 2019

[18] Q Wu L Zhou Y Chen and H Chen ldquoAn integratedapproach to green supplier selection based on the intervaltype-2 fuzzy best-worst and extended VIKOR methodsrdquoInformation Sciences vol 502 pp 394ndash417 2019

[19] V Torra and Y Narukawa ldquoOn hesitant fuzzy sets and deci-sionrdquo in Proceedings of the 18th IEEE International Conferenceon Fuzzy Systems pp 1378ndash1382 Jeju Island Korea 2009

[20] V Torra ldquoHesitant fuzzy setsrdquo International Journal of In-telligent Systems vol 25 no 6 pp 529ndash539 2010

[21] R M Rodrıguez L Martınez V Torra Z S Xu andF Herrera ldquoHesitant fuzzy sets state of the art and futuredirectionsrdquo International Journal of Intelligent Systemsvol 29 no 6 pp 495ndash524 2014

[22] B Zhu Z Xu and M Xia ldquoDual hesitant fuzzy setsrdquo Journalof Applied Mathematics vol 2012 Article ID 87962913 pages 2012

[23] Y Ju X liu and S Yang ldquoInterval-valued dual hesitant fuzzyaggregation operators and their applications to multiple at-tribute decision makingrdquo Journal of Intelligent amp FuzzySystems vol 27 no 3 pp 1203ndash1218 2014

[24] Y Zang X Zhao and S Li ldquoInterval-valued dual hesitantfuzzy heronian mean aggregation operators and their

application to multi-attribute decision makingrdquo InternationalJournal of Computational Intelligence and Applicationsvol 17 no 1 Article ID 1850005 2018

[25] J-j Peng J-q Wang X-h Wu H-y Zhang and X-h Chenldquoe fuzzy cross-entropy for intuitionistic hesitant fuzzy setsand their application in multi-criteria decision-makingrdquoInternational Journal of Systems Science vol 46 no 13pp 2335ndash2350 2015

[26] N Chen Z Xu and M Xia ldquoInterval-valued hesitant pref-erence relations and their applications to group decisionmakingrdquo Knowledge-Based Systems vol 37 pp 528ndash5402013

[27] N Chen and Z Xu ldquoProperties of interval-valued hesitantfuzzy setsrdquo Journal of Intelligent amp Fuzzy Systems vol 27no 1 pp 143ndash158 2014

[28] W Zeng D Li and Q Yin ldquoWeighted interval-valuedhesitant fuzzy sets and its application in group decisionmakingrdquo International Journal of Fuzzy Systems vol 12pp 1ndash12 2019

[29] Z Zhang ldquoInterval-valued intuitionistic hesitant fuzzy ag-gregation operators and their application in group decision-makingrdquo Journal of Applied Mathematics vol 2013 pp 1ndash332013

[30] M Xia and Z Xu ldquoHesitant fuzzy information aggregation indecision makingrdquo International Journal of ApproximateReasoning vol 52 no 3 pp 395ndash407 2011

[31] Z Zhang ldquoHesitant fuzzy power aggregation operators andtheir application to multiple attribute group decision mak-ingrdquo Information Sciences vol 234 pp 150ndash181 2013

[32] X Zhang and Z Xu ldquoHesitant fuzzy agglomerative hierar-chical clustering algorithmsrdquo International Journal of SystemsScience vol 46 no 3 pp 562ndash576 2015

[33] R R Yager ldquoPythagorean fuzzy subsetsrdquo in Proceeding of theJoint IFSA World Congress and NAFIPS Annual Meetingpp 57ndash61 Edmonton Canada 2013

[34] R R Yager and A M Abbasov ldquoPythagorean membershipgrades complex numbers and decision makingrdquo Interna-tional Journal of Intelligent Systems vol 28 no 5 pp 436ndash452 2013

[35] R R Yager ldquoPythagorean membership grades in multicriteriadecision makingrdquo IEEE Transactions on Fuzzy Systemsvol 22 no 4 pp 958ndash965 2014

[36] R R Yager ldquoProperties and applications of Pythagorean fuzzysetsrdquo Imprecision and Uncertainty in Information Represen-tation and Processing Studies in Fuzziness and Soft Com-puting Springer Cham Switzerland 2016

[37] X Zhang and Z S Xu ldquoExtension of TOPSIS to multiplecriteria decision making with Pythagorean fuzzy setsrdquo In-ternational Journal of Intelligent Systems vol 29 no 12pp 1061ndash1078 2015

[38] P Ren Z Xu and X Gou ldquoPythagorean fuzzy TODIMapproach to multi-criteria decision makingrdquo Applied SoftComputing vol 42 pp 246ndash259 2016

[39] Z Liu P Liu W Liu and J Pang ldquoPythagorean uncertainlinguistic partitioned Bonferroni mean operators and theirapplication in multi-attribute decision makingrdquo Journal ofIntelligent amp Fuzzy Systems vol 32 no 3 pp 2779ndash27902017

[40] F Teng Z Liu and P Liu ldquoSome powerMaclaurin symmetricmean aggregation operators based on Pythagorean fuzzylinguistic numbers and their application to group decisionmakingrdquo International Journal of Intelligent Systems vol 33no 9 pp 1949ndash1985 2018

Complexity 25

[41] X Peng and Y Yang ldquoFundamental properties of interval-valued Pythagorean fuzzy aggregation operatorsrdquo Interna-tional Journal of Intelligent Systems vol 31 no 5 pp 444ndash4872016

[42] K Rahman and S Abdullah ldquoGeneralized interval-valuedPythagorean fuzzy aggregation operators and their applica-tion to group decision-makingrdquo Granular Computing vol 1pp 1ndash11 2018

[43] K Rahman S Abdullah and A Ali ldquoSome induced aggre-gation operators based on interval-valued Pythagorean fuzzynumbersrdquo Granular Computing vol 1 pp 1ndash10 2018

[44] L Yi Y Qin and H Yun ldquoMultiple criteria decision makingwith probabilities in interval-valued Pythagorean fuzzy set-tingrdquo International Journal of Fuzzy Systems vol 20 no 2pp 558ndash571 2017

[45] D C Liang A P Darko and Z S Xu ldquoInterval-valuedPythagorean fuzzy extended Bonferroni mean for dealingwith heterogenous relationship among attributesrdquo Interna-tional Journal of Intelligent Systems pp 1ndash31 2018

[46] W F Liu and X He ldquoPythagorean hesitant fuzzy setsrdquo FuzzySystems and Mathematics vol 30 no 4 pp 107ndash115 2016 inChinese

[47] M S A Khan S Abdullah A Ali N Siddiqui and F AminldquoPythagorean hesitant fuzzy sets and their application togroup decision making with incomplete weight informationrdquoJournal of Intelligent amp Fuzzy Systems vol 33 no 6pp 3971ndash3985 2017

[48] G Wei M Lu X Tang and Y Wei ldquoPythagorean hesitantfuzzy Hamacher aggregation operators and their applicationto multiple attribute decision makingrdquo International Journalof Intelligent Systems vol 33 no 6 pp 1197ndash1233 2018

[49] H Dawood lteories of Interval Arithmetic MathematicalFoundations and Applications LAP Lambert AcademicPublishing Saarbrucken Germany 2011

[50] Z S Xu and Q L Da ldquoe uncertain OWA operatorrdquo In-ternational Journal of Intelligent Systems vol 17 no 6pp 569ndash575 2002

[51] H Garg ldquoNew exponential operational laws and their ag-gregation operators for interval-valued Pythagorean fuzzymulticriteria decision-makingrdquo International Journal of In-telligent Systems vol 33 no 3 pp 653ndash683 2018

