Extended VIKOR Method for Intuitionistic Fuzzy...

Research ArticleExtended VIKOR Method for Intuitionistic Fuzzy MultiattributeDecision-Making Based on a New Distance Measure

Xiao Luo1 and Xuanzi Wang2

1School of Air and Missile Defense Air Force Engineering University Xirsquoan 710051 China2School of Management Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Xiao Luo xiao hngz163com

Received 24 June 2017 Revised 18 September 2017 Accepted 27 September 2017 Published 29 October 2017

Academic Editor Anna M Gil-Lafuente

Copyright copy 2017 Xiao Luo and Xuanzi Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An intuitionistic fuzzy VIKOR (IF-VIKOR) method is proposed based on a new distance measure considering the waverof intuitionistic fuzzy information The method aggregates all individual decision-makersrsquo assessment information based onintuitionistic fuzzy weighted averaging operator (IFWA) determines the weights of decision-makers and attributes objectivelyusing intuitionistic fuzzy entropy calculates the group utility and individual regret by the new distance measure and then reachesa compromise solution It can be effectively applied to multiattribute decision-making (MADM) problems where the weights ofdecision-makers and attributes are completely unknown and the attribute values are intuitionistic fuzzy numbers (IFNs) Thevalidity and stability of this method are verified by example analysis and sensitivity analysis and its superiority is illustrated bythe comparison with the existing method

1 Introduction

Multiattribute decision-making (MADM) which has beenincreasingly studied and is of concern to researchers andadministrators is one of themost important parts of decisiontheory It aims to provide a comprehensive solution byevaluating and ranking alternatives based on conflictingattributes with respect to decision-makersrsquo (DMsrsquo) prefer-ences and has widely been used in engineering economicsand management Several classical MADM methods havebeen proposed by researchers in literature such as theTOPSIS (Technique for Order Preference by Similarity toIdeal Solution) method [1] the VIKOR (VIseKriterijumskaOptimizacija I Kompromisno Resenje in Serbian mean-ing multiattribute optimization and compromise solution)method [2] the PROMETHEE (Preference Ranking Organi-zation Method for Enrichment Evaluations) method [3] andthe ELECTRE (Elimination and Choice Translating reality)method [4] Among these methods VIKOR is shown to havesome advantages over others by several researchers In [5]Opricovic and Tzeng compared VIKORmethod and TOPSIS

method from the perspective of aggregation function andfound that although both methods calculate the distanceof an alternative to the ideal solution the former methodprovides a compromise solution bymutual concessions whilethe latter method obtains a solution with the farthest distancefrom the negative-ideal solution and the shortest distancefrom the positive ideal solution without considering therelative importance degrees of these distances FurthermoreOpricovic and Tzeng compared an extension of the VIKORmethod comprehensively with the TOPSIS ELECTRE andPROMETHEEmethod and revealed that the VIKORmethodis superior in meeting conflicting and noncommensurableattributes [6] At present the VIKORmethod has wide appli-cation inmany areas such as designmechanical engineeringand manufacturing [7ndash9] logistics and supply chain man-agement [10 11] structural construction and transportationengineering [12 13] and business management [14]

Since Zadah put forward the concept of fuzzy sets in1965 fuzzy sets especially fuzzy numbers (eg triangularfuzzy numbers and trapezoidal fuzzy numbers) have beenwidely researched and applied in MADM problems to deal

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 4072486 16 pageshttpsdoiorg10115520174072486

2 Mathematical Problems in Engineering

with uncertainty in actual decision-making process Sayadiet al extended the concept of VIKOR method to solveMADMproblems with interval fuzzy numbers [15] Kaya andKahrarnan developed an integrated VIKOR-AHP methodbased on triangular fuzzy numbers in forestation districtselection scenario [16] Ashtiani and Abdollahi Azgomiconstructed a trust modeling based on a combination offuzzy analytic hierarchy process and VIKOR method withtriangular fuzzy numbers [17] Liu et al applied the VIKORmethod to find a compromise priority ranking of failuremodes according to the risk factors in which the weights ofattributes and ratings of alternatives are linguistic variablesrepresented by trapezoidal or triangular fuzzy numbers [18]Ju and Wang proposed a new method to solve multiattributegroup decision-making (MAGDM) problems based on thetraditional idea of VIKOR method with trapezoidal fuzzynumbers [19] However none of the above-mentioned studiesconsidered hesitancy Due to the increasing complexity anduncertainty of economic environment decision-makersrsquo per-ception of things involves not only affirmation and negationbut also hesitation Therefore hesitation as one of the mostimportant pieces of uncertain information may have greatimpact on the final decision and should be consideredin decision-making process To deal with this Atanassovextended traditional fuzzy sets to intuitionistic fuzzy sets(IFSs) which consider membership nonmembership anddegree of hesitancy at the same time In practical applicationthe use of IFSs can depict the fuzziness and nonspecificityof problems by employing both membership function andnonmembership function Therefore it is considered to bemore effective than classical FSs with merely a member-ship function Recently intuitionistic fuzzy VIKOR (IF-VIKOR) method becomes a popular research topic becauseit aims to deal with the widespread uncertainty in decision-making process and some researchers focus on the IF-VIKOR method to solve MADM problems For instanceDevi [20] first extended VIKOR method in intuitionisticfuzzy environment to deal with robot selection problem inwhich the weights of attributes and ratings of alternatives arerepresented by triangular intuitionistic fuzzy sets (TIFSs) Luand Tang [21] employed the VIKOR method to evaluate autoparts supplier based on IFSs Roostaee et al extended VIKORmethod for group decision-making to solve the supplierselection problem under intuitionistic fuzzy environment[22] Wan et al put forward a new VIKOR method forMADM problem with TIFSs [23] in which the weights ofattributes and DMs are completely unknown Park et alextended VIKOR method for dynamic intuitionistic fuzzyMADM concerning university faculty evaluation problem[24] Based on similarity measure between IFSs Peng et alproposed a novel IF-VIKOR method to deal with multire-sponse optimization problem in which the importance ofeach response is given by an engineer as IFSs [25] Hashemiet al proposed a new compromise method based on classicalVIKOR model with interval-valued intuitionistic fuzzy sets(IVIFSs) illustrated by an application in reservoir floodcontrol operation [26] Mousavi et al presented a new groupdecision-making method for MADM problems based on IF-VIKOR in which the ratings of alternatives concerning each

attribute and the weights of criterions provided by DMs arerepresented as linguistic variables characterized by IFSs withmultijudge [27]

Distance is an important fundamental concept of IFSsand plays a significant role in VIKOR method The classicalcompromise programing is based on the distance measurewhich is determined by the closeness of a specific solutionto the idealinfeasible solution maximizing the group utilityandminimizing the individual regret simultaneously [28 29]Although some existing methods employ intuitionistic fuzzysubtraction and division operator to calculate ldquogroup utilityrdquoand ldquoindividual regretrdquo the result is in IFSs form and needsto be further sorted according to IFSsrsquo ranking rule To someextent the amount of computation is increased and theresults are not concise as the crisp value obtained by distancemeasure Also some researchers [30ndash32] reviewed existingdistance and similarity measures between IFSs and testedthem with an artificial benchmark The result showed thatmany existing distance measures generate counterintuitivecases and fail to capture waver the lack of knowledgewhich should have been an advantage of IFSs On the onehand existing distance measures are shown to have somelimitations with hindered effectiveness while on the otherhand the measure choice is crucial to IF-VIKOR methoddue to its significant influence on the result Therefore thispaperrsquos objective is to construct a new distance measurewhich can capture IFS information effectively Based on thisan IF-VIKOR method is further proposed to solve MADMproblems with completely unknown weights of DMs andattributes The main features and novelties of this paper areas follows

(1) Different from the VIKOR method based on classicalfuzzy sets such as interval fuzzy numbers [15] triangularfuzzy numbers [16ndash18] and trapezoidal fuzzy numbers [1819] the method proposed in this paper is based on IFSswhich adds a nonmembership on the basis of the traditionalmembership and is able to describe support opposition andneutrality in human cognition Such an extended definitionhelps more adequately to represent situations when decisionmaker abstains or hesitates from expressing their assess-ments

(2) Compared with the method [20 27] employingintuitionistic fuzzy operator to calculate ldquogroup utilityrdquo andldquoindividual regretrdquo our method generates crisp value resultin a more intuitive way with less computation AlthoughRoostaee et al [22] extended the VIKOR method to intu-itionistic fuzzy environment based on hamming distancethe method sometimes generates counterintuitive cases andtakes no consideration of waver in IFSs Instead our methodbased on a new distance measure could reflect intuitionisticfuzzy information effectively Also the new distance does notgenerate counterintuitive cases in the artificial benchmarktest proposed by [30]

(3) Compared with the VIKORmethod [20 23 26] basedon those extensions of IFSs IFSs possess sound theoreticalfoundation such as basic definition and operations compar-ison rules information fusion method and measures of IFS

Mathematical Problems in Engineering 3

The proposed IF-VIKORmethod can take advantage of thesetheories in construction and verification which is favorableto practical application

(4) Although IFSs have the advantage of being able toconsider waver most of existing IF-VIKOR methods areapplied to environment with less nonspecificity information[27 33] Through an example of ERP system selection ourpaper illustrates the effectiveness of proposedmethod in suchsituation Moreover an example of material handling selec-tion indicates that our method can obtain desired rankingresult superior in environment with low specificity

(5) In the MADM problem with intuitionistic fuzzynumbers (IFNs) subjective randomness exists because theweights of DMs or attributes are usually given artificially[22 27] To avoid this issue these weights are determinedobjectively using intuitionistic fuzzy entropy in the proposedmethod

(6) Compared with similar methods used for MADM [13 4] the proposedmethod based on the decision principle ofthe classical VIKOR method can maximize the group utilityand minimize the individual regret simultaneously makingthe decision result more reasonable Also the coefficientof decision mechanism can be changed according to actualrequirements to balance group utility and individual regretwhich can increase the flexibility of decision-making

