# Multiattribute Group Decision-Making Method Using a GeneticK-Means Clustering Algorithm

Authors: Liu Qingguo; Liu Xinxue; Wu Jian; Li Yaxiong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Multiattribute group decision-making (MAGDM) problems are characterized by the large number, uneven levels, and bounded rationality of decision-makers; multiple attributes and fuzziness of decision problems; and complex group behaviours. Considering these characteristics, we propose a MAGDM method using a geneticK-means clustering algorithm. First, we briefly review the traditional multiattribute decision-making method based on prospect theory (PT) and trapezoidal intuitionistic fuzzy numbers (TrIFNs) under the premise of human bounded rationality and uncertain decision environment. Then, the aggregation model of decision information given by decision-makers is established using the genetic K-means algorithm in order to determine optimal clustering results. Each clustering center represents decision information given by decision-makers in each cluster, and the weight of each clustering center is determined by considering the tightness of decision information within a cluster and the count of decision-makers in each cluster. Finally, the ranking of schemes is obtained according to the comparison rules of TrIFNs. We design comparison simulation experiments to test the proposed method and the simulation results demonstrate that the proposed method is apprehensible and feasible to solve MAGDM problems.


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## Body

## 1. Introduction

Multiattribute group decision-making (MAGDM) is a problem in which multiple decision-makers make decisions on multiple schemes under the premise of multiple attributes [1–3]. The basis of the MAGDM problem is multiattribute decision-making (MADM). MADM is characterized by bounded rationality of decision-makers as well as multiple attributes and fuzziness of decision problems. In addition to the characteristics of the MADM problem, the characteristics of the MAGDM problem include a large number and uneven levels of decision-makers and complex group behaviours.Prospect theory (PT) can accurately describe and explain decision-makers’ judgment and choice behaviours and thus can solve MADM problems under the condition of human bounded rationality [4, 5]. The problems of uncertainty of the decision-making conditions and fuzziness of decision information exist in many MADM problems, which are effectively solved by intuitionistic fuzzy decision-making methods. The trapezoidal intuitionistic fuzzy number (TrIFN) [6, 7] is usually used to solve MADM problems with uncertain decision conditions and fuzzy decision information. To solve MADM problems, a method combining PT with TrIFNs and various derived methods have been widely used. Reference [8] developed a prospect value determination method based on multiple reference points under a trapezoidal intuitionistic fuzzy environment. Reference [9] solved the MADM problem of wind energy by combining PT and TrIFNs. Reference [10] proposed an MAGDM method using the ITFN weighted geometric operator and hybrid geometric operator to obtain the collective overall values.MAGDM problems mainly include the determination of decision-maker’s weights, consensus analysis, aggregation of decision information, and ranking of schemes. Decision-makers’ weights can be determined by methods such as subjective, objective, and subjective-objective weighting [11–13]. Pang et al. [14] proposed an adaptive consensus method for MAGDM based on adaptive experts’ weights and explicit guidance rules. Sun and Ma [15] proposed an approach to consensus measurement of linguistic preference relations. The aggregation of decision information and ranking of schemes are the two aspects that can best embody the characteristics of MAGDM. Chen et al. [16] proposed aggregation operators based on interval-valued intuitionistic fuzzy numbers. Andrej [17] proposed a hybrid Delphi and aggregation–disaggregation procedure of MAGDM. Shakeel [18] proposed a cubic averaging aggregation operator. Under the premise of human bounded rationality and uncertain decision-making environment, the representative papers that systematically study MAGDM methods are as follows: Li and Chen [19] proposed a novel TOPSIS based on PT and TrIFNs, and the aggregation operator and ranking strategy can effectively obtain the final decision scheme. Yuan and Li [20] proposed an aggregation operator based on the Choquet integral and PT. In addition to the above studies, MAGDM problems can also be solved by methods using data mining and cluster analysis. Shankar and Kumar [21] used a K-means clustering algorithm to solve the MAGDM problem in the field of healthcare systems. Liu and Chen [22] proposed an improved K-means algorithm to solve the large group decision-making problem. Zhao [23] proposed an improved K-means clustering algorithm based on interval similarity measurement to solve the MAGDM problem. However, existing research has the following shortcomings: (a) decision information is still expressed in real numbers, interval numbers, and language values in some studies, thus leading to poor adaptability in processing uncertain and fuzzy information; (b) the preference of decision-makers is based on expected utility theory in some studies, which is inconsistent with the nature of human preference and judgment based on comparison; (c) some decision-making methods require decision-makers whose decision preference largely deviates from the group preference to change their decision preference, which incurs great cost; (d) the aggregation methods of decision information are too subjective, which cannot objectively reflect the decision information given by decision-makers; (e) the existing MAGDM methods based on the K-means clustering algorithm [24] are easily trapped in local optimum.Considering the above shortcomings of the MAGDM methods based on clustering algorithms, we propose a MAGDM method using a genetic K-means clustering algorithm on the basis of PT and TrIFNs. This paper is organized as follows. First, we briefly review the framework of the PT and TrIFN and their operational rules in Section2. Then, in Section 3, we propose the method to solve the MAGDM problem using the genetic K-means clustering algorithm and propose its details. Finally, in Section 4, we perform simulations to verify whether the proposed method is apprehensible and feasible to solve MAGDM problems. The proposed MAGDM method fully reflects the objectivity of decision-making, realizes the learning of clustering numbers automatically, and overcomes the shortcomings of MAGDM methods above.


