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Research ArticleDecision-Makerrsquos Risk Preference Based Intuitionistic FuzzyMultiattribute Decision-Making and Its Application in RobotEnterprises Investment

Liandong Zhou and QifengWang

Logistics and E-Commerce College Zhejiang Wanli University Ningbo 315100 China

Correspondence should be addressed to Qifeng Wang lhywqf163com

Received 29 June 2018 Accepted 4 September 2018 Published 24 September 2018

Guest Editor Carlos Llopis-Albert

Copyright copy 2018 Liandong Zhou and Qifeng Wang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

At present the utilization of hesitation information of intuitionistic fuzzy numbers is insufficient in many methods which wereproposed to solve the intuitionistic fuzzy multiple attribute decision-making problems And also there exist some flaws in theintuitionistic fuzzy weight vector constructions in many research papers In order to solve these insufficiencies this paper definedthree construction equations ofweight vectors based on the risk preferences of decision-makersThenwe developed an intuitionisticfuzzy dependent hybrid weighted operator (IFDHW) and proposed an intuitionistic fuzzy multiattribute decision-makingmethodFinally the effectiveness of this method is verified by a robot manufacturing investment example

1 Introduction

In 1986 the fuzzy theory of Zadeh [1] was extended to theintuitionistic fuzzy theory by Atanassov [2] Intuitionisticfuzzy sets contain three parts membership nonmembershipand hesitationWith these three parts intuitionistic fuzzy sets(IFSs) can describe the fuzzy nature world better than thetraditional fuzzy sets

Researchers have made great achievements in the studyof intuitionistic fuzzy information aggregation By usingthe IFS which are characterized by a membership func-tion and nonmembership functions Xu [3 4]developedintuitionistic fuzzy weighted averaging (IFWA) operatorintuitionistic fuzzy ordered weighted averaging (IFOWA)operator and intuitionistic fuzzy hybrid aggregation (IFHA)operator S Zeng [5] considered the probabilities and theOWA in the same formulation and proposed the Pythagoreanfuzzy probabilistic ordered weighted averaging (PFPOWA)operator Over the past decades researchers have devel-oped many operators to solve the multiple attribute groupdecision-making (MAGDM) problems Wei [6] proposedinduced intuitionistic fuzzy ordered weighted geometric (I-IFOWG) operator and induced interval-valued intuitionistic

fuzzy ordered weighted geometric (I-IIFOWG) to solve theMAGDM problems Su et al [7] extended the induced gen-eralized ordered weighted averaging (IGOWA) operator anddeveloped induced generalized intuitionistic fuzzy orderedweighted averaging (IG-IFOWA) operator Zeng et al [8]considered both ordered weighted average operator andinduced orderedweighted average and proposed pythagoreanfuzzy induced ordered weighted averaging weighted aver-age (PFIOWAWA) operator for MAGDM Combining intu-itionistic fuzzy operators and TOPSIS method togethermany multiattribute decision-making (MADM) methodshave been developed [9ndash11] Based on the TOPSIS methodintuitionistic fuzzy VIKOR methods were introduced [12 13]and the problem of choosing the best alternative due tothe incommensurability between attributes had been wellsolved Chatterjee et al [14] integrated the Analytic Hierar-chy Process and the VIKOR compromise-ranking methodtogether and constructed a flexible multicriteria decision-making (MCDM) framework Huang et al [15] extend theVIKOR method to MAGDM with interval neutrosophicnumbers (INNs) Meng et al [16] introduced the prospecttheory into MADM with interval-valued intuitionistic fuzzyinformation Qin et al [17] proposed a decision-making

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1720189 6 pageshttpsdoiorg10115520181720189

2 Mathematical Problems in Engineering

model by integrating VIKOR method and prospect theoryXie et al [18] applied prospect theory and grey relational anal-ysis to stochastic decision-making Li et al [19] aggregatedthe decision-making information in different natural statesby using the prospect theory