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

[53] V Torra and Y Narukawa Modeling Decisions InformationFusion and Aggregation Operators Springer Cham Swit-zerland 2007

[54] Z S Xu and X Cai ldquoNonlinear optimization models formultiple attribute group decision making with intuitionisticfuzzy informationrdquo International Journal of Intelligent Sys-tems vol 25 no 6 pp 489ndash513 2010

26 Complexity

[35 36] Zhang and Xu [37] Ren et al [38] Liu et al [39]and Teng et al [39] have studied several kinds of Pythag-orean fuzzy aggregation operators and applied them todecision-making problems Furthermore Peng and Yang[40] proposed the definition of interval-valued Pythagoreanfuzzy set and Rahamn et al [41 42] developed group de-cision-making with interval-valued Pythagorean fuzzy en-vironments Yi et al [43] applied it to multicriteria decision-making problems Also Liang et al [44] introduced theinterval-valued Pythagorean fuzzy extended Bonferronimean operators

Recently some scholars have tried to combine Py-thagorean fuzzy sets and hesitant fuzzy sets and all calledthem Pythagorean hesitant fuzzy sets (PHFSs) but theirconstruction methods are controversial LetA langx ΓA(x)ΨA(x)rang | x isin U1113864 1113865 be a Pythagorean fuzzy setin U where ΓA(x) μA(x) | μA(x) isin [0 1]1113864 1113865 andΨA(x) ]A(x) | ]A(x) isin [0 1]1113864 1113865 Liu and He [45] definedthat for any x any μA(x) isin ΓA(x) and any ]A(x) isin ΨA(x)(μA(x))2 + (]A(x))2 le 1 holds Khan et al [46] defined thatfor any x and for any μA(x) isin ΓA(x) there is ]A(x) isin ΨA(x)such that (μA(x))2 + (]A(x))2 le 1 Also for any]A(x) isin ΨA(x) there is μA(x) isin ΓA(x) such that(μA(x))2 + (]A(x))2 le 1 Wei et al[47] also defined anotherPythagorean fuzzy set P langx hP(x)rang | x isin U1113864 1113865 wherehP(x) is a set of some Pythagorean fuzzy elements in U ethree definitions have their own merits Considering thepairing of possible membership degrees and possible non-membership degrees it is more conducive to the aggregationof hesitant fuzzy numbers In this paper we choose Weirsquosdefinition as the definition of PHFS

From the above analysis we can see that PHFSs are moreconvenient to deal with fuzzy information than hesitantfuzzy sets or Pythagorean fuzzy sets However for realmultiattribute group decision-making (MAGDM) prob-lems it is difficult for decision makers to provide some exactand crisp fuzzy values to depict uncertain or insufficientalternatives because of the increasing complexity of socialand economic life e aim of this paper is to extend PHFSsto interval-valued Pythagorean hesitant fuzzy sets(IVPHFSs) and develop MAGDM approaches to interval-valued Pythagorean hesitant fuzzy environments based onnewly constructed aggregation operators In particular sincethere is no one-to-one correspondence between intervalnumbers and real numbers this paper directly uses intervalnumbers to define score functions and accuracy functionswhich can preserve interval-valued Pythagorean fuzzy in-formation as much as possible

e rest of this paper is organized as follows In Section2 we review some basic concepts and results of hesitantfuzzy sets interval-valued hesitant fuzzy sets and interval-valued Pythagorean fuzzy sets Section 3 proposes the def-inition of IVPHFS and discusses some basic operationallaws In particular we propose the concepts of the scorefunction and accuracy function which are both appeared asinterval values Some operators for aggregating interval-valued Pythagorean hesitant fuzzy information are studiedand developed in section 4 Section 5 shows the applicationto MAGDMs in interval-valued Pythagorean hesitant fuzzy

environments and illustrates the feasibility and applicabilityof the proposed method Concluding remarks are made inSection 6

2 Preliminaries

is section will briefly review the basic notations and resultsof hesitant fuzzy sets interval-valued Pythagorean fuzzy sets

Hereinafter without explanation let U be a nonemptyfinite set called the universe of discourse and D[x y] be theset of all closed subintervals of the interval [x y]

21 Hesitant Fuzzy Set (HFS) HFS was introduced by Torraand Narukawa [19] and Torra [20] which permits themembership being a set of possible values It is very suitablefor describing problems that are difficult to determine withonly one membership

Definition 1 (see [20]) A hesitant fuzzy set (HFS) E on Uis described as E langx hE(x)rang | x isin U1113864 1113865 where hE(x)

μE(x) | μE(x) isin [0 1]1113864 1113865 represents the set of possiblemembership degrees of E at x

For convenience Xia and Xu [30] proposed the conceptof a hesitant fuzzy element (HFE) denoted by h hE(x)where h c | c isin [0 1]1113864 1113865 ey gave some operations onHFEs

Definition 2 (see [19 20 30]) Let h h1 and h2 be three HFEsand λgt 0 then

(1) hC 1 minus c | c isin h1113864 1113865

(2) hλ cλ | c isin h1113864 1113865

(3) λh 1 minus (1 minus c)λ | c isin h1113966 1113967

(4) h1 oplus h2 c1 + c2 minus c1c2 | c1 isin h1 c2 isin h21113864 1113865

(5) h1 otimes h2 c1c2 | c1 isin h1 c2 isin h21113864 1113865

(6) h1 cup h2 max c1 c21113864 1113865 | c1 isin h1 c2 isin h21113864 1113865

(7) h1 cap h2 min c1 c21113864 1113865 | c1 isin h1 c2 isin h21113864 1113865

Furthermore to compare the HFEs the followingcomparison laws are given

Definition 3 (see [30]) For an HFE h S(h) 1|h|1113936c isin hc iscalled the score function of h with |middot| denoting the cardi-nality here and below For two HFEs h1 and h2 ifS(h1)gt S(h2) then h1 is superior to h2 denoted by h1 ≻ h2 ifS(h1) S(h2) then h1 is equivalent to h2 denoted byh1 h2

22 Interval-Valued Hesitant Fuzzy Set (IVHFS) Chen et al[26 27] generalized HFSs to interval-valued hesitant fuzzysets (IVPHFSs) in which the membership degree of eachobject of the universe is denoted by several possible intervalvalues

Definition 4 (see [26]) An interval-valued hesitant fuzzy set(IVHFS) 1113957E on U is described as 1113957E langx 1113957h1113957E(x)rang | x isin U1113966 1113967

2 Complexity


1113957E(x) μ+

1113957E(x)] isin D[0 1]1113966 1113967





(1) 1113957hC


(2) 1113957hλ





2 μ+1 + μ+


2 ]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2


2 μ+1 middot μ+

2 ] | 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h21113966 1113967


21113864 1113865max μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2


21113864 1113865min μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2









l(a) + l(b) 01113896 1113897 01113896 1113897 (1)








isin 0 1] x isin U[

(2)




1113957P x μminus

1113957P(x) μ+

1113957P(x)1113960 1113961 ]minus

1113957P(x) ]+

1113957P(x)1113960 11139611113924 1113925 | μminus

1113957P(x) μ+

1113957P(x)1113960 11139611113882

middot ]minus

1113957P(x) ]+

1113957P(x)1113960 1113961 isin D[0 1] μ+

1113957P(x)1113872 11138732

+ ]+

1113957P(x)1113872 11138732le 1 x isin U1113883

(3)

where [μminus

1113957P(x) μ+


1113957P(x) ]+



1113957P(x) μ+

1113957P(x)] []minus

1113957P(x) ]+



1 μ+1 ] []minus


2 μ+2 ] []minus

2 ]+2 ]rang be


(1) 1113957PC


(2) 1113957Pλ



1 minus (1 minus (]+)2)λ1113969

]rang

(3) λ1113957P lang[


1 minus (1 minus (μ+)2)λ1113969


(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

] []minus1]minus

2 ]+1]+

2 ]rang

Complexity 3


minus2 μ+

1μ+2 ] [



1113969

(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969

]rang


21113864 1113865max μ+1 μ+

21113864 1113865]

