# A Method for Multicriteria Group Decision Making with Different Evaluation Criterion Sets

Authors: Zhi-Ping Fan; Ming-Yang Li; Yang Liu; Tian-Hui You
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Multicriteria group decision making (MCGDM) with different evaluation criterion sets is a special kind of MCGDM problem, where criterion sets considered by multiple experts may be different, while research concerning this issue is still relatively scarce. The objective of this paper is to develop a method for MCGDM with different evaluation criterion sets. In the method, according to different criterion sets, several criterion subsets are first constructed, where each criterion subset includes the criteria concerned by the same group of experts. Then, with respect to each criterion subset, a ranking of alternatives is determined by normalizing the decision matrix and calculating the ranking value of each alternative with respect to the criterion subset. Next, according to the ranking of alternatives with respect to each criterion subset, a ranking possibility matrix of alternatives with respect to the criterion subset is constructed. Further, the weight of each criterion subset is determined according to weights of the experts and weights of the criteria in the criterion subset. Moreover, an overall ranking possibility matrix is constructed by aggregating the ranking possibility matrices and weights concerning different criterion subsets, and the final ranking result of alternatives is determined by solving a linear assignment model, where the elements in the overall ranking possibility matrix are regarded as the benefits on assigning each alternative into different ranking positions. Finally, a numerical example is given to illustrate the use of the proposed method.


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## Body

## 1. Introduction

Multicriteria group decision making (MCGDM) refers to the problem of classifying or ranking the alternatives based on the opinions provided by multiple experts concerning multiple criteria [1–3], which is a valuable research topic with extensive theoretical and practical backgrounds [4–8]. For example, to select a desirable product design scheme from several alternatives, it is necessary to make a decision according to the opinions concerning multiple criteria provided by multiple R&D experts such as financial experts and quality management experts [9]. For the recruitment of technicians, the desirable employee(s) will be selected by the human resource department and the technical department according to performances concerning multiple criteria [10–12]. For the determination of the winner in multiattribute procurement auction, the opinions of multiple financial experts and technique experts concerning the financial criteria and the technique criteria should be used [13, 14]. Therefore, how to solve the MCGDM problem is a valuable research topic.The MCGDM problem has attracted the attention of many scholars, and many methods have been proposed [15–27]. Tavana et al. [15] presented a framework to help decision-makers develop the group decision support system combining the analytic hierarchy process and the Delphi principles. Herrrera-Viedma et al. [16] proposed a MCGDM framework with linguistic preference relations. Li [17] developed a compromise ration method for fuzzy MCGDM problem. Merigό et al. [18] introduced the uncertain generalized probabilistic weighted averaging (UGPWA) operator to solve the MCGDM problem. Rosellό et al. [19] proposed a method for MCGDM under multi-granular linguistic assessment environment. Merigó et al. [20] introduced linguistic probabilistic weighted average (LPWA) operators to develop more efficient decision-making systems, and its main advantage is to consider subjective and objective information in the same formulation. Zavadskas et al. [21] extended the application of the MULTIMOORA method for group decision making in the uncertain environment and developed the interval-valued intuitionistic fuzzy MULTIMOORA method for MCGDM. Chen et al. [22] proposed a new fuzzy MCGDM method based on intuitionistic fuzzy sets and the evidential reasoning methodology. Chu et al. [23] investigated the consistency checking process, the consensus checking process, and the selection process with respect to additive intuitionistic fuzzy preference relation. Liu et al. [24, 25] considered the problems of ranking products according to the online reviews given by a large group of consumers and proposed the methods based on the sentiment analysis technique, the intuitionistic fuzzy set theory, and interval-valued intuitionistic fuzzy TOPSIS. Zhang et al. [26] proposed a consensus improving method based on the consensus criteria and introduced the whole group decision-making process based on the aggregation operators for the probabilistic linguistic term sets. Liu et al. [27] developed a novel decision-making method to solve MCGDM problems in which the experts have different priority levels and the criteria values are in the form of Pythagorean fuzzy uncertain linguistic variables.Prior studies have made significant contributions to MCGDM analysis. In most of the existing MCGDM studies, one criterion set is considered in group decision-making analysis. However, in reality, the criterion sets considered by multiple experts may be different since the experts are usually from different organizations or departments, and each expert may pay more attention to the criteria that relate to his/her own benefit or duty. For example, to determine the winner in multiattribute procurement auction, the opinions of multiple financial experts and technique experts should be considered, where the financial experts mainly pay attentions to the price, tax rate, and so on, while the technical experts mainly pay attentions to the technical indicators of goods from bidders. Currently, the studies on MCGDM method with different criterion sets are relatively scarce. Roy and Maji [28] presented a fuzzy soft set theoretic approach to solving the MCGDM problem. The approach involves construction of a comparison table from a fuzzy soft set in a parametric sense for decision making. Çaĝman and Enginoĝlu [10] constructed uni-int decision-making method by using the definitions of soft sets and uni-int decision function. In the method, the evaluation information concerning different criterion sets from two decision-makers is described using soft set, and the evaluation information of two decision-makers is disposed to reduce a large alternative set into the subset of alternatives. Feng et al. [29] extended the method proposed by Çaĝman and Enginoĝlu. They gave the definitions of satisfaction relations and developed a new algorithm to screen the desirable alternatives. Also, they pointed out that the uni-int decision making is a special case for the method proposed in their paper. Li et al. [30] proposed three rules for screening alternatives and gave the calculation process of screening alternatives based on soft sets theory. It is necessary to point out that, in the MCGDM problem with different evaluation criterion sets, a criterion may be concerned by one, two, or more experts. It is reasonable to consider different evaluation criterion sets concerned by multiple experts in group decision-making process, but different evaluation criterion sets are not considered in the existing MCGDM method. Therefore, it is necessary to develop a new decision-making method to solve MCGDM problem mentioned above. This is the research motivation of this paper.The objective of this paper is to develop a method for MCGDM with different evaluation criterion sets. In the method, according to the criterion sets considered by multiple experts, several criterion subsets are first constructed, where each criterion subset includes the criteria concerned by the same group of experts. Then, with respect to each criterion subset, a ranking of alternatives is determined by normalizing the decision matrix and calculating the ranking value of each alternative with respect to the criterion subset. According to the ranking of alternatives with respect to each criterion subset, a ranking possibility matrix of alternatives with respect to the criterion subset is constructed. Further, the weight of each criterion subset is determined according to the weights of experts and the weights of criteria in the criterion subset. Moreover, an overall ranking possibility matrix is constructed by aggregating the ranking possibility matrices and weights concerning different criterion subsets, and the final ranking result of alternatives is determined by solving a linear assignment model, where the elements in the overall ranking possibility matrix are regarded as the benefits on assigning each alternative into different ranking positions.The rest of this paper is arranged as follows. Section2 describes the MCGDM problem with different evaluation criterion sets. Section 3 presents a decision-making method for solving the MCGDM problem. In Section 4, a numerical example is used to illustrate the applicability of the proposed method. Finally, we summarize and highlight main features of the proposed method in Section 5.


