approach to multicriteria group decision making with z...

Research ArticleApproach to Multicriteria Group DecisionMaking with Z-Numbers Based on TOPSIS andPower Aggregation Operators

FengWang 1 and JunjunMao23

1School of Mathematics and Statistics Chaohu University Hefei 238000 China2School of Mathematical Science Anhui University Hefei 230601 China3Key Lab of Intelligent Computing amp Signal Processing of Ministry of Education Anhui University Hefei 230039 China

Correspondence should be addressed to Feng Wang wang18256625090163com

Received 8 January 2019 Revised 5 April 2019 Accepted 19 May 2019 Published 29 July 2019

Academic Editor Bonifacio Llamazares

Copyright copy 2019 FengWang and Junjun MaoThis 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

Decisions strongly rely on information so the information must be reliable yet most of the real-world information is impreciseand uncertain The reliability of the information about decision analysis should be measured Z-number which incorporates arestraint of evaluation on investigated objects and the corresponding degree of confidence is considered as a powerful tool tocharacterize this information In this paper we develop a novel approach based on TOPSIS (Technique for Order Preference bySimilarity to Ideal Solution) method and the power aggregation operators for solving the multiple criteria group decision making(MCGDM) problem where the weight information for decision makers (DMs) and criteria is incomplete In the MCGDM theevaluation information made by DMs is represented in the form of linguistic terms and the following calculation is performedusing Z-numbers First we establish an optimization model based on similarity measure to determine the weights of DMs and alinear programming model with partial weight information provided by DMs based on distance measure to determine the weightsof criteria Subsequently decision matrices from all the DMs are aggregated into a comprehensive evaluation matrix utilizingthe proposed ZWAPA operator or ZWGPA operator Then those considered alternatives are ranked in accordance with TOPSISidea and the feature of Z-evaluation Finally a practical example about supplier selection is given to demonstrate the detailedimplementation process of the proposed approach and the feasibility and validity of the approach are verified by comparisons withsome existing approaches

1 Introduction

Since the fuzzy set theory was coined by Zadeh [1] it hasbeen widely used in many application fields as a tool thatis capable of capturing the certainty of information anddepicting peoplersquos subjective thinking especially decision-making problems with uncertain imprecise even incompleteinformation After that the theory has been developingand type-2 fuzzy set [2] fuzzy multiset [3] intuitionisticfuzzy set [4ndash6] hesitant fuzzy set [7 8] and their intervalforms [9ndash11] various fuzzy numbers like triangular fuzzynumber [12] trapezoidal fuzzy number [13] etc appearsuccessively The classic fuzzy set and its generalizations and

extensions successfully address lots of challenges from thecomplexity of practical problems but they do not take wellinto consideration the reliability of the information that isprovided The reliability of the evaluation is an importantaspect in solving the decision making problem where itwill affect the final outcome of the evaluation Z-number anovel fuzzy number with confidence degree is pioneered byZadeh [14] to overcome this limitation Z-number is a 2-tuplefuzzy numbers that includes the restriction of the evaluationand the reliability of the judgment We use the first fuzzynumber to represent the uncertainty of evaluation and thesecond fuzzy number for a measure of certainty or otherrelated concepts such as sureness confidence reliability

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3014387 18 pageshttpsdoiorg10115520193014387

2 Mathematical Problems in Engineering

strength of truth or probability for the first fuzzy numberZ-number has more ability to describe the knowledge ofhuman since it considers the level of confidence of evaluatorsOwing to the inclusion of the reliability factor Z-numbersinvolve more uncertainties than fuzzy numbers They canprovide us with additional degree of freedom to represent theuncertainty and fuzziness of the real situation It provides anewperspective for us in the decisionmaking areaThereforeusing Z-numbers to model and process uncertainty of thereality is more reasonable and applicable than using fuzzynumbers

Traditionally the given evaluations in multiple criteriadecision making (MCDM) are real numbers As the com-plexity of the problem we investigate DMsrsquo opinions areusually expressed only in a fairly vague manner where itcannot be measured numerically Instead they normallyuse terms like ldquopoorrdquo ldquofairrdquo and ldquogoodrdquo to express thejudgment on an alternative which is closer to peoplersquosperception Then the problem of how to compute withterms or words arises It has gained enough attention bysome researchers and they use typically fuzzy number torepresent and compute linguistic terms which is able to seizethe impreciseness of this information and handle subjectivejudgment of human well In recent years linguistic variableswhose values are words or short sentences from naturalor artificial languages have been researched extensively andvarious related fruits [14ndash21] are achieved Because of thesuperiority of Z-number in expressing vague informationutilizing Z-number to characterize the issue with words is anew perspective It is noted that the two components of Z-numbers are mostly described in natural language [14] ForDMs the scale of the provided linguistic words is limitedso it is possible that two alternatives get the same semanticassessment under some criterion but small difference mayexist between them Z-number can distinguish this differencebecause expertsrsquo confidence degrees are different when givingthe criterion values The second component of Z-numberis precisely capable of playing this role At present theresearch involving Z-numbersmainly focuses on the decisionmaking issue with linguistic variables In most ill-defineddecision environment it is preferable for a DM to employlinguistic variables rather than real numbers when makingassessments

In decision making issue the core is to determine theranking order of alternatives Hence choosing an effectivemethod to rank Z-numbers gets very important when con-ducting decision making problem in Z-number informationKang et al [22] devised a method to transform a Z-numberinto a fuzzy number based on the fuzzy expectation of the sec-ond component of Z-number with Centroid Method whichhighlights the importance of the first part in a Z-numbercompared to the second and reduces successfully a Z-numberto a classical fuzzy number Due to that many researchersusually first converted Z-numbers to fuzzy numbers on thebasis of the method provided in [22] and then implementedthe related calculation on the fuzzy numbers that has beenconverted in the course of decision making For exampleauthors in [23] first integrated the preference values given by

DMs to a single value then converted the integrated result (Z-number) to generalized fuzzy number and finally calculatedthe spread and the fuzzy score value of standardized fuzzynumber to attain ranking order of Z-numbers Authors in[24] employed the sameway to transformZ-numbers to fuzzynumbers before ordering Z-numbers the whole course thatrank Z-numbers was analogous to [23] and the researchmade lots of analysis and discussion on different Z-numbercombinations to investigate the effectiveness of the proposedranking approach Kang and his team focused on solvingsupplier selection problem by utilizing the methodologypresented in that paper including two parts one changeda Z-number to a traditional fuzzy number according tothe fuzzy expectation the other worked out the optimalpriority weight for supplier selection with the improvedgenetic algorithm and the extended fuzzy AHP under Z-number environment in [25] In the next few years theydeveloped a new uncertainty measure of fuzzy set con-sidering the influence of fuzziness measure and the range(or cardinality) of the fuzzy set and proposed a methodof generating Z-number based on the OWA weights usingmaximum entropy considering the attitude (preference) ofthe decision maker in [26 27] In 2008 they proposed astable strategies analysis based on the utility of Z-numberand apply it to the evolutionary games in [28] The papertook full advantage of the structure of Z-numbers to simulatethe procedure of humans competition and cooperation Themethodology extends the classical evolutionary games intolinguistic-based ones which is quite applicable in the realsituations Authors in [29] integrated a Jaccard similaritymeasure of Z-numbers to solve MCGDM problem Afterlinguistic values of DMs are converted to Z-numbers themethod used the idea from Kang et al [22] to transform Z-numbers into trapezoidal fuzzy numbers (TpFNs) and thenapplied the Jaccard similarity measure between the expectedintervals of TpFNs and the ideal alternative to attain thefinal decision Authors in [30] put forward a multilayermethodology for ranking Z-numbers which consists of twolayers converting Z-numbers to generalised TpFNs as thefirst layer and using CPSmethod based on centroid point andspread to rank the generalised fuzzy numbers that have beenconverted as the second layerThemethodology is applicableto both positive and negative data values for alternativesand is considered as a generic decision making procedureDifferent from those just mentioned literatures Xiao [31]first converted Z-number to an interval-valued fuzzy set withfootprint of uncertainty and then defuzzified the intervalfuzzy sets as crisp numbers with the well-known K-Malgorithm [32] Authors in [33] converted directly Z-numbersto crisp numbers by performing the multiplication operationon the two parts of Z-numbers (represented by TFNs) whereboth TFNs were defuzzified to crisp numbers via the gradedmean integration and then multiplied into a crisp numberThere is no doubt that the approach seems a bit rough and theinformation of Z-numbers originally included is markedlyreduced

The above references always convert Z-numbers to classi-cal fuzzy numbers or generalized fuzzy numbers in handlingthe linguistic data before performing the real decision This

Mathematical Problems in Engineering 3

way can result in the loss of some valuable Z-informationand may give rise to the unreasonable phenomenon that twodifferent Z-numbers may be converted to the identical fuzzynumber Some researchers have been aware of the drawbackand made some improvements Zamri et al [34] applied thefuzzy TOPSIS to Z-numbers to handle uncertainty in theconstruction problem In that paper the distances of twoparts in Z-evaluations from the PIS and the NIS were calcu-lated respectively without any interaction The structure ofZ-numbers was not heavily damaged in the whole decisionprocess such that less useful information of Z-numbers waslost Peng and Wang [35] developed an innovative methodto address MCGDM problems where the weight informationfor DMs and criteria is incompletely known by introducinghesitant uncertain linguistic Z-numbers (HULZNs) Firstit defined operations and distance of HULZNs by meansof linguistic scale functions then developed two poweraggregation operators for HULZNs and finally incorporatedthe proposed operators and the VIKOR model to solve theERP system selection problemThemethod system combinedthe advantages of Z-numbers and linguisticmodels with deeptheory and rigorous logic Jiang et al [36] gave an improvedmethod for ranking generalized fuzzy numbers which tookinto consideration the weight of centroid points degrees offuzziness and the spreads of fuzzy numbers Subsequentlythis method was extended for ranking Z-numbers and thefinal step of the procedures revised the score function of Z-numbers in view of different status of the two parts of Z-numbers The methodology retains the fuzzy informationof the reliability instead of converting the reliability partof a Z-number to a crisp number as many existing meth-ods did and takes the full advantage that Z-number canexpress more vague information compared to other fuzzynumbers

In general there are not too many researches on Z-numbers so far and relevant achievements are limited Thispaper tries to gain some breakthrough in this aspect on thebasis of the aforementioned references Suppose that there isa collection of Z-numbers 119911119894 = (119860 119894 119877119894)119860 119894 is the main part of119911119894 and 119861119894 is the subsidiary factor 119861119894 can influence the rankingresult but cannot decide the result while 119860 119894 can Thereforethe weight of 119860 119894 should be larger than 119861119894 when processingthe decision making problem with Z-numbers Besides thecharacteristics of Z-number should attempt to be kept andthe information of Z-numbers should be retained as muchas possible In view of these considerations we will do someimproved work and develop another MCGDM approach byapplying the simple and practical TOPSIS method to thedecision information characterized by Z-numbers where theweight information of DMs is entirely unknown and theweight for criteria is partly known The decision procedurespresented in consideration of various data weights manageto hold the previous amount of information With thesepoints the remainder of this work is organized as followsSection 2 briefly reviews some necessary concepts such as Z-number and triangular fuzzy number (TFN) and defines theoperations of Z-numbers whose both parts are expressed byTFNs Section 3 defines the distance of Z-numbers from theperspective of TFNs and shows the properties the distance

measure possesses Section 4 develops two power aggregationoperators by extending the PA operator to Z-numbers anddiscusses comprehensively their properties In Section 5a MCGDM methodology in Z-information is proposedwhich contains two phases the first phase is to determinesuccessively the weight vectors for DMs and criteria byestablishing optimization models under the two circum-stances that the prior information is completely unknownand incompletely unknown respectively the second makesthe concrete decision procedures which combines the poweraggregation operators towards Z-numbers and the TOP-SIS method Section 6 shows the decision-making coursethrough a numerical example about supplier selection andconducts a careful comparison with the existing approachesto illustrate the validity and superiority of the suggestedapproach Finally the conclusions for the main points of thispaper are drawn in Section 7