Although there are many research achievements onintuitionistic fuzzy information aggregation and methods forsolving MADM and MAGDM problems there are still somedrawbacks and research gaps for further research(1)Intuitionistic fuzzy numbers (IFNs) contain threeparts membership nonmembership and hesitation There-fore when using the aggregation operator to rank IFNs thesethree aspects should be taken into account simultaneouslyBut most of the existing aggregation operators are only con-cerned about two parts membership and nonmembershipThe uncertainty (hesitation degree) of IFNs is often ignored(2)In the process of MADM common aggregation oper-ators often assume that attributes are independent from eachother Xu [20] proposed aweightingmethod based on normaldistribution The characteristic of this method is giving asmaller weight to the data that is too high or too low thereforethe effect of larger deviations on integration results can beeliminated as much as possible However there is a flaw inthismethod theweight is independent of the datawhich to beintegrated and cannot reflect the relationship between data Ifthe interaction factors of attributes are taken into account inthe aggregation operators decision-makers will be assisted toobtain more accurate decision results(3)Theexisting weight determination methods aremostlyfocused on subjective weighting [21] Some objective weightdetermination methods need to solve the linear or non-linear programming model [22 23] And computation ofthese methods is relatively cumbersome and is not suitablefor decision-making problems with lots of alternatives andattributes(4)Recently researchers made some progresses indecision-making with risk preferences Adding riskpreferences can affect the decision-makerrsquos psychologicalfactors into the decision-making process That can reducethe error of decision results and improve the quality of thedecision-making Liu J et al [24] proposed a new modelinvolved risk preferences of decision-makers based on theprospect theory and criteria reduction Wan et al [25]developed a new method with interval-valued intuitionisticfuzzy preference relations for solving group decision-makingproblems Y Lin et al [26] developed a method to determinerelative weights of decision-makers depending on preferenceinformation However the most existing research on riskpreferences focuses on priority weights and less researchesfrom the perspective of decision-makerrsquos attitude(5)Recently IFS has been applied to many decision-making fields such as supplier selection [10 27 28] andpattern recognition [29ndash31] But there is no application inrobot enterprises investment field

Considering all the problems listed above this paperfocused on the study of intuitionistic fuzzy multiattributedecision-making with decision-makersrsquo different attitudesAnd in order to give decision-makers most desirable resultswe combined intuitionistic fuzzy theory and risk preference

partition theory which is from expected utility theory Inrisk preference partition theory the risk preference attitudeof decision-makers can be divided into three categoriesrisk proneness risk aversion and risk neutralness Thenwe defined the weight vectors equations according to theattitudes of decision-makers based on the three parts of IFNsTheyrsquore objective weights and easy to be calculated By takinginteraction factors of attributes into account we defined theintuitionistic fuzzy dependent hybrid weighted operator andproposed a decision-making method The effectiveness ofthis method is verified by a robot enterprises investmentexample

2 Preliminaries

Atanassov [1] introduced nonmembership to the Zadeh fuzzysets (FS)[2] and defined IFS shown as follows

Definition 1 119883 = 1199091 1199092 sdot sdot sdot 119909119899 is an universe of discoursethen

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

is an IFS where for each element 119909 isin 119883 120583119860(119909) 119883 997888rarr[0 1] represents the membership and ]119860(119909) 119883 997888rarr [0 1]represents the nonmembership with the condition satisfying0 le 120583119860(119909)+]119860(119909) le 1forall119909 isin 119883 And120587119860(119909) = 1minus120583119860(119909)minus]119860(119909)is a degree that characterizes the uncertainty or hesitancy ofeach element 119909 isin 119883 in IFS set 119860 In particular if 120587119860(119909) = 0forall119909 isin 119883 then set 119860 degenerate into Zadeh fuzzy sets Themembership degree nonmembership degree and hesitationdegree of the IFS effectively extend the representation abilityof classical fuzzy sets For convenience we can define 120572 =(120583120572 ]120572) as an IFN where 120583120572 isin [0 1] ]120572 isin [0 1] and120583120572 + ]120572 le 1 Xu [4] introduced the IFNs operational lawsshown as follows

Definition 2 120572 = (120583120572 ]120572)and 120573 = (120583120573 ]120573) are two IFNs thenfive operational laws are as follows

120572 = (]120572 120583120572) (2)

120572 oplus 120573 = (120583120572 + 120583120573 minus 120583120572120583120573 ]120572]120573) (3)

120572 otimes 120573 = (120583120572120583120573 ]120572 + ]120573 minus ]120572]120573) (4)

120582120572 = (1 minus (1 minus 120583120572)120582 ]120572120582) 120582 gt 0 (5)

120572120582 = (120583120572120582 1 minus (1 minus ]120572)120582) 120582 gt 0 (6)

Based on operational laws of IFNs above a weighted averag-ing operator of IFNs is given by Xu [4]

Mathematical Problems in Engineering 3

Definition 3 120572119894 = (120583119894 ]119894)(119894 = 1 2 sdot sdot sdot 119899) is a set of IFNsand letting Θ be the set of intuitionistic fuzzy numbers then119868119865119882119860 Θ119899 997888rarr Θ is defined as follows

119868119865119882119860120596 (1205721 1205722 sdot sdot sdot 120572119899) = 119899sum119894=1