## 2. Problem Description

The following notations are used to denote the sets and variables in the MCGDM problem with different evaluation criterion sets, which are used throughout this paper.(i)A={A1,A2,⋯,Am}: the set of m alternatives, where Ai denotes the ith alternative, i=1,2,⋯,m.(ii)E={E1,E2,⋯,Eq}: the set of q experts, where Ek denotes the kth expert, k=1,2,⋯,q.(iii)b=(b1,b2,⋯,bq): the weight vector of experts, where bk denotes the weight or importance degree of expert Ek, k=1,2,⋯,q.(iv)C={C1,C2,⋯,Cn}: the set of all criteria considered by the experts, where Cj denotes the jth criterion, j=1,2,⋯,n.(v)Ck: the set of criteria concerned by the expert Ek, k=1,2,⋯,q, Ck⊂C, C1∪C2∪⋯∪Cq=C. If Ck=⌀, then it means that Ek does not concern any criteria and it is not necessary for Ek to participate the decision analysis as an expert. Thus, we assume Ck≠⌀ in this paper.(vi)wk=(w1k,w2k,⋯,wnk): the weight vector of criteria provided by Ek, where wjk denotes the weight or importance degree of evaluation criterion Cj. If Cj∉Ck, i.e., Ek does not concern criterion Cj, then wjk=0; if Cj∈Ck, then 0<wjk≤1. According to the above description, we know that b1,b2,⋯,bq have been used to represent the differences among the weights. Thus, w1k,w2k,⋯,wnk are only used to reflect the differences among the weights of different criteria in Ek’s mind, k=1,2,⋯,q. For this, the sum of w1k,w2k,⋯,wnk should be unified, i.e., ∑j=1nwjk=1, k=1,2,⋯,q. The w1k,w2k,⋯,wnk can be obtained either directly from the expert or indirectly using the existing procedures such as AHP [31, 32].(vii)V=[vij]m×n: the decision matrix, where vij denotes the value of alternative Ai with respect to criterion Cj, i=1,2,⋯,m, j=1,2,⋯,n.Without loss of generality, the criteria can be classed into two types: the benefit criterion and the cost criterion [33–35]. The benefit criterion means that the larger the value of an alternative with respect to the criterion is, the better the alternative will be; and the cost criterion means that the smaller the value of an alternative with respect to the criterion is, the better the alternative will be. Let CB and CC be the sets of benefit and cost criteria, respectively, CB∩CC=⌀, CB∪CC=C.The problem addressed in this paper is how to rank alternatives or select the most desirable alternative(s) among the setA using weight vector b, weight vector wk, and decision matrix V.To illustrate the MCGDM problem with different evaluation criterion sets more clearly, Figure1 is given as an example, where two experts E1 and E2 are considered. In Figure 1, the criterion sets concerned by E1 and E2 are different, i.e., the criterion set concerned by E1 is {C1,C2,C3,C4,C5,C6} and the criterion set concerned by E2 is {C4,C5,C6,C7,C8}. According to different criterion sets concerned by the two experts, three criterion subsets can be constructed, i.e., the criterion subset {C1,C2,C3} concerned by E1, the criterion subset {C7,C8} concerned by E2, and the criterion subset {C4,C5,C6} concerned by both E1 and E2. To determine the ranking of alternatives, it is necessary to consider the performance of each alternative with respect to the above three criterion subsets.Figure 1
The MCGDM problem with two different evaluation criterion sets.