2 Preliminaries

This section mainly introduces some basic concepts relat-ing to Z-number which will be needed in subsequentsections

Definition 1 (see [14]) A Z-numbers is an ordered pairof fuzzy numbers denoted by 119885 = (119860 119877) which isassociated with a real-valued uncertain variable 119883 Thefirst component of Z-number 119860 is a restriction on thevalues that 119883 can take The second component 119877 is ameasure of reliability of the first component such as con-fidence sureness strength of belief and probability orpossibility

A Z-number gives information about not only uncertainvariable but also the reliability of the information The firstcomponent 119860 is the principal part of a Z-number while thesecond one 119877 is just an explanation for 119860 Typically 119860 and119877 are perception-based values which are described in naturallanguage For example ldquotravel time by car from Berkeley toSan Francisco about 30 min usuallyrdquo ldquodegree of Robertrsquoshonesty high not surerdquo and ldquotomorrowrsquos weather is warmlikelyrdquo For simplicity these natural languages are presentedby linguistic terms that is ldquo30min usuallyrdquo ldquohigh not surerdquoand ldquowarm likelyrdquo

Due to the complexity of real problems many domainexperts give their opinions via fuzzy numbers For instancein a new product launch price forecast one expert may givehis opinion like this the lowest price is 2 dollars the mostpossible price may be 3 dollars and the highest price willnot exceed 4 dollars Thus we can use a triangular fuzzynumber (2 3 4) to express the expertrsquos idea Throughoutthis paper the linguistic variables from the two componentsof Z-numbers 119860 and 119877 are represented by triangular fuzzynumbers Accordingly the triangular fuzzy number is definedbelow

Definition 2 (see [12]) A triangular fuzzy number 119860 can bedenoted by a triplet (1198861 1198862 1198863) whose membership function120583119860(119909) is determined as follows


1a 2a 3aO

1

x

(x)A

Figure 1 Triangular fuzzy number

120583119860 (119909) =

0 119909 isin (minusinfin 1198861)119909 minus 11988611198862 minus 1198861 119909 isin [1198861 1198862]1198863 minus 1199091198863 minus 1198862 119909 isin [1198862 1198863]0 119909 isin (1198863 +infin)(1)

For convenience we signify the triangular fuzzy numberwith TFN for short in what follows The TFN 119860 = (1198861 1198862 1198863)expressed by (1) can be shown in Figure 1

Lacking of typical properties for Z-number forces us tosimplify its representation In order to take advantage of Z-numberrsquos trait and the relation of the two components inZ-numberrsquos structure we introduce Kang et alrsquos conversionthinking [22]

Let a Z-number 119911 = (119860 119877) and convert the reliability 119877of 119885 into a crisp value using Centroid Method

120572 = int119909120583119877 (119909) 119889119909int 120583119877 (119909) 119889119909 (2)

where int denotes an algebraic integrationSince 119877 = (1199031 1199032 1199033) (6) becomes the following equation

owing to the membership function 120583119877(119909) of a TFN120572= int11990321199031 119909 ((119909 minus 1199031) (1199032 minus 1199031)) 119889119909 + int11990331199032 119909 ((1199033 minus 119909) (1199033 minus 1199032)) 119889119909(12) (1199033 minus 1199031) 1199032and 1

(3)

Subsequently take the centroid value 120572 of the reliability119877 as the weight of the restriction 119860 and add the weight fromthe second part 119877 to the first part 119860 Thereby the weightedZ-number can be written as 119911120572 = ⟨119909 120583119860120572(119909)⟩ | 120583119860120572(119909) =120572120583119860(119909) 119909 isin 119883

Because both parts of Z-number are characterized byTFNs in the whole article the partial operations of Z-numbers will follow those of TFNs In light of the structureand meaning of Z-numbers we define some operational lawsconcerning Z-numbers combining with the operations ofTFNs

Definition 3 Assume that 1199111 = ((1198861198971 1198861198981 1198861199061 ) (1199031198971 1199031198981 1199031199061 )) 1199112 =((1198861198972 1198861198982 1198861199062 ) (1199031198972 1199031198982 1199031199062 )) are any two Z-numbers then nor-malize their second components as1199111 = ((1198861198971 1198861198981 1198861199061 ) (11990311989711199031 1199031198981 1199031 1199031199061 1199031))1199112 = ((1198861198972 1198861198982 1198861199062 ) (11990311989721199032 1199031198982 1199032 1199031199062 1199032))

where 1199031 = 1199031198971 + 1199031198981 + 1199031199061 1199032 = 1199031198972 + 1199031198982 + 1199031199062(1) 1199111 + 1199112 = ((12057211198861198971 + 12057221198861198972 12057211198861198981 + 12057221198861198982 12057211198861199061 +12057221198861199062 ) (max11990311989711199031 11990311989721199032max1199031198981 1199031 1199031198982 1199032

max1199031199061 1199031 1199031199062 1199032))(2) 11991111199112 = ((1205721120572211988611989711198861198972 120572112057221198861198981 1198861198982 1205721120572211988611990611198861199062 ) (min1199031198971119903111990311989721199032min1199031198981 1199031 1199031198982 1199032 min1199031199061 1199031 1199031199062 1199032))(3) 1205821199111 = ((12058212057211198861198971 12058212057211198861198981 12058212057211198861199061 ) (11990311989711199031 1199031198981 1199031 1199031199061 1199031))120582 gt 0(4) 1199111205821 = (((12057211198861198971)120582 (12057211198861198981 )120582 (12057211198861199061 )120582) (11990311989711199031 1199031198981 1199031 1199031199061 1199031)) 120582 gt 0According to (2) 120572119894 = int119909120583119877119894(119909)119889119909 int 120583119877119894(119909)119889119909 and 1 119877119894 =(119903119897119894 119903119898119894 119903119906119894 ) 119894 = 1 2In Definition 3 the reliability measure of Z-numbers is

unified in a scale of 0 to 1 in order to be compared easily bynormalizing them In the operations of the first componentsof Z-numbers they are given their respective weights 120572from the degradation value of the corresponding secondcomponents by (2) in order that the supplementary role of theconfidence component to the restriction component in a Z-number is highlighted In Section 4 two power aggregationoperators on Z-numbers are developed and construction ofthe operators will involve the operations of Definition 3 Inthe last application part of this paper we will conduct a largeamount of calculations where the four operation rules abovewill be used frequently

Next we give the ranking rule for Z-numbers used fordecision-making

Definition 4 Assume that 119911119894 = (119860 119894 119877119894) are arbitrary two Z-numbers and 119860 119894 119877119894 are all normal fuzzy numbers 119894 = 1 2

If 1198601 ≻ 1198602 then 1199111 ≻ 1199112If 1198601 = 1198602 thus we have the following(i) 1198771 ≻ 1198772 then 1199111 ≻ 1199112(ii) 1198771 = 1198772 then 1199111 = 1199112For the preference orders of the fuzzy numbers 119860 119894 119877119894(119894 =1 2) there have been varieties of existing methods to deter-

mine them

3 Distance Measure of Z-Numbers

There have been numerous distance formulas for previousfuzzy numbers such as Hamming distance Euclidean dis-tance and Hausdorff distance With the aid of the conceptof cross-entropy in information theory we construct a newdistance formula to measure the discrimination uncertaininformation of TFNs before defining the distance measure ofZ-numbers


Definition 5 Let 119860 = (1198861 1198862 1198863) and 119861 = (1198871 1198872 1198873) be TFNsused for describing the uncertain information of two objectsthen their cross-entropy is defined by

119862119864 (119860 119861) = 13 3sum119894=1 100381610038161003816100381610038161003816100381610038161003816119886119894 log2 119886119894(12) (119886119894 + 119887119894) 100381610038161003816100381610038161003816100381610038161003816 (4)

It is noticeable that (4) is meaningless if 119886119894 = 0 or 119887119894 = 0thereupon we make the following extension

Remark 6 For the cases 119886119894 = 0 and 119887119894 = 0 we reverse theposition of 119886119894 and 119887119894 of (4) that is turn 119886119894 log2(119886119894(12)(119886119894+119887119894))into 119886119894 log2(119887119894(12)(119886119894 + 119887119894)) that is 119886119894 if 119886119894 = 0 and 119887119894 = 0naturally the distance between them should be 0 hence wetake the cross entropy 119886119894 log2(119886119894(12)(119886119894 + 119887119894)) = 0 at thispoint

The cross entropy measure in fuzzy sets is able to dis-criminate effectively different fuzzy information so it can beconsidered as an information distance Despite its asymmetryof (4) it is necessary to symmetrize the cross-entropy so as tobecome a real distance measure as follows119889 (119860 119861) = 12 [119862119864 (119860 119861) + 119862119864 (119861 119860)] (5)

Clearly the distance measure of (5) satisfies the threebasic properties that ordinary distance measure in fuzzyenvironment has (see Theorem 7)

Theorem 7 Let 119860 119861 be two arbitrary TFNs and 119860 =(1198861 1198862 1198863) 119861 = (1198871 1198872 1198873) then(1) 119889(119860 119861) = 119889(119861 119860)(2) 119889(119860 119861) ⩾ 0(3) 119889(119860 119861) = 0 lArrrArr 119860 = 119861

Proof Items (1) and (2) apparently hold so we primarilyprove item (3)119889(119860 119861) = 0 lArrrArr |119886119894 log2(119886119894(12)(119886119894 + 119887119894))| = 0 and|119887119894 log2(119887119894(12)(119886119894 + 119887119894))| = 0 119894 = 1 2 3 Let 119891(119909) =119909 log2(119909(12)(119909 + 119886)) (119886 is a constant) then 119891(119909) is amonotonic function Thus if 119886119894 log2(119886119894(12)(119886119894 + 119887119894)) = 0 or119887119894 log2(119887119894(12)(119886119894+119887119894)) = 0 the solution is uniqueApparentlythe only solution is 119886119894 = 119887119894 119894 = 1 2 3 that is 119860 = 119861

In this paper both components of Z-number are charac-terized by TFNs so the distance between Z-numbers can bebroken down into the distance for the twoTFNsThus we cangive the distance formula for Z-numbers as follows