120596119899120572119899= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(7)

then 119868119865119882119860 is called intuitionistic fuzzy weighted averagingoperator where 120596 = (1205961 1205962 sdot sdot sdot 120596119899)119879 is the weight vectorof 120572119894(119894 = 1 2 sdot sdot sdot 119899) with the condition satisfying 120596119894 isin[0 1](119894 = 1 2 sdot sdot sdot 119899) sum119899119894=1 120596119894 = 1 Obviously by using theIFWAoperator to aggregate IFNs the aggregated value is alsoan IFN Thus the loss of information is avoided

3 Intuitionistic Fuzzy Dependent HybridWeighted Operator

The IFWA operator only considers the importance of IFNsby using the weight vector but the risk attitude informationinside the IFNs is also very important In order to exploitthe risk attitude information inside the IFNs we need tointroduce similarity degree defined by Xu [32]

Definition 4 1205721 = (1205831205721 ]1205721) and 1205722 = (1205831205722 ]1205722)are any twoIFNs 1205722 = (]1205722 1205831205722) is the complement of 1205722 then

119878 (1205721 1205722) = 05 1205721 = 1205722 = 1205722

119889 (1205721 1205722)119889 (1205721 1205722) + 119889 (1205721 1205722) others(8)

is the similarity degree between 1205721 and 1205722where119889 (1205721 1205722) = 12 (100381610038161003816100381610038161205831205721 minus 1205831205722 10038161003816100381610038161003816 + 10038161003816100381610038161003816]1205721 minus ]1205722

10038161003816100381610038161003816 + 100381610038161003816100381610038161205871205721 minus 1205871205722 10038161003816100381610038161003816) (9)

is the standard Hamming distance between 1205721 and 1205722Definition 5 120572119894 = (120583119894 ]119894)(119894 = 1 2 sdot sdot sdot 119899) is a set of IFNs thenthe average of the IFNs is defined as

120572 = 1119899 (1205721 oplus 1205722 oplus sdot sdot sdot oplus 120572119899)= (1 minus 119899prod

119894=1

(1 minus 120583119894)1119899 119899prod119894=1

]1198941119899)

(10)

In order to reflect the preferences of decision-makers wedivide the risk attitude information into three kinds riskproneness risk aversion and risk neutralness Then we rede-fine the weight equations to extract risk attitude informationthat inside the IFNs

Definition 6 120572119894 = (120583119894 ]119894)(119894 = 1 2 sdot sdot sdot 119899) is a set of IFNs thenthree kinds of risk attitudes are introduced by using differentweight equations

(i) Risk proneness weight equation for 120572119894 is defined as

120596119894 = 120587119894sum119899119894=1 120587119894 (11)

with the condition satisfying sum119899119894=1 120596119894 = 1 (119894 = 1 2 sdot sdot sdot 119899)120596119894 isin [0 1] The degree of hesitancy is 120587119894 = 1 minus 120583119894 minus ]119894Obviously the greater the degree of hesitancy the greaterthe corresponding weight Risk proneness decision-makersconsider hesitancy as advantage

(ii) Risk aversion weight equation for 120572119894 is defined as

120596119894 = 1 minus 120587119894sum119899119894=1 (1 minus 120587119894) (12)

with the condition satisfying sum119899119894=1 120596119894 = 1 (119894 = 1 2 sdot sdot sdot 119899)120596119894 isin [0 1] Because of 1 minus 120587119894 = 120583119894 + ]119894 the greater the degreeof hesitancy the smaller the corresponding weight Riskaversiondecision-makers consider hesitancy as disadvantage

(iii) Risk neutralness weight equation for 120572119894 is based onsimilarity degree and average of IFNs defined as follows

120596119894 = 119878 (120572120590(119894) 120572)sum119899119894=1 119878 (120572120590(119894) 120572) 119894 = 1 2 sdot sdot sdot 119899 (13)

with the condition satisfying sum119899119894=1 120596119894 = 1(119894 = 1 2 sdot sdot sdot 119899)120596119894 isin [0 1] where 119878(120572120590(119894) 120572) is the similarity degree between120572120590(119894) and 120572 is calculated by (8) 120572 is the average valueof 120572119894 which is calculated by (10) And 120572120590(119894) is ith largestof 120572119894 and with the condition satisfying 120572120590(119894minus1) ge 120572120590(119894)(119894 = 1 2 sdot sdot sdot 119899) The weight calculations depend on themembership nonmembership and hesitation of IFNs If theintuitionistic fuzzy value is closer to the average value theweight value will be greater If the intuitionistic fuzzy valueis far away from the average value the weight value willbe smaller It can represents the risk neutralness decision-makersrsquo attitude

In order to aggregate risk attitude information both theimportance of IFNs and the risk factors brought by thehesitation of IFNs should be taken into consideration Weproposed an intuitionistic fuzzy dependent hybrid weightedoperator (IFDHW) defined as follows