## 2. Brief Review of the Prospect Theory and Intuitionistic Trapezoidal Fuzzy Number

The traditional method to solve MADM problems is based on PT and TrIFNs, which lays the foundation of our proposed MADM method. On the basis of the PT framework, the decision information is expressed in the TrIFN form. In this section, the PT and the TrIFN are briefly reviewed, respectively.


### 2.1. Framework of PT

The basic unit of PT is the prospect, which is denoted asf=x1,p1;x2,p2;…;xn,pn. The reference point, which is used to describe the attributes of MADM, is given according to the subjective feelings of the decision-maker. xi is the gain or loss value of the ith attribute value compared to the ith attribute value of the reference point. pi is the probability of xi. The most important feature of PT is that it reflects the value of change rather than the absolute value, which is also consistent with the fact that decision-makers are more sensitive to change. The prospect is evaluated by the prospect value pvf as follows:(1)pvf=∑i=1nwpivxi,where wpi and vxi are the weight function and the value function of the ith attribute, respectively. wpi and vxi are given by(2)vxi=xiα,xi≥0,−δ−xiβ,xi<0,(3)wpi=piγpiγ+1−piγ1/γ,where α and β are concave and convex degrees of xi, respectively, with α<1 and β<1; δ indicates that the loss curve is steeper than the gain curve, and δ>1 indicates that decision-makers do not accept losses; γ is the fitting parameter.


