WenqiJiangand Jennifer Shang

1.Schoolof Economic and Management,Nanjing University of Science and Technology,Nanjing 210094,China; 2.Katz Graduate Schoolof Business,University of Pittsburgh,Pittsburgh PA 15260,USA

Multi-criteria group decision making with fuzzy data: an extension ofthe VIKOR method

WenqiJiang1,*and Jennifer Shang2

1.Schoolof Economic and Management,Nanjing University of Science and Technology,Nanjing 210094,China; 2.Katz Graduate Schoolof Business,University of Pittsburgh,Pittsburgh PA 15260,USA

The VIKOR method is a multi-criteria decision making aid,which employs linear normalization to offer compromise solutions and has been successfully applied to various group decision making problems.However,the conventional VIKOR techniques used to integrate group judgments and the information loss arising from defuzzification are problematic and distort final outcomes. An improved integration method,which is optimization-based,is proposed.And it can handle fuzzy criteria values and weights. The precondition for accurately defuzzifying triangular fuzzy numbers is identified.Several effective defuzzification procedures are proposed to improve the extant VIKOR,and a comprehensive evaluation framework is offered to aid multi-criteria group decision making.Finally,a numerical example is provided to illustrate the practicability ofthe proposed method.

multi-criteria group decision making,triangular fuzzy numbers,VIKOR,compromise solution.

1.Introduction

Multi-criteria decision making(MCDM)is fast growing and has been employed in many areas of business,such as contractorevaluation[1],repair-to-ordersystem[2],environmental corporate policy[3],supply chain[4],boron selection[5],water management[6],service selection[7], transplantation[8],garment matching[9],credit decision making[10],information management[11],emergency decision[12],supplier selection[13],and urban distribution centers[14].Many MCDM methods,such as multiplicative exponential weighting,simple additive weighting,and technique for ordering preference by similarity to idealsolution(TOPSIS),and analytic hierarchy process (AHP),have been used to assess,rank and select the most favorable alternative(s)from a setof options,which are often characterized by multiple and conflicting criteria.

The increasingly complex MCDM problems make it difficult for an individual decision maker to have exact knowledge and sufficient experiences to consider all factors simultaneously[15].Thus,the group decision making technology,which studies how to integrate the wisdom of multiple decision makers into one group judgment,is frequently employed in the realworld.Recently,multiple criteria group decision making(MCGDM)has received much attention.In classical MCGDM problems,the values and weights of the evaluation criteria are crisp,whereas in the world full of uncertainty and inexactness,it is unrealistic and often impossible to assign a crisp value forone’s judgmentdue to the ill-defined and vague information.In 1970, Bellman and Zadeh started the firstattemptatconstructing a conceptualframework to address decision making in the fuzzy environments[16].Researchers have since proposed severalapproaches to tackle problems of stochastic nature, rather than using exactnumbers to approximate subjective judgmentand linguistic variables.

It is hard to obtain the solutions to satisfy all criteria simultaneously for problems of practical sizes with non-commensurable and conflicting criteria.The VIKOR method(named from Serbian:Vlsekriterijumska Optimizacija I Kompromisno Resenje)[17,18],developed to work with complex multiple criteria optimization systems, can be conveniently applied to the MCDM and MCGDM problems,such as those discussed in[19–31].The outcome of the VIKOR method is a set of non-inferior solutions or compromise solutions[17,18].However,different integration methods of group fuzzy evaluation may result in differentintegrated matrices,and form different priorities.Thus,it is importantto develop an effective and reliable method to integrate group information and defuzzify the fuzzy numbers when the VIKOR method is applied to MCGDM.

The rest of the paper is organized as follows.In Section 2,we briefly discuss the operationallaw offuzzy num-bers and the VIKORmethod.Section 3 studies the impacts of integration methods on group decision making.The enhancementofthe VIKOR method is presented in Section 4 to address the fuzzy MCGDM problems.Section 5 offers a numerical example to illustrate the implementation process and the validity of the proposed model.Concluding remarks and future research are given in Section 6.