[min ]minus1 ]minus

21113864 1113865 min ]+1 ]+

21113864 1113865]rang


21113864 1113865 min μ+1 μ+

21113864 1113865] [max ]minus1 ]minus

21113864 1113865

max ]+1 ]+

21113864 1113865]rang



S(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

minus ]minus( )

2minus ]+

( 11138572

1113960 1113961 (4)


H(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

+ ]minus( )

2+ ]+

( 11138572

1113960 1113961 (5)


(1) If S(1113957P1)gt S(1113957P2) then 1113957P1 ≻ 1113957P2

(2) If S(1113957P1) S(1113957P2) then





P x hP(x)1113866 1113867 | x isin U1113864 1113865 (6)


P(x) μ+P(x)1113858 1113859 isin D[0 1]1113864

1113957]P(x) ]minusP(x) ]+

P(x)1113858 1113859 isin D[0 1] μ+P(x)( 1113857

2+ ]+

P(x)( 11138572 le 11113967

(7)









1 ]1113957]1 []minus1 ]+

1 ]1113864 1113865 and 1113957P2 lang1113957μ21113957]2rang| 1113957μ2 1113864

[μminus2 μ+

2 ]1113957]2 []minus2 ]+


(1) 1113957PC


(2) 1113957Pλ



1113882

1 minus (1 minus (]+)2)λ1113969


(3) λ 1113957P [


1 minus (1 minus (μ+)2)λ1113969

]1113882


(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

1113882

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

]

[]minus1]minus

2 ]+1]+



2 μ+1μ

+2 ][

(]minus

1 )1113968 2 + (]minus


(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969



21113864 1113865max μ+1 μ+

21113864 1113865]1113864

[min ]minus1 ]minus

21113864 1113865min ]+1 ]+



21113864 1113865min μ+1 μ+

21113864 1113865]1113864

[max ]minus1 ]minus

21113864 1113865max ]+1 ]+







μ+( 1113857

λ1113874 1113875

2+

1 minus 1 minus ]+( )2( 1113857λ

1113969

1113874 11138752

μ+( 1113857


( 11138572

1113872 1113873λ

le μ+( 1113857

2λ+ 1 minus μ+

( 11138572λ

1

(8)


As for 1113957P1 oplus 1113957P2

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ ]+

1( 11138572 ]+

2( 11138572

le μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

1113872 1113873 1 minus ]+2( 1113857

21113872 1113873

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

minus μ+2( 1113857

2

+ μ+1( 1113857

2 μ+2( 1113857

2 1

(9)

4 Complexity



2 en (max μ+

1 μ+21113864 1113865)2 (μ+


1 1113864






λ ( 1113957P1otimes 1113957P2)

λ


λ2 1113957P

(λ1+λ2)

(4) ( 1113957PC

)λ (λ 1113957P)C and λ( 1113957PC

) ( 1113957Pλ)C

(5) ( 1113957P1)Coplus( 1113957P2)

C ( 1113957P1 otimes 1113957P2)C and ( 1113957P1)


( 1113957P1 oplus 1113957P2)C




λ 1113957P1 oplus λ 1113957P2


21113872 1113873

λ1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus1( 1113857

λ ]+

1( 1113857λ

1113876 11138771113898 1113899 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113896 1113897

oplus


21113872 1113873

λ1113970

1 minus 1 minus μ+2( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus2( 1113857

λ ]+

2( 1113857λ

1113876 11138771113898 1113899 1113957μ2 1113957]21113866 1113867 isin 1113957P211138681113868111386811138681113896 1113897


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

λ 1113957P1 oplus 1113957P21113872 1113873

λ

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

λ


21113872 1113873 1 minus μminus

2( 11138572

1113872 1113873

1113969

1 minus 1 minus μ+1( 1113857

21113872 1113873 1 minus μ+

2( 11138572

1113872 1113873

1113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

(10)



λ

λ1 1113957Poplus λ2 1113957P


1113872 1113873λ1

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1

1113970

1113890 1113891 ]minus( )

λ1 ]+( 1113857

λ11113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

oplus


1113872 1113873λ2

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ2

1113970

1113890 1113891 ]minus( )

λ2 ]+( 1113857

λ21113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ1+λ2( )

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1+λ2( )

1113970

1113890 1113891 ]minus( )

λ1+λ2( ) ]+( 1113857

λ1+λ2( )1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

λ1 + λ2( 1113857 1113957P

(11)


λ2 1113957P(λ1+λ2)

Complexity 5

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)


) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+


2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)



and λgt 0 then



λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C


cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)



6 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity


1113957E(x) μ+

1113957E(x)] isin D[0 1]1113966 1113967





(1) 1113957hC


(2) 1113957hλ





2 μ+1 + μ+


2 ]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2


2 μ+1 middot μ+

2 ] | 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h21113966 1113967


21113864 1113865max μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2


21113864 1113865min μ+1 μ+

21113864 1113865]1113864

| 1113957μ1 isin 1113957h1 1113957μ2 isin 1113957h2









l(a) + l(b) 01113896 1113897 01113896 1113897 (1)








isin 0 1] x isin U[

(2)




1113957P x μminus

1113957P(x) μ+

1113957P(x)1113960 1113961 ]minus

1113957P(x) ]+

1113957P(x)1113960 11139611113924 1113925 | μminus

1113957P(x) μ+

1113957P(x)1113960 11139611113882

middot ]minus

1113957P(x) ]+

1113957P(x)1113960 1113961 isin D[0 1] μ+

1113957P(x)1113872 11138732

+ ]+

1113957P(x)1113872 11138732le 1 x isin U1113883

(3)

where [μminus

1113957P(x) μ+


1113957P(x) ]+



1113957P(x) μ+

1113957P(x)] []minus

1113957P(x) ]+



1 μ+1 ] []minus


2 μ+2 ] []minus

2 ]+2 ]rang be


(1) 1113957PC


(2) 1113957Pλ



1 minus (1 minus (]+)2)λ1113969

]rang

(3) λ1113957P lang[


1 minus (1 minus (μ+)2)λ1113969


(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

] []minus1]minus

2 ]+1]+

2 ]rang

Complexity 3


minus2 μ+

1μ+2 ] [



1113969

(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969

]rang


21113864 1113865max μ+1 μ+

21113864 1113865]

[min ]minus1 ]minus

21113864 1113865 min ]+1 ]+

21113864 1113865]rang


21113864 1113865 min μ+1 μ+

21113864 1113865] [max ]minus1 ]minus

21113864 1113865

max ]+1 ]+

21113864 1113865]rang



S(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

minus ]minus( )

2minus ]+

( 11138572

1113960 1113961 (4)


H(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

+ ]minus( )

2+ ]+

( 11138572

1113960 1113961 (5)


(1) If S(1113957P1)gt S(1113957P2) then 1113957P1 ≻ 1113957P2

(2) If S(1113957P1) S(1113957P2) then





P x hP(x)1113866 1113867 | x isin U1113864 1113865 (6)


P(x) μ+P(x)1113858 1113859 isin D[0 1]1113864

1113957]P(x) ]minusP(x) ]+

P(x)1113858 1113859 isin D[0 1] μ+P(x)( 1113857

2+ ]+

P(x)( 11138572 le 11113967

(7)