Definition 8 For two arbitrary Z-numbers 1199111 = (1198601 1198771)1199112 = (1198602 1198772) where 119860 119894 = (119886119897119894 119886119898119894 119886119906119894 ) 119877119894 = (119903119897119894 119903119898119894 119903119906119894 ) 119894 =1 2 then119889 (1199111 1199112) = 13 [119889 (1198601 1198602) + 119889 (1198771 1198772) + 119889 (11986012057211 11986012057222 )] (6)

where 120572119894 = int119909120583119877119894(119909)119889119909 int 120583119877119894(119909)119889119909 and 1 119860120572119894119894 = (120572119894119886119897119894 120572119894119886119898119894 120572119894119886119906119894 ) 119894 = 1 2From (5) and (6) we easily know that 119889(sdot sdot) satisfies the

following properties

If 1199111 1199112 are two Z-numbers then(1) 119889(1199111 1199112) ⩾ 0(2) 119889(1199111 1199112) = 119889(1199112 1199111)(3) 119889(1199111 1199112) = 0 lArrrArr 1199111 = 1199112Through the distance measure between Z-numbers in

Definition 8 we notice that the influence factor of the firstcomponent of Z-number is higher than the second

4 Power Aggregation Operator for Z-Numbers

Power average (PA) operator was pioneered by Yager [37] in2001 As an effective data aggregation tool it allows inputvalues to support each other and provide more versatilityThis section manages to apply PA operator in the situationwhere the input arguments are Z-numbers and derive twoaggregation operators with desirable properties based on thePA operator

Definition 9 PA operator is a mapping R119899 997888rarr R that isgiven by the following 119899-variate function

119875119860 (1198861 1198862 119886119899) = sum119899119894=1 (1 + 119879 (119886119894)) 119886119894sum119899119894=1 (1 + 119879 (119886119894)) (7)

where 119879(119886119894) = sum119899119895=1119895 =119894 sup(119886119894 119886119895) and sup(119886119894 119886119895) is called thesupport for 119886119894 from 119886119895 which satisfies the following threeproperties

(1) sup(119886119894 119886119895) isin [0 1](2) sup(119886119894 119886119895) = sup(119886119895 119886119894)(3) sup(119886119894 119886119895) ⩾ sup(119886119904 119886119905) if |119886119894 minus 119886119895| lt |119886119904 minus 119886119905|Motivated by the PA operator and the geometric average

operator (GA) Xu andYager [38] developed a power geomet-ric (PG) operator as follows

Definition 10 PG operator is a mapping R119899 997888rarr R that isgiven by the following 119899-variate function

119875119866 (1198861 1198862 119886119899) = 119899prod119894=1

119886(1+119879(119886119894))sum119899119894=1(1+119879(119886119894))119894 (8)

where 119879(119886119894) is the same as that of Definition 9

Combining the PA operator and the weighted arithmeticaverage (WAA) operator we can deduce a weighted arith-metic power average operator with Z-numbers ZWAPA

Definition 11 Let 1199111 1199112 119911119899 be a collection of Z-numberslet Ω be the family of all Z-numbers and 119908119894 corresponds tothe weight of 119911119894 with 119908119894 isin [0 1] and sum119899119894=1 119908119894 = 1 then theZWAPAoperator is thismappingΩ119899 997888rarr Ω which is definedas

119885119882119860119875119860 (1199111 1199112 119911119899) = sum119899119894=1 119908119894 (1 + 119879 (119911119894)) 119911119894sum119899119894=1 119908119894 (1 + 119879 (119911119894)) (9)

where 119879(119911119894) = sum119899119895=1119895 =119894 119908119895 sup(119911119894 119911119895) and sup(119911119894 119911119895) is thesupport for 119911119894 from 119911119895 with the following conditions


(1) sup(119911119894 119911119895) ⩾ 0(2) sup(119911119894 119911119895) = sup(119911119895 119911119894)(3) sup(119911119894 119911119895) ⩾ sup(119911119904 119911119905) if 119889(119911119894 119911119895) lt 119889(119911119904 119911119905) where119889(sdot sdot) is the distance measure between Z-numbers

From Definition 11 it can be deduced that the supportmeasure sup(sdot sdot) can be used to measure the proximity ofa preference value in the form of Z-number provided by aDM to another The closer the two preferences 119911119894 and 119911119895 thesmaller the distance of both and the more they support eachother

It is evident that the aggregated result using ZWAPAoperator is still a Z-number Moreover it can be easily provedthat ZWAPA operator has the following properties

Theorem 12 Let 1199111 1199112 119911119899 be a collection of Z-numbers ifsup(119911119894 119911119895) = 119896 for all 119894 = 119895 then

119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)

where 119896 is a constantTheorem 12 implies that when all the supports between Z-

numbers are equal the ZWAPA operator reduces to theWAAoperator

Theorem 13 (commutativity) Let 1199111 1199112 119911119899 be a collectionof Z-numbers and let 11991110158401 11991110158402 1199111015840119899 be any permutation of1199111 1199112 z119899 If the weights1199081 1199082 119908119899 are irrelevant to theposition of arguments then119885119882119860119875119860 (1199111 1199112 119911119899) = 119885119882119860119875119860(11991110158401 11991110158402 1199111015840119899) (11)

Theorem 14 (idempotency) Let 119911119894(119894 = 1 2 119899) be a set ofZ-numbers if 119911119894 = 119911 for all 119894 then119885119882119860119875119860 (1199111 1199112 119911119899) = 119911 (12)

Theorem 15 (boundedness) Let 1199111 1199112 119911119899 be a set of Z-numbers then119911119898119894119899 ⩽ 119885119882119860119875119860 (1199111 1199112 119911119899) ⩽ 119911119898119886119909 (13)

where 119911119898119894119899 = (119860119898119894119899 119877119898119894119899) 119860119898119894119899 = (119886119897119898119894119899 119886119898119898119894119899 119886119906119898119894119899) 119877119898119894119899 =(119903119897119898119894119899 119903119898119898119894119899 119903119906119898119894119899) and 119911119898119886119909 = (119860119898119886119909 119877119898119886119909) 119860119898119886119909 =(119886119897119898119886119909 119886119898119898119886119909 119886119906119898119886119909) 119877119898119886119909 = (119903119897119898119886119909 119903119898119898119886119909 119903119906119898119886119909)Combining PA operator with the weighted geometric

average (WGA) operator we can obtain a weighted geometricpower average operator in Z-numbers case ZWGPA

Definition 16 Let 1199111 1199112 119911119899 be a collection of Z-numberslet Ω be the set of all Z-numbers and 119908119894 is the weight of 119911119894with 119908119894 isin [0 1] and sum119899119894=1 119908119894 = 1 then the ZWGPA operatoris a mappingΩ119899 997888rarr Ω which is expressed as

119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)

where 119879(119911119894) is the same as the counterpart of Definition 11

In compliance with the operations of Z-numbers andDefinition 16 the resulting value determined by ZWGPAoperator is still a Z-number

Being similar to ZWAPA operator ZWGPA operator alsopossesses the following properties


119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)

where 119896 is a constantTheorem 17 indicates that when all the support degrees

are equal the ZWGPA operator reduces to the WGA oper-ator Furthermore ZWGPA operator has also the threeproperties commutativity idempotency and boundedness

Obviously the ZWAPA and ZWGPA operators are twononlinear weighted aggregation tools where the weight119908119894(1 + 119879(119911119894))sum119899119894=1 119908119894(1 + 119879(119911119894)) (not the weight 119908119894 of 119911119894119894 = 1 2 119899) depends upon the input arguments and allowsthese values being aggregated to support and reinforce eachother which are able to retain the input information well andtake fully into account the interrelation among the inputs

5 A MCGDMMethod with Z-Numbers

This section will build a framework for group decisionmaking under Z-number environment that is DMsrsquo eval-uation values are expressed with Z-numbers This processcontains the determination of weights of DMs and criteria byestablishing optimization models and the ranking of all thealternatives by means of the thought of TOPSIS

Before presenting elaborate operation we need to drawthe outline of MCGDM problem Suppose alternatives set119874 = 1199001 1199002 119900119898 criteria set 119862 = 1198881 1198882 119888119899 whoseweight vector is 119908 = (1199081 1199082 119908119899)T with 119908119895 isin [0 1] andsum119899119895=1 119908119895 = 1 and decision-makers set 119863 = 1198891 1198892 119889119902whose weight vector is 120596 = (1205961 1205962 120596119902)T with 120596119896 isin [0 1]andsum119902119896=1 120596119896 = 1 Generally the set119862 is divided into two types119862119887 and 119862119888 where 119862119887 denotes the set of benefit criteria and119862119888 represents the set of cost criteria besides 119862119887 cap 119862119888 = and 119862119887 cup 119862119888 = 119862 Furthermore the evaluation informationof 119900119894(119894 = 1 2 119898) with respect to 119888119895(119895 = 1 2 119899) isdenoted by 119911119896119894119895 It is a Z-number and transformed by thesemantic information of the 119896th DM 119889119896(119896 = 1 2 119902) with119911119896119894119895 = (119860119896119894119895 119877119896119894119895) Eventually the decision matrix 119885119896 = (119911119896119894119895)119898times119899for the DM 119889119896 is produced51 Weight-Determining Method for DMs and Criteria inMCGDM The determination of DMsrsquo weights and criteriais an important research topic in MCGDM because theywill have vital impacts on the final decision Different DMspossess different knowledge backgrounds and professionaldegrees so they cannot be assigned optional weights orgiven equal weight Moreover an alternative or object is


evaluated viamultiple criterion indexes and these criteria playdifferent roles in the final decision Therefore they shouldalso be given different weights When weights informationis completely unknown or partly unknown we need toexcavate fully clues from the decision matrices provided byDMs

In previous studies DMsrsquo weights are subjectively givenon the basis of experience with some subjective randomnessIn this research an objective approach for working outDMsrsquo weights will be adopted Similarity measure a meansthat distinguishes diverse specialization degrees of DMsfor decision-making problems will be utilized under Z-number information If the overall similarity of evaluationvalues in the decision matrix 119885119896 = (119911119896119894119895)119898times119899 given bythe 119896th DM 119889119896 is greater than the overall similarity ofthe decision matrix 119885119897 = (119911119897119894119895)119898times119899 from the 119897th DM 119889119897indicating that 119889119896 provides less inconsistent and conflictingdecision information than 119889119897 among DMs and plays a

relatively important role in decision-making process then119889119896 should be endowed bigger weight than 119889119897 In contrastdecision values a DM provides are figured out as smalleroverall similarity then this DM will be endowed smallerweight

This article does not put forward similarity measure forZ-numbers but distance measure has been suggested Incompliance with the relation between distance and similaritywe can transformdistancemeasure into similaritymeasure byusing 119904 (119911119896119894119895 119911119897119894119895) = max

119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)

According to the analysis above the assignment principle ofDMsrsquo weight vector120596 is tomake the total similarity that DMsrsquoevaluations form reaches the maximum Motivated by Xursquoswork [39] we establish the following programming model toobtain the weights of DMs

(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)

To solve the above model we construct Lagrange function

119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)

where 120585 is a Lagrangemultiplier Let all the partial derivativesof the function 119871 be 0 we have120597119871120596119895 = 1119898119899119902 119902sum

119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)

By solving the above equations we can get

120596119896 = sum119902119897=1sum119899119895=1sum119898119894=1 119904 (119911119896119894119895 119911119897119894119895)radicsum119902119896=1 (sum119902119897=1sum119899119895=1sum119898119894=1 119904 (119911119896119894119895 119911119897119894119895))2 (20)