Definition 7 120572119894 = (120583119894 ]119894)(119894 = 1 2 sdot sdot sdot 119899)is a set of IFNsand letting Θ be the set of intuitionistic fuzzy numbers then119868119865119863119867119882 Θ119899 997888rarr Θ is defined as follows


120596119894120590(119894)= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(14)

then IFDHW is an intuitionistic fuzzy dependent hybridweighted operator where 119894 = 119899119908119894120572119894 120590(119894minus1) ge 120590(119894) for alli and119882 = (1199081 1199082 sdot sdot sdot 119908119899)119879 is the weight vector of 120572119894 with theconditionsum119899119894=1 119908119894 = 1 (119894 = 1 2 sdot sdot sdot 119899) 119908119894 isin [0 1] n is calledthe balancing coefficient 120596119894 is decided by decision-makersrsquothree kinds of attitude risk proneness risk aversion and riskneutralness 120596119894 can be calculated by using (11)-(13)


4 Intuitionistic Fuzzy MultipleAttribute Decision-Making Method Basedon Decision-Makerrsquos Risk Attitude

For solving a MADM problem with intuitionistic fuzzyinformation let us suppose that 119860 = 1198601 1198602 sdot sdot sdot 119860119899 (119894 =1 2 sdot sdot sdot n) is a set of n alternatives to be selected 119866 =1198661 1198662 sdot sdot sdot 119866119898 (119895 = 1 2 sdot sdot sdot 119898) is a set of m attributes andwhose weight vector is119882 = (1199081 1199082 sdot sdot sdot 119908119898)119879 where119908119895 is theweight for attribute 119866119895 with the condition sum119899119894=1119908119894 = 1 (119894 =1 2 sdot sdot sdot 119899) 119908119894 isin [0 1] D = (120572119894119895)mtimes119899 = ((120583119894119895 ]119894119895))mtimes119899 is thedecision matrix where 120572119894119895 is provided by decision-maker foralternative 119860 119894 with respect to attribute 119866119895

In the following three steps we will use the IFDHWoperator to solve MADM problems by developing a methodbased on decision-makerrsquos risk attitude

Step 1 The decision-maker gives the decision matrix 119863 =(120572119894119895)119899times119898 according to the actual situation with weight vector119882 = (1199081 1199082 sdot sdot sdot 119908119898)119879 Meanwhile decision-maker choosesthe appropriate risk weight equation to calculate 120596119894 accordingto the decision-makerrsquos risk preference

Step 2 Utilize the IFDHW operator and calculate overallvalues 119903119860119894 for all the alternatives 119860 119894(119894 = 1 2 sdot sdot sdot 119899) by using(14)

Step 3 Based on aggregated value 119903119860119894 for all 119860 119894 (119894 =1 2 sdot sdot sdot 119899) the score values 119878(120572119894) are calculated and rankedThe best one(s) of all the alternatives 119860 119894 would be selected

5 Illustrated Example

There is an investment company who want to invest in oneof the robot manufacturing enterprises 119860 119894(119894 = 1 2 3 4)The investment company has determined five attributes119866119895(119895 = 1 2 3 4 5) to evaluate the robot manufacturingenterprises production capacity technological innovationability marketing ability management ability risk aversionability 119908 = (025 02 02 01 025)119879 is the weight vectorof these attributes The intuitionistic fuzzy decision matrixis provided by the company which is listed in Table 1 Theinvestment decision-making steps are shown as follows

Step 1 Using weight equation to calculate alternativesrsquo riskweight vector we take the one kind of decision-makerrsquosattitude for an example Using risk proneness weight equation120596119894 = (1 minus 120583119894 minus ]119894)sum119899119894=1(1 minus 120583119894 minus ]119894) to calculate four robotsmanufacturing enterprisesrsquo risk weight vector

1205961 = (01667 03333 01667 01667 01667)119879 1205962 = (01250 01250 01250 03750 02500)119879 1205963 = (01000 03000 02000 03000 01000)119879 1205964 = (001667 01667 0000 01667 05000)119879

(15)

Step 2 Use the IFDHW operator First calculate 119894 =119899119908119894120572119894 taking alternative A1 for example 1 = (040540503) 2 = (07000 01000) 3 = (07000 02000) 4 =(03183 05946) and 5 = (06464 02530) Then calculateoverall values 119903119860119894 for every 119860 119894(119894 = 1 2 sdot sdot sdot 119899) by using (14)

1199031198601 = (06038 02308) 1199031198602 = (05356 02753) 1199031198603 = (05146 02321) 1199031198604 = (05033 02669)

(16)