2.Preliminaries

2.1 Operationallaws of triangular fuzzy numbers

Among the various shapes of fuzzy numbers,the triangular fuzzy number(TFN)setexpressed by(a1,a2,a3) is popularly employed to representfuzzy information.We summarize some of the importantproperties of operations on TFNs.

Definition 1A fuzzy setin a universe of discourse X is characterized as a membership functionμ~A(x)that assigns each element x in X a realnumber in the interval [0,1].

Definition 2is defined by a triplet(a1,a2,a3)that satisfies a1?a2?a3,i.e.,the left value is less than the medium value,which is in turn less than the rightvalue.

The membership functionμ~A(x)is given by

Definition 3Let~A=(a1,a2,a3)and~B=(b1,b2,b3) be two TFNs with a1,b1,k≥0.The operationallaws associated with these TFNs are defined as follows.

(i)Addition of TFNs(+):

(ii)Subtraction of TFNs(–):

(iii)Multiplication of TFNs(×):

(iv)Division of TFNs(/):

(v)Multiplication by a scalar number k:

(vi)Operator max:

(vii)Operator min:

Definition 4The centroid defuzzification value mof a TFN=(a1,a2,a3)is

Based on Yager index[31],m(~A)can be simplified as

2.2 VIKOR method

VIKOR has been applied to several MCDM problems. Shemshadi et al.[19]used the VIKOR method to extract objective weights based on Shannon entropy concept.Bazzazietal.[22]modified the VIKOR method to incorporate deterministic data,interval numbers and linguistic terms. Zandi et al.[32]proposed a fuzzy ELECTRE method based on the VIKORmethod,while Liu etal.[23]used the induced ordered weighted averaging standardized distance operator to reflect the complex attitudinalcharacter of the decision maker.Then again,Kaya etal.[25]performed the VIKORand AHP methods undera fuzzy environment,and Chen etal.[26]provided a rationaland systematic process for developing the best alternative and compromise solution for each selection criterion.

Usually,the MCDM problems in our society belong to fuzzy MCDM problems,it is necessary to discuss the application of the fuzzy VIKOR(FVIKOR)in fuzzy conditions.Yucenur et al.[27]used the FVIKOR method to determine the best feasible solution.Likewise,Liu et al. [28]studied the FVIKOR method with trapezoidal or triangular fuzzy numbers,and Park et al.[33]extended the VIKOR method for dynamic intuitionistic fuzzy MCDM through two aggregation operators.Zhang etal.[34]developed the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitantfuzzy set information. Equally,Sayadi et al.[35]used the VIKOR method to solve MCDM problems with interval number.Opricovic [31]and Ju et al.[36]solved the MCGDM problems, where both the criteria values and weights are linguistic information.Finally,Liu et al.[24]presented an interval 2-tuple linguistic VIKOR method for an environmentwith uncertain and incomplete information.

The above papers emphasize the application of the VIKOR method in different fuzzy conditions,however, they ignore two critical problems:how to integrate each individualcriteria weightor value more accurately and effectively,and how to calculate the comprehensive values by applying the VIKOR method when there are negative numbers in triangularnumbers(the specific reasons willbe described in Section 3).The paper intends to analyze the above difficulties deeply,and proposes feasible and optimalapproaches.

The VIKOR method can use an aggregate function to determine the closeness ofan elementto the idealsolution, such as thatfound in(4).

Let1?p?∞,i∈(1,...,m),and rijbe the value of the j th criterion for the i th alternative,then

The VIKOR method employs D1i(as Si)and D∞i(as Ri)to attain measures and rankings,where Siis the group utility of decision alternative Ai,and Riis the individual regret of the opponent.The compromise solution Aiis a feasible solution thatis closestto the idealsolution,where compromise means an agreement established by mutual concessions.

3.Information aggregation on FVIKOR outcome

Suppose there are p decision makers participating in an MCGDM problem.The decision matrix given by the k th decision maker(DMk)is depicted aswhich there are m alternatives A1,...,Am;and n criteria c1,...,cn.Also,is the rating of alternative Aiwith respect to the criterion cj,whose weight given by DMkis

are crisp,it is easy to integrate a decision maker’s judgment by common methods such as additive weighting.Otherwise,the problem belongs to fuzzy

Among the various VIKOR methods applied to the crisp-number MCGDMproblems,we find there existfour main differences.