1 ]1113957]1 []minus1 ]+

1 ]1113864 1113865 and 1113957P2 lang1113957μ21113957]2rang| 1113957μ2 1113864

[μminus2 μ+

2 ]1113957]2 []minus2 ]+


(1) 1113957PC


(2) 1113957Pλ



1113882

1 minus (1 minus (]+)2)λ1113969


(3) λ 1113957P [


1 minus (1 minus (μ+)2)λ1113969

]1113882


(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

1113882

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

]

[]minus1]minus

2 ]+1]+



2 μ+1μ

+2 ][

(]minus

1 )1113968 2 + (]minus


(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969



21113864 1113865max μ+1 μ+

21113864 1113865]1113864

[min ]minus1 ]minus

21113864 1113865min ]+1 ]+



21113864 1113865min μ+1 μ+

21113864 1113865]1113864

[max ]minus1 ]minus

21113864 1113865max ]+1 ]+







μ+( 1113857

λ1113874 1113875

2+

1 minus 1 minus ]+( )2( 1113857λ

1113969

1113874 11138752

μ+( 1113857


( 11138572

1113872 1113873λ

le μ+( 1113857

2λ+ 1 minus μ+

( 11138572λ

1

(8)


As for 1113957P1 oplus 1113957P2

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ ]+

1( 11138572 ]+

2( 11138572

le μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

1113872 1113873 1 minus ]+2( 1113857

21113872 1113873

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

minus μ+2( 1113857

2

+ μ+1( 1113857

2 μ+2( 1113857

2 1

(9)

4 Complexity



2 en (max μ+

1 μ+21113864 1113865)2 (μ+


1 1113864






λ ( 1113957P1otimes 1113957P2)

λ


λ2 1113957P

(λ1+λ2)

(4) ( 1113957PC

)λ (λ 1113957P)C and λ( 1113957PC

) ( 1113957Pλ)C

(5) ( 1113957P1)Coplus( 1113957P2)

C ( 1113957P1 otimes 1113957P2)C and ( 1113957P1)


( 1113957P1 oplus 1113957P2)C




λ 1113957P1 oplus λ 1113957P2


21113872 1113873

λ1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus1( 1113857

λ ]+

1( 1113857λ

1113876 11138771113898 1113899 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113896 1113897

oplus


21113872 1113873

λ1113970

1 minus 1 minus μ+2( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus2( 1113857

λ ]+

2( 1113857λ

1113876 11138771113898 1113899 1113957μ2 1113957]21113866 1113867 isin 1113957P211138681113868111386811138681113896 1113897


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

λ 1113957P1 oplus 1113957P21113872 1113873

λ

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

λ


21113872 1113873 1 minus μminus

2( 11138572

1113872 1113873

1113969

1 minus 1 minus μ+1( 1113857

21113872 1113873 1 minus μ+

2( 11138572

1113872 1113873

1113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

(10)



λ

λ1 1113957Poplus λ2 1113957P


1113872 1113873λ1

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1

1113970

1113890 1113891 ]minus( )

λ1 ]+( 1113857

λ11113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

oplus


1113872 1113873λ2

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ2

1113970

1113890 1113891 ]minus( )

λ2 ]+( 1113857

λ21113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ1+λ2( )

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1+λ2( )

1113970

1113890 1113891 ]minus( )

λ1+λ2( ) ]+( 1113857

λ1+λ2( )1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

λ1 + λ2( 1113857 1113957P

(11)


λ2 1113957P(λ1+λ2)

Complexity 5

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)


) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+


2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)



and λgt 0 then



λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C


cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)



6 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity


minus2 μ+

1μ+2 ] [



1113969

(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969

]rang


21113864 1113865max μ+1 μ+

21113864 1113865]

[min ]minus1 ]minus

21113864 1113865 min ]+1 ]+

21113864 1113865]rang


21113864 1113865 min μ+1 μ+

21113864 1113865] [max ]minus1 ]minus

21113864 1113865

max ]+1 ]+

21113864 1113865]rang



S(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

minus ]minus( )

2minus ]+

( 11138572

1113960 1113961 (4)


H(1113957P) 12

μminus( 1113857

2+ μ+

( 11138572

+ ]minus( )

2+ ]+

( 11138572

1113960 1113961 (5)


(1) If S(1113957P1)gt S(1113957P2) then 1113957P1 ≻ 1113957P2

(2) If S(1113957P1) S(1113957P2) then





P x hP(x)1113866 1113867 | x isin U1113864 1113865 (6)


P(x) μ+P(x)1113858 1113859 isin D[0 1]1113864

1113957]P(x) ]minusP(x) ]+

P(x)1113858 1113859 isin D[0 1] μ+P(x)( 1113857

2+ ]+

P(x)( 11138572 le 11113967

(7)









1 ]1113957]1 []minus1 ]+

1 ]1113864 1113865 and 1113957P2 lang1113957μ21113957]2rang| 1113957μ2 1113864

[μminus2 μ+

2 ]1113957]2 []minus2 ]+


(1) 1113957PC


(2) 1113957Pλ



1113882

1 minus (1 minus (]+)2)λ1113969


(3) λ 1113957P [


1 minus (1 minus (μ+)2)λ1113969

]1113882


(4) 1113957P1 oplus 1113957P2 lang[



2 )21113969

1113882

(μ+1 )2 + (μ+

2 )2 minus (μ+1 )2(μ+

2 )21113969

]

[]minus1]minus

2 ]+1]+



2 μ+1μ

+2 ][

(]minus

1 )1113968 2 + (]minus


(]minus2 )2

(]+1 )2 + (]+

2 )2 minus (]+1 )2(]+

2 )21113969



21113864 1113865max μ+1 μ+

21113864 1113865]1113864

[min ]minus1 ]minus

21113864 1113865min ]+1 ]+



21113864 1113865min μ+1 μ+

21113864 1113865]1113864

[max ]minus1 ]minus

21113864 1113865max ]+1 ]+







μ+( 1113857

λ1113874 1113875

2+

1 minus 1 minus ]+( )2( 1113857λ

1113969

1113874 11138752

μ+( 1113857


( 11138572

1113872 1113873λ

le μ+( 1113857

2λ+ 1 minus μ+

( 11138572λ

1

(8)


As for 1113957P1 oplus 1113957P2

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ ]+

1( 11138572 ]+

2( 11138572

le μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

1113872 1113873 1 minus ]+2( 1113857

21113872 1113873

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

2+ 1 minus μ+

1( 11138572

minus μ+2( 1113857

2

+ μ+1( 1113857

2 μ+2( 1113857

2 1

(9)

4 Complexity



2 en (max μ+

1 μ+21113864 1113865)2 (μ+


1 1113864






λ ( 1113957P1otimes 1113957P2)

λ


λ2 1113957P

(λ1+λ2)

(4) ( 1113957PC

)λ (λ 1113957P)C and λ( 1113957PC

) ( 1113957Pλ)C

(5) ( 1113957P1)Coplus( 1113957P2)

C ( 1113957P1 otimes 1113957P2)C and ( 1113957P1)


( 1113957P1 oplus 1113957P2)C




λ 1113957P1 oplus λ 1113957P2


21113872 1113873

λ1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus1( 1113857

λ ]+

1( 1113857λ

1113876 11138771113898 1113899 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113896 1113897

oplus


21113872 1113873

λ1113970

1 minus 1 minus μ+2( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus2( 1113857

λ ]+

2( 1113857λ

1113876 11138771113898 1113899 1113957μ2 1113957]21113866 1113867 isin 1113957P211138681113868111386811138681113896 1113897