## 3. The Method

To solve the above MCGDM problem with different evaluation criterion sets, a method is proposed in this section. The resolution procedure of the method is shown in Figure2. In the method, according to the criterion sets concerned by different experts, several criterion subsets are first constructed, where each criterion subset includes the criteria concerned by the same group of experts. Then, with respect to each criterion subset, a ranking of alternatives is determined by normalizing the decision matrix and calculating the ranking value of each alternative with respect to the criterion subset. According to the ranking of alternatives with respect to each criterion subset, a ranking possibility matrix of alternatives is constructed with respect to the criterion subset. Further, the weight of each criterion subset is determined according to the weights of experts and the weights of criteria in the criterion subset. Moreover, an overall ranking possibility matrix is constructed by aggregating the ranking possibility matrices and the weights concerning different criterion subsets, and the final ranking results of alternatives is determined by solving a linear assignment model, where the elements in the overall ranking possibility matrix are regarded as the benefits on assigning each alternative into different ranking positions.Figure 2
The resolution procedure for solving the MCGDM problem with different evaluation criterion sets.


### 3.1. Constructing Criterion Subsets Concerned by Different Groups of Experts

According to the resolution procedure shown in Figure2, several criterion subsets are first constructed, where each criterion subset includes the criteria concerned by the same group of experts.LetP(E) be the power set of E, i.e., P(E)={⌀,{E1},{E2},⋯,{Eq},{E1,E2},⋯,{E1,E2,⋯,Eq}}; let P~(E)=P(E)∖⌀; i.e., P~(E) is the subset of P(E) by deleting ⌀ from P(E). Let E′ denote a subset of experts (i.e., E′∈P~(E)); let C(E′) denote the criterion subset concerned by the experts in E′. For example, if E′={E1}, then C(E′)=C({E1}), and C(E′) denotes the criterion set, in which the criteria are only concerned by E1. If E′={E1,E2}, then C(E′)=C({E1,E2}), and C(E′) denotes the criterion set, in which the criteria are concerned by E1 and E2. If E′={E1,E2,⋯,Eq}, then C(E′)=C({E1,E2,⋯,Eq}), and C(E′) denotes the criterion set, in which the criteria are concerned by all the experts. To determine C(E′) concerning each E′ (E′∈P~(E)), the process for constructing criterion subsets considered by different groups of experts is given below.Firstly, an indicator matrixT=[tkj]q×n is constructed, tkj=0 or 1, where tkj=1 if the criterion Cj is concerned by Ek (i.e., Cj∈Ck); otherwise tkj=0, k=1,2,⋯,q,j=1,2,⋯,n. An example on the construction of indicator matrix T=[tkj]q×n is given below.Example 1.
Consider a MCGDM problem with three experts (E1,E2,E3), and there are eight criteria concerned by the experts, i.e., C={C1,C2,⋯,C8}. The criteria concerned by E1, E2, and E3 are C1={C1,C2,C5,C7,C8}, C2={C2,C3,C4,C6,C7,C8}, and C3={C2,C3,C6,C7}, respectively. Thus, indicator matrix T=[tkj]3×8 can be constructed, as shown in Table 1.Table 1
The indicator matrixT=[tkj]3×8.
C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 E 1 1 1 0 0 1 0 1 1 E 2 0 1 1 1 0 1 1 1 E 3 0 1 1 0 0 1 1 0Then, according to the constructed indicator matrixT=[tkj]q×n, an indicator vector tj=[t1j,t2j,…,tqj]T concerning criterion Cj can be determined, j=1,2,⋯,n. Thus, T=[tkj]q×n can be further represented by T=[tkj]q×n=[t1,t2,⋯,tn], where vector tj is the indicator vector concerning criterion Cj, j=1,2,⋯,n.Further, lete(E′)=[e1,e2,…,eq]T denote the indicator vector corresponding to set C(E′), ek=0 or 1, where ek=1 denotes that expert Ek belongs to the set E′; otherwise ek=0, k=1,2,⋯,q.The following Example2 is given to illustrate the construction of indicator vector e(E′)=[e1,e2,…,eq]T.Example 2.
Consider the MCGDM problem shown in Example1. The subset of experts is P~(E)={{E1},{E2},{E3},{E1,E2},{E1,E3},{E2,E3},{E1,E2,E3}}. Obviously, there are seven indicator vectors e(E′)=[e1,e2,…,eq]T, i.e.,(1)eE1=100,eE2=010,eE3=001,eE1,E2=110,eE1,E3=101,eE2,E3=011,eE1,E2,E3=111.Property 1.
In a MCGDM problem withq experts, there are 2q-1 indicator vectors e(E′)=[e1,e2,…,eq]T.Proof.
Since there are two possible values ofek,ek=0 or 1, there are 2q possible combinations of the q binary variables. Since e({⌀})=(0,0,…,0) is not considered, there are 2q-1 possible indicator vectors e(E′)=[e1,e2,…,eq]T.
Next, according to the indicator vectortj=[t1j,t2j,⋯,tqj]T and e(E′)=[e1,e2,…,eq]T, the distance between tj and e(E′) can be calculated using (2), i.e.,(2)dtj,eE′=∑k=1qtkj-ek2,j=1,2,⋯,n,E′∈P~EIt can be seen from (2) that Cj∈C(E′) if d(tj,e(E′))=0; otherwise Cj∉C(E′).