The remainder of this paper is organized as followsSection 2 reviews some basic concepts of IFSs In Section 3 anew distancemeasure for IFSs is proposed and a comparativeanalysis is conducted In Section 4 a new distance based IF-VIKOR method is developed to deal with MADM problemIn Section 5 two application examples are demonstrated tohighlight the applicability and superiority of proposed IF-VIKORmethodThe final section summarizes themain workof this paper with a discussion of implications for futureresearch

2 Preliminaries

In this section we briefly review some basic notions andtheories related to IFSs

21 Intuitionistic Fuzzy Set

Definition 1 An intuitionistic fuzzy set 119860 in the universe ofdiscourse119883 is defined as follows [34]119860 = ⟨119909 119906119860 (119909) V119860 (119909)⟩ | 119909 isin 119883 (1)where 119906119860 119883 rarr [0 1] and V119860 119883 rarr [0 1] arecalled membership function and nonmembership functionof 119909 to 119860 respectively 119906119860(119909) represents the lowest boundof degree of membership derived from proofs of supporting119909 while V119860(119909) represents the lowest bound of degree ofnonmembership derived from proofs of opposing 119909 For any119909 isin 119883 0 le 119906119860+V119860 le 1The function120587119860(119909) = 1minus119906119860(119909)minusV119860(119909)is called hesitance index and 120587119860(119909) isin [0 1] represents thedegree of hesitancy of 119909 to 119860 Specially if 120587119860(119909) = 0 where119909 isin 119883 is known absolutely the intuitionistic fuzzy set 119860degenerates into a fuzzy set

Definition 2 For convenience of computation let 119886 = (119906119886 V119886)be an intuitionistic fuzzy number (IFN) Then the scorefunction of 119886 is defined as follows [35]

119904 (119886) = (119906119886 minus V119886) (2)

And the accuracy function of 119886 is defined as follows [36]ℎ (119886) = (119906119886 + V119886) (3)

Let 1198861 and 1198862 be two IFNs then Xu and Yager [37]proposed the following rules for ranking of IFNs

(1) 119904(1198861) lt 119904(1198862) then 1198861 is smaller than 1198862 denoted by1198861 lt 1198862(2) 119904(1198861) gt 119904(1198862) then 1198862 is smaller than 1198861 denoted by1198861 gt 1198862(3) 119904(1198861) = 119904(1198862) then

(i) ℎ(1198861) = ℎ(1198862) then 1198861 is equal to 1198862 denoted by1198861 = 1198862(ii) ℎ(1198861) lt ℎ(1198862) then 1198861 is smaller than 1198862

denoted by 1198861 lt 1198862(iii) ℎ(1198861) gt ℎ(1198862) then 1198862 is smaller than 1198861

denoted by 1198861 gt 1198862Definition 3 Let 119886119894 (119894 = 1 2 119899) be the set of IFNs where119886119894 = (119906119886119894 V119886119894) The intuitionistic fuzzy weighted averaging(IFWA) operator can be defined as follows [33]

IFWA119908 (1198861 1198862 119886119899) = 11990811198861 oplus 11990821198862 oplus sdot sdot sdot oplus 119908119899119886119899= (1 minus 119899prod

119894=1

(1 minus 119906119886119894)119908119894 119899prod119894=1

V119886119894119908119894) (4)

where 119908119894 is the weight of 119886119894 (119894 = 1 2 119899) 119908119894 isin [0 1] andsum119899119894=1 119908119894 = 1Definition 4 Let 119860 be an intuitionistic fuzzy set in theuniverse of discourse 119883 Intuitionistic fuzzy entropy is givenas follows [38]

119864 (119860) = 1119899119899sum119894=1

min (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909)max (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909) (5)

22 Distance Measure between IFSs

Definition 5 For any119860 119861 119862 isin IFSs(119883) let119889 be amapping119889 IFSs(119883) times IFSs(119883) rarr [0 1] If 119889(119860 119861) satisfies the followingproperties [39]

(DP1) 0 le 119889(119860 119861) le 1(DP2) 119889(119860 119861) = 0 if and only if 119860 = 119861(DP3) 119889(119860 119861) = 119889(119861 119860)


(DP4) if 119860 sube 119861 sube 119862 then 119889(119861 119862) le 119889(119860 119862) and 119889(119860 119861) le119889(119860 119862)then 119889(119860 119861) is a distance measure between IFSs 119860 and 119861

Let 119883 be the universe of discourse where 119883 = 1199091 1199092 119909119899 and let 119860 and 119861 be two IFSs in the universe ofdiscourse 119883 where 119860 = ⟨119909119894 119906119860(119909119894) V119860(119909119894)⟩ | 119909119894 isin 119883

and 119861 = ⟨119909119894 119906119861(119909119894) V119861(119909119894)⟩ | 119909119894 isin 119883 Several widely useddistance measures are reviewed as follows

In [40] Szmidt and Kacprzyk proposed four distancesbetween IFSs using the well-known hamming distanceEuclidean distance and their normalized counterparts asfollows

119889119867 (119860 119861) = 12119899sum119894=1

[1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)1003816100381610038161003816] 119889119899119867 (119860 119861) = 12119899

119899sum119894=1

[1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)1003816100381610038161003816] 119889119864 (119860 119861) = radic 12

119899sum119894=1

[(119906119860 (119909119894) minus 119906119861 (119909119894))2 + (V119860 (119909119894) minus V119861 (119909119894))2 + (120587119860 (119909119894) minus 120587119861 (119909119894))2]119889119899119864 (119860 119861) = radic 12119899

119899sum119894=1

[(119906119860 (119909119894) minus 119906119861 (119909119894))2 + (V119860 (119909119894) minus V119861 (119909119894))2 + (120587119860 (119909119894) minus 120587119861 (119909119894))2]

(6)

Wang and Xin [39] pointed out that Szmidt andKacprzykrsquos distancemeasures [40] have some good geometricproperties but there are some limitations in the applicationTherefore they proposed several new distances as follows

1198891 (119860 119861) = 1119899sdot 119899sum119894=1

[1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)10038161003816100381610038164+ max (1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816)2 ]

1198891198752 (119860 119861) = 1119901radic119899sdot 119901radic 119899sum119894=1

(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)10038161003816100381610038162 )119901

(7)

On the basis of the Hausdorff metric Grzegorzewski[41] put forward some approaches for computing distances

between IFSs and these suggested distances are also gener-alizations of the well-known hamming distance Euclideandistance and their normalized counterparts

119889ℎ (119860 119861)= 119899sum119894=1

max 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 119897ℎ (119860 119861)= 1119899119899sum119894=1

max 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 119890ℎ (119860 119861)= radic 119899sum119894=1

max (119906119860 (119909119894) minus 119906119861 (119909119894))2 (V119860 (119909119894) minus V119861 (119909119894))2119902ℎ (119860 119861)= radic 1119899

119899sum119894=1

max (119906119860 (119909119894) minus 119906119861 (119909119894))2 (V119860 (119909119894) minus V119861 (119909119894))2

(8)

Yang and Chiclana [42] suggested that the 3D interpreta-tion of IFSs could provide different contradistinction resultsto the ones obtained with their 2D counterparts [41] andintroduced several extended 3D Hausdorff based distances

119889119890ℎ (119860 119861) = 119899sum119894=1

max 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)1003816100381610038161003816 119897119890ℎ (119860 119861) = 1119899

119899sum119894=1

max 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)1003816100381610038161003816


119890119890ℎ (119860 119861) = radic 119899sum119894=1

max (119906119860 (119909119894) minus 119906119861 (119909119894))2 (V119860 (119909119894) minus V119861 (119909119894))2 (120587119860 (119909119894) minus 120587119861 (119909119894))2

119902119890ℎ (119860 119861) = radic 1119899119899sum119894=1

max (119906119860 (119909119894) minus 119906119861 (119909119894))2 (V119860 (119909119894) minus V119861 (119909119894))2 (120587119860 (119909119894) minus 120587119861 (119909119894))2(9)

3 New Distance Measure between IFSs

31 Analysis on Existing Distance Measures between IFSsAlthough various types of distance measures between IFSswere proposed over the past several years and most of themsatisfies the distance axioms (Definition 5) some unreason-able cases can still be found [31 32] For example considerthree single-element IFSs 119860 = (119909 04 02) 119861 = (119909 05 03)and 119862 = (119909 05 02) In this case the distance between119860 and119861 calculated by some existing measures (eg 119889119899119867 119889119899119864 119897ℎ 119897119890ℎ11988911198891198752 ) is equal to or greater than that between119860 and119862 whichdoes not seem to be reasonable since IFSs A B and C areordered as 119862 gt 119861 gt 119860 according to the score function andaccuracy function given in Definition 2 indicating that thedistance between119860 and 119861 is smaller than that between119860 and119862 For one reason different distance measures have differentfocus which may be suitable for different applications Foranother this could mean the definition of distance measuresis too weak Specifically the definitionmay bemore completeand reasonable if it has considered the following two aspectsFirst Definition 5 just provides the value constraints when119889(119860 119861) = 0 and the other endpoint 119889(119860 119861) = 1 has notbeen discussed Second the distance measure between IFSscould be better convinced if it satisfies the requirement of thetriangle inequality

Another point worth noting is that there are two facets ofuncertainty of intuitionistic fuzzy information one of whichis related to fuzziness while the other is related to lack ofknowledge or nonspecificity [43] First IFSs are an extensionof FSs and hence are associated with fuzzinessThis fuzzinessis the first kind of uncertainty that mainly reflects two typesof distinct and specific information degree of membershipand degree of nonmembership In IFSs there exists anotherkind of uncertainty that can be called lack of knowledgeor nonspecificity and might also be called waver [30] Thiskind of uncertainty reflects a type of nonspecific informationdegree of hesitancy indicating how much we do not knowabout membership and nonmembership These two kinds ofuncertainty are obviously different the degree of hesitancywith its own uncertainty because it represents a state of ldquoboththis and thatrdquo However most of the existing distance mea-sures only consider the differences between numerical valuesof the IFS parameters and ignore the waver of intuitionisticfuzzy information As an illustration consider two patternsrepresented by IFSs 119860 = (119909 01 02) and 119861 = (119909 01 02) inwhich both hesitance indexes of 119860 and 119861 are equal to 07 Inthis case the distance between IFSs 119860 and 119861 calculated by allaforementioned measures is equal to zero denoting that thetwo patterns are the same This is true from the perspective

of numerical value because the three parameters of IFSs 119860and 119861 have the same value However from the perspectiveof intuitionistic fuzzy information the conclusion drawn isunreasonable The hesitance index 120587(119909) = 1 minus 119906(119909) minus V(119909)in both IFSs A and B is equal to 07 but it just means that inIFSs A and B the proportion of lack of knowledge is 07 andit cannot represent that the patterns 119860 and 119861 are the sameIn other words since we have a lack of knowledge we shouldnot draw the conclusion that the two patterns are the same