### 2.2. TrIFN and Its Operational Rules

Leta˜=a1,a2,a3,a4,b1,b2,b3,b4;μa˜,va˜ be a TrIFN. Parameters a1, a2, a3, a4, b1, b2, b3, and b4 are real numbers that satisfy the condition b1≤a1≤b2≤a2≤b3≤a3≤b4≤a4. The membership function and nonmembership function are, respectively,(4)μa˜x=x−a1a2−a1,μa˜,a1≤x<a2,μa˜,a2≤x≤a3,a4−xa4−a3,μa˜,a3≤x≤a4,0,xequalsany other value,(5)va˜x=b2−x+va˜x−b1b2−b1,b1≤x<b2,va˜,b2≤x≤b3,x−b3+va˜b4−xb4−b3,b3≤x≤b4,0,xequalsany other value,where μa˜∈0,1, va˜∈0,1, and 0≤μa˜+va˜≤1.Before reviewing operational rules, we seta˜1=a11,a12,a13,a14,b11,b12,b13,b14;μa˜1,va˜1 and a˜2=a21,a22,a23,a24,b21,b22,b23,b24;μa˜2,va˜2 as two TrIFNs. The operational rules of TrIFNs are defined as follows:(6)a˜1⊕a˜2=a11+a21,a12+a22,a13+a23,a14+a24,b11+b21,b12+b22,b13+b23,b14+b24;μa˜1+μa˜2−μa˜1μa˜2,va˜1va˜2,(7)λa˜1=λa11,λa12,λa13,λa14,λb11,λb12,λb13,λb14;1−1−μa˜1λ,va˜1λ,λ≥0,λa14,λa13,λa12,λa11,λb14,λb13,λb12,λb11;1−1−μa˜1−λ,va˜1−λ,λ<0,(8)a˜1λ=a11λ,a12λ,a13λ,a14λ,b11λ,b12λ,b13λ,b14λ;μa˜1λ,1−1−va˜1λ,λ≥0,aijλ>0,bijλ>0,(9)a˜1⊗a˜2=a11a21,a12a22,a13a23,a14a24,b11b21,b12b22,b13b23,b14b24;μa˜1μa˜2,va˜1+va˜2−va˜1va˜2.The expected value ofa˜1 is as follows:(10)EVa˜1=18μa˜1a11+a12+a13+a14+1−va˜1b11+b12+b13+b14.The score function and accuracy function ofa˜1 are(11)Sa˜1=EVa˜1⋅μa˜1−va˜1,(12)Aa˜1=EVa˜1⋅μa˜1+va˜1,where EVa˜1 is the expected value of a˜1.The rules for comparinga˜1 and a˜2 are as follows: If Sa˜1>Sa˜2, a˜1>a˜2. If Sa˜1=Sa˜2 and Aa˜1>Aa˜2, a˜1>a˜2. If Sa˜1=Sa˜2 and Aa˜1=Aa˜2, a˜1=a˜2.Hamming distance is used as the distance measured betweena˜1 and a˜2 as follows:(13)da˜1,a˜2=18∑j=14μa˜1a1j−μa˜2a2j+1−va˜1b1j−1−va˜2b2j.The values of attribute and weight are given by decision-makers in the form of TrIFN.


## 2.1. Framework of PT

The basic unit of PT is the prospect, which is denoted asf=x1,p1;x2,p2;…;xn,pn. The reference point, which is used to describe the attributes of MADM, is given according to the subjective feelings of the decision-maker. xi is the gain or loss value of the ith attribute value compared to the ith attribute value of the reference point. pi is the probability of xi. The most important feature of PT is that it reflects the value of change rather than the absolute value, which is also consistent with the fact that decision-makers are more sensitive to change. The prospect is evaluated by the prospect value pvf as follows:(1)pvf=∑i=1nwpivxi,where wpi and vxi are the weight function and the value function of the ith attribute, respectively. wpi and vxi are given by(2)vxi=xiα,xi≥0,−δ−xiβ,xi<0,(3)wpi=piγpiγ+1−piγ1/γ,where α and β are concave and convex degrees of xi, respectively, with α<1 and β<1; δ indicates that the loss curve is steeper than the gain curve, and δ>1 indicates that decision-makers do not accept losses; γ is the fitting parameter.