Step 3 Sort the alternatives by calculating the score functions119878(119903119860119894) (119894 = 1 2 3 4) for every alternatives based on overallvalues 1199031198601 1199031198602 1199031198603 and 1199031198604 If two or more score values areequal then we can use accuracy function 119867(119903119860119894 ) to get theranking results

119878 (1199031198601) = 03729119878 (1199031198602) = 02603119878 (1199031198603) = 02825119878 (1199031198604) = 02364

(17)

According to the results of 119878(119903119860119894 ) (119894 = 1 2 3 4) and thusA1 ≻ 1198603 ≻ 1198602 ≻ 1198604 where ldquo≻rdquo denotes ldquobe superior tordquotherefore for a decision-maker in the risk proneness attitudeA1 is the best investment company Using the same steps theranking results in other two cases are shown in Table 2

Table 2 shows that rest on different risk attitudes the bestalternatives can be different For risk proneness attitude thatthe best investment company is 1198601 for risk aversion attitudeit is 1198602 and for risk neutralness attitude it is1198603 This rankingmethod can reflect the impact of risk factors on the rankingresults and can also choose the best alternative according todifferent risk attitudes of decision-makers

6 Conclusions

In this paper we want to solve the MADM problems whendecision-makers take different risk attitude The innovationsof this paper are listed as follows(1) We introduced three risk preference attitudes ofdecision-makers to MADM field Three risk preference atti-tudes are from risk preference partition theory which iscontained in expected utility theory(2) Considering the hesitation information of IFNswe defined three equations for constructing weight vectorsaccording to different decision-makersrsquo attitudes The weightvectors are subjective and easy to calculate for solving theMADM problems with lots of alternatives and attributes(3) This decision-making method can provide decision-making basis for many different fields when decision-makerswant to check if there are any differences while they are


Table 1 Intuitionistic fuzzy decision matrix for investment

G1 G2 G3 G4 G5A1 (0504) (0701) (0702) (0405) (0504)A2 (0603) (0504) (0702) (0304) (0602)A3 (0702) (0601) (0503) (0403) (0504)A4 (0504) (0504) (0604) (0603) (0403)

Table 2 Ranking results of different risk attitudes

Risk attitudes Ranking resultsRisk proneness A1 ≻ 1198603 ≻ 1198602 ≻ 1198604Risk aversion A2 ≻ 1198603 ≻ 1198601 ≻ 1198604Risk neutralness A3 ≻ 1198602 ≻ 1198601 ≻ 1198604

in different attitudes Then they can get the most desirablealternative(s)

In the future we should study the accuracy of this pro-posed method Meanwhile we can also extend the proposedmethod to solve the MAGDM problems

Data Availability

The intuitionistic fuzzy data used to support the findings ofthis study are included within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This paper is supported by the Key Research Institute ofPhilosophy and Social Science of Zhejiang Province (Mod-ern Port Service Industry and Creative Culture ResearchCenter) (nos 16JDGH067 15JDLG01YB) Research Projectof Philosophy and Social Science of Zhejiang Province (no18NDJC283YB) and Soft Science Project of Ningbo (nos2017A10085 2017A10068)

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 pp338ndash353 1965

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

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

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

[5] S Zeng ldquoPythagorean fuzzy multiattribute group decisionmaking with probabilistic information and OWA approachrdquoInternational Journal of Intelligent Systems vol 32 no 11 pp1136ndash1150 2017

[6] G Wei ldquoSome induced geometric aggregation operators withintuitionistic fuzzy information and their application to groupdecision makingrdquo Applied Soft Computing vol 10 no 2 pp423ndash431 2010

[7] Z Su G Xia M Chen and L Wang ldquoInduced generalizedintuitionistic fuzzy OWA operator for multi-attribute groupdecision makingrdquo Expert Systems with Applications vol 39 no2 pp 1902ndash1910 2012

[8] S Zeng Z Mu and T Balezentis ldquoA novel aggregation methodfor Pythagorean fuzzy multiple attribute group decision mak-ingrdquo International Journal of Intelligent Systems vol 33 no 3pp 573ndash585 2018

[9] C Q Tan ldquoA multi-criteria interval-valued intuitionistic fuzzygroup decision making with Choquet integral-based TOPSISrdquoExpert Systems with Applications vol 38 no 4 pp 3023ndash30332011

[10] F E Boran S Genc M Kurt and D Akay ldquoA multi-criteriaintuitionistic fuzzy group decisionmaking for supplier selectionwith TOPSISmethodrdquoExpert Systems with Applications vol 36no 8 pp 11363ndash11368 2009

[11] S Zeng and Y Xiao ldquoA method based on topsis and distancemeasures for hesitant fuzzy multiple attribute decisionmakingrdquoTechnological and Economic Development of Economy vol 24no 3 pp 969ndash983 2018