As a powerful tool in modeling uncertain informationIFSs have the advantage of being able to consider wavermainly because the hesitant index can be used to describe aneutrality state of ldquoboth this and thatrdquo Moreover it shouldbe noted that the ultimate goal of distance measure is tomeasure the difference of information carried by IFSs ratherthan the numerical value of IFSs themselves Therefore inour view the distance measure should take into accountthe characteristics of intuitionistic fuzzy information bothfuzziness and nonspecificity and have its own target

32 Intuitive Distance for IFSs Inspired by the characteristicsof intuitionistic fuzzy information we introduce a notioncalled intuitive distance to compare the information betweentwo IFSs and compute the degree of difference and theaxiomatic definition of intuitive distance is as follows

Definition 6 For any119860 119861 119862119863 isin IFSs(119883) let119889 be amapping119889 IFSs(119883) times IFSs(119883) rarr [0 1] 119889(119860 119861) is said to be anintuitive distance between 119860 and 119861 if 119889(119860 119861) satisfies thefollowing properties

(P1) 0 le 119889(119860 119861) le 1(P2) 119889(119860 119861) = 0 if and only if 119860 = 119861 and 120587119860(119909) = 120587119861(119909) =0(P3) 119889(119860 119861) = 1 if and only if both A and B are crisp sets

and 119860 = 119861119862(P4) 119889(119860 119861) = 119889(119861 119860)(P5) 119889(119860 119862) le 119889(119860 119861)+119889(119861 119862) for any119860 119861 119862 isin IFSs(119883)(P6) if 119860 sube 119861 sube 119862 then 119889(119861 119862) le 119889(119860 119862) and 119889(119860 119861) le119889(119860 119862)(P7) if 119860 = 119861 and 119862 = 119863 then

(i) 119889(119860 119861) gt 119889(119862119863) when 120587119860(119909) + 120587119861(119909) gt120587119862(119909) + 120587119863(119909)(ii) 119889(119860 119861) lt 119889(119862119863) when 120587119860(119909) + 120587119861(119909) lt120587119862(119909) + 120587119863(119909)


(iii) 119889(119860 119861) = 119889(119862119863) when 120587119860(119909) + 120587119861(119909) =120587119862(119909) + 120587119863(119909)The above definition is a development for Definition 5

On the one hand some properties are introduced to makethe definition stronger (P3) is a more precise condition forthe endpoint of distance 119889(119860 119861) = 1 (P5) is a new strongproperty condition which requires distance to meet triangleinequality On the other hand the definition highlightsthe characteristics of intuitionistic fuzzy information by

reinforcing the existing property condition (P2) as well asadding a new property condition (P7) The intuitive distancenot only meets the most general properties of traditionaldistance but also satisfies a special property as long as thehesitant index is not zero even if two IFSs are equal in valuethe distance between them is not zero And the higher thehesitant index the greater the distance

Then we propose a new distance measure Let 119860 =⟨119909 119906119860(119909) V119860(119909)⟩ 119861 = ⟨119909 119906119861(119909) V119861(119909)⟩ be two IFSs in theuniverse of discourse119883 and denote119883 = 1199091 1199092 119909119899

119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)] 1205791 (119909119894) = 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894)) minus (119906119861 (119909119894) + 1 minus V119861 (119909119894))10038161003816100381610038162 1205792 (119909119894) = 120587119860 (119909119894) + 120587119861 (119909119894)2 1205793 (119909119894) = max(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)10038161003816100381610038162 )

(10)

The structure of 119889119871(119860 119861) is mainly related to twoaspects One is to take the differences of IFS parametersinto account which is similar to the traditional distance Itincludes differences betweenmemberships119906119860(119909119894) and119906119861(119909119894)nonmemberships V119860(119909119894) and V119861(119909119894) hesitance indexes 120587119860(119909119894)and 120587119861(119909119894) and the differences between median values ofintervals (119906119860(119909119894)+1minusV119860(119909119894))2 and (119906119861(119909119894)+1minusV119861(119909119894))2Thesecond is to satisfy the new requirements of intuitive distancetaking the degree of hesitancy into account 1205792(119909119894) aims toreflect the waver of uncertain information The higher thedegree of nonspecificity of intuitionistic fuzzy informationthe greater the possibility of existing differences betweenthem Then we have the following theorem

Theorem 7 119889119871(119860 119861) is an intuitive distance between IFSs 119860and 119861 in the universe of discourse119883 = 1199091 1199092 119909119899Proof It is easy to see that 119889119871(119860 119861) satisfies (P4) and (P7) ofDefinition 6 Therefore we shall prove that 119889119871(119860 119861) satisfies(P1) (P2) (P3) (P5) and (P6)

(P1) Let 119860 and 119861 be two IFSs we can write the relationalexpression as follows

0 le 1205791 (119909119894) + 1205792 (119909119894) = (1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816+ 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816+ 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894)) minus (119906119861 (119909119894) + 1 minus V119861 (119909119894))1003816100381610038161003816+ 120587119860 (119909119894) + 120587119861 (119909119894)) times (2)minus1 = (1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816+ 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816+ 1003816100381610038161003816(119906119860 (119909119894) minus 119906119861 (119909119894)) minus (V119860 (119909119894) minus V119861 (119909119894))1003816100381610038161003816 + 2

minus 119906119860 (119909119894) minus 119906119861 (119909119894) minus V119860 (119909119894) minus V119861 (119909119894)) times (2)minus1le (1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816+ 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 2minus 119906119860 (119909119894) minus 119906119861 (119909119894) minus V119860 (119909119894) minus V119861 (119909119894)) times (2)minus1= (2 1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 minus (119906119860 (119909119894) + 119906119861 (119909119894))+ 2 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 minus (V119860 (119909119894) + V119861 (119909119894)) + 2)times (2)minus1 le (2 (119906119860 (119909119894) + 119906119861 (119909119894))minus (119906119860 (119909119894) + 119906119861 (119909119894)) + 2 (V119860 (119909119894) + V119861 (119909119894))minus (V119860 (119909119894) + V119861 (119909119894)) + 2) times (2)minus1= 119906119860 (119909119894) + 119906119861 (119909119894) + V119860 (119909119894) + V119861 (119909119894) + 22= (119906119860 (119909119894) + V119860 (119909119894)) + (119906119861 (119909119894) + V119861 (119909119894)) + 22le 1 + 1 + 22 = 2

(11)

It is known that

0 le 1205793 (119909119894) = max(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)10038161003816100381610038162 ) le 1

(12)


Taking (11) and (12) into account it is not difficult to find that

0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)

And therefore we have

0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)

(P2) Let 119860 and 119861 be two IFSs the following relationalexpression can be written

119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)

Thus 119889119871(119860 119861) satisfies (P2) of Definition 6

(P3) Let 119860 and 119861 be two IFSs Taking (11) and (12) intoaccount the following relational expressions can bewritten

119889119871 (119860 119861) = 11205791 (119909119894) + 1205792 (119909119894) = 21205793 (119909119894) = 1119906119860 (119909119894) + 119906119861 (119909119894) + V119860 (119909119894) + V119861 (119909119894) + 22 = 2max(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)10038161003816100381610038162 ) = 1

119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)


(P5) Let 119860 119861 and 119862 be three IFSs the distances betweenA and B B and C and A and C are the following

119889119871 (119860 119861) = 13119899119899sum119894=1

[1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894)) minus (119906119861 (119909119894) + 1 minus V119861 (119909119894))10038161003816100381610038162+ 120587119860 (119909119894) + 120587119861 (119909119894)2 +max(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)10038161003816100381610038162 )]

119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1

[1003816100381610038161003816119906119860 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119862 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894)) minus (119906119862 (119909119894) + 1 minus V119862 (119909119894))1003816100381610038161003816 + 120587119860 (119909119894) + 120587119862 (119909119894)2+max(1003816100381610038161003816119906119860 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119862 (119909119894)1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119862 (119909119894)10038161003816100381610038162 )]

(17)


It is obvious that

1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816119906119861 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816 ge 1003816100381610038161003816119906119860 (119909119894)minus 119906119861 (119909119894) + 119906119861 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816 = 1003816100381610038161003816119906119860 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119861 (119909119894) minus V119862 (119909119894)1003816100381610038161003816 ge 1003816100381610038161003816V119860 (119909119894)minus V119861 (119909119894) + V119861 (119909119894) minus V119862 (119909119894)1003816100381610038161003816 = 1003816100381610038161003816V119860 (119909119894) minus V119862 (119909119894)1003816100381610038161003816 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894)) minus (119906119861 (119909119894) + 1 minus V119861 (119909119894))1003816100381610038161003816+ 1003816100381610038161003816(119906119861 (119909119894) + 1 minus V119861 (119909119894))minus (119906119862 (119909119894) + 1 minus V119862 (119909119894))1003816100381610038161003816 = 1003816100381610038161003816119906119860 (119909119894) + V119861 (119909119894)minus 119906119861 (119909119894) minus V119860 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816119906119861 (119909119894) + V119862 (119909119894) minus 119906119862 (119909119894)minus V119861 (119909119894)1003816100381610038161003816 ge 1003816100381610038161003816119906119860 (119909119894) + V119861 (119909119894) minus 119906119861 (119909119894) minus V119860 (119909119894)+ 119906119861 (119909119894) + V119862 (119909119894) minus 119906119862 (119909119894) minus V119861 (119909119894)1003816100381610038161003816 = 1003816100381610038161003816119906119860 (119909119894)

minus V119860 (119909119894) + V119862 (119909119894) minus 119906119862 (119909119894)1003816100381610038161003816= 1003816100381610038161003816(119906119860 (119909119894) + 1 minus V119860 (119909119894))minus (119906119862 (119909119894) + 1 minus V119862 (119909119894))1003816100381610038161003816 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816120587119861 (119909119894) minus 120587119862 (119909119894)1003816100381610038161003816 ge 1003816100381610038161003816120587119860 (119909119894)minus 120587119861 (119909119894) + 120587119861 (119909119894) minus 120587119862 (119909119894)1003816100381610038161003816 = 1003816100381610038161003816120587119860 (119909119894)minus 120587119862 (119909119894)1003816100381610038161003816