### 3.2. Ranking Alternatives with respect to Each Criterion Subset

First, according to the decision matrixV=[vij]m×n, the normalized decision matrix V-=[v-ij]m×n can be constructed, where v-ij denotes the normalized value of vij. It can be calculated by the following (3), i.e.,(3)v-ij=vij-vjminvjmax-vjmin,i=1,2,⋯,m;j∈CBvjmax-vijvjmax-vjmin,i=1,2,⋯,m;j∈CCwhere vjmin=min⁡{vij∣i=1,2,⋯,m} and vjmax=max⁡{vij∣i=1,2,⋯,m}, j=1,2,⋯,n.Then, the ranking value of alternativeAi with respect to the criterion subset C(E′) can be calculated, i.e., (4)VE′Ai=∑Cj∈CE′Ek∈E′bkwjkv-ij,i=1,2,⋯,mAccording to the ranking values,VE′(A1), VE′(A2),…,VE′(Am), the ranking order of alternatives (A1,A2,⋯,Am) with respect to the criterion subset C(E′) can be determined. Let rE′=[risE′]m×m denote the ranking possibility matrix concerning the criterion subset C(E′), where risE′ denotes the possibility that alternative Ai is ranked at the sth position with respect to the criterion subset C(E′), i,s=1,2,⋯,m. As for the determination of the value of risE′, the following two possible cases are considered.① If and only ifAi itself with respect to criterion subset C(E′) is ranked at the kth position, then risE′ can be determined by(5)risE′=1,s=k0,s≠k,i=1,2,⋯,m② Ifg alternatives including Ai are ranked at the kth position, then risE′ can be given by(6)risE′=1g,s=k,k+1,⋯,k+g-10,otherwise,i=1,2,⋯,mTo illustrate the determination ofrE′=[risE′]m×m, the following Example 2 is given.Example 2.
There are 5 alternatives (A1,A2,A3,A4,A5) in a MCGDM problem, and the ranking order of alternatives with respect to the evaluation criteria set C(E′) is A2≻A3≻A1≻A5≻A4. The ranking possibility matrix can be constructed as (7)rE′=risE′5×5=A1A2A3A4A51st2nd3rd4th5th010000.500.5000.500.5000000100010


### 3.3. Determining the Weight of Ranking Result Concerning Each Criterion Subset

To determine the final ranking result of alternatives by aggregating the ranking possibility matrices concerning different criterion subsets, it is necessary to determine the weight of ranking result concerning each criterion subset. Since each criterion subset includes the criteria concerned by a group of experts, it is reasonable that the weight of ranking result concerning each criterion subset is calculated by both the weights of experts and the weights of evaluation criteria concerned by different experts. The weight of criterion subsetC(E′) can be obtained by(8)wE′=∑Cj∈CE′Ek∈E′bkwjk,E′∈P~E