(i)Differentmethods used to aggregate group opinions

Currently,three approaches are available to aggregate group judgments for VIKOR:

·Let the minimum of all decision makers’minimal judgments be the left value,and the maximum of all decision makers’maximaljudgments be the rightvalue.Furthermore,let the average of all decision makers’middle judgments be the medium value[19,20,28].That is,the aggregate fuzzy valuecan be expressed as

It is obvious that if the left value of some TFN is very small,or the rightvalue is very large,the aggregate rating will be greatly influenced.Thus,a single decision maker could dominate the final outcome and generate biased results.

·The aggregate value is the average of individuals’judgments[25–27,29,36–38]:

The average may approximate a group’s opinion fairly only when their variances are small.Otherwise,the results may be inappropriately influenced by extreme judgments.

·The third approach is to combine differentoperators, as found in[21,23,33].

where wk∈[0,1],k∈(1,2,...,p)

(ii)Differentpositive and negative idealsolutions

The positive and negative ideals are the maximal and minimal values in an MCDM problem.The ideal may be the maximal(or minimal)values of the left,the medium and the rightvalues in the TFNs,as expressed below:

(iii)Information loss during the defuzzification of TFNs

Definition 5Let m(TFN)be the defuzzified value of a TFN.Information loss may take place in the defuzzification procedure of the TFN,in which m(TFN)cannotaccurately representallthe information provided by the TFN.

Different defuzzification processes will generate different values and information loss.The difference from(see(4))under the criterion cjmay have negative number where rijis a TFN and

may be less than zero althoughthus the division law cannot be used to compute Dpias the computational results willviolate Definition 2.The defuzzification procedure(Definitions 3 and 4)therefore becomes necessary.

(iv)Differentdefuzzification methods for TFNs

Information loss may affectfinalresults and change the compromise solutions.Itis importantto avoid(or reduce) information loss so as to improve the accuracy of Siand Ri.One can reduce information loss by selecting an appropriate defuzzification method.

4.The proposed method

Consider an MCGDM problem with p decision makers

and TFNs.Its decision matrix given byare the fuzzy judgments and weights underthe criterion cj.

To address the issues discussed in Section 3, which are how to aggregate group opinions and how to select an appropriate defuzzification method if it is necessary to defuzzify TFNs,we propose a method to tackle fuzzy MCGDM problems when both the criteria values and weights are triangular.The improved VIKOR model is presented in Fig.1 and detailed below.

Step 1Normalize the TFNs

The criteria can in generalbe classified into two groups: benefit criterion set B and cost criterion set C.Differenttypes ofcriteria have differentnormalization methods,

Fig.1 The proposed VIKORmethod

which can be depicted as follows:

The corresponding normalized decision matrix isSimilarly,the normalized criteria weights can be described as

Step 2Aggregate group judgments

To avoid the effectof extreme opinions,we need to employ an effective aggregation method to integrate all decision makers’judgments.Thus,we formulate an optimization modelto derive the group value as follows:

which in turn can be simplified as model(13)

By using Lingo 9.0,we are able to optimize the nonlinear programming model.We define the resulting inte-

Step 3Selectthe maximaland minimal TFNs

Step 4

may not satisfy Definition 2 if any of the elements in~Fijis negative,as which invalidates the division law.Such dilemma can be resolved by defuzzification butithas notbeen addressed in literature.

Defuzzification will cause information loss(Definition 5).To mitigate information loss,one can employ(3) to defuzzify if the variance of the three elements in a TFN is smaller than the threshold number set by the group in advance.Otherwise,(2)can be applied directly.

Step 5Compute Si,Ri,Qi

where Siis the group utility value and Riis the regretvalue of alternative Ai,respectively.

where v is a weight and Qiis the composite value of Aibased on Siand Ri.We can then sort Qiin the ascending order,and consider the alternative with the smallest Qias the bestalternative.