120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)

Therefore we have 119889119871(119860 119861) + 119889119871(119861 119862) ge 119889119871(119860 119862)(P6) Let 119860 119861 and 119862 be three IFSs if 119860 sube 119861 sube 119862 then

we have 119906119860(119909119894) le 119906119861(119909119894) le 119906119862(119909119894) V119860(119909119894) ge V119861(119909119894) geV119862(119909119894) The following equations are given out

119889119871 (119860 119861) = 13119899119899sum119894=1

[119906119861 (119909119894) minus 119906119860 (119909119894) + V119860 (119909119894) minus V119861 (119909119894) + 119906119861 (119909119894) minus 119906119860 (119909119894) + V119860 (119909119894) minus V119861 (119909119894) + 120587119860 (119909119894) + 120587119861 (119909119894)2+max(119906119861 (119909119894) minus 119906119860 (119909119894) V119860 (119909119894) minus V119861 (119909119894) 1003816100381610038161003816120587119860 (119909119894) minus 120587119861 (119909119894)10038161003816100381610038162 )] = 13119899sdot 119899sum119894=1

[119906119861 (119909119894) minus 119906119860 (119909119894) + V119860 (119909119894) minus V119861 (119909119894) + 119906119861 (119909119894) minus 119906119860 (119909119894) + V119860 (119909119894) minus V119861 (119909119894) + 2 minus 119906119860 (119909119894) minus 119906119861 (119909119894) minus V119860 (119909119894) minus V119861 (119909119894)2+max(119906119861 (119909119894) minus 119906119860 (119909119894) V119860 (119909119894) minus V119861 (119909119894) 1003816100381610038161003816119906119861 (119909119894) minus 119906119860 (119909119894) + V119861 (119909119894) minus V119860 (119909119894)10038161003816100381610038162 )] = 13119899sdot 119899sum119894=1

[119906119861 (119909119894) minus 3119906119860 (119909119894) + V119860 (119909119894) minus 3V119861 (119909119894)2 +max (119906119861 (119909119894) minus 119906119860 (119909119894) V119860 (119909119894) minus V119861 (119909119894))] 119889119871 (119860 119862) = 13119899

119899sum119894=1



[119906119862 (119909119894) minus 3119906119860 (119909119894) + V119860 (119909119894) minus 3V119862 (119909119894)2 +max (119906119862 (119909119894) minus 119906119860 (119909119894) V119860 (119909119894) minus V119862 (119909119894))]

(19)

We know that119906119862 (119909119894) minus 3V119862 (119909119894) ge 119906119861 (119909119894) minus 3V119861 (119909119894) 119906119862 (119909119894) minus 119906119860 (119909119894) ge 119906119861 (119909119894) minus 119906119860 (119909119894) V119860 (119909119894) minus V119862 (119909119894) ge V119860 (119909119894) minus V119861 (119909119894)

(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)

Similarly it is easy to prove that

119889119871 (119861 119862) le 119889119871 (119860 119862) (22)



Table 1 Test intuitionistic fuzzy sets (IFSs)

Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)

Table 2 The comparison of distance measures (counterintuitive cases are in bold italic type)

Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500

That is to say 119889119871(119860 119861) is an intuitive distance betweenIFSs 119860 and 119861 since it satisfies (P1)ndash(P7)33 A Comparison of Distance Measures for IFSs Based onan Artificial Benchmark When a new distance measure isproposed it is always accompanied with explanations ofovercoming counterintuitive cases of other methods andthese cases are usually illustrated by single-element IFSs In[30] Li et al summarized counterintuitive cases proposed byprevious literature and constituted an artificial benchmarkwith six different pairs of single-element IFSs which hasbeen widely used in the test of distance and similaritymeasures [31 44ndash49] Although these cases cannot representall counterintuitive situations they are typical and represen-tative In [31] Papakostas et al suggested that any proposedmeasures should be tested by the artificial benchmark toavoid counterintuitive cases To illustrate the effectiveness ofthe proposed distancemeasure all the test IFSs of the artificialbenchmark are applied to compare the proposed distancemeasure to the widely used distance measures In additionin order to reflect the characteristics of intuitionistic fuzzyinformation a new pair of single-element IFSs as illustratedin Section 31 is added to the artificial benchmark test setTable 1 shows the test IFSs of extended artificial benchmark

Table 2 provides a comprehensive comparison of thedistance measures for IFSs with counterintuitive cases It isapparent that the property condition (P3) is not met by 119889119899119867119889119899119864 119897ℎ 119897119890ℎ because the distances calculated by thesemeasuresare equal to 1 when 119860 = (119909 1 0) 119861 = (119909 0 0) Similarlythe property condition (P3) is also not satisfied by 119889119899119867 119897119890ℎwhen 119860 = (119909 0 0) 119861 = (119909 05 05) The distance measures119889119899119867 119897119890ℎ and 1198891198752 119901 = 1 indicate that the distances of the1st test IFSs and the 2nd test IFSs are identical which doesnot seem to be reasonable Distance measures 119897ℎ 1198891 and1198891198752 119901 = 1 claim that the distances of the 4th test IFSs andthe 5th test IFSs show the same value of 01 which indicatesthat there are not sufficient abilities to distinguish positivedifference from negative difference The distance of 3rd test

IFSs is equal to or greater than the distance of the 7th testIFSs when 119889119899119867 119889119899119864 119897ℎ 119897119890ℎ 1198891 and 1198891198752 119901 = 1 are usedwhich does not seem to be reasonable since IFSs A B andC are ordered as 119862 gt 119861 gt 119860 according to the score functionand accuracy function given in Definition 2 indicating thatthe distance between 119860 and 119861 is smaller than that between 119860and 119862 Furthermore all of these existing distances claim thatthe distance of the 6th test IFSs is equal to zero which doesnot seem to be reasonable As a mathematical tool IFSs candescribe the uncertain information greatly because it adds ahesitant index to describe the state of ldquoboth this and thatrdquo Inthis case we indeed cannot confirm that there is no differencebetween the information carried by IFSs 119860 and 119861 becausethe hesitance index includes some nonspecificity informationand the proportion of support and opposition is not sure

Based on analysis in Table 2 it is deduced that the existingdistance measures with their own measuring focus can meetall or most of properties condition of distance measurebetween IFSs however most distance measures show coun-terintuitive cases and may fail to distinguish IFSs accuratelyin some practical applications Besides the proposed distancemeasure is the only one that has no aforementioned coun-terintuitive cases as illustrated in Table 2 Furthermore theproposed distance conforms to all the property requirementsof the intuitive distance and the potential difference broughtby hesitance index is considered

4 IF-VIKOR Method for MADM

On the basis of new distance measure this section presentstepwise algorithm for proposed IF-VIKOR method

For a MADM problem with 119899 alternatives 119860 119894 (119894 =1 2 119898) the performance of the alternative119860 119894 concerningthe attribute119862119895 (119895 = 1 2 119899) is assessed by a decision orga-nization with several decision-makers 119863119902 (119902 = 1 2 119897)The corresponding weights of attributes are denoted by119908119895 (119895 = 1 2 119899) 0 le 119908119895 le 1sum119899119895=1 119908119895 = 1 and the weightsof DMs are denoted by 120582119902 (119902 = 1 2 119897) 0 le 120582119902 le 1


sum119897119902=1 120582119902 = 1 Inspired by the classical VIKOR method andits extensions the intuitive distance based VIKOR methodcan be given for MADM problem with intuitionistic fuzzyinformation it includes seven steps

Step 1 Generate assessment information Assume that DMs119863119902 (119902 = 1 2 119897) provide their opinion of the alternatives119860 119894 concerning each attribute 119862119895 by using IFNs 119909119902119894119895 =(119906119902119894119895 V119902119894119895 120587119902119894119895) or linguistic values represented by IFNsThen theassessments given by119863119902 can be expressed as

119863(119902) =[[[[[[[[[

119909(119902)11 119909(119902)12 sdot sdot sdot 119909(119902)1119899119909(119902)21 119909(119902)22 sdot sdot sdot 119909(119902)2119899 d

119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)

Step 2 Acquire the weights of DMs According to the degreeof fuzziness and nonspecificity of assessments provided byDMs in this step DM weight 120582119902 (119902 = 1 2 119897) can beacquired by intuitionistic fuzzy entropy measure objectivelyThe lower the degree of fuzziness and nonspecificity is thesmaller the entropy is and the bigger the weight of DM isand vice versa By using (5) the intuitionistic fuzzy entropyof assessments provided by119863119902 can be obtained as follows

119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)

Then the weight of DM119863119902 can be defined as follows

120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)

where 119897 is the number of DMs

Step 3 Establish the aggregated intuitionistic fuzzy decisionmatrix By using (4) all individual decisionmatrixes119863(119902) canbe converted into an aggregated decision matrix as follows

119863 = [[[[[[[

11990911 11990912 sdot sdot sdot 119909111989911990921 11990922 sdot sdot sdot 1199092119899 d1199091198981 1199091198982 sdot sdot sdot 119909119898119899