### 3.4. Determining the Final Ranking Result of Alternatives

Based on the ranking possibility matrixrE′=[risE′]m×m and weight wE′ concerning criterion subset C(E′), the overall ranking possibility matrix P=[pis]m×m can be constructed, where pis denotes the overall possibility that alternative Ai is ranked at the sth position, and pis can be calculated by(9)pis=∑E′∈P~EwE′·risE′Based on the overall ranking possibility matrixP=[pis]m×m, the final ranking result of alternatives can be determined. In this paper, we want to obtain a clear alternative ranking results, i.e., each alternative is only ranked at one position, and each position only contains one alternative. Thus, a linear assignment model is constructed as follows:(10a)maxZ=∑i=1m∑s=1mpisxis(10b)s.t.∑i=1mxis=1,s=1,2,⋯,m(10c)∑s=1mxis=1,i=1,2,⋯,m(10d)x=0or1,i=1,2,⋯,m,s=1,2,⋯,mIn the model ((10a)–(10d)), (10a) is to maximize the sum of ranking possibilities. Equations (10b) and (10c) are equation constrains, i.e., (10b) is to guarantee that each position only contains one alternative, and (10c) is to guarantee that each alternative is ranked at one position. If xis=1, then the alternative Ai is ranked at the sth position. Obviously, model ((10a)–(10d)) is an assignment model. It can be converted into a classical assignment model and be solved by Kuhn-Munkres algorithm [36] or Bertsekas algorithm [37]. According to the solution to model ((10a)–(10d)), the final ranking results of alternatives can be determined.In summary, the procedure for solving the MCGDM problem with different evaluation criterion sets is given as follows.Step 1.
Construct the nonempty power setP~(E) based on the expert set E, and determine criterion subset C(E′) concerning each E′ (E′∈P~(E)).Step 2.
Construct the indicator matrixT=[tkj]q×n, and determine the indicator vector tj concerning criterion Cj.Step 3.
Construct the indicator vectore(E′) corresponding to set C(E′).Step 4.
Calculate the distance betweentj and e(E′) using (2), and determine whether the criterion Cj belongs to the criterion subset C(E′).Step 5.
Construct the ranking possibility matrixrE′ concerning the criterion subset C(E′) using (3)–(6).Step 6.
Calculate the weightwE′ of criterion subset C(E′) using (8).Step 7.
Construct the overall ranking possibility matrixP using (9).Step 8.
Construct the linear assignment model ((10a)–(10d)) based on the overall ranking possibility matrix P.Step 9.
Determine the final ranking result by solving the model ((10a)–(10d)).


## 3.1. Constructing Criterion Subsets Concerned by Different Groups of Experts

According to the resolution procedure shown in Figure2, several criterion subsets are first constructed, where each criterion subset includes the criteria concerned by the same group of experts.LetP(E) be the power set of E, i.e., P(E)={⌀,{E1},{E2},⋯,{Eq},{E1,E2},⋯,{E1,E2,⋯,Eq}}; let P~(E)=P(E)∖⌀; i.e., P~(E) is the subset of P(E) by deleting ⌀ from P(E). Let E′ denote a subset of experts (i.e., E′∈P~(E)); let C(E′) denote the criterion subset concerned by the experts in E′. For example, if E′={E1}, then C(E′)=C({E1}), and C(E′) denotes the criterion set, in which the criteria are only concerned by E1. If E′={E1,E2}, then C(E′)=C({E1,E2}), and C(E′) denotes the criterion set, in which the criteria are concerned by E1 and E2. If E′={E1,E2,⋯,Eq}, then C(E′)=C({E1,E2,⋯,Eq}), and C(E′) denotes the criterion set, in which the criteria are concerned by all the experts. To determine C(E′) concerning each E′ (E′∈P~(E)), the process for constructing criterion subsets considered by different groups of experts is given below.Firstly, an indicator matrixT=[tkj]q×n is constructed, tkj=0 or 1, where tkj=1 if the criterion Cj is concerned by Ek (i.e., Cj∈Ck); otherwise tkj=0, k=1,2,⋯,q,j=1,2,⋯,n. An example on the construction of indicator matrix T=[tkj]q×n is given below.Example 1.
Consider a MCGDM problem with three experts (E1,E2,E3), and there are eight criteria concerned by the experts, i.e., C={C1,C2,⋯,C8}. The criteria concerned by E1, E2, and E3 are C1={C1,C2,C5,C7,C8}, C2={C2,C3,C4,C6,C7,C8}, and C3={C2,C3,C6,C7}, respectively. Thus, indicator matrix T=[tkj]3×8 can be constructed, as shown in Table 1.Table 1
The indicator matrixT=[tkj]3×8.
C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 E 1 1 1 0 0 1 0 1 1 E 2 0 1 1 1 0 1 1 1 E 3 0 1 1 0 0 1 1 0Then, according to the constructed indicator matrixT=[tkj]q×n, an indicator vector tj=[t1j,t2j,…,tqj]T concerning criterion Cj can be determined, j=1,2,⋯,n. Thus, T=[tkj]q×n can be further represented by T=[tkj]q×n=[t1,t2,⋯,tn], where vector tj is the indicator vector concerning criterion Cj, j=1,2,⋯,n.Further, lete(E′)=[e1,e2,…,eq]T denote the indicator vector corresponding to set C(E′), ek=0 or 1, where ek=1 denotes that expert Ek belongs to the set E′; otherwise ek=0, k=1,2,⋯,q.The following Example2 is given to illustrate the construction of indicator vector e(E′)=[e1,e2,…,eq]T.Example 2.
Consider the MCGDM problem shown in Example1. The subset of experts is P~(E)={{E1},{E2},{E3},{E1,E2},{E1,E3},{E2,E3},{E1,E2,E3}}. Obviously, there are seven indicator vectors e(E′)=[e1,e2,…,eq]T, i.e.,(1)eE1=100,eE2=010,eE3=001,eE1,E2=110,eE1,E3=101,eE2,E3=011,eE1,E2,E3=111.Property 1.
In a MCGDM problem withq experts, there are 2q-1 indicator vectors e(E′)=[e1,e2,…,eq]T.Proof.
Since there are two possible values ofek,ek=0 or 1, there are 2q possible combinations of the q binary variables. Since e({⌀})=(0,0,…,0) is not considered, there are 2q-1 possible indicator vectors e(E′)=[e1,e2,…,eq]T.
Next, according to the indicator vectortj=[t1j,t2j,⋯,tqj]T and e(E′)=[e1,e2,…,eq]T, the distance between tj and e(E′) can be calculated using (2), i.e.,(2)dtj,eE′=∑k=1qtkj-ek2,j=1,2,⋯,n,E′∈P~EIt can be seen from (2) that Cj∈C(E′) if d(tj,e(E′))=0; otherwise Cj∉C(E′).