[12] K Chatterjee M B Kar and S Kar ldquoStrategic Decisions UsingIntuitionistic Fuzzy Vikor Method for Information System (IS)Outsourcingrdquo in Proceedings of the International Symposium onComputational and Business Intelligence IEEE Computer Societypp 123ndash126 2013

[13] H Liao and Z Xu ldquoA VIKOR-based method for hesitantfuzzy multi-criteria decision makingrdquo Fuzzy Optimization andDecision Making vol 12 no 4 pp 373ndash392 2013

[14] K Chatterjee and S Kar ldquoUnified Granular-number-basedAHP-VIKOR multi-criteria decision frameworkrdquo GranularComputing vol 2 no 3 pp 199ndash221 2017

[15] Y Huang G Wei and C Wei ldquoVIKOR Method for Inter-val Neutrosophic Multiple Attribute Group Decision-MakingrdquoInformation vol 8 no 4 p 144 2017

[16] F Meng C Tan and X Chen ldquoAn approach to Atanassovrsquosinterval-valued intuitionistic fuzzy multi-attribute decisionmaking based on prospect theoryrdquo International Journal ofComputational Intelligence Systems vol 8 no 3 pp 591ndash6052015

[17] J D Qin X W Liu and W Pedrycz ldquoAn extended VIKORmethod based on prospect theory formultiple attribute decisionmaking under interval type-2 fuzzy environmentrdquo Knowledge-Based Systems vol 86 pp 116ndash130 2015

[18] N Xie Z Li and G Zhang ldquoAn intuitionistic fuzzy softset method for stochastic decision-making applying prospecttheory and grey relational analysisrdquo Journal of Intelligent ampFuzzy Systems Applications in Engineering and Technology vol33 no 1 pp 15ndash25 2017


[19] Peng Li Yingjie Yang and Cuiping Wei ldquoAn IntuitionisticFuzzy Stochastic Decision-Making Method Based on Case-Based Reasoning and ProspectTheoryrdquoMathematical Problemsin Engineering vol 2017 Article ID 2874954 13 pages 2017

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

[21] X Guo Z Yuan and B Tian ldquoSupplier selection based onhierarchical potential support vector machinerdquo Expert Systemswith Applications vol 36 no 3 pp 6978ndash6985 2009

[22] A Zouggari and L Benyoucef ldquoSimulation based fuzzy TOPSISapproach for group multi-criteria supplier selection problemrdquoEngineering Applications of Artificial Intelligence vol 25 no 3pp 507ndash519 2012

[23] G Wang S H Huang and J P Dismukes ldquoProduct-drivensupply chain selection using integrated multi-criteria decision-making methodologyrdquo International Journal of Production Eco-nomics vol 91 no 1 pp 1ndash15 2004

[24] J Liu S-F Liu P Liu X-Z Zhou and B Zhao ldquoA newdecision support model in multi-criteria decision making withintuitionistic fuzzy sets based on risk preferences and criteriareductionrdquo Journal of the Operational Research Society vol 64no 8 pp 1205ndash1220 2013

[25] S Wan F Wang and J Dong ldquoA Three-Phase Method forGroup Decision Making with Interval-Valued IntuitionisticFuzzy Preference Relationsrdquo IEEE Transactions on Fuzzy Sys-tems vol 26 no 2 pp 998ndash1010 2018

[26] Y Lin and Y Wang ldquoGroup decision making with consistencyof intuitionistic fuzzy preference relations under uncertaintyrdquoIEEECAA Journal of Automatica Sinica vol 5 no 3 pp 741ndash748 2018

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

[28] G Buyukozkan and F Gocer ldquoApplication of a new com-bined intuitionistic fuzzy MCDMapproach based on axiomaticdesign methodology for the supplier selection problemrdquoApplied Soft Computing vol 52 pp 1222ndash1238 2017

[29] W L Hung and M S Yang ldquoOn the J -divergence of intu-itionistic fuzzy sets with its application to pattern recognitionrdquoInformation Sciences vol 178 no 6 pp 1641ndash1650 2008

[30] S-H Cheng S-M Chen and T-C Lan ldquoA New SimilarityMeasure between Intuitionistic Fuzzy Sets for Pattern Recog-nition Based on the Centroid Points of Transformed FuzzyNumbersrdquo in Proceedings of the IEEE International Conferenceon Systems Man and Cybernetics IEEE pp 1125ndash1129 2015

[31] H Nguyen ldquoA novel similaritydissimilarity measure for intu-itionistic fuzzy sets and its application in pattern recognitionrdquoExpert Systems with Applications vol 45 pp 97ndash107 2016