]]]]]]] (26)

where 119909119894119895 = (119906119894119895 V119894119895 120587119894119895) 119906119894119895 = 1 minus prod119897119902=1(1 minus 119906119902119894119895)120582119902 V119894119895 =prod119897119902=1(V119902119894119895)120582119902 120587119894119895 = 1 minus 119906119894119895 minus V119894119895Step 4 Acquire the weights of attributes Similar to Step 2the unknown weight of attribute 119908119895 (119895 = 1 2 119899) canbe determined by entropy measure to effectively reduce thesubjective randomness By using (5) we can obtain theentropy with respect to 119862119895

119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)

Then the weight of attribute119862119895 can be defined as follows119908119895 = 1 minus 119864119895119899 minus sum119899119895=1 119864119895 (28)

where 119899 is the number of attributes

Step 5 Find the best and worst value The best value 119909+119895 andthe worst value 119909minus119895 for each attribute 119862119895 can be defined asfollows

119909+119895 = max119894=12119898

119909119894119895 for benefit attribute 119862119895min119894=12119898

119909119894119895 for cos t attribute 119862119895(119895 = 1 2 119899)

119909minus119895 = min119894=12119898

119909119894119895 for benefit attribute 119862119895max119894=12119898


(29)

Step 6 Compute the values 119878119894 119877119894 and119876119894 Three key values ofIF-VIKOR method the group utility value 119878119894 the individualregret value 119877119894 and the compromise value 119876119894 are computedin light of the intuitive distance measure for each alternative

119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))

119876119894 = 120574( 119878119894 minus 119878lowast119878minus minus 119878lowast) + (1 minus 120574) ( 119877119894 minus 119877lowast

119877minus minus 119877lowast) (30)

where 119878minus = max119894 119878119894 119878lowast = min119894 119878119894 119877minus = max119894 119877119894 119877lowast =min119894 119877119894 120574 is the coefficient of decision mechanism Thecompromise solution can be elected by majority (120574 gt 05)consensus (120574 = 05) or veto (120574 lt 05)Step 7 Rank the alternatives and derive the compromise solu-tion Sort 119878119894 119877119894 and119876119894 in ascending order and generate threeranking lists 119878[sdot] 119877[sdot] and 119876[sdot] Then the alternative 119860(1) thatranks the best in 119876[sdot] (minimum value) and fulfills followingtwo conditions simultaneously would be the compromisesolution

Condition 1 (acceptable advantage) One has

119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)

where 119860(1) and 119860(2) are the top two alternatives in 119876119894


Condition 2 (acceptable stability) The alternative119860(1) shouldalso be the best ranked by 119878119894 and 119877119894

If the above conditions cannot be satisfied simultaneouslythere exist multiple compromise solutions

(1) alternatives 119860(1) and 119860(2) if only Condition 2 is notsatisfied

(2) alternatives 119860(1) 119860(2) 119860(119906) if Condition 1 is notsatisfied where 119860(119906) is established by the relation119876(119860(119906)) minus 119876(119860(1)) lt 1(119898 minus 1) for the maximum

5 Application Examples

51 ERP Selection Problem In recent years enterpriseresource planning (ERP) system has become a powerful toolfor enterprises to improve their operating performance andcompetitiveness However ERP projects report an unusuallyhigh failure rate and sometimes imperil implementersrsquo coreoperation due to their high costs and wide range of configu-ration Consequently selecting the ERP system which fit theenterprise would be a critical step to success Consideringa situation that one high-tech manufacturing enterprise istrying to implement ERP system with four alternatives 11986011198602 1198603 and 1198604 three DMs of 1198631 1198632 1198633 are employed toevaluate these alternatives from five main aspects as follows1198621 Functionality and reliability which involve suitabilityaccuracy security functionality compliance maturity recov-erability fault tolerance and reliability compliance1198622 Usability and efficiency which involve understand-ability learnability operability attractiveness usability com-pliance time behavior resource behavior and efficiencycompliance1198623 Maintainability and portability which involve analyz-ability changeability testability coexistence interoperationmaintainability and portability compliance

Table 3 Linguistic terms for rating the alternatives with IFNs

Linguistic variables IFNsExtremely poor (EP) (005 095 000)Poor (P) (020 070 010)Medium poor (MP) (035 055 010)Medium (M) (050 040 010)Medium good (MG) (065 025 010)Good (G) (080 010 010)Extremely good (EG) (095 005 000)

1198624 Supplier services which involve the quality of trainingstaff technical support and follow-up services the level ofimplementation and standardization customer satisfactionsupplier credibility and strength1198625 The enterprise characteristics which involve enter-prisemanagement employee support comprehensive invest-ment cost internal rate of return benefit cost ratio anddynamic payback period

Since the weights of attributes and DMs are completelyunknown the best alternative would be selected with theinformation given above In the following the proposedIF-VIKOR method is applied to solve this problem Theoperation process according to the algorithm developed inSection 4 is given below

Step 1 Each DM assesses alternative 119860 119894 concerning attribute119862119895 with linguistic rating variables in Table 3 Table 4 showsthe assessments by three decision-makers

Step 2 By using (24) and (25) the weights of DMs can beobtained as 1205821 = 03220 1205822 = 03244 1205823 = 03536Step 3 Byusing (26) we can establish the aggregated decisionmatrix as follows

119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)

Step 4 By using (27) and (28) the weights of attributes areobtained as 1199081 = 01950 1199082 = 02129 1199083 = 01980 1199084 =01966 1199085 = 01976Step 5 By using (29) the best and the worst values of allattribute ratings can be calculated and we have 119909+1 = (0668902193) 119909+2 = (08000 01000) 119909+3 = (08355 01223) 119909+4 =(08465 01077) 119909+5 = (07562 01383) 119909minus1 = (05542 03438)119909minus2 = (06495 02331) 119909minus3 = (03500 05500) 119909minus4 = (0500004000) 119909minus5 = (05423 03491)

Step 6 Without loss of generality let 120574 = 05 By using (30)the values of 119878119894119877119894 and119876119894 for each alternative can be obtainedas listed in Table 5

Step 7 From Table 5 we have 1198763 lt 1198761 lt 1198764 lt 1198762which means 1198603 (minimum value) ranks best in termsof 119876 In addition 1198761 minus 1198763 = 03298 ge 025 and1198603 is also the best ranked by 119878119894 and 119877119894 which showsthat 1198603 is the unique compromise solution for this prob-lem


Table 4 Rating of the alternatives from DMs

Attributes DM1 DM3 DM4A1 A2 A3 A4 A1 A2 A3 A4 A1 A2 A3 A4

C1 MG M MG MP G MP M MG M G M GC2 G MP G G G MG M M G G G MGC3 MP MG M M MP M G MG MP MG EG MGC4 MG M EG M G M MG MG MG M G MC5 M G MG MG M G MG P MG MG G MG

Table 5 The values of 119878 119877 and 119876 for all alternatives by the proposed IF-VIKOR method

Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698

Table 6 Linguistic terms for rating the alternatives with trapezoidalfuzzy numbers

Linguistic variables Trapezoidal fuzzy numbersExtremely poor (EP) (0 0 1 2)Poor (P) (1 2 2 3)Medium poor (MP) (2 3 4 5)Medium (M) (4 5 5 6)Medium good (MG) (5 6 7 8)Good (G) (7 8 8 9)Extremely good (EG) (8 9 10 10)

Liu et al [18] developed the VIKOR method for MADMproblem based on trapezoidal fuzzy numbers which is oneof the most commonly used fuzzy numbers In this methodthe weights of DMs or attributes are given artificially andthe linguistic variables are represented by trapezoidal fuzzynumbers shown in Table 6 A trapezoidal fuzzy number119860 canbe denoted as (1198861 1198862 1198863 1198864) where 1198861 and 1198864 are called lowerand upper limits of 119860 and 1198862 and 1198863 are two most promisingvalues When 1198862 and 1198863 are the same value the trapezoidalfuzzy number degenerated a triangular fuzzy number Weapply themethod [18] in the ERP selection problem to explainthe difference between the proposed method and traditionalVIKOR method with trapezoidal fuzzy numbers Assumethat the weights of DMs and attributes are 1205821 = 032201205822 = 03244 1205823 = 03536 and 1199081 = 01950 1199082 = 021291199083 = 01980 1199084 = 01966 1199085 = 01976 respectivelyThrough calculation of linguistic assessment information andtrapezoidal fuzzy numbers shown inTables 4 and 6 the valuesof 119878119877 and119876 for all alternatives are obtained as inTable 7Theresult shows that the ranking order of alternatives obtainedby the method [18] is 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 whichis in harmony with our proposed method Although theresults obtained from the twoVIKORmethods are consistentthe trapezoidal fuzzy numbers are only characterized bya membership function while IFNs are characterized bya membership function and a nonmembership function

which closely resembles the thinking habit of human beingsunder the situation of being uncertain and hesitant Fur-thermore the trapezoidal fuzzy numbers need four valuesto determine the membership distribution while the IFNsonly need two values membership and nonmembershipand the degree of hesitancy can be automatically generatedby 1 minus membership and nonmembership In generalthe computation of trapezoidal fuzzy numbers is large andthe application of IFNs is relatively simple In addition themethod presented by [18] needs the weight information ofDMs and attributes predetermined whereas theseweights aredetermined objectively using intuitionistic fuzzy entropy inthe proposed method which avoids subjective randomnessto some extent

In [22] Roostaee et al put forward a hamming distancebased IF-VIKOR method To further illustrate its effective-ness the proposed method is compared to the IF-VIKORmethod presented by [22] Computational results for thehamming distance based IF-VIKOR method are shown inTable 8 The result indicates that these two methods reach aconsensus that the third ERP system should be implementedby the high-tech manufacturing enterprise

In this example it should be noted that the assessmentsprovided by DMs are in low degree of hesitancy As shownin Tables 3 and 4 the maximum hesitance index withrespect to linguistic values is 01 denoting the low degreeof nonspecificity (lack of knowledge) for assessments Inthis case both methods can effectively evaluate and sortthe alternatives However the hamming distance based IF-VIKOR method is not always capable of obtaining validresults especially in MADM problem with high degree ofnonspecificity It might generate some counterintuitive casesso that unreasonable results might be obtainedThe followingexample further illustrates such cases

