,3

1.Jiangsu Key Laboratory of Urban ITS,Southeast University,Nanjing 210096,China; 2.Jiangsu Province Collaborative Innovation Center of Modern Urban Traf fi c Technologies,Nanjing 210096,China; 3.Departmentof Civiland Environmental Engineering,University of Washington,Seattle 352700,USA

Solution to multiple attribute group decision making problems with two decision makers

FangweiZhang1,2,*,WeiWang1,2,and Xuedong Hua1,2,3

1.Jiangsu Key Laboratory of Urban ITS,Southeast University,Nanjing 210096,China; 2.Jiangsu Province Collaborative Innovation Center of Modern Urban Traf fi c Technologies,Nanjing 210096,China; 3.Departmentof Civiland Environmental Engineering,University of Washington,Seattle 352700,USA

A kind of multiple attribute group decision making (MAGDM)problem is discussed from the perspective of statistic decision-making.Firstly,on the basis of the stability theory,a new idea is proposed to solve this kind of problem.Secondly,a concrete method corresponding to this kind of problem is proposed. The main tool of our research is the technique of the jackknife method.The main advantage of the new method is that it can identify and determine the reliability degree ofthe existed decision making information.Finally,a traf fi c engineering example is given to show the effectiveness ofthe new method.

multiple attribute group decision making(MAGDM), stability theory,jackknife method,credibility degree,traf fi c engineering.

1.Introduction

Recently,the multiple attribute group decision making (MAGDM)theories have been deeply researched[1-4]. A lot of methods and techniques have been proposed to solve different kinds of MAGDM problems.Overall,the characteristics of these research achievements are multidiscipline combination and comprehensive practicability. However,because ofthe variety ofthe decision making environments,some studies on MAGDM problems are neither systematical nor thorough.For example,there is a kind of MAGDM problem,in which there are two decision makers only[5-7],and the decision making power of them is not provided[8-10].If the information of the two decision makers does not conform with each other,it is usually dif fi cult to compare the decision making information of them.In addition,it is also dif fi cult to make a persuasive decision.This kind of problem is thought to be a statistical problem in essence[11-13].In order to effectively solve it,a new method,which integrates the MAGDMproblem and the statisticaltheory,is proposed.

Here,some related researches would be introduced.To solve the MAGDM problems in which the attribute values are expressed in intuitionistic fuzzy numbers,and the weight information about the experts and the attributes is unknown,Xu developed two nonlinear optimization models.These two models provided us a very beautiful way to treat the MAGDM problems[14].With respect to MAGDMproblems with linguistic information of attribute values,Weiproposed a group decision making method.In this method,some new aggregation operators were proposed[15].In the next year,Wei et al.researched a kind of MAGDM problem,in which both the attribute weights and the expertweights took the form of realnumbers,and the attribute values took the form of interval intuitionistic trapezoidalfuzzy numbers.In their research,some new aggregation operators were proposed and some desirable properties of these operators were studied[16].Zhou et al.researched the uncertain linguistic MAGDMproblems, in which the attributes and experts were at different priority levels[17].In their research,motivated by the idea of prioritized aggregation operators,some prioritized aggregation operators for aggregating uncertain linguistic information were developed,and then,they were applied to develop some models for uncertain linguistic MAGDM problems.In the same year,Wei et al.researched a kind of MAGDM problem,in which the attribute values took the form of triangular fuzzy information[18].In their research,some power aggregation operators for aggregating triangular fuzzy information were developed,some models for MAGDM problems with triangular fuzzy information were proposed.Further,Wei researched a kind of MAGDM problem,in which the attribute values took theform of linguistic variables.In their research,some power aggregation operators foraggregating 2-tuple linguistic information were proposed[19].Besides,Zhu investigated a kind of dynamic multi-attribute decision making problem, with the research tool of advantage retention degree.The thoughtof their work is similar to this paper[20].