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

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model by integrating VIKOR method and prospect theoryXie et al [18] applied prospect theory and grey relational anal-ysis to stochastic decision-making Li et al [19] aggregatedthe decision-making information in different natural statesby using the prospect theory

Although there are many research achievements onintuitionistic fuzzy information aggregation and methods forsolving MADM and MAGDM problems there are still somedrawbacks and research gaps for further research(1)Intuitionistic fuzzy numbers (IFNs) contain threeparts membership nonmembership and hesitation There-fore when using the aggregation operator to rank IFNs thesethree aspects should be taken into account simultaneouslyBut most of the existing aggregation operators are only con-cerned about two parts membership and nonmembershipThe uncertainty (hesitation degree) of IFNs is often ignored(2)In the process of MADM common aggregation oper-ators often assume that attributes are independent from eachother Xu [20] proposed aweightingmethod based on normaldistribution The characteristic of this method is giving asmaller weight to the data that is too high or too low thereforethe effect of larger deviations on integration results can beeliminated as much as possible However there is a flaw inthismethod theweight is independent of the datawhich to beintegrated and cannot reflect the relationship between data Ifthe interaction factors of attributes are taken into account inthe aggregation operators decision-makers will be assisted toobtain more accurate decision results(3)Theexisting weight determination methods aremostlyfocused on subjective weighting [21] Some objective weightdetermination methods need to solve the linear or non-linear programming model [22 23] And computation ofthese methods is relatively cumbersome and is not suitablefor decision-making problems with lots of alternatives andattributes(4)Recently researchers made some progresses indecision-making with risk preferences Adding riskpreferences can affect the decision-makerrsquos psychologicalfactors into the decision-making process That can reducethe error of decision results and improve the quality of thedecision-making Liu J et al [24] proposed a new modelinvolved risk preferences of decision-makers based on theprospect theory and criteria reduction Wan et al [25]developed a new method with interval-valued intuitionisticfuzzy preference relations for solving group decision-makingproblems Y Lin et al [26] developed a method to determinerelative weights of decision-makers depending on preferenceinformation However the most existing research on riskpreferences focuses on priority weights and less researchesfrom the perspective of decision-makerrsquos attitude(5)Recently IFS has been applied to many decision-making fields such as supplier selection [10 27 28] andpattern recognition [29ndash31] But there is no application inrobot enterprises investment field

Considering all the problems listed above this paperfocused on the study of intuitionistic fuzzy multiattributedecision-making with decision-makersrsquo different attitudesAnd in order to give decision-makers most desirable resultswe combined intuitionistic fuzzy theory and risk preference

partition theory which is from expected utility theory Inrisk preference partition theory the risk preference attitudeof decision-makers can be divided into three categoriesrisk proneness risk aversion and risk neutralness Thenwe defined the weight vectors equations according to theattitudes of decision-makers based on the three parts of IFNsTheyrsquore objective weights and easy to be calculated By takinginteraction factors of attributes into account we defined theintuitionistic fuzzy dependent hybrid weighted operator andproposed a decision-making method The effectiveness ofthis method is verified by a robot enterprises investmentexample

2 Preliminaries

Atanassov [1] introduced nonmembership to the Zadeh fuzzysets (FS)[2] and defined IFS shown as follows

Definition 1 119883 = 1199091 1199092 sdot sdot sdot 119909119899 is an universe of discoursethen

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

is an IFS where for each element 119909 isin 119883 120583119860(119909) 119883 997888rarr[0 1] represents the membership and ]119860(119909) 119883 997888rarr [0 1]represents the nonmembership with the condition satisfying0 le 120583119860(119909)+]119860(119909) le 1forall119909 isin 119883 And120587119860(119909) = 1minus120583119860(119909)minus]119860(119909)is a degree that characterizes the uncertainty or hesitancy ofeach element 119909 isin 119883 in IFS set 119860 In particular if 120587119860(119909) = 0forall119909 isin 119883 then set 119860 degenerate into Zadeh fuzzy sets Themembership degree nonmembership degree and hesitationdegree of the IFS effectively extend the representation abilityof classical fuzzy sets For convenience we can define 120572 =(120583120572 ]120572) as an IFN where 120583120572 isin [0 1] ]120572 isin [0 1] and120583120572 + ]120572 le 1 Xu [4] introduced the IFNs operational lawsshown as follows

Definition 2 120572 = (120583120572 ]120572)and 120573 = (120583120573 ]120573) are two IFNs thenfive operational laws are as follows

120572 = (]120572 120583120572) (2)

120572 oplus 120573 = (120583120572 + 120583120573 minus 120583120572120583120573 ]120572]120573) (3)

120572 otimes 120573 = (120583120572120583120573 ]120572 + ]120573 minus ]120572]120573) (4)