Step 6Select the compromise solution based on Qi,Si,Ri

Name the alternative with the minimum Qias A1.A1will be the unique,best alternative,if the following two conditions are satisfied:

(i)Condition 1:Acceptable advantage(improvement exceeds a certain level)

where A2is the alternative which is ranked second by Qi, and m is the numberof alternatives.

(ii)Condition 2:Acceptable stability

Alternative A1must also be placed at the top when ranked by Siand Ri.

If either condition described above is notsatisfied,then there exista setof compromise solutions:

(i)If only Condition 1 is satisfied,then alternatives A1and A2are compromise solutions.

(ii)If Condition 1 is not satisfied,then alternatives A1,...,AMare all compromise solutions.Note that the index M can be determined through Q(AM)?Q(A1)＜1/(m?1).

5.Numericalexamples

For the sake of comparison,we draw on the data given by Fatih Emre Boran(2009)to help an automotive company selectthe mostappropriate supplier for one of its key manufacturing elements.Four executives were invited to participate in decision making to selectfrom five suppliers.

Four evaluation criteria are thought-out:

c1:productquality,

c2:relationship closeness,

c3:delivery performance,

c4:price.

The judgmentmatrices are given in Tables 1–4.

Table 1 MCDMmatrix of DM1

Table 2 MCDMmatrix of DM2

Table 3 MCDMmatrix of DM3

Table 4 MCDMmatrix of DM4

The decision making process described below follows the steps given in Fig.1.

Step 1Normalize the TFNs

The results are shown in Tables 5–8.

Table 5 Normalization matrix of DM1

Table 6 Normalization matrix of DM2

Table 7 Normalization matrix of DM3

Table 8 Normalization matrix of DM4

Step 2Aggregate group judgments(see Table 9).

Table 9 Group decision matrix

Note:The actual elementis 10?5times of each element in this table.

The criteria weights ofthe fourdecision makers are displayed in Table 10.

Table 10 Criteria weights of the decision makers

The resulting weights are(0.162,0.256 5,0.446 5), (0.083 75,0.225 25,0.450 75),(0.19,0.294,0.542 75)and (0.095 5,0.224 75,0.465 5)respectively.

Step 3Selectthe maximaland minimalTFNs(see Table 11)

Table 11 The maximal and minimalvalues

Note:The actual elementis 10?5times of each element in this table.

Step 4Calculate

Table 12 Weighted criteria

Step 5Calculate Si,Ri,Qi(see Tables 13 and 14)

Table 13 Values of Siand Ri

Since there are negative numbers inwe should defuzzify them.The centroid defuzzification method is used as their variances are 2.308,1.265,0.830 and 0.489,respectively.

Table 14 Values of Qi

Step 6The compromise solution in Table 8 is A5,as the value of Q(A1)?Q(A5)for each column of v is less than 0.25,and A5is also the best alternative since S5and R5are 4.144 and 1.435 correspondingly.

The computational results based on the traditional approaches((5)and(6))can be obtained.Compared with them,we find the maximal values of each criterion have changed as shown in Tables 7.

Since the lower Qiis preferred(see Step 6),alternatives A5and A1are thus ranked first and second,respectively.From Tables 8,we can determine the differences of Q(A1)?Q(A5)and presentthem in Table 15.

Table 15 Comparisons based on Q(A1)?Q(A5)

Table 15 shows that regardless of the value of v,the compromise solutions are A1and A5,when the average method is used to integrate group evaluation.In contrast,when the optimal and the maximal/minimal methods are applied,the compromise solution becomes A5.Although the results given by the maximal/minimal method are the same as the optimalmethod,they are often closer to 1/(m?1)than those derived by the optimal method. Therefore,the differences(i.e.Q(A1)?Q(A5))derived from the proposed optimal method are more pronounced, and the optimization method can offer a higher degree of the discriminating powerin contrasting alternatives.

In the paper,the deviations of integrated group judgments and group criteria weights are fewer than those of the other methods,therefore itis proven thatthe proposed method is more effective.

Applying the integration method in(5),we willfind the computationalresults in Table 16.