Then normalize 120596119896 as 120603119896 to ensure that 120603119896 satisfies 0 ⩽120603119896 ⩽ 1 and sum119902119896=1 120603119896 = 1 Now (20) is standardized as

120603119896 = sum119902119897=1sum119899119895=1sum119898119894=1 119904 (119911119896119894119895 119911119897119894119895)sum119902119896=1sum119902119897=1sum119899119895=1sum119898119894=1 119904 (119911119896119894119895 119911119897119894119895) 119896 = 1 2 119902 (21)

It is (21) that is weight coefficient of DMs By inspectingthe formula structure we can find out that the bigger the

overall similarity degree that a DM corresponds to thebigger the weighting value he or she can be endowed Hencethe result decided by (21) accords with the aforementionedrequirement about the weights assignment

When it comes to criterion weights after aggregatingpreference information of all the DMs by utilizing aggre-gation operators we apply the well-known maximizingdeviation method [40] to Z-numbers and still establishan optimization model to acquire weights of criteria fromthe aggregated matrix 119885 = (119911119894119895)119898times119899 If there are markeddifferences between preference values of alternatives undera criterion 119888119895 then 119888119895 has a strong power that distinguishesdistinct alternatives and the criterion is considered relativelyimportant in choosing the best alternative It will accountfor a relatively high weight On the contrary if there aremore similar performance values under one criterion thiscriterion will be assigned a small weight In this paper thedeviation between any two alternativesrsquo preference valuesis measured by the distance measure 119889(sdot sdot) of Z-numbersIn this way the optimal criterion weight distribution 119908shouldmake the total deviation of all alternatives with respectto all criteria maximized However the information aboutcriterion weights is not entirely unknown in many actualcircumstances Evaluators usually have a subjective judgmentor limitation on weight coefficients of criteria but the givenweighting information is often inadequate and imprecise andpartial weights only fall within the range set by them outof the complexity of practical situation In this case we aresupposed to combine the subjective weighting method andthe objective weighting method


Assume that the set of known weighting information isdenoted by 119882 We construct another programming model

with some constraints utilizing the maximizing deviationidea of the reference [39]

(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)

Table 1 Linguistic terms and the corresponding TFNs for restric-tion

Linguistic terms TFNsVery Low (VL) (0 0 1)Low (L) (0 1 3)Medium Low (ML) (1 3 5)Medium (M) (3 5 7)Medium High (MH) (5 7 9)High (H) (7 9 10)Very High (VH) (9 10 10)

Table 2 Linguistic terms and the correspondingTFNs for reliability

Linguistic terms TFNsNot Sure (NS) (0 0 1)Not Very Sure (NVS) (1 3 5)Sure (S) (5 7 9)Very Sure (VS) (9 10 10)

where 119882 is a set of constraint conditions that criterionweights 119908must satisfy in terms of the real situation

Model (M-2) is a linear programming model We canuse the specialized LINGO software to work out the optimalsolution of the model (M-2)

52 Approach to MCGDM with Z-Number Information Inthis research the evaluations made by DMs take the form oflinguistic variables and are represented by TFNsThe expertsare asked to specify ratings for alternatives over evaluationfactors using seven linguistic values varying from ldquoVery Lowrdquoto ldquoVery Highrdquo as restriction part and the linguistic scaleranging from ldquoNot Surerdquo to ldquoVery Surerdquo as reliability partThetransformations between linguistic values and TFNs abouttwo parts of each evaluation are shown in Tables 1 and2 respectively The corresponding two scales of TFNs arepresented graphically in Figures 2 and 3 respectively Byreferring to linguistic variables of Tables 1 and 2 DMs givetheir preference ratings on alternatives under all criteria Eachpreference involves two components the restriction ratingfrom Table 1 and the matching reliability measure using theshort sentences of Table 2

Based on the above-mentioned knowledge preparationand the evaluation information given by DMs for decision

0 1 3 5 7 9 10

VL L ML M MH H VH

Figure 2 Various triangular fuzzy representations for restriction

0 1 3 5 7 9 10

NS NVS S VS

Figure 3 Various triangular fuzzy representations for reliability

problem we integrate the suggested power aggregation oper-ators and the popular TOPSISmethod to develop aMCGDMapproach in Z-number context TOPSIS proposed byHwangand Yoon [41] is a kind ofmethod to solveMADMproblemswhich aims at choosing the alternative with the shortestdistance from the positive ideal solution (PIS) and the farthestdistance from the negative ideal solution (NIS) TOPSIS issimple in the operation and swift in the calculation It is abletomake full use of the information in the original data reduceeffectively the information loss and improve the precisionin the final results In addition it is suitable for both smallsample data and large sampleTherefore themethod is widelyused for tackling the ranking problems in real situations Inthis paper the main procedures are demonstrated in Figure 4including 3 stages and the detailed process is presentedbelow

Step 1 (construct decision matrixes) According to the trans-formation between words and TFNs in Tables 1 and 2 all


Work out the criterion weights by the maximizing deviation method

Normalize all the evaluation data

Aggregate decision matrices from each DM to the comprehensive matrix by the proposed operators

Rank alternatives by the TOPSIS method

Select the optimal alternative

Calculate the weight of every evaluation data associated with support degree and DMs weight

Form MCGDM problem by survey research

Work out the DMs weights based on similarity measure

Determine criteria as evaluation index for alternatives

Organize experts group for decision making

Construct decision matrix of each DMChoose the considered alternatives

Structure decision-making hierarchy

①

Set weight information for criteria according to the practical situation

Evaluate alternatives according to criteria

②

③

Figure 4 The flow diagram of the proposed approach for MCGDM A Decision preparation B Decision implementation C Decisionoperation

the evaluations provided by DMs are converted into Z-numbers and a set of decision matrixes 119885119896 = (119911119896119894119895)119898times119899 areobtained where the preference value 119911119896119894119895 = (119860119896119894119895 119877119896119894119895) denotesthe numerical evaluation (including the restriction 119860119896119894119895 andreliability 119877119896119894119895) of the alternative 119900119894(119894 = 1 2 119898) withrespect to the criterion 119888119895(119895 = 1 2 119899) provided by the 119896thDM 119889119896(119896 = 1 2 119902)Step 2 (normalize the evaluation information) In order toeliminate the influence of different dimensions and the differ-ence in priority that different types of criteria may bring wenormalize all preference values to the same magnitude gradeand make all the research objects comparable Concretelylet 119885119896 = (119911119896119894119895)119898times119899 be normalized as 119885119896 = (119896119894119895)119898times119899 where119896119894119895 = (119860119896119894119895 119877119896119894119895)119860119896119894119895 =

( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged

Step 3 (calculate the supports) Set

sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)

where 119889(119896119894119895 119897119894119895)(119894 = 1 2 119898 119895 = 1 2 119899 119896 119897 =1 2 119902) can be computed by (6) Obviously the supportdegree sup(119896119894119895 119897119894119895) completely satisfies support conditions(1)-(3) listed in Definition 11

Step 4 (calculate the comprehensive support with DMsrsquoweights and weighting coefficient for each performancevalue) Utilize (21) to acquire DMsrsquo weights 1206031 1206032 120603119902and calculate the comprehensive weighted support degree119879(119896119894119895) of 119896119894119895 by the other performance value 119897119894119895(119897 = 119896)

119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)

and compute the weights associated with the performancevalue 119896119894119895 120585119896119894119895 = 120603119896 (1 + 119879 (119896119894119895))sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) (26)

where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1


Step 5 (aggregate the evaluation values for all DMs) Utilizethe ZWAPA operator (9) or ZWGPA operator (14) to aggre-gate all the individual decision matrix 119885119896 = (119896119894119895)119898times119899 as thecollective decision matrix 119885 = (119894119895)119898times119899 That is let 120585119896119894119895 beweight of 119896119894119895 (119896 = 1 2 3) in Step 4 and utilize the WAAor WGA operator to aggregate decision evaluations from allDMs119894119895 = 119885119882119860119875119860(1119894119895 2119894119895 119902119894119895)

= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)

Step 6 (compute the criterion weights) Based on the inte-grated decisionmatrix119885 construct themodel (M-2) in termsof a practical problem and obtain the weight vector 119908 =(1199081 1199082 119908119899)T of criteria by solving the (M-2)

Step 7 (determine the PIS 119900+ and theNIS 119900minus for alternatives onthe integrated matrix 119885) The PIS 119900+ numerically expressedby + here + = +1 +2 +119899 where+119895 = (119860+119895 +119895 ) = (max

119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)

TheNIS 119900minus expressed by minus here minus = minus1 minus2 minus119899 whereminus119895 = (119860minus119895 minus119895 ) = (min119894119860 119894119895min

119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)

Step 8 (compute the relative closeness degree) According tothe distance formula Eq (5) compute the weighted distance119863+119894 between the restraint components119860 119894sdot of the alternative 119900119894and the corresponding PIS 119860+119895 of 119900+

119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)

and the weighted distance 119863minus119894 associated with the alternative119900119894119863minus119894 = 119899sum

119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)

Analogously we can get the weighted distance 119889+119894 and 119889minus119894about the confidence component 119894sdot of the alternative 119900119894 fromthe PIS +119895 and the NIS minus119895 respectively

119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)

and then calculate the relative closeness of the alternative 119900119894to the ideal alternative 119900+ in form of (1198621119894 1198622119894 ) which is a pairof numbers derived from the two components of Z-numbers

(1198621119894 1198622119894 ) = ( 119863minus119894119863+119894 + 119863minus119894 119889minus119894119889+119894 + 119889minus119894 ) 119894 = 1 2 119898 (34)

Step 9 (determine the ranking order of all the alternatives)First we define such a priority relation between any two pairs(1198621120572 1198622120572) and (1198621120573 1198622120573)

If 1198621120572 lt 1198621120573 then (1198621120572 1198622120572) lt (1198621120573 1198622120573)If1198621120572 = 1198621120573 thenwe have the following(i) if1198622120572 = 1198622120573 then(1198621120572 1198622120572) = (1198621120573 1198622120573) (ii) if1198622120572 lt 1198622120573 then (1198621120572 1198622120572) lt (1198621120573 1198622120573)Rank all alternatives 119900119894(119894 = 1 2 119898) through the above

ordering rule on (1198621119894 1198622119894 ) Notice that the first part of each pairof closeness coefficients plays a leading role when comparedwith each other A desirable alternative should be as close tothe PIS and as far from theNIS as possible As a result its twocloseness coefficients in the pair will be as large as possibleTherefore we can use the closeness coefficients to prioritizealternatives and choose the best one The larger the twocloseness degrees the better the corresponding alternativeThe best alternative is the one with the greatest relativecloseness to the ideal solution

6 Application in Supplier Selection

The selection of suppliers is an important issue as com-panyrsquos face in the business management where raw materialsand components represent a significant percentage of thetotal product cost The importance of supplier selection hasincreased also from outsourcing initiatives in which compa-nies rely more on suppliers to improve the quality of theirproducts to reduce their costs or to focus on a specific part oftheir operations Thus supplier selection constitutes a strate-gic decision [42] Numerous companies take this work as theone of key tasks in outside cooperative trade Therefore howto select a satisfying supplier is crucial Recently researchesinvolving this issue have triggered enough attention andhave obtained fruitful achievements In this section we applythe suggested algorithm in this background and make somesimple comparisons and discussion against several existingdecision-making approaches after deriving consequences ofthis problem