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

λ 1113957P1 oplus 1113957P21113872 1113873

λ

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

λ


21113872 1113873 1 minus μminus

2( 11138572

1113872 1113873

1113969

1 minus 1 minus μ+1( 1113857

21113872 1113873 1 minus μ+

2( 11138572

1113872 1113873

1113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

(10)



λ

λ1 1113957Poplus λ2 1113957P


1113872 1113873λ1

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1

1113970

1113890 1113891 ]minus( )

λ1 ]+( 1113857

λ11113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

oplus


1113872 1113873λ2

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ2

1113970

1113890 1113891 ]minus( )

λ2 ]+( 1113857

λ21113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ1+λ2( )

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1+λ2( )

1113970

1113890 1113891 ]minus( )

λ1+λ2( ) ]+( 1113857

λ1+λ2( )1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

λ1 + λ2( 1113857 1113957P

(11)


λ2 1113957P(λ1+λ2)

Complexity 5

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)


) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+


2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)



and λgt 0 then



λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C


cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)



6 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity



2 en (max μ+

1 μ+21113864 1113865)2 (μ+


1 1113864






λ ( 1113957P1otimes 1113957P2)

λ


λ2 1113957P

(λ1+λ2)

(4) ( 1113957PC

)λ (λ 1113957P)C and λ( 1113957PC

) ( 1113957Pλ)C

(5) ( 1113957P1)Coplus( 1113957P2)

C ( 1113957P1 otimes 1113957P2)C and ( 1113957P1)


( 1113957P1 oplus 1113957P2)C




λ 1113957P1 oplus λ 1113957P2


21113872 1113873

λ1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus1( 1113857

λ ]+

1( 1113857λ

1113876 11138771113898 1113899 1113957μ1 1113957]11113866 1113867 isin 1113957P111138681113868111386811138681113896 1113897

oplus


21113872 1113873

λ1113970

1 minus 1 minus μ+2( 1113857

21113872 1113873

λ1113970

1113890 1113891 ]minus2( 1113857

λ ]+

2( 1113857λ

1113876 11138771113898 1113899 1113957μ2 1113957]21113866 1113867 isin 1113957P211138681113868111386811138681113896 1113897


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

λ 1113957P1 oplus 1113957P21113872 1113873

λ

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

λ


21113872 1113873 1 minus μminus

2( 11138572

1113872 1113873

1113969

1 minus 1 minus μ+1( 1113857

21113872 1113873 1 minus μ+

2( 11138572

1113872 1113873

1113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883


21113872 1113873

λ1 minus μminus

2( 11138572

1113872 1113873λ

1113970

1 minus 1 minus μ+1( 1113857

21113872 1113873

λ1 minus μ+

2( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus1]

minus2( 1113857

λ ]+

1]+2( 1113857

λ1113876 11138771113898 1113899 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113896 1113897

(10)



λ

λ1 1113957Poplus λ2 1113957P


1113872 1113873λ1

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1

1113970

1113890 1113891 ]minus( )

λ1 ]+( 1113857

λ11113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

oplus


1113872 1113873λ2

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ2

1113970

1113890 1113891 ]minus( )

λ2 ]+( 1113857

λ21113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ1+λ2( )

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ1+λ2( )

1113970

1113890 1113891 ]minus( )

λ1+λ2( ) ]+( 1113857

λ1+λ2( )1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

λ1 + λ2( 1113857 1113957P

(11)


λ2 1113957P(λ1+λ2)

Complexity 5

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)


) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+


2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)



and λgt 0 then



λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C


cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)



6 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity

1113957PC

1113874 1113875λ

]minus( )

λ ]+( 1113857

λ1113876 1113877


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 11138911113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897


1113872 1113873λ

1113970

1 minus 1 minus μ+( 11138572

1113872 1113873λ

1113970

1113890 1113891 ]minus( )

λ ]+( 1113857

λ1113876 11138771113898 1113899 | 1113957μ 1113957]1113866 1113867 isin 1113957P1113896 1113897

(λ 1113957P)C

(12)


) ( 1113957Pλ)C

1113957PC

1 oplus 1113957PC

2 ]minus1 ]+

11113858 1113859 μminus1 μ+


2 ]+21113858 1113859 μminus

2 μ+21113858 11138591113866 1113867 1113957μ2 1113957]21113866 1113867 isin 1113957P2

11138681113868111386811138681113966 1113967

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 1113877 μminus1μ

minus2 μ+

1μ+21113858 11138591113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 2

11138681113868111386811138681113882 1113883

μminus1μ

minus2 μ+

1μ+21113858 1113859

]minus1( 1113857

2+ ]minus

2( 11138572


2 ]minus2( 1113857

21113969

]+1( 1113857

2+ ]+

2( 11138572

minus ]+1( 1113857

2 ]+2( 1113857

21113969

1113876 11138771113884 1113885 1113957μi 1113957]i1113866 1113867 isin 1113957Pi i 1 211138681113868111386811138681113882 1113883

1113957P1 otimes 1113957P21113872 1113873C

(13)

Similarly 1113957PC

1 otimes 1113957PC

2 ( 1113957P1 oplus 1113957P2)C holds

1113957P1oplus 1113957P21113872 1113873oplus 1113957P3

μminus1( 1113857

2+ μminus

2( 11138572


2 μminus2( 1113857

21113969

μ+1( 1113857

2+ μ+

2( 11138572

minus μ+1( 1113857

2 μ+2( 1113857

21113969

1113876 1113877 ]minus1]

minus2 ]+

1]+21113858 11138591113884 1113885 1113957μ11113957]11113866 1113867isin 1113957P1 1113957μ21113957]21113866 1113867isin 1113957P2

11138681113868111386811138681113882 1113883

oplus μminus3 μ+

31113858 1113859 ]minus3 ]+

31113858 11138591113866 1113867 1113957μ31113957]31113866 11138671113868111386811138681113868 isin 1113957P31113966 1113967

μminus1( 1113857

2+ μminus

2( 11138572

+ μminus3( 1113857

2minus μminus

1( 11138572 μminus

2( 11138572


2 μminus3( 1113857

2minus μminus

2( 11138572 μminus

3( 11138572

+ μminus1( 1113857

2 μminus2( 1113857

2 μminus3( 1113857

21113969

111387611138841113882

μ+1( 1113857

2+ μ+

2( 11138572

+ μ+3( 1113857

2minus μ+

1( 11138572 μ+

2( 11138572

minus μ+1( 1113857

2 μ+3( 1113857

2minus μ+

2( 11138572 μ+

3( 11138572

+ μ+1( 1113857

2 μ+2( 1113857

2 μ+3( 1113857

21113969

1113877

]minus1]

minus2]

minus3 ]+

1]+2]

+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 123

1113868111386811138681113868

μminus1 μ+

11113858 1113859 ]minus1 ]+

11113858 11138591113866 1113867 1113957μ11113957]11113866 1113867isin 1113957P111138681113868111386811138681113966 1113967oplus

μminus2( 1113857

2+ μminus

3( 11138572


2 μminus3( 1113857

21113969

μ+2( 1113857

2+ μ+

3( 11138572

minus μ+2( 1113857

2 μ+3( 1113857

21113969

1113876 111387711138841113882

]minus2]

minus3 ]+

2]+31113858 11138591113885 1113957μi1113957]i1113866 1113867isin 1113957Pi i 23

1113868111386811138681113868 1113883

1113957P1 oplus 1113957P2oplus 1113957P31113872 1113873

(14)



and λgt 0 then



λ ( 1113957P1cap 1113957P2)

λ

(3) ( 1113957P1)C cup ( 1113957P2)

C ( 1113957P1 cap 1113957P2)C and ( 1113957P1)