## 3.2. Ranking Alternatives with respect to Each Criterion Subset

First, according to the decision matrixV=[vij]m×n, the normalized decision matrix V-=[v-ij]m×n can be constructed, where v-ij denotes the normalized value of vij. It can be calculated by the following (3), i.e.,(3)v-ij=vij-vjminvjmax-vjmin,i=1,2,⋯,m;j∈CBvjmax-vijvjmax-vjmin,i=1,2,⋯,m;j∈CCwhere vjmin=min⁡{vij∣i=1,2,⋯,m} and vjmax=max⁡{vij∣i=1,2,⋯,m}, j=1,2,⋯,n.Then, the ranking value of alternativeAi with respect to the criterion subset C(E′) can be calculated, i.e., (4)VE′Ai=∑Cj∈CE′Ek∈E′bkwjkv-ij,i=1,2,⋯,mAccording to the ranking values,VE′(A1), VE′(A2),…,VE′(Am), the ranking order of alternatives (A1,A2,⋯,Am) with respect to the criterion subset C(E′) can be determined. Let rE′=[risE′]m×m denote the ranking possibility matrix concerning the criterion subset C(E′), where risE′ denotes the possibility that alternative Ai is ranked at the sth position with respect to the criterion subset C(E′), i,s=1,2,⋯,m. As for the determination of the value of risE′, the following two possible cases are considered.① If and only ifAi itself with respect to criterion subset C(E′) is ranked at the kth position, then risE′ can be determined by(5)risE′=1,s=k0,s≠k,i=1,2,⋯,m② Ifg alternatives including Ai are ranked at the kth position, then risE′ can be given by(6)risE′=1g,s=k,k+1,⋯,k+g-10,otherwise,i=1,2,⋯,mTo illustrate the determination ofrE′=[risE′]m×m, the following Example 2 is given.Example 2.
There are 5 alternatives (A1,A2,A3,A4,A5) in a MCGDM problem, and the ranking order of alternatives with respect to the evaluation criteria set C(E′) is A2≻A3≻A1≻A5≻A4. The ranking possibility matrix can be constructed as (7)rE′=risE′5×5=A1A2A3A4A51st2nd3rd4th5th010000.500.5000.500.5000000100010


## 3.3. Determining the Weight of Ranking Result Concerning Each Criterion Subset

To determine the final ranking result of alternatives by aggregating the ranking possibility matrices concerning different criterion subsets, it is necessary to determine the weight of ranking result concerning each criterion subset. Since each criterion subset includes the criteria concerned by a group of experts, it is reasonable that the weight of ranking result concerning each criterion subset is calculated by both the weights of experts and the weights of evaluation criteria concerned by different experts. The weight of criterion subsetC(E′) can be obtained by(8)wE′=∑Cj∈CE′Ek∈E′bkwjk,E′∈P~E