This case study is about the supplier selection problemconcerning an automobile manufacturing company Due toits activity the company needs a certain amount of raw


Supplier selection

PriceCost 2cTechnological capability 3c

Partnership 4c

Supplier 2o Supplier 3o Supplier 4o Supplier 5oSupplier 1o

Quality 1cOn-time delivery

5c

Supplier 6o

Figure 5 Hierarchical structure for the supplier selection

Table 3 All evaluations made by 11988911198881 1198882 1198883 1198884 11988851199001 (VLNVS) (MLVS) (MHS) (HNVS) (MHNS)1199002 (LS) (MHNVS) (MNS) (HVS) (MNVS)1199003 (MHNVS) (MLS) (HVS) (MNVS) (VLNVS)1199004 (HS) (HNS) (MHVS) (MLNS) (VHS)1199005 (MNVS) (MVS) (MHNS) (HNS) (VHS)1199006 (MLNS) (MLNS) (HS) (HNVS) (LS)

Table 4 All evaluations made by 11988921198881 1198882 1198883 1198884 11988851199001 (MNVS) (MLNS) (MHS) (MHNVS) (HVS)1199002 (MHS) (MS) (MHNVS) (HNS) (MVS)1199003 (MHNS) (MVS) (HNVS) (MNVS) (MHNVS)1199004 (MHNVS) (LNVS) (VHNS) (MNS) (MLNS)1199005 (MHVS) (VHNS) (MVS) (HNS) (VHNS)1199006 (HNS) (HNVS) (MLNVS) (HS) (MHVS)

materials such as iron wire and tire and has to coordinatewith a number of suppliers Through collecting plenty ofinformation on material producing companies there are sixqualified suppliers being able to supply the required rawmaterials denoted by 1199001 1199002 1199006 To promote stable andhealthy development of themanufacturing company it is verynecessary to use a reasonable algorithm for selecting trust-worthy supplier(s) For this purpose the company invitessome experts and organizes a professional team to evaluateperformances of the potential suppliers by the considerationof various factors The team consists of three experts (DMs)denoted by 1198891 1198892 1198893 They define five criteria to evaluatethese alternatives including 1198881 quality 1198882 pricecost 1198883 tech-nological capability 1198884 partnership and 1198885 on-time deliveryAfter a heated discussion partial information regarding theweights of criteria is provided as119882 = 1199081 ⩾ 1199082 1199081 + 1199082 lt05 1199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082 151199082 ⩾ 1199085 1199083 gt 015 1199084 ⩽1199085 151199084 ⩾ 1199085 like the format SH Kim BS Ahn and

ZS Xu previously mentioned in [43 44] Further each DMgives their respective evaluative ratings of suppliers withrespect to those setting criteria by employing linguistic termsof Table 1 (for restriction) and Table 2 (for reliability) It isnoted that each criterion value given by experts is actually thecombination of two linguistic ratings The entire evaluationsmade by the three DMs are listed in Tables 3ndash5 and thehierarchical structure of this problem is displayed in Figure 5

61 Illustration of the Proposed Approach Now we takeadvantage of the proposed approach and implement its stepsstep by step to make the final choice of suppliers

Step 1 By the conversion relation of words and TFNs inTables 1 and 2 the evaluation information provided by1198891 1198892 1198893 is turned into three numerical decision matrices1198851 = (1199111119894119895)6times5 1198852 = (1199112119894119895)6times5 1198853 = (1199113119894119895)6times5


Table 5 All evaluations made by 11988931198881 1198882 1198883 1198884 11988851199001 (MHS) (MS) (HNVS) (MNVS) (MS)1199002 (HNVS) (MHNVS) (HNS) (MNS) (MLNVS)1199003 (MVS) (MLNS) (HS) (MHS) (MHNS)1199004 (MHVS) (VLNS) (VHS) (MHS) (MNS)1199005 (HNVS) (VHNVS) (MLNS) (HVS) (VHNVS)1199006 (MHS) (HNS) (MLNS) (VHNVS) (HNS)

Step 2 Evidently only 1198882 is a cost-type index among allthe criteria and the rest are benefit index In the light ofnormalization formula Eq (23) let all the evaluation values of1198851 1198852 1198853 become1198851 1198852 1198853 where119885119896 = (119896119894119895)6times5 119896 = 1 2 3Step 3 By (24) compute the support degrees between anytwo normalizedmatrices1198851 1198852 1198853 and obtain three supportmatrices between diverse pairwise decision matrices as fol-lows

sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)

Step 4 Utilize (21) to figure out DMsrsquo weights 1206031 =03358 1206032 = 03284 1206033 = 03358 and by (25) calculate all thecomprehensive support 119879(1119894119895) 119879(2119894119895) 119879(3119894119895)with the weightsFinally figure out the weight 120585119896119894119895 in relation to 119896119894119895 by (26) andget the following three weighting matrixes 120585119896 = (120585119896119894119895)6times5 119896 =1 2 3

1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)

Step 5 Aggregate the evaluation information provided by allthe DMs using ZWGPA operator and get the comprehensiveevaluation matrix 119885 (see Table 6)

Step 6 Use (6) to compute all the distances between Z-numbers of the aggregated matrix 119885 According to the partlyknown criteria information discussed by experts constructthe model (M-2) as


Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)

Solve this model by LINGO software and get the optimalweight vector of criteria 119908 = (031 016 015 016 023)TStep 7 From the collective decision matrix 119885 incompliance with (28) and (29) find the positive andnegative ideal performance values concerning each cri-terion and constitute the PIS + = ((05589 08564 13315)(9 10 10)) ((00155 06866 26278) (9 10 10)) ((0734709311 12207) (9 10 10)) ((07558 10000 13314) (9 1010)) ((08995 10000 11220) (9 10 10)) the NIS minus =((00526 00778 05759) (5 7 9)) ((00042 00397 03967)(1 3 5)) ((01933 04489 07745) (1 3 5)) ((02343 0503308920) (1 3 5)) ((00583 00749 04712) (1 3 5))Step 8 In accordance with (30)ndash(33) calculate respectivelythe distance119863+119894 119863minus119894 119889+119894 119889minus119894 about all the suppliers

119863+1 = 02352119863+2 = 02527119863+3 = 01788119863+4 = 01163119863+5 = 01201119863+6 = 02004119863minus1 = 02335119863minus2 = 01934119863minus3 = 02926119863minus4 = 14011119863minus5 = 04045

119863minus6 = 02332119889+1 = 12388119889+2 = 12271119889+3 = 09303119889+4 = 16459119889+5 = 03902119889+6 = 15102119889minus1 = 28569119889minus2 = 28914119889minus3 = 30674119889minus4 = 22448119889minus5 = 42687119889minus6 = 24489(38)

Then work out the relative closeness pair (1198621119894 1198622119894 ) of thesupplier 119900119894(119894 = 1 2 119898) by (34)(11986211 11986221) = (05003 06975) (11986212 11986222) = (04335 07021) (11986213 11986223) = (06207 07673) (11986214 11986224) = (08234 05770) (11986215C25) = (07711 09162) (11986216 11986226) = (05378 06185)

(39)

Step 9 With all the closeness pairs for suppliers in Step 8 incompliance with the rule used for comparing different two-tuple closenesses that the two large closenesses indicate agood performance for an alternative we easily get the rankingorder of the considered suppliers by the first values of the sixpairs of closeness degrees 1199004 ≻ 1199005 ≻ 1199003 ≻ 1199006 ≻ 1199001 ≻ 1199002 Thusthe most desirable supplier is 1199004 with the largest closenessvalue

In Step 5 if we adopt the ZWAPA operator to integratedifferent DMsrsquo evaluation information and the other stepsremain unchanged until Step 8 the eventual relative closenessvalues of all the suppliers become(11986211 11986221) = (05242 06975) (11986212 11986222) = (04944 07021) (11986213 11986223) = (05513 07673)


(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)

Therefore we have 11986214 gt 11986215 = 11986216 gt 11986213 gt 11986211 gt 11986212 obviouslythe priority order of these suppliers can be all determinedexcept that of 1199115 and 1199116 (in fact the two closenesses are slightdifferent but they are equal when they kept four decimals)Owing to 11986225 gt 11986226 1199005 ≻ 1199006 through the ranking rule foralternatives in Step 9 Then the global priority is 1199004 ≻ 1199005 ≻1199006 ≻ 1199003 ≻ 1199001 ≻ o2 It follows that though the eventual priorityorder for suppliers changes a little (just the position of 1199003 and1199006 is exchanged) the best supplier is still 1199004 which does notaffect the selection of the global suppliers

62 Comparative Analysis to Existing Approaches In orderto verify the feasibility and validity of the recommendedapproach we conduct a comparative analysis with someexisting approaches for the illustrative example describedabove

In the process of decision making with Z-numbers manyresearchers try to defuzzify the preference values expressedby Z-numbers such as [23 24 33]They use classic defuzzifiedmethod to handle two components of Z-numbers because thetwo components are denoted by classic fuzzy numbers andthen convert the whole Z-numbers to crisp numbers or con-struct the score function for each object even average themwhen ordering Z-numbers However these ways of dealingwith Z-numbers inevitably involve the loss and distortion oforiginal information If we use themethod system in [23 24]the order of the suppliers will be 1199002 ≻ 1199005 ≻ 1199006 ≻ 1199001 ≻1199004 ≻ 1199003 if we use the methodology in [33] the priority orderof the suppliers is 1199003 ≻ 1199005 ≻ 1199004 ≻ 1199006 ≻ 1199001 ≻ 1199002 It canbe noted that different defuzzified methods will get differentranking outcomes of alternatives Sometimes it is difficult toidentify their advantages and disadvantages toward specificproblem among these methods For the illustrative examplein this paper we can find that the supplier 1199004 is the bestthrough carefully inspecting evaluation ratings of 6 suppliersprovided by 3 experts while the results derived from thosemethods as in [23 24 33] are not Additionally 1199003 is superiorto 1199001 by comparing their respective preference rating and thematching confidence level but the ranking order in [23 24]is converse Therefore simple defuzzified way to tackle Z-numbers is not very convincing Jiang et al [36] overcomethe drawback of these literature and present an improvedmethod for ranking Z-numbers where they construct a pairof score functions about 119860 and 119877 (suppose 119911 = (119860 119877))instead of transforming 119911 into a fuzzy number by converting119861 to a crisp number The method reduces the loss of usefuldecision information to some extent but it regards 119860 and 119877as two independent components and uses the same meansto tackle the two aspects of information However we knowthat there is a certain relationship between 119860 and 119877 119877 is themeasure of the confidence extent that 119860 is given so the two

parts of Z-numbers cannot be considered separately and theirinterrelationship should be reflected in the decision-makingprocess