Table 16 Group decision matrix by using(5)

Similarly,the integrated weights become(0.12,0.256, 0.5),(0.048,0.225,0.6),(0.19,0.294,0.6)and(0.08, 0.225,0.6)respectively,and the maximaland minimalvalues are shown in Table 17.

Table 17 The maximal and minimalvalues in Table 16

Table 18 Weighted criteria by using(5)

Table 19 Values of Siand Riby using(5)

Table 20 Values of Qiby using(5)

The compromise solution is also A5.If we apply the integration method in(6),the results can be summarized in Table 21.

Table 21 Group decision matrix by using(6)

The integrated weights are(0.162,0.256,0.446),(0.084, 0.225,0.451),(0.19,0.294,0.543)and(0.096,0.225, 0.465)respectively,the maximal and minimal values are shown in Table 22,the weighted criteria is in Table 23,the values of Siand Riand Qiare in Tables 24 and 25.

Table 22 The maximaland minimal values in Table 21

Table 23 Weighted criteria by using(6)

Table 24 Values of Siand Riby using(6)

Table 25 Values of Qiby using(6)

The compromise solution is A1and A5.If we apply the integration method in(5),the results can be summarized in Tables 26 and 27.

Table 26 Deviation ofevaluation values by using(5)

Table 27 Deviation of criteria weights by using(5)

Ifwe apply the integration method in(6),the results can be summarized in Tables 28 and 29.

Table 28 Deviation ofevaluation values by using(6)

Table 29 Deviation ofcriteria weights by using(6)

If we apply the proposed method,the results can be summarized in Tables 30 and 31.

Table 30 Deviation of evaluation values by using our method

Table 31 Deviation of criteria weights by using our method

The deviations of the evaluation values with the three methods are 4.043 039,1.796 858 and 1.796 respectively, and the deviations of the criteria weights with the three methods are 0.344 381,0.130 607 and 0.130 402 respectively.The deviations of the evaluation values and criteria weights are minimalby the proposed method in the paper.

6.Conclusions

The VIKOR method is an effective decision making aid for tackling MCDM problems.Much MCDM attention has been directed toward business applications,but little is given to the applications of the VIKOR method to MCGDM problems.

Different from the traditional MCDM,there are various decision judgments given by multiple decision makers in MCGDM problems.To apply the VIKOR method to these problems,we need to aggregate group opinions. The traditionalintegration methods ignore the influence of bigger/smaller judgments on the composite results,which may allow single judgmentto dominate the end result,and reduce the advantage and effectiveness of group decision making.

In this paper,we discuss the disadvantages of the traditionalapproach and the impactof choosing specific methods on compromise solutions.We improve the existing VIKOR method so as to tackle the challenges in judgments integration,which is often encountered in MCDGM problems.The proposed method is applicable to MCDGM problemswith triangularfuzzy numbers,as wellas to those with interval numbers or trapezoidal fuzzy numbers.It is feasible and more effective than the maximal/minimal method and the average method,as evidenced by the numericalexample.

On the other hand,the process of integrating TFNs by operationallaws is necessary to obtain Si,Ri,and Qi.The defuzzification process is called for when the condition in Step 4 is met(i.e.negative elements in TFN),which has not been taken into account in the literature.We also identify the necessary condition for defuzzifying TFNs,and offer guidelines to selectappropriate defuzzification methods to avoid or reduce information loss.

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Biographies

Wenqi Jiangwas born in 1975.He is a Ph.D. and an associate professor in Nanjing University of Science and Technology.He had published many papers which include multi-attribute decision approach,the method of allocating the limited service resource in systems engineering and electronic technology and system engineering.His research interests are decision analysis and service management.

E-mail:wqjiang@ustc.edu

Jennifer Shangwas born in 1960.She is a professor in Pittusburgh University in USA.She has published many papers in Marketing Science and European Journal of Operation Research,etc.Her research interests are multiple attribute decision making analysis.

E-mail:SHANG@katz.pitt.edu

10.1109/JSEE.2015.00085

Manuscript received June 06,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(71271116).