## 3.4. Determining the Final Ranking Result of Alternatives

Based on the ranking possibility matrixrE′=[risE′]m×m and weight wE′ concerning criterion subset C(E′), the overall ranking possibility matrix P=[pis]m×m can be constructed, where pis denotes the overall possibility that alternative Ai is ranked at the sth position, and pis can be calculated by(9)pis=∑E′∈P~EwE′·risE′Based on the overall ranking possibility matrixP=[pis]m×m, the final ranking result of alternatives can be determined. In this paper, we want to obtain a clear alternative ranking results, i.e., each alternative is only ranked at one position, and each position only contains one alternative. Thus, a linear assignment model is constructed as follows:(10a)maxZ=∑i=1m∑s=1mpisxis(10b)s.t.∑i=1mxis=1,s=1,2,⋯,m(10c)∑s=1mxis=1,i=1,2,⋯,m(10d)x=0or1,i=1,2,⋯,m,s=1,2,⋯,mIn the model ((10a)–(10d)), (10a) is to maximize the sum of ranking possibilities. Equations (10b) and (10c) are equation constrains, i.e., (10b) is to guarantee that each position only contains one alternative, and (10c) is to guarantee that each alternative is ranked at one position. If xis=1, then the alternative Ai is ranked at the sth position. Obviously, model ((10a)–(10d)) is an assignment model. It can be converted into a classical assignment model and be solved by Kuhn-Munkres algorithm [36] or Bertsekas algorithm [37]. According to the solution to model ((10a)–(10d)), the final ranking results of alternatives can be determined.In summary, the procedure for solving the MCGDM problem with different evaluation criterion sets is given as follows.Step 1.
Construct the nonempty power setP~(E) based on the expert set E, and determine criterion subset C(E′) concerning each E′ (E′∈P~(E)).Step 2.
Construct the indicator matrixT=[tkj]q×n, and determine the indicator vector tj concerning criterion Cj.Step 3.
Construct the indicator vectore(E′) corresponding to set C(E′).Step 4.
Calculate the distance betweentj and e(E′) using (2), and determine whether the criterion Cj belongs to the criterion subset C(E′).Step 5.
Construct the ranking possibility matrixrE′ concerning the criterion subset C(E′) using (3)–(6).Step 6.
Calculate the weightwE′ of criterion subset C(E′) using (8).Step 7.
Construct the overall ranking possibility matrixP using (9).Step 8.
Construct the linear assignment model ((10a)–(10d)) based on the overall ranking possibility matrix P.Step 9.
Determine the final ranking result by solving the model ((10a)–(10d)).


## 4. Illustrative Example

In this section, an example is used to illustrate the use of the proposed method. Company BS is a food machinery manufacturing enterprise. To meet the increasing personalized demands of customers, Company BS decides to purchase a three axis vertical integrated cutting CNC machine. In order to improve the purchasing efficiency and save the purchasing cost, Company BS purchases the machine by the way of online reverse auction. The manufacture department (E1) and the financial department (E2) are responsible for the purchasing task. The criteria concerned by the manufacture department (E1) include C1: positioning accuracy (mm), C2: maximum load (kg), C3: mean time to failure (h), C4: degree of standardization of parts (0-10, the greater criterion value, the degree of standardization is greater), and C5: reliable service life (h), and C6: delivery time (month). The criteria concerned by the financial department (E2) include C5: reliable service life (h), C6: delivery time (month), C7: price ($), and C8:down payment ratio (percent). The importance degrees of the manufacture department (E1) and the financial department (E2) are same in the decision making of reverse auction, i.e., b=(b1,b2)=(0.5,0.5). The weight vector of criterion set concerned by the manufacture department (E1) is w1=(0.3,0.2,0.1,0.05,0.2,0.15,0,0), and the weight vector of criterion set concerned by the financial department (E2) is w2=(0,0,0,0,0.25,0.15,0.45,0.15).There are six CNC machine manufacturers bidding online; the bidding products are denoted asA1,A2,A3,A4,A5,A6. Based on the actual situation of manufacturers and the performance of bidding products, the decision matrix for bidding products is shown as Table 2.Table 2
The decision matrixV=[vij]6×8 for bidding products.
C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 7 A 1 0.005 500 850 8.6 30500 1.5 530000 50 A 2 0.01 550 925 8.2 26500 2 420000 40 A 3 0.008 600 960 9.0 28500 3 450000 35 A 4 0.008 450 720 9.2 25800 2 480000 40 A 5 0.015 400 650 8.0 24000 1.5 380000 30 A 6 0.012 480 710 8.4 23500 1 40000 60To determine the best bidding products, the method proposed in this paper is used. Some computation processes and results are presented below.Firstly,E={E1,E2}, so the nonempty power set P~(E) are constructed, i.e., P~(E)={{E1},{E2},{E1,E2}}, the criterion subset concerning {E1} is C({E1})={C1,C2,C3,C4}, the criterion subset concerning {E2} is C({E2})={C7,C8}, and the criterion subset concerning {E1,E2} is C({E1,E2})={C5,C6}.Then, using (3), the normalized decision matrix V-=[v-ij]6×8 for bidding products is constructed, as shown in Table 3.Table 3
The normalized decision matrixV-=[v-ij]6×8 for bidding products.
C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 7 A 1 1 0.5 0.6452 0.5 1 0.75 0 0.3333 A 2 0.5 0.75 0.8871 0.1667 0.4286 0.5 0.7333 0.6667 A 3 0.7 1 1 0.8333 0.7143 0 0.5333 0.8333 A 4 0.7 0.25 0.2258 1 0.3286 0.5 0.3333 0.6667 A 5 0 0 0 0 0.0714 0.75 1 1 A 6 0.3 0.4 0.1935 0.3333 0 1 0.8667 0Using (4), the ranking value of each alternative with respect to the criterion subsets C({E1}), C({E2}), C({E1,E2}) is calculated, and the ranking orders of alternatives A1,A2,⋯,A6 with respect to the criterion subsets C({E1}), C({E2}), C({E1,E2}) can be determined as follows.For the criterion subsetC({E1})={C1,C2,C3,C4}, V{E1}(A1)=0.2448, V{E1}(A2)=0.1985, V{E1}(A3)=0.2758, V{E1}(A4)=0.1663, V{E1}(A5)=0, and V{E1}(A6)=0.1030. We can obtain that the ranking order is A3≻A1≻A2≻A4≻A6≻A5.For the criterion subsetC({E2})={C7,C8}, V{E2}(A1)=0.0250, V{E2}(A2)=0.2150, V{E2}(A3)=0.1825, V{E2}(A4)=0.1250, V{E2}(A5)=0.3, and V{E2}(A6)=0.1950. We can obtain that the ranking order is A5≻A2≻A6≻A3≻A4≻A1.For the criterion subsetC({E1,E2})={C5,C6}, V{E1,E2}(A1)=0.3375, V{E1,E2}(A2)=0.1714, V{E1,E2}(A3)=0.1607, V{E1,E2}(A4)=0.1489, V{E1,E2}(A5)=0.1286, and V{E1,E2}(A6)=0.15. We can obtain that the ranking order is A1≻A2≻A3≻A6≻A4≻A5.Using (8), the weights of criterion subsets C({E1}), C({E2}), C({E1,E2}) can be obtained, i.e., w{E1}=0.325, w{E2}=0.3, and w{E1,E2}=0.375.Further, using (9), the overall ranking possibility matrix P=[pis]6×6 is constructed, i.e.,(11)P=0.3750.3250000.300.6780.3250000.32500.3750.3000000.3250.67500.300000.7000.30.3750.3250Based on matrixP, a linear assignment model is built as follows:(12)maxZ=∑i=16∑s=16pisxiss.t.∑i=1mxis=1,s=1,2,⋯,6∑s=1mxis=1,i=1,2,⋯,6x=0or1,i=1,2,⋯,6,s=1,2,⋯,6By solving the above model, the optimal solution can be obtained, i.e.,(13)X=xis6×6=100000010000001000000010000001000100Therefore, the final ranking result of alternatives isA1≻A2≻A3≻A6≻A4≻A5. Obviously, A1 is the best bidding product.