Recently Zamri et al [34] andMohamad et al [29] devel-oped a fuzzy TOPSIS method to solve MCGDM problemswith Z-numbers which are relatively close to our approachSo we might as well conduct a comparison in the choice ofthose suppliers In [34] both original evaluation informationand expertsrsquo weights are simply averaged and then the averageevaluation values are endowed the averageweightsThat leadsto a challenge that all the information related to experts failsto be used fully The relative closeness in the last step abinary structure is composed of the closeness coefficientsof two components of Z-numbers to the PIS and the NISrespectively However the way of ranking alternatives is tocalculate and compare the means of the two closeness coeffi-cients in the relative closeness for an alternative As we knowthe first component is more important than the second onein a Z-number Apparently it is unreasonable that the finaloutcomes are obtained by the simple unweighted average ofthe two parts in other words two components of Z-numbersare deemed as equally important Authors in [29] transformdirectly Z-numbers into trapezoidal fuzzy numbers (TpFNs)and use the TpFNs to depict the uncertainty of objects insteadof the corresponding Z-numbers which will lead to thedamage of the feature of Z-numbers More importantly itconverts a TpFN to an expected interval number as a vectorwhen constructing Jaccard similarity measure of two vectorsbut it is not reasonable that interval numbers are seen asvectors So it is not very accurate that the paper utilizes thesimilarity measure to measure the proximity between eachalternative and the ideal one in order to rank the alternativesIf we adopt the decision-making methods in [29 34] in thesupplier selection the final ranking results are 1199003 ≻ 1199001 ≻ 1199004 ≻1199002 ≻ 1199005 ≻ 1199006 and 1199004 ≻ 1199005 ≻ 1199006 ≻ 1199002 ≻ 1199003 ≻ 1199001 respectivelyWe easily notice that the supplier 1199004 is not the most favouredone in the first preference orders and the second ranking issimilar to ours but the poorest supplier should be 1199002 not 1199001Hence our approachhas some advantages over thosemethodsmentioned above at least

(1) In the treatment of Z-numbers we take the differenceof the first component and the second one into accountWhen the distance measure between Z-numbers is definedthe first component plays a larger portion than the secondone because the latter is merely a measure of reliability of theformer When the priority of alternatives is compared thebinary relative closeness degrees play respectively differentrolesThe first number stemmed from the first component ofZ-numbers is the main force

(2) The distance measure under Z-number context thatwe structure is derived from the cross-entropy of informationscience which has higher ability that discriminates differ-ent uncertain information than those conventional distanceforms due to its specific construction In the condition ofhigh uncertainty depicted by Z-numbers the constructeddistance formula can measure relatively accurate discrim-ination measurement Based on this distance measure weutilize the maximizing deviation method to determine thecriteria weights with DMsrsquo subjective information about


criteria and the assignment of DMsrsquo weights is not subjectivebut a programming model is established in light of theprinciple that all the DMsrsquo evaluations on alternatives getthe most similar while DMsrsquo weights are given subjectivelyin many relevant articles Moreover we put forward ZWAPAand ZWGPA operators to aggregate each expert evaluationinformation In the aggregation process all the performancevalues of each alternative are endowed different weights asso-ciated with DMsrsquo weights and the supports which adequatelyconsiders the interaction among various kinds of evaluationinformation and grasp effectively the characteristic of eachalternative under each criterion This is a more versatile dataaggregation tool that allows DMsrsquo evaluation informationsupports and reinforces each other It can reduce the informa-tion loss which always happens in the process of informationaggregation

(3) In the whole process of decision making the struc-ture of Z-numbers is retained well namely the informa-tion denoted by Z-numbers is not defuzzified into ordi-nary fuzzy numbers or crisp numbers which holds theinformation of Z-numbers and keeps the advantage of Z-numbers in describing the uncertainty of things Whenwe take advantage of the TOPSIS method to prioritize Z-numbers the PIS and NIS still keep the feature of Z-numberunchanged so that alternatives can be compared in termsof the ranking rule of Z-structure Those objects that arecompared are not integrated into crisp numbers but twotuples originated from the two components of Z-numbersconsequently the information of Z-numbers included is lesslost which makes the ranking result of alternatives morepersuasive

In addition the classic TOPSISmethod is extended to thesituation with Z-numbers which avoids using the extendedoperators to integrate multiple criterion values for alterna-tives and enhances the operability of the decision-makingalgorithm and thereby reduces the amount of calculationand the loss and distortion of the original information sothat the eventual calculated results are more reliable andconvincing

7 Conclusions

Thereliability of information is an important issue in decisionmaking However the reliability of the knowledge fromexperts is not efficiently taken into consideration in thepast decades After the notion of Z-number was introducedby Zadeh in 2011 a door has been opened to handle thisissue Z-number has more capability of describing uncertaininformation with both restraint and reliability which ispaid much attention recently This paper puts forward amethodology forMCGDMwhere all the decision evaluationsare expressed by Z-numbers First the four operations aboutZ-numbers are defined according to the trait of Z-numbersand the operational laws of TFNs In view of different status ofthe two components of a Z-number we transform the secondcomponent into a real number through Centroid Method asthe weight of the first component Consequently the distancebetween Z-numbers includes two parts the distance between

the first components and the distance between the weightedfirst components Next the efficient PA operator is general-ized to integrate the information depicted by Z-numbers Asa result the ZWAPA operator and the ZWGPA operator arepresented and they are proved to be the analogous propertiesto the PAoperator and the PGoperator respectively Based onthese theoretical preparations a MCGDMmethod can be setup consisting mainly of two stages weight-determining stageand decision procedure-conducting stageThe determinationof weights is divided into two parts one for DMs and theother for criteria Being different from most literature DMsrsquoweights are not given subjectively but an objective methodis established in this article based on the consensus that theallocation of expertsrsquo weights shouldmake the evaluations onalternatives they make be as similar as possible The weightsof criteria are gained based on the thought that the weight ofa criterion can maximize the difference of evaluation valuesunder this criterion Through the two principles we canbuild two programming models to obtain the correspondingweight vectors In the second stage we formally give decisionprocedures inspired by PA aggregation thought in [37] andTOPSIS theory in [41] towards the MCGDM problem withZ-numbers When the evaluation information from differentexperts is aggregated the weight of each evaluation valueis constructed based on the support measure in which themutual support of these evaluation values is taken intoaccount sufficiently When ranking alternatives (the perfor-mances of these alternatives are described by Z-numbers) wedonot distort the basic structure of Z-numbers namely everyZ-number is not converted to a crisp value whether it wasthe PISs andNISs or the final relative closenessesMeanwhilewe also give the rule that uses the binary closeness degrees todetermine the priority of alternatives in consideration of thedifferent importance of the two components in a Z-numberLast this article shows a course to solve one applicationexample step by step according to the proposed operationprocedures and verifies the practicability and availability ofour method system by comparing it against other methods indepth

Data Availability

The text data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors are highly grateful to any anonymous reviewerfor their careful reading and insightful comments andtheir constructive opinions and suggestions will generate animproved version of this article The work is partly sup-ported by the National Natural Science Foundation of China(No 61673020) the Provincial Nature Science Research KeyProject for Colleges and Universities of Anhui Province


(No KJ2013A033) and the Talent Support Key Projectfor Outstanding Young of Colleges and Universities (NogxyqZD2016453)

References

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

[2] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo Information Sciencesvol 8 pp 199ndash249 1975

[3] R R Yager ldquoOn the theory of bagsrdquo International Journal ofGeneral Systems Methodology Applications Education vol 13no 1 pp 23ndash37 1987

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

[5] K T Atanassov ldquoMore on intuitionistic fuzzy setsrdquo Fuzzy Setsand Systems vol 33 no 1 pp 37ndash45 1989

[6] L Fei H Wang L Chen et al ldquoA new vector valued similaritymeasure for intuitionistic fuzzy sets based on OWA operatorsrdquoIranian Journal of Fuzzy Systems vol 16 no 3 pp 113ndash126 2019

[7] V Torra andYNarukawa ldquoOn hesitant fuzzy sets and decisionrdquoin Proceedings of the IEEE International Conference on FuzzySystems pp 1378ndash1382 Jeju-do Republic of Korea August 2009

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

[9] D Wu and J M Mendel ldquoUncertainty measures for intervaltype-2 fuzzy setsrdquo Information Sciences vol 177 no 23 pp5378ndash5393 2007

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

[11] J-Q Wang J-T Wu H-Y Zhang and X-H Chen ldquoInterval-valued hesitant fuzzy linguistic sets and their applicationsin multi-criteria decision-making problemsrdquo Information Sci-ences vol 288 pp 55ndash72 2014

[12] D Dubois and H Prade ldquoOperations on fuzzy numbersrdquoInternational Journal of Systems Science vol 9 no 6 pp 613ndash626 1978

[13] S Abbasbandy and T Hajjari ldquoA new approach for rankingof trapezoidal fuzzy numbersrdquo Computers amp Mathematics withApplications vol 57 no 3 pp 413ndash419 2009

[14] L A Zadeh ldquoA note on Z-numbersrdquo Information Sciences vol181 no 14 pp 2923ndash2932 2011

[15] Y-H Chen T-C Wang and C-Y Wu ldquoMulti-criteria decisionmaking with fuzzy linguistic preference relationsrdquo AppliedMathematical Modelling vol 35 no 3 pp 1322ndash1330 2011

[16] R M Rodrıguez A Labella and L Martınez ldquoAn overview onfuzzy modelling of complex linguistic preferences in decisionmakingrdquo International Journal of Computational IntelligenceSystems vol 9 pp 81ndash94 2016

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

[18] Z S Xu ldquoUncertain linguistic aggregation operators basedapproach to multiple attribute group decision making underuncertain linguistic environmentrdquo Information Sciences vol168 no 1ndash4 pp 171ndash184 2004

[19] G W Wei X F Zhao R Lin and H J Wang ldquoUncertainlinguistic Bonferroni mean operators and their application

to multiple attribute decision makingrdquo Applied MathematicalModelling vol 37 no 7 pp 5277ndash5285 2013

[20] J H Park M G Gwak and Y C Kwun ldquoUncertain linguisticharmonic mean operators and their applications to multipleattribute group decision makingrdquo Computing vol 93 no 1 pp47ndash64 2011

[21] Z-P Tian JWang J-QWang andH-Y Zhang ldquoA likelihood-based qualitative flexible approachwith hesitant fuzzy linguisticinformationrdquo Cognitive Computation vol 8 no 4 pp 670ndash6832016

[22] B Y Kang D Wei Y Li and Y Deng ldquoA method of convertingZ-number to classical fuzzy numberrdquo Journal of Information andComputational Science vol 9 no 3 pp 703ndash709 2012

[23] D Mohamada S A Shaharanib and N H Kamisc ldquoA Z-number-based decision making procedure with ranking fuzzynumbersrdquo in Proceedings of the International Conference onQuantitative Sciences and Its Applications pp 160ndash166 2014

[24] D Mohamada S A Shaharanib and N H Kamisc ldquoOrderingof Z-numbersrdquo in Proceedings of the 24th National Symposiumon Mathematical Sciences pp 1ndash6 2017

[25] B Y Kang Y Hu Y Deng and D Zhou ldquoA new methodologyof multicriteria decision-making in supplier selection based onZ-Numbersrdquo Mathematical Problems in Engineering vol 2016Article ID 8475987 17 pages 2016