52 Material Handling Selection Problem To illustrate thesuperiority of the proposed methods a comparison betweenthe proposed IF-VIKOR method and the hamming distancebased IF-VIKOR method in a material handling selectionproblem is made


Table 7 The values of 119878 119877 and 119876 for all alternatives by the trapezoidal fuzzy numbers based VIKOR method

Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052

Table 8 The values of 119878 119877 and 119876 for all alternatives by the hamming distance based IF-VIKOR method

Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726

Table 9 Rating of the alternatives from decision organization

C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)

Table 10The compromise values and ranking results obtained by the hamming distance based IF-VIKORmethod and the proposed methodin this paper

Alternative Hamming distance based IF-VIKOR method The method proposed in this paperThe values of 119876119894 Ranking orders The values of 119876119894 Ranking orders

A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1

Suppose that a manufacturing company is consideringimplementing a material handling system After preliminaryscreening four alternatives of 1198601 1198602 1198603 and 1198604 remainto be further evaluated Several DMs from the companyrsquostechnical committee are arranged to evaluate and selectthe appropriate alternative They assess the four alternativesaccording to four conflicting attributes including investmentcost 1198621 operation time 1198622 expansion possibility 1198623 andcloseness tomarket demand1198624 Due to the lack of experiencetime constraints and other factors the ratings of alternativesconcerning each attribute provided by DMs are representedas IFNs with high degree of hesitancy listed in Table 9 Byusing (27) and (28) the weights of attributes are obtainedas 1199081 = 02426 1199082 = 02490 1199083 = 02593 1199084 =02490 Without losing generality let 120574 = 05 Then thevalues of 119878119894 119877119894 and 119876119894 for each alternative can be calculatedComputational results obtained by the hamming distancebased IF-VIKOR method and the proposed method in thispaper are listed in Table 10 The ranking of alternativescalculated by the hamming distance based IF-VIKORmethodis 1198601 ≻ 1198602 ≻ 1198603 ≻ 1198604 reaching a conclusion that1198601 (minimum value) is the best choice and 1198604 is the worstchoice for the material handling selection problem Howeveraccording to the score function and accuracy function each

attribute rating of 1198601 is lower than or equal to that of 1198604indicating that 1198604 is superior to 1198601 which is contradicted tothe ranking results obtained by hamming distance based IF-VIKORmethod Instead our proposedmethod can overcomethe drawback of traditional method and obtain a validranking result as 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 which meansthat material handling system 1198604 is the best choice for themanufacturing company

53 Sensitivity Analysis In intuitionistic fuzzy VIKORmethod 120574 the coefficient of decisionmechanism is critical tothe ranking results Hence a sensitivity analysis is conductedin order to assess the stability of our method in theseexamples For each 120574 from 0 to 1 at 01 intervals we calculatethe corresponding compromise solution to investigate theinfluence of different 120574 on the ranking result

Table 11 shows the sensitivity analysis of ERP selectionexample For all the tested values of 120574 three different rankingresults are generated including 1198603 ≻ 1198604 ≻ 1198601 ≻ 11986021198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 and 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 While theranking result is indeed affected by 1205741198603 is always the optimalsolution Table 12 shows the sensitivity analysis of materialhandling system selection example For all the tested valuesof 120574 the ranking result remains 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 and


Table 11 Rating of the alternatives for different 120574 values (ERP selection example)

120574 1198761 1198762 1198763 1198764 Ranking Optimal solution0 01676 10000 0 01397 1198603 ≻ 1198604 ≻ 1198601 ≻ 1198602 119860301 02000 09813 00000 02257 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860302 02325 09626 00000 03117 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860303 02649 09439 00000 03978 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860304 02974 09252 00000 04838 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860305 03298 09065 00000 05698 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860306 03622 08878 00000 06559 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860307 03947 08691 00000 07419 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860308 04271 08504 00000 08279 1198603 ≻ 1198601 ≻ 1198604 ≻ 1198602 119860309 04596 08317 00000 09140 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 11986031 04920 08130 00000 10000 1198603 ≻ 1198601 ≻ 1198602 ≻ 1198604 1198603

Table 12 Rating of the alternatives for different 120574 values (material handling selection example)

120574 1198761 1198762 1198763 1198764 Ranking Optimal solution0 03224 06299 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860401 03199 06496 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860402 03174 06694 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860403 03149 06891 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860404 03124 07089 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860405 03099 07286 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860406 03074 07484 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860407 03049 07682 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860408 03025 07879 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 119860409 03000 08077 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 11986041 02975 08274 10000 0 1198604 ≻ 1198601 ≻ 1198602 ≻ 1198603 1198604the optimal solution is 1198604 The sensitivity analysis illustrateshow the decision-making strategy would affect the result andalso it indicates that the conclusion arrived from our methodis stable and effective

6 Conclusion

Since the VIKOR method is an effective MADM methodto reach a compromise solution and IFSs are an effectivetool to depict fuzziness and nonspecificity in assessmentinformation we combine them and develop a new intuitivedistance based IF-VIKOR method This new method aims atMADM problems with unknown weights of both the DMsand attributes in intuitionistic fuzzy environment It aggre-gates assessment information by intuitionistic fuzzy weightedaveraging operator generates weights of DMs and attributesby intuitionistic fuzzy entropy objectively calculates thegroup utility and individual regret based on intuitive distancemeasure and finally reaches the compromise solution Twoapplication examples of ERP and material handing selectionproblem further illustrate each step of this method Com-paredwith the hamming distancemeasure used in traditionalIF-VIKOR method the new intuitive distance measure inthis method focuses on the fuzziness and nonspecificityof intuitionistic fuzzy information reflecting not only the

difference among the values of intuitionistic fuzzy sets butalso the waver of intuitionistic fuzzy sets Both the artificialbenchmark test and application examples demonstrate itseffectiveness and superiority to traditional method Alsothe determination of weights of DMs and attributes usingintuitionistic fuzzy entropy can avoid subjective randomnessand sensitivity analysis shows the stability of the proposedmethod For future work the comparison of the proposedVIKOR method with other MADM methods under intu-itionistic fuzzy environment such as the TOPSIS methodthe PROMETHEE method and the ELECTRE method isworthy of further study and exploration It would also beinteresting to apply the proposed VIKOR method to otherMADM problems such as investment decision and supplierselection

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China under Grant 61273275


References

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[2] S Opricovic Multicriteria Optimization of Civil EngineeringSystems Faculty of Civil Engineering Belgrade Serbia 1998

[3] B Mareschal and J P Vincke ldquoPROMETHEE a new fam-ily of outranking methods in multi criteria analysisrdquo BransJpoperational Research vol 84 pp 477ndash490 1984

[4] R Benayoun B Roy and B Sussman ldquoELECTRE ldquoUnemethode pour guider le choix en presence de points de vuemultiplesrdquo RevFranaise InformatRecherche Operationnelle vol3 pp 31ndash56 1969

[5] S Opricovic and G H Tzeng ldquoCompromise solution byMCDM methods a comparative analysis of VIKOR and TOP-SISrdquo European Journal of Operational Research vol 156 no 2pp 445ndash455 2004

[6] S Opricovic and G Tzeng ldquoExtended VIKORmethod in com-parison with outranking methodsrdquo European Journal of Opera-tional Research vol 178 no 2 pp 514ndash529 2007

[7] L Anojkumar M Ilangkumaran and V Sasirekha ldquoCompara-tive analysis of MCDM methods for pipe material selection insugar industryrdquo Expert Systems with Applications vol 41 no 6pp 2964ndash2980 2014

[8] P P Mohanty and S Mahapatra ldquoA compromise solution byvikormethod for ergonomically designed product with optimalset of design characteristicsrdquo Procedia Materials Science vol 6pp 633ndash640 2014

[9] G Vats and R Vaish ldquoPiezoelectric material selection fortransducers under fuzzy environmentrdquo Journal of AdvancedCeramics vol 2 no 2 pp 141ndash148 2013

[10] K-H Peng and G-H Tzeng ldquoA hybrid dynamic MADMmodel for problem-improvement in economics and businessrdquoTechnological and Economic Development of Economy vol 19no 4 pp 638ndash660 2013

[11] G Akman ldquoEvaluating suppliers to include green supplierdevelopment programs via fuzzy c-means and VIKOR meth-odsrdquo Computers amp Industrial Engineering vol 86 pp 69ndash822015

[12] B Vahdani SMMousavi HHashemiMMousakhani and RTavakkoli-Moghaddam ldquoA new compromise solution methodfor fuzzy group decision-making problems with an applicationto the contractor selectionrdquo Engineering Applications of Artifi-cial Intelligence vol 26 no 2 pp 779ndash788 2013

[13] F Mohammadi M K Sadi F Nateghi A Abdullah and MSkitmore ldquoA hybrid quality function deployment and cyber-netic analytic network processmodel for project manager selec-tionrdquo Journal of Civil Engineering and Management vol 20 no6 pp 795ndash809 2014

[14] W-H Tsai W Hsu and T W Lin ldquoNew financial servicedevelopment for banks in Taiwan based on customer needs andexpectationsrdquo The Service Industries Journal vol 31 no 2 pp215ndash236 2011

[15] M K Sayadi M Heydari and K Shahanaghi ldquoExtension ofVIKOR method for decision making problem with intervalnumbersrdquo Applied Mathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol33 no 5 pp 2257ndash2262 2009

[16] T Kaya and C Kahrarnan ldquoFuzzy multiple criteria forestrydecision making based on an integrated VIKOR and AHPapproachrdquo Expert Systems with Applications vol 38 no 6 pp7326ndash7333 2011

[17] M Ashtiani and M Abdollahi Azgomi ldquoTrust modeling basedon a combination of fuzzy analytic hierarchy process and fuzzyVIKORrdquo Soft Computing vol 20 no 1 pp 399ndash421 2016