120582120572 = (1 minus (1 minus 120583120572)120582 ]120572120582) 120582 gt 0 (5)

120572120582 = (120583120572120582 1 minus (1 minus ]120572)120582) 120582 gt 0 (6)

Based on operational laws of IFNs above a weighted averag-ing operator of IFNs is given by Xu [4]




120596119899120572119899= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(7)





119878 (1205721 1205722) = 05 1205721 = 1205722 = 1205722

119889 (1205721 1205722)119889 (1205721 1205722) + 119889 (1205721 1205722) others(8)


10038161003816100381610038161003816 + 100381610038161003816100381610038161205871205721 minus 1205871205722 10038161003816100381610038161003816) (9)



119894=1

(1 minus 120583119894)1119899 119899prod119894=1

]1198941119899)

(10)




120596119894 = 120587119894sum119899119894=1 120587119894 (11)



120596119894 = 1 minus 120587119894sum119899119894=1 (1 minus 120587119894) (12)








120596119894120590(119894)= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(14)












1205961 = (01667 03333 01667 01667 01667)119879 1205962 = (01250 01250 01250 03750 02500)119879 1205963 = (01000 03000 02000 03000 01000)119879 1205964 = (001667 01667 0000 01667 05000)119879

(15)


1199031198601 = (06038 02308) 1199031198602 = (05356 02753) 1199031198603 = (05146 02321) 1199031198604 = (05033 02669)

(16)


119878 (1199031198601) = 03729119878 (1199031198602) = 02603119878 (1199031198603) = 02825119878 (1199031198604) = 02364

(17)



6 Conclusions




G1 G2 G3 G4 G5A1 (0504) (0701) (0702) (0405) (0504)A2 (0603) (0504) (0702) (0304) (0602)A3 (0702) (0601) (0503) (0403) (0504)A4 (0504) (0504) (0604) (0603) (0403)





Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018











120596119899120572119899= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(7)





119878 (1205721 1205722) = 05 1205721 = 1205722 = 1205722

119889 (1205721 1205722)119889 (1205721 1205722) + 119889 (1205721 1205722) others(8)


10038161003816100381610038161003816 + 100381610038161003816100381610038161205871205721 minus 1205871205722 10038161003816100381610038161003816) (9)



119894=1

(1 minus 120583119894)1119899 119899prod119894=1

]1198941119899)

(10)




120596119894 = 120587119894sum119899119894=1 120587119894 (11)



120596119894 = 1 minus 120587119894sum119899119894=1 (1 minus 120587119894) (12)








120596119894120590(119894)= (1 minus 119899prod

119894=1

(1 minus 120583119894)120596119894 119899prod119894=1

]119894120596119894)

(14)












1205961 = (01667 03333 01667 01667 01667)119879 1205962 = (01250 01250 01250 03750 02500)119879 1205963 = (01000 03000 02000 03000 01000)119879 1205964 = (001667 01667 0000 01667 05000)119879

(15)


1199031198601 = (06038 02308) 1199031198602 = (05356 02753) 1199031198603 = (05146 02321) 1199031198604 = (05033 02669)

(16)


119878 (1199031198601) = 03729119878 (1199031198602) = 02603119878 (1199031198603) = 02825119878 (1199031198604) = 02364

(17)



6 Conclusions




G1 G2 G3 G4 G5A1 (0504) (0701) (0702) (0405) (0504)A2 (0603) (0504) (0702) (0304) (0602)A3 (0702) (0601) (0503) (0403) (0504)A4 (0504) (0504) (0604) (0603) (0403)





Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018


















1205961 = (01667 03333 01667 01667 01667)119879 1205962 = (01250 01250 01250 03750 02500)119879 1205963 = (01000 03000 02000 03000 01000)119879 1205964 = (001667 01667 0000 01667 05000)119879

(15)


1199031198601 = (06038 02308) 1199031198602 = (05356 02753) 1199031198603 = (05146 02321) 1199031198604 = (05033 02669)

(16)


119878 (1199031198601) = 03729119878 (1199031198602) = 02603119878 (1199031198603) = 02825119878 (1199031198604) = 02364

(17)



6 Conclusions




G1 G2 G3 G4 G5A1 (0504) (0701) (0702) (0405) (0504)A2 (0603) (0504) (0702) (0304) (0602)A3 (0702) (0601) (0503) (0403) (0504)A4 (0504) (0504) (0604) (0603) (0403)





Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018










G1 G2 G3 G4 G5A1 (0504) (0701) (0702) (0405) (0504)A2 (0603) (0504) (0702) (0304) (0602)A3 (0702) (0601) (0503) (0403) (0504)A4 (0504) (0504) (0604) (0603) (0403)





Data Availability




Acknowledgments


References









































Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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