C cap ( 1113957P2)C

( 1113957P1 cup 1113957P2)C


cap 1113957P3 1113957P1 cap ( 1113957P2 cap 1113957P3)



6 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity





S( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 minus 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

minus ]+( 1113857

21113873

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2minus ]minus

( )2

1113872 1113873⎤⎥⎥⎥⎥⎥⎥⎥⎦

(15)


H( 1113957P) 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

1113957μ2 + 1113957]21113872 1113873 1

2| 1113957P|1113944

1113957μ1113957]lang rangisin 1113957P

μminus( 1113857

21113872

⎡⎢⎢⎢⎢⎢⎢⎢⎣

+ ]minus( )

2)

12| 1113957P|

1113944

1113957μ1113957]lang rangisin 1113957P

μ+( 1113857

2+ ]+

( 11138572

1113872 1113873]

(16)





(2) If P(S( 1113957P1)gt S( 1113957P2)) 05 then



1j] 1113957]1j 1113966

[]minus1j ]+


μ+2j] 1113957]2j []minus



2j μ+1j le μ+


2j and ]+1j ge ]+

2jthen 1113957P1 ≼ 1113957P2


S 1113957P11113872 1113873 12m

1113944

m

j1μminus1j1113872 1113873

2minus ]+

1j1113872 11138732 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

21113876 1113877

S 1113957P21113872 1113873 12m

1113944

m

j1μminus2j1113872 1113873

2minus ]+

2j1113872 11138732 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

21113876 1113877

(17)

Suppose

L≜1113936

mj1 μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

1j1113872 11138732


21113874 1113875

1113936mj1 μ+

1j1113872 11138732

minus ]minus1j1113872 1113873

2+ ]+

1j1113872 11138732


2+ μ+

2j1113872 11138732

minus ]minus2j1113872 1113873

2+ ]+

2j1113872 11138732


21113874 1113875

(18)



2j μ+1j le μ+


2j and]+1j ge ]+

2j (μ+2j)

2 minus (μminus1j)

2 + (]+1j)

2 minus (]minus2j)

2 ge (μ+1j)

2 minus (μminus2j)

2+

(]+2j)

2 minus (]minus1j)








ni1ωi 1 and λgt 0

Complexity 7



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity



1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

1113945n

i1]minus

i( 1113857ωi 1113945

n

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899


⎭

(19)



ωn

n

1113945n

i1μminus

i( 1113857ωi 1113945

n

i1μ+

i( 1113857ωi ⎤⎦⎡⎣1113898

⎧⎨

⎩

middot

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

middot

1 minus 1113945

n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎭

(20)




1113957Pλn1113874 1113875

1λ

1 minus 1113945n

i11 minus μminus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867


⎫⎪⎪⎬

⎪⎪⎭

(21)

where λgt 0




1λ


ωn1113872 1113873



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

n

i11 minus ]minus

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]+

i( 11138572λ

1113872 1113873ωi⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣


⎫⎪⎪⎬

⎪⎪⎭

(22)



8 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity





IVPHFWA 1113957P11113957P2

1113957P31113872 1113873 ω11113957P1 oplusω1

1113957P2 oplusωn1113957P3

1 minus 11139453

i11 minus μminus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot 1113945

3

i1]minus

i( 1113857ωi 1113945

3

i1]+

i( 1113857ωi⎡⎣ ⎤⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123

⎫⎬

⎭




IVPHFWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pω11 otimes 1113957P


ω33

11139453

i1μminus

i( 1113857ωi 1113945

3

i1μ+

i( 1113857ωi⎡⎣ ⎤⎦1113898

1 minus 11139453

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1 minus 11139453

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎭




GIVPHFWA31113957P1

1113957P21113957P31113872 1113873 ω1

1113957P31 oplusω2

1113957P32 oplusω3

1113957P331113874 1113875

13

1 minus 1113945

3

i11 minus μminus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus μ+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972



i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭




GIVPHFWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957P11113872 1113873ω1 otimes 3 1113957P21113872 1113873

ω2 otimes 3 1113957P31113872 1113873ω3

1113872 1113873


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972


3


i( 11138572

1113872 11138733

1113874 1113875ωi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 11139453

i11 minus ]minus

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

i( 11138572times3

1113872 1113873ωi⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899 | 1113957μi 1113957]i1113866 1113867isin 1113957Pi i 123⎫⎪⎪⎬

⎪⎪⎭



⎭

(24)

Complexity 9


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity


S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01344 00838]

S IVPHFWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01569 00611]


1113957P21113957P31113872 11138731113872 1113873 [minus 01188 00957]


1113957P21113957P31113872 11138731113872 1113873 [minus 01649 00401]


1113957P31113872 1113873le S IVPHFWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05518

P SIVPHFWG 1113957P11113957P2

1113957P31113872 1113873le S GIVPHFWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05840


1113957P21113957P31113872 1113873le S IVPHFWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05876


1113957P31113872 1113873≼ IVPHFWA 1113957P11113957P2

1113957P31113872 1113873

IVPHFWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFWA 1113957P1

1113957P21113957P31113872 1113873

(25)



ni1ωi 1 then

1113945

n

i1xωi

i le 1113944n

i1ωixi (26)



ni1ωi 1 then



1113957Pn1113872 1113873

(27)



i ] []minusi ]+


1113945

n

i1μminus

i( 11138572

1113872 1113873ωi le 1113944

n

i1ωi μminus

i( 11138572

1 minus 1113944n


i( 11138572

1113872 1113873le 1 minus 1113945n

i11 minus μminus

i( 11138572

1113872 1113873ωi

(28)

So 1113937ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2)ωi

1113969

Similarly we have

1113945

n

i1μ+

i( 1113857ωi le

1 minus 1113945n

i11 minus μ+

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]minus

i( 1113857ωi le

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

1113945

n

i1]+

i( 1113857ωi le

1 minus 1113945n

i11 minus ]+

i( 11138572

1113872 1113873ωi

11139741113972

(29)



ωn





1113957Pn1113872 1113873

(30)


ni1ωi 1 then


1113957Pn1113872 1113873≼GIVPHFWAλ1113957P1


GIVPHFWGλ1113957P1



(31)


i ] []minusi ]+


ni1((μminus



i )2λ)1λ




ni1(μ

minusi )ωi

1113937ni1((μminus

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

Similarly 1113937

ni1(μ+

i )ωi

1113937ni1((μ+

i )2)ωi

1113969

le


i )2λ)ωi )1λ1113969

holds

10 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity

Also



i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

le



i( 11138572

1113872 1113873λ

1113874 1113875⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113944

n


i( 11138572

1113872 1113873λ

⎛⎝ ⎞⎠

1λ11139741113972

le

1 minus 1113945

n

i11 minus ]minus

i( 11138572

1113872 1113873λ

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus ]minus

i( 11138572

1113872 1113873ωi

11139741113972

(32)

Similarly


i )2)λ)ωi )1λ1113969

le


i )2)ωi

1113969







ni1ωi 1 then


1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(33)


1 minus 1113945n

i11 minus μminus

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μminus

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ2

11139741113972

1 minus 1113945n

i11 minus μ+

i( 11138572λ11113872 1113873

ωi⎛⎝ ⎞⎠

1λ111139741113972

le

1 minus 1113945n

i11 minus μ+

i( 11138572λ21113872 1113873

ωi⎛⎝ ⎞⎠

1λ211139741113972

(34)




i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(35)

Similarly



i( 11138572

1113872 1113873λ1

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ111139741113972

ge



i( 11138572

1113872 1113873λ2

1113874 1113875ωi

⎛⎝ ⎞⎠

1λ211139741113972

(36)