[18] H-C Liu L Liu N Liu and L-X Mao ldquoRisk evaluation infailuremode and effects analysis with extendedVIKORmethodunder fuzzy environmentrdquo Expert Systems with Applicationsvol 39 no 17 pp 12926ndash12934 2012

[19] Y Ju and A Wang ldquoExtension of VIKOR method for multi-criteria group decision making problem with linguistic infor-mationrdquoAppliedMathematical Modelling Simulation and Com-putation for Engineering and Environmental Systems vol 37 no5 pp 3112ndash3125 2013

[20] K Devi ldquoExtension of VIKOR method in intuitionistic fuzzyenvironment for robot selectionrdquo Expert Systems with Applica-tions vol 38 no 11 pp 14163ndash14168 2011

[21] S Lu and J Tang ldquoResearch on evaluation of auto parts suppliersby VIKOR method based on intuitionistic language multi-criteriardquoKeyEngineeringMaterials vol 467-469 pp 31ndash35 2011

[22] R Roostaee M Izadikhah F H Lotfi and M Rostamy-Malkhalifeh ldquoA multi-criteria intuitionistic fuzzy group deci-sion making method for supplier selection with vikor methodrdquoInternational Journal of Fuzzy System Applications vol 2 no 1pp 1ndash17 2012

[23] S-P Wan Q-Y Wang and J-Y Dong ldquoThe extended VIKORmethod for multi-attribute group decision making with trian-gular intuitionistic fuzzy numbersrdquo Knowledge-Based Systemsvol 52 pp 65ndash77 2013

[24] J H Park H J Cho and Y C Kwun ldquoExtension of the VIKORmethod to dynamic intuitionistic fuzzy multiple attribute deci-sion makingrdquo Computers amp Mathematics with Applications AnInternational Journal vol 65 no 4 pp 731ndash744 2013

[25] J-P PengW-C Yeh T-C Lai and C-P Hsu ldquoSimilarity-basedmethod for multiresponse optimization problems with intu-itionistic fuzzy setsrdquo Proceedings of the Institution of MechanicalEngineers Part B Journal of Engineering Manufacture vol 227no 6 pp 908ndash916 2013

[26] H Hashemi J Bazargan S M Mousavi and B VahdanildquoAn extended compromise ratio model with an application toreservoir flood control operation under an interval-valued intu-itionistic fuzzy environmentrdquo Applied Mathematical ModellingSimulation andComputation for Engineering and EnvironmentalSystems vol 38 no 14 pp 3495ndash3511 2014

[27] S M Mousavi B Vahdani and S Sadigh Behzadi ldquoDesigninga model of intuitionistic fuzzy VIKOR in multi-attribute groupdecision-making problemsrdquo Iranian Journal of Fuzzy Systemsvol 13 no 1 pp 45ndash65 2016

[28] M Gul E Celik N Aydin A Taskin Gumus and A F GunerildquoA state of the art literature review of VIKOR and its fuzzyextensions on applicationsrdquoApplied Soft Computing vol 46 pp60ndash89 2016

[29] H Liao and Z Xu ldquoA VIKOR-based method for hesitantfuzzy multi-criteria decision makingrdquo Fuzzy Optimization andDecisionMaking A Journal ofModeling andComputationUnderUncertainty vol 12 no 4 pp 373ndash392 2013

[30] Y Li D L Olson and Z Qin ldquoSimilarity measures betweenintuitionistic fuzzy (vague) sets a comparative analysisrdquoPatternRecognition Letters vol 28 no 2 pp 278ndash285 2007

[31] G A Papakostas A G Hatzimichailidis and V G KaburlasosldquoDistance and similarity measures between intuitionistic fuzzysets a comparative analysis from a pattern recognition point ofviewrdquo Pattern Recognition Letters vol 34 no 14 pp 1609ndash16222013


[32] Z S Xu and J Chen ldquoAn overview of distance and similaritymeasures of intuitionistic fuzzy setsrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no4 pp 529ndash555 2008

[33] Z Xu ldquoIntuitionistic fuzzy aggregation operatorsrdquo IEEE Trans-actions on Fuzzy Systems vol 15 no 6 pp 1179ndash1187 2007

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

[35] S M Chen and J M Tan ldquoHandling multicriteria fuzzydecision-making problems based on vague set theoryrdquo FuzzySets and Systems vol 67 no 2 pp 163ndash172 1994

[36] D H Hong and C-H Choi ldquoMulticriteria fuzzy decision-making problems based on vague set theoryrdquo Fuzzy Sets andSystems vol 114 no 1 pp 103ndash113 2000

[37] 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

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

[39] W Q Wang and X L Xin ldquoDistance measure between intu-itionistic fuzzy setsrdquo Pattern Recognition Letters vol 26 no 13pp 2063ndash2069 2005

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

[41] P Grzegorzewski ldquoDistances between intuitionistic fuzzy setsandor interval-valued fuzzy sets based on the Hausdorffmetricrdquo Fuzzy Sets and Systems vol 148 no 2 pp 319ndash3282004

[42] Y Yang and F Chiclana ldquoConsistency of 2D and 3D distances ofintuitionistic fuzzy setsrdquo Expert Systems with Applications vol39 no 10 pp 8665ndash8670 2012

[43] G Beliakov M Pagola and TWilkin ldquoVector valued similaritymeasures for Atanassovrsquos intuitionistic fuzzy setsrdquo InformationSciences vol 280 pp 352ndash367 2014

[44] S-M Chen and C-H Chang ldquoA novel similarity measurebetween Atanassovrsquos intuitionistic fuzzy sets based on trans-formation techniques with applications to pattern recognitionrdquoInformation Sciences vol 291 pp 96ndash114 2015

[45] F E Boran and D Akay ldquoA biparametric similarity measureon intuitionistic fuzzy sets with applications to pattern recog-nitionrdquo Information Sciences vol 255 pp 45ndash57 2014

[46] P Intarapaiboon ldquoA hierarchy-based similarity measure forintuitionistic fuzzy setsrdquo SoftComputing vol 20 no 5 pp 1909ndash1919 2016

[47] Y Song XWang L LeiWQuan andWHuang ldquoAn evidentialview of similarity measure for Atanassovrsquos intuitionistic fuzzysetsrdquo Journal of Intelligent amp Fuzzy Systems Applications inEngineering and Technology vol 31 no 3 pp 1653ndash1668 2016

[48] S Ngan ldquoAn activation detection based similarity measure forintuitionistic fuzzy setsrdquo Expert Systems with Applications vol60 pp 62ndash80 2016

[49] Y Song X Wang L Lei and A Xue ldquoA novel similarity mea-sure on intuitionistic fuzzy sets with its applicationsrdquo AppliedIntelligence vol 42 no 2 pp 252ndash261 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society



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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with uncertainty in actual decision-making process Sayadiet al extended the concept of VIKOR method to solveMADMproblems with interval fuzzy numbers [15] Kaya andKahrarnan developed an integrated VIKOR-AHP methodbased on triangular fuzzy numbers in forestation districtselection scenario [16] Ashtiani and Abdollahi Azgomiconstructed a trust modeling based on a combination offuzzy analytic hierarchy process and VIKOR method withtriangular fuzzy numbers [17] Liu et al applied the VIKORmethod to find a compromise priority ranking of failuremodes according to the risk factors in which the weights ofattributes and ratings of alternatives are linguistic variablesrepresented by trapezoidal or triangular fuzzy numbers [18]Ju and Wang proposed a new method to solve multiattributegroup decision-making (MAGDM) problems based on thetraditional idea of VIKOR method with trapezoidal fuzzynumbers [19] However none of the above-mentioned studiesconsidered hesitancy Due to the increasing complexity anduncertainty of economic environment decision-makersrsquo per-ception of things involves not only affirmation and negationbut also hesitation Therefore hesitation as one of the mostimportant pieces of uncertain information may have greatimpact on the final decision and should be consideredin decision-making process To deal with this Atanassovextended traditional fuzzy sets to intuitionistic fuzzy sets(IFSs) which consider membership nonmembership anddegree of hesitancy at the same time In practical applicationthe use of IFSs can depict the fuzziness and nonspecificityof problems by employing both membership function andnonmembership function Therefore it is considered to bemore effective than classical FSs with merely a member-ship function Recently intuitionistic fuzzy VIKOR (IF-VIKOR) method becomes a popular research topic becauseit aims to deal with the widespread uncertainty in decision-making process and some researchers focus on the IF-VIKOR method to solve MADM problems For instanceDevi [20] first extended VIKOR method in intuitionisticfuzzy environment to deal with robot selection problem inwhich the weights of attributes and ratings of alternatives arerepresented by triangular intuitionistic fuzzy sets (TIFSs) Luand Tang [21] employed the VIKOR method to evaluate autoparts supplier based on IFSs Roostaee et al extended VIKORmethod for group decision-making to solve the supplierselection problem under intuitionistic fuzzy environment[22] Wan et al put forward a new VIKOR method forMADM problem with TIFSs [23] in which the weights ofattributes and DMs are completely unknown Park et alextended VIKOR method for dynamic intuitionistic fuzzyMADM concerning university faculty evaluation problem[24] Based on similarity measure between IFSs Peng et alproposed a novel IF-VIKOR method to deal with multire-sponse optimization problem in which the importance ofeach response is given by an engineer as IFSs [25] Hashemiet al proposed a new compromise method based on classicalVIKOR model with interval-valued intuitionistic fuzzy sets(IVIFSs) illustrated by an application in reservoir floodcontrol operation [26] Mousavi et al presented a new groupdecision-making method for MADM problems based on IF-VIKOR in which the ratings of alternatives concerning each

attribute and the weights of criterions provided by DMs arerepresented as linguistic variables characterized by IFSs withmultijudge [27]