Based on the above research achievements,a train of thoughts,which are used for solving the proposed kind of MAGDM problem,have formed.With this in view,this paper is structured as follows.In Section 2,the researched problem is introduced.In Section 3,the problem is analyzed.In Section 4,a new decision making method is proposed.In Section 5,an example is discussed to demonstrate the implementation of the new method.In Section 6, conclusions and projective of the study are presented.

2.The problem

In this section,the given problem would be introduced by mathematicallanguage.The details are as follows.Let X={x1,x2,...,xm}be the discrete set of m feasible alternatives,and F={f1,f2,...,fn}be the fi nite set of attributes,let yij=fj(xi)(i=1,2,...,m;j= 1,2,...,n)be the value of the alternative xiunder the attribute fj.And the situation that the attribute values are given in the form of real numbers is only considered. In this paper,the attribute value yijis given by two experts,and the decision making power of them is not provided.For convenience,the two experts are denoted as d1and d2.D is denoted as D={d1,d2},and yijtis denoted as the attribute value that is provided by the expert dt(t∈T),T={1,2}.Besides,it is denoted that M={1,2,...,m},N={1,2,...,n}.

The decision matrix,which is of attribute set F,with regard to the set X,and provided by expert dt(t∈T),is expressed as

Here,itis supposed thatthe decision matrix Y has been normalized[21,22].

The attribute weightvector is denoted as

where wj(j∈N)re fl ects the relative importance degree ofthe attribute fj.In this paper,the research situation is defi ned as that wjis given as the realnumber.On the above conditions,the problem is how to sortand rank the alternatives.The given problem would be analyzed in the following section.

3.Problem analysis

3.1 Classicalsolutions to the given problem

In order to solve the given two-people MAGDMproblem, the classicalsolutions are as follows.

Firstly,by using the simple additive weighting method [23-25],the decision matrices which are proposed by the two decision makers could be calculated,and two comprehensive attribute value vectors could be got.Ifthe decision matrix is proposed by the fi rst decision maker,the comprehensive attribute value vector would be denoted as Z1, while the vectorwould be denoted as Z2when the decision matrix is proposed by the second decision maker.Details are as follows:

Secondly,for any alternative,take xi(i∈M)as an example.

could be denoted.

Then,a vector is got,which is

Thirdly,by comparing the z?i(i∈M),the optimal alternative would be selected and all the alternatives would be ranked.

This approach has a lot of advantages,however,it also has many disadvantages at the same time.According to the above steps,the biggestde fi ciency is thatthe decisionmaking power of the two decision makers are valued virtually the same.Because there is no information aboutthis decision-making power,the dealing seems unreasonable.

3.2 New train of thought for the given problem

In this subsection,by analyzing the given problem,a new train of thoughtis given.

It is thoughtthatthe core of solving the given problem is to determine the decision-making power of the two decision makers.Besides,the decision-making power of the two decision makers is thought to be associated with the degree of extreme tendency of their values[26,27].This problem would be analyzed in detail,and ways would be soughtto solve the problem as follows.

Firstly,as a whole,the decision-making information, which is given by the decision makers,is believed to follow the principle ofindependence,objectivity and impartiality. The extreme tendency decision-making information,if it exists,may only exist in a/a few local location/locations, and under one attribute/a few attributes.

Secondly,the decision-making power of the two decision makers,could be indexed to representthe trustdegree of the corresponding attribute value[28].Taking yijt(i∈M,j∈N,t∈T)as an example,in regard to fi,if yijtis subjective and extreme,it would be thoughtthat the trust degree of dtfor Wjshould be lower.

Based on the above two points,a new train of thought has formed.The new thought is borrowed from the jackknife method.That is,each attribute would be cut out in turn,and the decision making would also be carried out according to the rest of the attribute values in turn.Finally,by dealing with the above decision-making results, the decision-making goal could be obtained.It is worth mentioning thatthis decision-making solution would have a trustdegree of the fi nal decision results.In the nextsection,this solution would be proposed in details.