Complexity 11



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity



1113957P2 1113957Pn1113872 1113873≼GIVPHFWAλ21113957P1

1113957P2 1113957Pn1113872 1113873


1113957P2 1113957Pn1113872 1113873≽GIVPHFWGλ21113957P2

1113957P2 1113957Pn1113872 1113873(37)







1 minus 1113945n

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

n

i1]minusσ(i)1113872 1113873

κi 1113945

n

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


(38)




κn

σ(n)

1113945n

i1μminusσ(i)1113872 1113873

κi 1113945

n

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113898

1 minus 1113945n

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

⎧⎪⎪⎨

⎪⎪⎩


(39)






1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎪⎪⎬

⎪⎪⎭

(40)

12 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity

where λgt 0




1λ

λ 1113957Peuro


euroσ(n)1113874 1113875

κn

1113874 1113875


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972


n


σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899



⎫⎪⎪⎬

⎪⎪⎭

(41)






[04 07] [02 06]lang rang

(42)




S 1113957P11113872 1113873 12

(03)2

minus (04)2

1113872 111387312

(05)2

minus (02)2

1113872 11138731113876 1113877 [minus 0035 0045]

S 1113957P21113872 1113873 14

(01)2

minus (06)2

+(01)2

minus (07)2

1113872 111387314

(04)2

minus (03)2

+(04)2

minus (05)2

1113872 11138731113876 1113877


S 1113957P31113872 1113873 16

(03)2

minus (06)2

+(01)2

minus (06)2

+(04)2

minus (06)2

1113872 111387316

(04)2

minus (02)2

+(03)2

minus (02)2

+(07)2

minus (02)2

1113872 11138731113876 1113877

[minus 01367 01033]

(43)




Complexity 13

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity

IVPHFOWA 1113957P11113957P2


1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1]minusσ(i)1113872 1113873

κi 1113945

3

i1]+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[03097 05865] [02756 06259]lang rang

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873 1113957Pκ1σ(1) otimes 1113957P


κ3σ(3)

11139453

i1μminusσ(i)1113872 1113873

κi 1113945

3

i1μ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 11139453

i11 minus ]minus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎪⎪⎭



[02325 05380] [03456 06607]lang rang

GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873 κ1 1113957P

3σ(1) oplus κ2 1113957P

3σ(2) oplus κ3 1113957P

3σ(3)1113874 1113875

13

1 minus 11139453

i11 minus μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873

13

3 1113957Pσ(1)1113872 1113873κ1 otimes 3 1113957Pσ(2)1113872 1113873

κ2 otimes 3 1113957Pσ(3)1113872 1113873κ3

1113872 1113873


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

middot

1 minus 1113945

3

i11 minus ]minus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus ]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


⎭




(44)

14 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity


S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01389 00796]

S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873 [minus 01670 00556]


1113957P21113957P31113872 11138731113872 1113873 [minus 01197 01001]


1113957P21113957P31113872 11138731113872 1113873 [minus 01776 00356]

(45)

en

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P11113957P2

1113957P31113872 11138731113872 11138731113872 1113873 05591

P S IVPHFOWG 1113957P11113957P2

1113957P31113872 11138731113872 1113873le S GIVPHFOWA31113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 06038


1113957P21113957P31113872 11138731113872 1113873le S IVPHFOWA 1113957P1

1113957P21113957P31113872 11138731113872 11138731113872 1113873 05958

(46)


1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

IVPHFOWG 1113957P11113957P2

1113957P31113872 1113873≼GIVPHFOWA31113957P1

1113957P21113957P31113872 1113873

GIVPHFOWG31113957P1

1113957P21113957P31113872 1113873≼ IVPHFOWA 1113957P1

1113957P21113957P31113872 1113873

(47)















(48)




ni1ωi 1 Suppose κ





_1113957Pσ(n)

1 minus 1113945n

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113898

⎧⎪⎪⎨

⎪⎪⎩

1113945n

i1_]

minus

σ(i)1113872 1113873κi

1113945n

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899 | _1113957μσ(i)


⎭ (49)

Complexity 15






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity






κn

1113945n


κi 1113945

n

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945n


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭

(50)







_1113957Pσ(n)1113874 1113875λ

1113888 1113889

1λ

1 minus 1113945

n

i11 minus _μminus

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945

n

i11 minus _μ+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899


(51)






1λ


κn

1113874 1113875



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972



σ(i)1113872 11138732

1113874 1113875λ

1113888 1113889

κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945n


σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

i11 minus euro]+

σ(i)1113872 11138732λ

1113874 1113875κi

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899


⎫⎬

⎭

(52)


16 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity









[05000 08190] [00703 04305]lang rang

euro1113957P1 1113957P11113872 11138733times015

[05817 07320] [01349 02747]lang rang

euro1113957P2 1113957P21113872 11138733times03


euro1113957P3 1113957P31113872 11138733times03


[02205 05552] [02552 07219]lang rang

S_1113957P11113874 1113875 [minus 01984 minus 00567] S

_1113957P21113874 1113875 [minus 02267 minus 00278]

S_1113957P31113874 1113875 [minus 00242 01750] S

euro1113957P11113874 1113875 [01315 02588]

Seuro1113957P21113874 1113875 [minus 01884 00187] S

euro1113957P31113874 1113875 [minus 02493 00300]

(54)


_1113957P2)) 1P(S(

_1113957P2)le S(_1113957P3)) 05009 and P(S(

euro1113957P3)leS(euro1113957P2))



_1113957Pσ(1) _1113957P3

_1113957Pσ(2) _1113957P2

_1113957Pσ(3) _1113957P1

euro1113957Pσ(1) euro1113957P1

euro1113957Pσ(2) euro1113957P2

euro1113957Pσ(3) euro1113957P3

(55)

Complexity 17


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity


IVPHFHA 1113957P11113957P2

1113957P31113872 1113873 κ1_1113957Pσ(1) oplus κ1

_1113957Pσ(2) oplus κn_1113957Pσ(3)

1 minus 1113945

3

i11 minus _μminus

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus _μ+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113945

3

i1_]

minus

σ(i)1113872 1113873κi

1113945

3

i1_]

+

σ(i)1113872 1113873κi⎡⎣ ⎤⎦1113899

⎧⎪⎪⎨

⎪⎪⎩

| 1113957μ_ σ(i) 1113957]_ σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




IVPHFHG 1113957P11113957P2

1113957P31113872 1113873 euro1113957Pσ(1)1113874 1113875

κ1otimes euro1113957Pσ(2)1113874 1113875

κ2otimes euro1113957Pσ(3)1113874 1113875

κ3

1113945

3


κi 1113945

3

i1euroμ+σ(i)1113872 1113873

κi⎡⎣ ⎤⎦

1 minus 1113945

3


σ(i)1113872 11138732

1113874 1113875κi

11139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732

1113874 1113875κi

11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎨

⎪⎪⎩


⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHA31113957P1

1113957P21113957P31113872 1113873 κ1

_1113957Pσ(1)1113874 11138753oplus κ2

_1113957Pσ(2)1113874 11138753oplus κ3

_1113957Pσ(3)1113874 11138753

1113888 1113889

13

1 minus 11139453

i11 minus _μminus

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 11139453

i11 minus _μ+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| 1113957μ_σ(i) 1113957]_

σ(i)1113924 1113925 isin _1113957Pσ(i) i 1 2 3⎫⎪⎪⎬

⎪⎪⎭




GIVPHFHG31113957P1

1113957P21113957P31113872 1113873

13


κ2otimes 3 euro1113957Pσ(3)1113874 1113875

κ31113874 1113875



σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972


3


σ(i)1113872 11138732

1113874 11138753

1113888 1113889

κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩

1 minus 1113945

3


σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

1 minus 1113945

3

i11 minus euro]+

σ(i)1113872 11138732times3

1113874 1113875κi

⎛⎝ ⎞⎠

1311139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899





(56)