Distance is an important fundamental concept of IFSsand plays a significant role in VIKOR method The classicalcompromise programing is based on the distance measurewhich is determined by the closeness of a specific solutionto the idealinfeasible solution maximizing the group utilityandminimizing the individual regret simultaneously [28 29]Although some existing methods employ intuitionistic fuzzysubtraction and division operator to calculate ldquogroup utilityrdquoand ldquoindividual regretrdquo the result is in IFSs form and needsto be further sorted according to IFSsrsquo ranking rule To someextent the amount of computation is increased and theresults are not concise as the crisp value obtained by distancemeasure Also some researchers [30ndash32] reviewed existingdistance and similarity measures between IFSs and testedthem with an artificial benchmark The result showed thatmany existing distance measures generate counterintuitivecases and fail to capture waver the lack of knowledgewhich should have been an advantage of IFSs On the onehand existing distance measures are shown to have somelimitations with hindered effectiveness while on the otherhand the measure choice is crucial to IF-VIKOR methoddue to its significant influence on the result Therefore thispaperrsquos objective is to construct a new distance measurewhich can capture IFS information effectively Based on thisan IF-VIKOR method is further proposed to solve MADMproblems with completely unknown weights of DMs andattributes The main features and novelties of this paper areas follows

(1) Different from the VIKOR method based on classicalfuzzy sets such as interval fuzzy numbers [15] triangularfuzzy numbers [16ndash18] and trapezoidal fuzzy numbers [1819] the method proposed in this paper is based on IFSswhich adds a nonmembership on the basis of the traditionalmembership and is able to describe support opposition andneutrality in human cognition Such an extended definitionhelps more adequately to represent situations when decisionmaker abstains or hesitates from expressing their assess-ments

(2) Compared with the method [20 27] employingintuitionistic fuzzy operator to calculate ldquogroup utilityrdquo andldquoindividual regretrdquo our method generates crisp value resultin a more intuitive way with less computation AlthoughRoostaee et al [22] extended the VIKOR method to intu-itionistic fuzzy environment based on hamming distancethe method sometimes generates counterintuitive cases andtakes no consideration of waver in IFSs Instead our methodbased on a new distance measure could reflect intuitionisticfuzzy information effectively Also the new distance does notgenerate counterintuitive cases in the artificial benchmarktest proposed by [30]

(3) Compared with the VIKORmethod [20 23 26] basedon those extensions of IFSs IFSs possess sound theoreticalfoundation such as basic definition and operations compar-ison rules information fusion method and measures of IFS







2 Preliminaries





119904 (119886) = (119906119886 minus V119886) (2)








119894=1

(1 minus 119906119886119894)119908119894 119899prod119894=1

V119886119894119908119894) (4)


119864 (119860) = 1119899119899sum119894=1

min (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909)max (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909) (5)









119889119867 (119860 119861) = 12119899sum119894=1


119899sum119894=1


119899sum119894=1


119899sum119894=1


(6)


1198891 (119860 119861) = 1119899sdot 119899sum119894=1



(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)10038161003816100381610038162 )119901

(7)



119889ℎ (119860 119861)= 119899sum119894=1




119899sum119894=1


(8)


119889119890ℎ (119860 119861) = 119899sum119894=1


119899sum119894=1



119890119890ℎ (119860 119861) = radic 119899sum119894=1


119902119890ℎ (119860 119861) = radic 1119899119899sum119894=1

















119889119871 (119860 119861) = 13119899119899sum119894=1


(10)






(11)

It is known that


(12)



0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of










2 Preliminaries





119904 (119886) = (119906119886 minus V119886) (2)








119894=1

(1 minus 119906119886119894)119908119894 119899prod119894=1

V119886119894119908119894) (4)


119864 (119860) = 1119899119899sum119894=1

min (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909)max (119906119860 (119909119894) V119860 (119909)) + 120587119860 (119909) (5)









119889119867 (119860 119861) = 12119899sum119894=1


119899sum119894=1


119899sum119894=1


119899sum119894=1


(6)


1198891 (119860 119861) = 1119899sdot 119899sum119894=1



(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)10038161003816100381610038162 )119901

(7)



119889ℎ (119860 119861)= 119899sum119894=1




119899sum119894=1


(8)


119889119890ℎ (119860 119861) = 119899sum119894=1


119899sum119894=1



119890119890ℎ (119860 119861) = radic 119899sum119894=1


119902119890ℎ (119860 119861) = radic 1119899119899sum119894=1

















119889119871 (119860 119861) = 13119899119899sum119894=1


(10)






(11)

It is known that


(12)



0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









119889119867 (119860 119861) = 12119899sum119894=1


119899sum119894=1


119899sum119894=1


119899sum119894=1


(6)


1198891 (119860 119861) = 1119899sdot 119899sum119894=1



(1003816100381610038161003816119906119860 (119909119894) minus 119906119861 (119909119894)1003816100381610038161003816 + 1003816100381610038161003816V119860 (119909119894) minus V119861 (119909119894)10038161003816100381610038162 )119901

(7)



119889ℎ (119860 119861)= 119899sum119894=1




119899sum119894=1


(8)


119889119890ℎ (119860 119861) = 119899sum119894=1


119899sum119894=1



119890119890ℎ (119860 119861) = radic 119899sum119894=1


119902119890ℎ (119860 119861) = radic 1119899119899sum119894=1

















119889119871 (119860 119861) = 13119899119899sum119894=1


(10)






(11)

It is known that


(12)



0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





119890119890ℎ (119860 119861) = radic 119899sum119894=1


119902119890ℎ (119860 119861) = radic 1119899119899sum119894=1

















119889119871 (119860 119861) = 13119899119899sum119894=1


(10)






(11)

It is known that


(12)



0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









119889119871 (119860 119861) = 13119899119899sum119894=1


(10)






(11)

It is known that


(12)



0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






0 le 1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894) le 3 (13)


0 le 119889119871 (119860 119861) = 13119899119899sum119894=1

[1205791 (119909119894) + 1205792 (119909119894) + 1205793 (119909119894)]le 33 = 1

(14)


119889119871 (119860 119861) = 0119906119860 (119909119894) = 119906119861 (119909119894) V119860 (119909119894) = V119861 (119909119894)

120587119860 (119909119894) + 120587119861 (119909119894) = 0119860 (119909) = 119861 (119909) 120587119860 (119909) = 120587119861 (119909) = 0

(15)




119860 (119909) = (1 0 0) 119861 (119909) = (0 1 0)or 119860 (119909) = (0 1 0) 119861 (119909) = (1 0 0)

(16)



119889119871 (119860 119861) = 13119899119899sum119894=1


119889119871 (119861 119862) = 13119899119899sum119894=1


119889119871 (119860 119862) = 13119899sdot 119899sum119894=1


(17)


It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





It is obvious that



120587119860 (119909119894) + 120587119861 (119909119894) + 120587119861 (119909119894) + 120587119862 (119909119894)2 ge 120587119860 + 1205871198622 (18)



119889119871 (119860 119861) = 13119899119899sum119894=1




119899sum119894=1




(19)


(20)

It means that

119889119871 (119860 119861) le 119889119871 (119860 119862) (21)


119889119871 (119861 119862) le 119889119871 (119860 119862) (22)




Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Test IFSs1 2 3 4 5 6 7119860 = (119906119860 V119860) (1 0) (0 0) (04 02) (03 03) (03 04) (01 02) (04 02)119861 = (119906119861 V119861) (0 0) (05 05) (05 03) (04 04) (04 03) (01 02) (05 02)


Test IFSs1 2 3 4 5 6 7119889119871(119860 119861) 08333 05000 01667 01667 02000 02333 01833119889119899119867(119860 119861) 10000 10000 02000 02000 01000 0 01000119889119899119864(119860 119861) 10000 08660 01732 01732 01000 0 01000119897ℎ(119860 119861) 10000 05000 01000 01000 01000 0 01000119897119890ℎ(119860 119861) 10000 10000 02000 02000 01000 0 010001198891(119860 119861) 07500 05000 01000 01000 01000 0 007501198891198752 (119860 119861) 119901 = 1 05000 05000 01000 01000 01000 0 00500











119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119863(119902) =[[[[[[[[[


119909(119902)1198981 119909(119902)1198982 sdot sdot sdot 119909(119902)119898119899

]]]]]]]]] (23)


119864119902 = 119898sum119894=1

119899sum119895=1

min (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895max (119906(119902)119894119895 V(119902)119894119895 ) + 120587(119902)119894119895 (24)


120582119902 = 1 minus 119864119902119897 minus sum119897119902=1 119864119902 (25)



119863 = [[[[[[[


]]]]]]] (26)


119864119895 = 1119898119898sum119894=1

min (119906119894119895 V119894119895) + 120587119894119895max (119906119894119895 V119894119895) + 120587119894119895 (27)




119909+119895 = max119894=12119898



119909minus119895 = min119894=12119898



(29)


119878119894 = 119899sum119895=1

119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 )) 119877119894 = max

119895119908119895(119889(119909+119895 119909119894119895)119889 (119909+119895 119909minus119895 ))





119876(119860(2)) minus 119876 (119860(1)) ge 1119898 minus 1 (31)















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of

















119863 =[[[[[[[[[

(06689 02193) (08000 01000) (03500 05500) (07081 01857) (05592 03388)(06062 02717) (06495 02331) (06071 02912) (05000 04000) (07562 01383)(05542 03438) (07308 01568) (08355 01223) (08465 01077) (07128 01808)(06495 02331) (06719 02168) (06074 02908) (05546 03434) (05423 03491)

]]]]]]]]] (32)









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









Value A1 A2 A3 A4119878 05470 06657 03911 07175119877 01979 02127 01948 01974119876 03298 09065 00000 05698










Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Value A1 A2 A3 A4119878 03827 04629 03231 05632119877 02135 02204 01973 01996119876 02736 07911 00000 05052


Value A1 A2 A3 A4119878 04495 05896 03295 06569119877 01980 02129 01950 01976119876 02671 08972 00000 05726


C1 C2 C3 C4A1 (01 02) (04 02) (05 02) (02 04)A2 (02 01) (03 03) (03 05) (05 05)A3 (00 03) (03 04) (02 03) (05 02)A4 (04 04) (05 02) (05 02) (02 04)



A1 0 1 03099 2A2 04029 2 07286 3A3 05046 3 10000 4A4 06312 4 0 1










6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









6 Conclusion





Acknowledgments



References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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