[26] B Y Kang Y Deng K Hewage et al ldquoA method of measuringuncertainty for Z-numberrdquo IEEETransactions on Fuzzy Systemsvol 27 pp 731ndash738 2019

[27] B Y Kang Y Deng K Hewage and R Sadiq ldquoGeneratingZ-number based on OWA weights using maximum entropyrdquoInternational Journal of Intelligent Systems vol 33 no 8 pp1745ndash1755 2018

[28] B Y Kang G Chhipi-Shrestha Y Deng K Hewage andR Sadiq ldquoStable strategies analysis based on the utility ofZ-number in the evolutionary gamesrdquo Applied MathematicsComputation vol 324 pp 202ndash217 2018

[29] D Mohamad and S Z Ibrahim ldquoDecision making procedurebased on jaccard similarity measure with Z-numbersrdquo Per-tanika Journal of Science amp Technology vol 25 no 2 pp 561ndash574 2017

[30] A S Abu Bakar and A Gegov ldquoMulti-layer decision methodol-ogy for ranking Z-numbersrdquo International Journal of Computa-tional Intelligence Systems vol 8 no 2 pp 395ndash406 2015

[31] Z-QXiao ldquoApplication of Z-numbers inmulti-criteria decisionmakingrdquo in Proceedings of the International Conference onInformative and Cybernetics for Computational Social Systems(ICCSS rsquo14) pp 91ndash95 IEEE Qingdao China October 2014

[32] N N Karnik and J M Mendel ldquoCentroid of a type-2 fuzzy setrdquoInformation Sciences vol 132 no 1ndash4 pp 195ndash220 2001

[33] B Y Kang D Wei Y Li and Y Deng ldquoDecision makingusing Z-numbers under uncertain environmentrdquo Journal ofComputational Information Systems vol 8 no 7 pp 2807ndash28142012

[34] N Zamri F Ahmad A N M Rose et al ldquoA fuzzy TOPSISwith Z-numbers approach for evaluation on accident at theconstruction siterdquo in Recent Advances on Soft Computing andData Mining pp 41ndash50 2017

[35] H G Peng and J Q Wang ldquoHesitant uncertain linguistic Z-numbers and their application in multi-criteria group decision-making problemsrdquo International Journal of Fuzzy Systems vol19 pp 1300ndash1316 2017


[36] W Jiang C Xie Y Luo and Y Tang ldquoRanking Z-numbers withan improved ranking method for generalized fuzzy numbersrdquoJournal of Intelligent amp Fuzzy Systems Applications in Engineer-ing and Technology vol 32 no 3 pp 1931ndash1943 2017

[37] R R Yager ldquoThe power average operatorrdquo IEEE Transactions onSystems Man and Cybernetics Systems vol 31 no 6 pp 724ndash731 2001

[38] Z S Xu and R R Yager ldquoPower-geometric operators and theiruse in group decision makingrdquo IEEE Transactions on FuzzySystems vol 18 no 1 pp 94ndash105 2010

[39] Z S XuUncertainMultiple Attribute DecisionMakingMethodsand Applications Springer Beijing China 2004

[40] Y M Wang ldquoUsing the method of maximizing deviations tomake decision formulti-indicesrdquo Journal of Systems Engineeringand Electronics vol 7 pp 24ndash26 1998

[41] C L Hwang and K Yoon Multiple Attribute Decision MakingMethods and Applications vol 186 Springer Berlin Germany1981

[42] S Hamdan and A Cheaitou ldquoSupplier selection and orderallocation with green criteria an MCDM and multi-objectiveoptimization approachrdquo Computers amp Operations Research vol81 pp 282ndash304 2017

[43] S H Kim and B S Ahn ldquoInteractive group decision makingprocedure under incomplete informationrdquo European Journal ofOperational Research vol 116 no 3 pp 498ndash507 1999

[44] Z S Xu ldquoIntuitionistic preference relations and their applica-tion in group decision makingrdquo Information Sciences vol 177no 11 pp 2363ndash2379 2007

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strength of truth or probability for the first fuzzy numberZ-number has more ability to describe the knowledge ofhuman since it considers the level of confidence of evaluatorsOwing to the inclusion of the reliability factor Z-numbersinvolve more uncertainties than fuzzy numbers They canprovide us with additional degree of freedom to represent theuncertainty and fuzziness of the real situation It provides anewperspective for us in the decisionmaking areaThereforeusing Z-numbers to model and process uncertainty of thereality is more reasonable and applicable than using fuzzynumbers

Traditionally the given evaluations in multiple criteriadecision making (MCDM) are real numbers As the com-plexity of the problem we investigate DMsrsquo opinions areusually expressed only in a fairly vague manner where itcannot be measured numerically Instead they normallyuse terms like ldquopoorrdquo ldquofairrdquo and ldquogoodrdquo to express thejudgment on an alternative which is closer to peoplersquosperception Then the problem of how to compute withterms or words arises It has gained enough attention bysome researchers and they use typically fuzzy number torepresent and compute linguistic terms which is able to seizethe impreciseness of this information and handle subjectivejudgment of human well In recent years linguistic variableswhose values are words or short sentences from naturalor artificial languages have been researched extensively andvarious related fruits [14ndash21] are achieved Because of thesuperiority of Z-number in expressing vague informationutilizing Z-number to characterize the issue with words is anew perspective It is noted that the two components of Z-numbers are mostly described in natural language [14] ForDMs the scale of the provided linguistic words is limitedso it is possible that two alternatives get the same semanticassessment under some criterion but small difference mayexist between them Z-number can distinguish this differencebecause expertsrsquo confidence degrees are different when givingthe criterion values The second component of Z-numberis precisely capable of playing this role At present theresearch involving Z-numbersmainly focuses on the decisionmaking issue with linguistic variables In most ill-defineddecision environment it is preferable for a DM to employlinguistic variables rather than real numbers when makingassessments

In decision making issue the core is to determine theranking order of alternatives Hence choosing an effectivemethod to rank Z-numbers gets very important when con-ducting decision making problem in Z-number informationKang et al [22] devised a method to transform a Z-numberinto a fuzzy number based on the fuzzy expectation of the sec-ond component of Z-number with Centroid Method whichhighlights the importance of the first part in a Z-numbercompared to the second and reduces successfully a Z-numberto a classical fuzzy number Due to that many researchersusually first converted Z-numbers to fuzzy numbers on thebasis of the method provided in [22] and then implementedthe related calculation on the fuzzy numbers that has beenconverted in the course of decision making For exampleauthors in [23] first integrated the preference values given by

DMs to a single value then converted the integrated result (Z-number) to generalized fuzzy number and finally calculatedthe spread and the fuzzy score value of standardized fuzzynumber to attain ranking order of Z-numbers Authors in[24] employed the sameway to transformZ-numbers to fuzzynumbers before ordering Z-numbers the whole course thatrank Z-numbers was analogous to [23] and the researchmade lots of analysis and discussion on different Z-numbercombinations to investigate the effectiveness of the proposedranking approach Kang and his team focused on solvingsupplier selection problem by utilizing the methodologypresented in that paper including two parts one changeda Z-number to a traditional fuzzy number according tothe fuzzy expectation the other worked out the optimalpriority weight for supplier selection with the improvedgenetic algorithm and the extended fuzzy AHP under Z-number environment in [25] In the next few years theydeveloped a new uncertainty measure of fuzzy set con-sidering the influence of fuzziness measure and the range(or cardinality) of the fuzzy set and proposed a methodof generating Z-number based on the OWA weights usingmaximum entropy considering the attitude (preference) ofthe decision maker in [26 27] In 2008 they proposed astable strategies analysis based on the utility of Z-numberand apply it to the evolutionary games in [28] The papertook full advantage of the structure of Z-numbers to simulatethe procedure of humans competition and cooperation Themethodology extends the classical evolutionary games intolinguistic-based ones which is quite applicable in the realsituations Authors in [29] integrated a Jaccard similaritymeasure of Z-numbers to solve MCGDM problem Afterlinguistic values of DMs are converted to Z-numbers themethod used the idea from Kang et al [22] to transform Z-numbers into trapezoidal fuzzy numbers (TpFNs) and thenapplied the Jaccard similarity measure between the expectedintervals of TpFNs and the ideal alternative to attain thefinal decision Authors in [30] put forward a multilayermethodology for ranking Z-numbers which consists of twolayers converting Z-numbers to generalised TpFNs as thefirst layer and using CPSmethod based on centroid point andspread to rank the generalised fuzzy numbers that have beenconverted as the second layerThemethodology is applicableto both positive and negative data values for alternativesand is considered as a generic decision making procedureDifferent from those just mentioned literatures Xiao [31]first converted Z-number to an interval-valued fuzzy set withfootprint of uncertainty and then defuzzified the intervalfuzzy sets as crisp numbers with the well-known K-Malgorithm [32] Authors in [33] converted directly Z-numbersto crisp numbers by performing the multiplication operationon the two parts of Z-numbers (represented by TFNs) whereboth TFNs were defuzzified to crisp numbers via the gradedmean integration and then multiplied into a crisp numberThere is no doubt that the approach seems a bit rough and theinformation of Z-numbers originally included is markedlyreduced

The above references always convert Z-numbers to classi-cal fuzzy numbers or generalized fuzzy numbers in handlingthe linguistic data before performing the real decision This





2 Preliminaries







1a 2a 3aO

1

x

(x)A


120583119860 (119909) =





120572 = int119909120583119877 (119909) 119889119909int 120583119877 (119909) 119889119909 (2)



(3)




where 1199031 = 1199031198971 + 1199031198981 + 1199031199061 1199032 = 1199031198972 + 1199031198982 + 1199031199062(1) 1199111 + 1199112 = ((12057211198861198971 + 12057221198861198972 12057211198861198981 + 12057221198861198982 12057211198861199061 +12057221198861199062 ) (max11990311989711199031 11990311989721199032max1199031198981 1199031 1199031198982 1199032






mine them





119862119864 (119860 119861) = 13 3sum119894=1 100381610038161003816100381610038161003816100381610038161003816119886119894 log2 119886119894(12) (119886119894 + 119887119894) 100381610038161003816100381610038161003816100381610038161003816 (4)
















119875119860 (1198861 1198862 119886119899) = sum119899119894=1 (1 + 119879 (119886119894)) 119886119894sum119899119894=1 (1 + 119879 (119886119894)) (7)





119875119866 (1198861 1198862 119886119899) = 119899prod119894=1

119886(1+119879(119886119894))sum119899119894=1(1+119879(119886119894))119894 (8)




119885119882119860119875119860 (1199111 1199112 119911119899) = sum119899119894=1 119908119894 (1 + 119879 (119911119894)) 119911119894sum119899119894=1 119908119894 (1 + 119879 (119911119894)) (9)







119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)









119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)





119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)












119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018












2 Preliminaries







1a 2a 3aO

1

x

(x)A


120583119860 (119909) =





120572 = int119909120583119877 (119909) 119889119909int 120583119877 (119909) 119889119909 (2)