18 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity




let M(k) ( 1113957P(k)


(k)

ij lang1113957μ(k)ij 1113957](k)






(k)


(k)


1113957P(k)

ij

1113957P(k)


1113957P(k)

ij1113874 1113875C


⎧⎪⎪⎨

⎪⎪⎩(57)

where ( 1113957P(k)






(k)


(k)




(1)


ij1113874 1113875

1 minus 1113945

l

k11 minus μminus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus μ+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972



1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎭

(58)

or


(1)


ij1113874 1113875


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972


l


1113957P(k)

ij

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

ξk

⎛⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎠

1λ111397411139731113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

1 minus 1113945

l

k11 minus ]minus

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

1 minus 1113945

l

k11 minus ]+

1113957P(k)

ij

1113888 1113889

2λ⎛⎝ ⎞⎠

ξk

⎛⎜⎝ ⎞⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899 | 1113957μ(k)

1113957Pij

1113957](k)

1113957Pij

1113884 1113885 isin 1113957P(k)


⎫⎪⎪⎪⎬

⎪⎪⎪⎭

(59)

Complexity 19




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity




1 minus 1113945n

j11 minus _μminus

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus _μ+

1113957Piσ(j)

1113874 11138752λ

1113888 1113889

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎨

⎪⎪⎩



1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972


n


1113957Piσ(j)

1113874 11138752

1113888 1113889

λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113899

| _1113957μ1113957Piσ(j)

_1113957]1113957Piσ(j)


⎭

(60)




1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972



1113957Piσ(j)

1113888 1113889

2⎛⎝ ⎞⎠

λ

⎛⎜⎝ ⎞⎟⎠

κj

⎛⎜⎜⎝ ⎞⎟⎟⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

1 minus 1113945n


1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

1 minus 1113945n

j11 minus euro]+

1113957Piσ(j)

1113888 1113889

2λ⎛⎝ ⎞⎠

κj

⎛⎝ ⎞⎠

1λ11139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ 1113899

| euro1113957]1113957Piσ(j)

euro1113957]1113957Piσ(j)


⎭

(61)






1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus μ+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ]minus

1113957Pij

1113874 1113875ntimesωj( 1113857

]+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭


μminus

1113957Pij

1113874 1113875ntimesωj( 1113857

μ+

1113957Pij

1113874 1113875ntimesωj( 1113857

⎡⎣ ⎤⎦


1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

1 minus 1 minus ]+

1113957Pij

1113888 1113889

2⎛⎝ ⎞⎠

ntimesωj( 111385711139741113972

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦1113898 1113899

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

| 1113957μ1113957Pij

1113957]1113957Pij


⎫⎪⎪⎬

⎪⎪⎭

(62)

20 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity



(k)


(k)


(Tables 4ndash7)


(k)



(k)


S 1113957P1( 1113857 [02736 02833]

S 1113957P2( 1113857 [02142 02988]

S 1113957P3( 1113857 [02398 02741]

(63)





C1 C2 C3


lang[06 08] [01 03]rang

lang[06 07] [03 03]rang

lang[07 09] [01 01]rang

Y2lang[01 01] [08 09]rang

lang[02 04] [05 06]rang

lang[04 05] [05 05]rang

lang[06 07] [01 02]rang

lang[06 07] [01 02]rang

lang[08 09] [01 01]rang

Y3lang[02 03] [06 06]rang



C1 C2 C3

Y1

lang[07 09] [01 02]rang

lang[07 08] [01 02]rang

lang[06 08] [01 01]rang

lang[04 06] [03 05]rang



lang[06 08] [01 02]rang


lang[07 09] [01 01]rang

lang[01 03] [06 07]rang

lang[02 02] [07 08]rang

lang[03 04] [06 06]rang


C1 C2 C3



Y2lang[04 06] [03 04]rang


lang[03 05] [05 05]rang

lang[01 02] [07 07]rang

lang[03 04] [05 06]rang

Y3lang[08 09] [01 01]rang

lang[05 07] [01 02]rang

lang[03 06] [04 04]rang

lang[04 05] [04 05]rang

lang[03 05] [04 05]rang

lang[09 09] [01 01]rang

Complexity 21







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity







C1 C2 C3


lang[07 09] [01 01]rang

Y2lang[08 09] [01 01]rang

lang[05 06] [02 04]rang



Y3lang[06 06] [02 03]rang



C1 C2 C3


Y2lang[03 04] [04 06]rang



lang[03 04] [05 06]rang

Y3lang[01 01] [08 09]rang

lang[01 02] [05 07]rang




C1 C2 C3


lang[01 01] [06 08]rang

lang[04 06] [03 05]rang




lang[07 09] [01 01]rang


lang[03 04] [06 06]rang

Table 7 IVPHFDM N

C1 C2 C3

Y1

lang[03553 04562] [04579 06046]rang

lang[03559 04562] [04579 06014]rang

lang[03553 04562] [04404 06014]rang

lang[03801 05702] [03631 05396]rang

lang[04751 06655] [02778 03778]rang

lang[03801 05702] [03836 05614]rang

lang[04751 06655] [02913 03870]rang

lang[08648 08744] [01226 02079]rang

lang[08648 08722] [01363 02079]rang

lang[08654 08818] [01226 01862]rang

Y2

lang[06387 07283] [04172 04696]rang

lang[03979 04791] [04643 06207]rang

lang[06403 07288] [04172 04511]rang

lang[04562 04959] [04643 05790]rang

lang[07619 08622] [01814 03798]rang

lang[07625 08630] [01568 03451]rang

lang[07620 08630] [01568 03451]rang

lang[06419 07985] [02351 03016]rang

lang[06566 07988] [01530 02351]rang

lang[06419 07983] [02447 03150]rang

lang[06566 07983] [01588 02447]rang

lang[06419 07983] [02351 03104]rang

lang[06566 07983] [01530 02414]rang

Y3

lang[04767 04767] [05428 06111]rang

lang[04767 05567] [04758 06616]rang

lang[04767 04767] [04998 06014]rang

lang[04767 05567] [04447 06475]rang

lang[06566 07995] [02247 02247]rang

lang[06998 08742] [01466 01466]rang

lang[06567 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[06566 07985] [02247 02351]rang

lang[06998 08738] [01466 01530]rang

lang[08094 08095] [02913 02997]rang

lang[08094 08095] [02997 03029]rang

lang[08094 08095] [02913 02913]rang

22 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity











λ 01 λ 10 λ 20Y1 [minus 04036 minus 03452] [03207 03240] [03519 03530]

Y2 [00264 00736] [02865 03506] [03352 03988]

Y3 [minus 00184 00027] [02979 03073] [03388 03396]


λ 30 λ 40 λ 50Y1 [03634 03641] [03694 03699] [03730 03734]

Y2 [03533 04179] [03627 04279] [03684 04339]

Y3 [03543 03544] [03624 03624] [03673 03673]


03

Scor

e fun

ctio

n 02

07

14

21

28

35

42

49

56

63

λ




70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

y1y2y3




λ

03

Scor

e fun

ctio

n

02

07

14

21

28

35

42

49

56

63

70

77

84

91

98

105

112

119

01

00

ndash01

ndash02

ndash03

y1y2y3


Complexity 23




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity




6 Conclusions



Abbreviations









element









Data Availability




Acknowledgments


References







24 Complexity




































Complexity 25















26 Complexity




































Complexity 25















26 Complexity















26 Complexity

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