## 5. Conclusions

This paper presents a novel method for solving the MCGDM problem in which evaluation criterion sets concerned by multiple experts are different. In the method, an approach is first given to construct criterion subsets based on the criterion sets considered by different experts. Then, the ranking possibility matrix with respect to each criterion subset is built, and the weight of each criterion subset is measured according to the weights of experts and the weights of criteria in the criterion subset. Further, by aggregating the ranking possibility matrices and the weights concerning different criterion subsets, the overall ranking possibility matrix is built and the final ranking result of alternatives is determined by a linear assignment model. The major contributions of this paper are discussed as follows.First, this paper focuses on the MCGDM problem with different evaluation criterion sets, which is a new research topic with a lot of practical backgrounds. In the problem, multiple experts from different organizations or departments take part in the decision-making process and each expert can pay more attention to the criteria that relate to his/her own benefit or duty.Second, this paper presents a method for solving the MCGDM problem with different evaluation criterion sets. The method includes four aspects:(1) constructing criterion subsets concerned by different groups of experts; (2) ranking alternatives with respect to each criterion subset; (3) determining the weight of ranking result concerning each criterion subset; (4) determining the final ranking result of alternatives. It is a new idea for aggregating individual preferences or opinions in group decision-making analysis and lays a good foundation for studying on MCGDM problems with different evaluation criterion sets.It is important to highlight that, since the proposed method is new and different from the existing methods, it can give the decision-maker one more choice for identifying the appropriate method for solving the MCGDM problem with different evaluation criterion sets. In addition to supplementing the existing methods, the proposed method is also important for developing and enriching theories and methods of MCGDM.The study also has some limitations, which may serve as avenues for future research. First, in this study, we only consider the situation that the values of alternatives with respect to criteria are in the form of crisp numbers. But, in reality, the values of alternatives with respect to criteria may be different forms, such as interval number, fuzzy number, or stochastic variable. Thus, it is necessary to develop some new methods for solving MCGDM problems in which the criterion sets considered by different experts are different and the criterion values are in different formats. Besides, to support DMs to facilitate the use of the method proposed in this paper, the decision support system needs to be developed.



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*Source: 7189451-2018-10-21.xml*