(3)




where 1199031 = 1199031198971 + 1199031198981 + 1199031199061 1199032 = 1199031198972 + 1199031198982 + 1199031199062(1) 1199111 + 1199112 = ((12057211198861198971 + 12057221198861198972 12057211198861198981 + 12057221198861198982 12057211198861199061 +12057221198861199062 ) (max11990311989711199031 11990311989721199032max1199031198981 1199031 1199031198982 1199032






mine them





119862119864 (119860 119861) = 13 3sum119894=1 100381610038161003816100381610038161003816100381610038161003816119886119894 log2 119886119894(12) (119886119894 + 119887119894) 100381610038161003816100381610038161003816100381610038161003816 (4)
















119875119860 (1198861 1198862 119886119899) = sum119899119894=1 (1 + 119879 (119886119894)) 119886119894sum119899119894=1 (1 + 119879 (119886119894)) (7)





119875119866 (1198861 1198862 119886119899) = 119899prod119894=1

119886(1+119879(119886119894))sum119899119894=1(1+119879(119886119894))119894 (8)




119885119882119860119875119860 (1199111 1199112 119911119899) = sum119899119894=1 119908119894 (1 + 119879 (119911119894)) 119911119894sum119899119894=1 119908119894 (1 + 119879 (119911119894)) (9)







119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)









119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)





119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)












119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018









1a 2a 3aO

1

x

(x)A


120583119860 (119909) =





120572 = int119909120583119877 (119909) 119889119909int 120583119877 (119909) 119889119909 (2)



(3)




where 1199031 = 1199031198971 + 1199031198981 + 1199031199061 1199032 = 1199031198972 + 1199031198982 + 1199031199062(1) 1199111 + 1199112 = ((12057211198861198971 + 12057221198861198972 12057211198861198981 + 12057221198861198982 12057211198861199061 +12057221198861199062 ) (max11990311989711199031 11990311989721199032max1199031198981 1199031 1199031198982 1199032






mine them





119862119864 (119860 119861) = 13 3sum119894=1 100381610038161003816100381610038161003816100381610038161003816119886119894 log2 119886119894(12) (119886119894 + 119887119894) 100381610038161003816100381610038161003816100381610038161003816 (4)
















119875119860 (1198861 1198862 119886119899) = sum119899119894=1 (1 + 119879 (119886119894)) 119886119894sum119899119894=1 (1 + 119879 (119886119894)) (7)





119875119866 (1198861 1198862 119886119899) = 119899prod119894=1

119886(1+119879(119886119894))sum119899119894=1(1+119879(119886119894))119894 (8)




119885119882119860119875119860 (1199111 1199112 119911119899) = sum119899119894=1 119908119894 (1 + 119879 (119911119894)) 119911119894sum119899119894=1 119908119894 (1 + 119879 (119911119894)) (9)







119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)









119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)





119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)












119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018










119862119864 (119860 119861) = 13 3sum119894=1 100381610038161003816100381610038161003816100381610038161003816119886119894 log2 119886119894(12) (119886119894 + 119887119894) 100381610038161003816100381610038161003816100381610038161003816 (4)
















119875119860 (1198861 1198862 119886119899) = sum119899119894=1 (1 + 119879 (119886119894)) 119886119894sum119899119894=1 (1 + 119879 (119886119894)) (7)





119875119866 (1198861 1198862 119886119899) = 119899prod119894=1

119886(1+119879(119886119894))sum119899119894=1(1+119879(119886119894))119894 (8)




119885119882119860119875119860 (1199111 1199112 119911119899) = sum119899119894=1 119908119894 (1 + 119879 (119911119894)) 119911119894sum119899119894=1 119908119894 (1 + 119879 (119911119894)) (9)







119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)









119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)





119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)












119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018













119885119882119860119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119908119894119911119894 (10)









119885119882119866119875119860 (1199111 1199112 119911119899) = 119899prod119894=1

119911119908119894(1+119879(119911119894))sum119899119894=1 119908119894(1+119879(119911119894))119894 (14)





119885119882119866119875119860 (1199111 1199112 119911119899) = 119899sum119894=1

119911119908119894119894 (15)












119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018













119894119895119896119897119889 (119911119896119894119895 119911119897119894119895) minus 119889 (119911119896119894119895 119911119897119894119895) (16)


(M-1)max 119878 (120596) = 119902sum

119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896st 120596119896 ⩾ 0 119902sum

119896=1

1205962119896 = 1 119896 = 1 2 119902 (17)


119871 (120596 120585) = 119902sum119896=1

( 1119898119899119902 119902sum119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895))120596119896+ 1205852 ( 119902sum

119896=1

1205962119896 minus 1) (18)


119897=1

119899sum119895=1

119898sum119894=1

119904 (119911119896119894119895 119911119897119894119895) + 120585120596119896 = 0120597119871120585 = 12 ( 119902sum

119896=1

1205962119896 minus 1) = 0 (19)











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018











(M-2)max119863 (119908) = 119899sum

119895=1

119908119895 119898sum119894=1

119898sum119897=1

119889 (119911119894119895 119911119897119895)st 119908 isin 119882 119908119895 ⩾ 0 119899sum

119895=1

119908119895 = 1 119895 = 1 2 119899 (22)









0 1 3 5 7 9 10

VL L ML M MH H VH


0 1 3 5 7 9 10

NS NVS S VS

















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018





















①



②

③



( 119886119896119897119894119895max119894 119886119896119906119894119895 119886119896119898119894119895

max119894 119886119896119898119894119895 119886119896119906119894119895max119894 119886119896119897119894119895 ) 119888119895 isin 119862119887

(min119894 119886119896119897119894119895119886119896119906119894119895 min119894 119886119896119898119894119895119886119896119898119894119895 min119894 119886119896119906119894119895119886119896119897119894119895 ) 119888119895 isin 119862119888 (23)

and 119877119896119894119895 is unchanged


sup (119896119894119895 119897119894119895) = max119894119895119896119897

119889 (119896119894119895 119897119894119895) minus 119889 (119896119894119895 119897119894119895) (24)



119879 (119896119894119895) = 119902sum119897=1119897 =119896

120603119897 sup (119896119894119895 119897119894119895) 119896 = 1 2 119902 (25)


where 120585119896119894119895 ⩾ 0sum119902119896=1 120585119896119894119895 = 1



= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018










= sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) 119896119894119895sum119902119896=1 120603119896 (1 + 119879 (119896119894119895)) = 119902sum119896=1

120585119896119894119895119896119894119895119894119895 = 119885119882119866119875119860(1119894119895 2119894119895 119902119894119895)= 119902prod119896=1

(119896119894119895)120603119896(1+119879(119896119894119895))sum119902119896=1 120603119896(1+119879(119896119894119895)) = 119902prod119896=1

(119896119894119895)120585119896119894119895(27)



119894119860 119894119895max

119894119894119895)

= ((max119894119886119897119894119895max119894119886119898119894119895 max

119894119886119906119894119895)

(max119894119903119897119894119895max119894119903119898119894119895 max119894119903119906119894119895)) 119895 = 1 2 119899

(28)


119894119894119895)

= ((min119894119886119897119894119895min119894119886119898119894119895 min119894119886119906119894119895)

(min119894119903119897119894119895min119894119903119898119894119895 min119894119903119906119894119895)) 119895 = 1 2 119899

(29)


119863+119894 = 119899sum119895=1

119908119895119889 (119860 119894119895 119860+119895) 119894 = 1 2 119898 (30)


119895=1

119908119895119889 (119860 119894119895 119860minus119895) 119894 = 1 2 119898 (31)


119889+119894 = 119899sum119895=1

119908119895119889 (119894119895 +119895 ) 119894 = 1 2 119898 (32)

119889minus119894 = 119899sum119895=1

119908119895119889 (119894119895 minus119895 ) 119894 = 1 2 119898 (33)










Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018









Supplier selection


Partnership 4c



5c

Supplier 6o











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018











sup (1198851 1198852)

=(((((((

09938 04572 09965 09974 0677209920 09834 07083 05212 0989006154 09529 09883 10000 0995309832 02837 05704 08585 0846209753 09124 06135 10000 0545406042 00677 09935 09978 09764

)))))))

sup (1198852 1198853)

=((((((

09964 04932 09919 09963 0993109972 09965 05839 06774 0996306053 08599 09998 09940 0671109968 04452 05305 07129 0840909967 09257 08483 05275 0536605244 09624 08527 09971 05846))))))

sup (1198851 1198853)

=((((((

09936 09811 09976 09931 0662409924 09953 08260 07447 0996209963 08383 09855 09988 0665609942 08015 09862 08184 0955109965 09916 06647 05323 0999808061 00000 08572 09982 05874))))))

(35)


1205851 =((((((

03364 03492 03339 03366 0320103359 03441 03461 03354 0336703466 03389 03406 03410 0345203372 03417 03424 03422 0340403358 03352 03259 03488 0348503473 02893 03389 03390 03458))))))

1205852 =((((((

03273 03014 03278 03268 0335703275 03218 03204 03206 0327503172 03315 03272 03275 0336103287 03021 03125 03257 0321403277 03250 03321 03402 0304303220 03592 03322 03277 03375))))))

1205853 =((((((

03371 03507 03362 03364 0345103371 03328 03358 03465 0334703463 03322 03379 03353 0321303355 03542 03484 03331 0340403364 03381 03441 03122 0348203411 03615 03301 03366 03244))))))

(36)




Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018









Table6Com

prehensiv

eevaluativem

atrix

119885forsup

pliers

119888 1119888 2

119888 3119888 4

119888 5119900 1( (0

052600778

05759 )(5

79))((0011

20082116

643)(9101

0))((0473

30710410

385)(579)

)((04711

07294113

07)(135))

((04690062

3009493)(

91010))

119900 2((00694

045090920

9)(579))

((00062005

0705824)(

579))((

046720702

810365)(1

35))((0532

30798411

617)(9101

0))((02072

042060695

1)(91010)

)119900 3

((04188070

2211834)

(91010))

((00093009

1517343)(

91010))((0

699409311

12207)(9

1010))((0

360206013

10000)(5

79))((0058

30074904

712)(135)

)119900 4((0

558908564

13315)(5

79))((0015

50686626

278)(135)

)((07347

090931177

1)(91010)

)((02343

050330892

0)(579))

((03064055

2507904)(

579))

119900 5((0480

30755712

343)(910

10))((0005

90040603

967)(9101

0))((02425

049490820

2)(91010)

)((069930

967513314)

(91010))

((08995100

0011220)(

579))

119900 6((03183

063311120

)(579))

((00042003

9705859)(

135))((

019330448

907745)(5

79))((0755

81000013

205)(579)

)((00640

038570713

6)(91010)

)


max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018









max 119863 (119908)= 024121199081 + 007231199082 + 011251199083+ 015381199084 + 028611199085

st

1199081 ⩾ 11990821199081 + 1199082 lt 051199081 minus 1199082 ⩽ 1199083 ⩽ 1199081 + 1199082151199082 ⩾ 11990851199083 gt 0151199084 ⩽ 1199085151199084 ⩾ 11990855sum119895=1119908119895 = 1

(37)





(39)




(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018









(11986214 11986224) = (05641 05770) (11986215C25) = (05539 09162) (11986216 11986226) = (05539 06185) (40)












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018












7 Conclusions



Data Availability




Acknowledgments




References






















































Journal of












Journal of







Volume 2018






Volume 2018










References






















































Journal of












Journal of







Volume 2018






Volume 2018

























Journal of












Journal of







Volume 2018






Volume 2018








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