4.Decision method

In the process of estimating the uncertainties of an event, sometimes standard and classicalmethods are unavailable or inconvenient.In this case,a usefulstatisticaltoolwhich is called the jackknife method[29,30]may be resorted to.The jackknife method was proposed in 1956 by Quenouille.The core ofthe jackknife method is simply a leaveout-one procedure.In this paper,the jackknife method is used as a tool to estimate the trust degree of the decision making information,which is provided by the two decision makers.Essentially,this process is equivalent to identifying the decision making power ofthe two decision makers. The new decision making method is as follows.

Step 1For the convenience of the research,denote

Then,one would denote

Step 2By cutting outfrom Y1,and cutting outfrom Y2,two new matrices

could be got.Meanwhile,

could be got.

By using the simple additive weighting method,could be calculated,and the result is denoted asIn the same wayc ould be gotby calculating

Step 3By using(2),the vector

Step 4Following the similar processes of Step 2 and Step 3,the optimalalternativecould be got too.Accordingly,the valuesη2,η3,...,ηncould also be got.

Step 5By counting the times thateach alternative ap-

subsets.The m subsets are denoted asAccordingly,the set{η1,η2,...,ηn}could be divided into m subsets too.The m subsets are denoted as

Step 6Takerandomly,and add the elements of(i∈M)together.The calculation result is denoted as?ηi.Similarly,a vector(?η1,?η2,...,?ηm)could be got.Denote

By comparing the valuesρi(i∈M),the optimal attribute X?could be got,and the trust degree of X?could also be got,which is

5.Application example

In this section,a lane layoutproblem is introduced.There are three public transportlane layoutalternatives X1,X2, X3to be chosen,and the corresponding attributes are the saturation fl ow rate S,the maximum sum of critical-lane volumes V,the average over fl ow delay of vehicles De, and the economic costs E.The weightvector for these attributes is

In order to solve this problem,two experts are invited to estimate the attribute values with different theories.By normalization,the two decision-making matrices could be obtained as

By using the method given in this paper,this problem could be solved.The speci fi c steps are as follows:

Step 1For the convenience of research,denote

Then,one would denoteStep 2By cutting outfrom Y1,and cutting outfrom Y2,two new matrices

could be got.

Meanwhile,

could be got.

By using the simple additive weighting method, could be got.

Step 3By using(2),

could be got,then the optimalalternativecould be got too,which is x3with a trustdegree 0.68.

Step 4Following the similar processes of Step 2 and Step 3,one could get

Accordingly,the degree of trust for them could be got, which are 0.77,0.75,and 0.80,respectively.

Step 5By counting the times thateach alternative ap-

Step 6For each(i=1,2,3),the elements ofcould be added together.By comparing these calculation results,the optimal alternative x1with a trust degree 50.49%,is fi nally got.

6.Conclusions

From the above example,it could be found that the new method is effective in solving the given kind of MAGDM problem.The main advantage ofthis method is thatitdeals with the decision-making power of the decision makers well.Thus,this new method may have a better application prospect.

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Biographies

Fangwei Zhang was born in 1980.He received his M.S.degree of engineering in system theory from Liaocheng Unversity in 2003,and Ph.D.degree in controlscience and engineering from Naval Aeronautical Engineering Institute.He is recently a postdoctor in Southeast University.His research interests include multiple attribute decision making theory,optimization and its applications.

E-mail:fangweizhang@aliyun.com

Wei Wang was born in 1959.He received his M.S. degree and Ph.D.degree both in transportation engineering from Southeast University,in 1985 and 1989 respectively.His research interests include transportation planning and management,optimization and its applications.

E-mail:wangwei@seu.edu.cn

Xuedong Hua was born in 1987.He received his B.S.degree of transportation engineering from Southeast University in 2010.His research interests include optimization and its applications transportation planning and management.Currently,he is pursuing his Ph.D.degree in transportation engineering in Southeast University.

E-mail:5874175@qq.com

10.1109/JSEE.2015.00038

Manuscriptreceived March 15,2014.

*Corresponding author.

This work was supported by the National Key Basic Research Program of China(973 Program)(2012CB725402),the National High-Tech R&D Program of China(863 Program)(SS2014AA110303),and the Science Foundation for Post-doctoral Scientists of Jiangsu Province(1301011A).