## 2.2. TrIFN and Its Operational Rules

Leta˜=a1,a2,a3,a4,b1,b2,b3,b4;μa˜,va˜ be a TrIFN. Parameters a1, a2, a3, a4, b1, b2, b3, and b4 are real numbers that satisfy the condition b1≤a1≤b2≤a2≤b3≤a3≤b4≤a4. The membership function and nonmembership function are, respectively,(4)μa˜x=x−a1a2−a1,μa˜,a1≤x<a2,μa˜,a2≤x≤a3,a4−xa4−a3,μa˜,a3≤x≤a4,0,xequalsany other value,(5)va˜x=b2−x+va˜x−b1b2−b1,b1≤x<b2,va˜,b2≤x≤b3,x−b3+va˜b4−xb4−b3,b3≤x≤b4,0,xequalsany other value,where μa˜∈0,1, va˜∈0,1, and 0≤μa˜+va˜≤1.Before reviewing operational rules, we seta˜1=a11,a12,a13,a14,b11,b12,b13,b14;μa˜1,va˜1 and a˜2=a21,a22,a23,a24,b21,b22,b23,b24;μa˜2,va˜2 as two TrIFNs. The operational rules of TrIFNs are defined as follows:(6)a˜1⊕a˜2=a11+a21,a12+a22,a13+a23,a14+a24,b11+b21,b12+b22,b13+b23,b14+b24;μa˜1+μa˜2−μa˜1μa˜2,va˜1va˜2,(7)λa˜1=λa11,λa12,λa13,λa14,λb11,λb12,λb13,λb14;1−1−μa˜1λ,va˜1λ,λ≥0,λa14,λa13,λa12,λa11,λb14,λb13,λb12,λb11;1−1−μa˜1−λ,va˜1−λ,λ<0,(8)a˜1λ=a11λ,a12λ,a13λ,a14λ,b11λ,b12λ,b13λ,b14λ;μa˜1λ,1−1−va˜1λ,λ≥0,aijλ>0,bijλ>0,(9)a˜1⊗a˜2=a11a21,a12a22,a13a23,a14a24,b11b21,b12b22,b13b23,b14b24;μa˜1μa˜2,va˜1+va˜2−va˜1va˜2.The expected value ofa˜1 is as follows:(10)EVa˜1=18μa˜1a11+a12+a13+a14+1−va˜1b11+b12+b13+b14.The score function and accuracy function ofa˜1 are(11)Sa˜1=EVa˜1⋅μa˜1−va˜1,(12)Aa˜1=EVa˜1⋅μa˜1+va˜1,where EVa˜1 is the expected value of a˜1.The rules for comparinga˜1 and a˜2 are as follows: If Sa˜1>Sa˜2, a˜1>a˜2. If Sa˜1=Sa˜2 and Aa˜1>Aa˜2, a˜1>a˜2. If Sa˜1=Sa˜2 and Aa˜1=Aa˜2, a˜1=a˜2.Hamming distance is used as the distance measured betweena˜1 and a˜2 as follows:(13)da˜1,a˜2=18∑j=14μa˜1a1j−μa˜2a2j+1−va˜1b1j−1−va˜2b2j.The values of attribute and weight are given by decision-makers in the form of TrIFN.


## 3. Proposed MAGDM Method

Aggregation of decision information of decision-makers in one group by using the geneticK-means clustering algorithm is the key to the proposed MAGDM method. The decision information set F˜l, which is formed by decision information (provided by decision-makers) of the lth decision scheme, is partitioned into different clusters and each cluster center represents the decision information of each cluster. The weight of each clustering center is determined by considering the tightness of decision information within a cluster and the count of decision-makers in each cluster. Then the aggregation results are taken as the basis for ranking. In the following section, the aggregation of decision information by using the genetic K-means clustering algorithm and the calculation steps of the proposed method are introduced.


### 3.1. Aggregation of Decision Information by UsingK-Means Clustering Algorithm

TheK-means clustering algorithm is sensitive to the initial cluster centers, cannot determine the optimal number of clusters, and is easily trapped in local optima. Genetic algorithms can overcome the shortcomings of K-means clustering algorithms and realize the automatic learning of the cluster count. The decision information expressed by TrIFNs is clustered by a genetic K-means clustering algorithm. First, we review the conventional K-means clustering algorithm as follows. k pieces of decision information in F˜l are randomly selected as the initial cluster centers. Then, each piece of the remaining decision information is assigned to the cluster center of the nearest distance. The cluster centers are recalculated to minimize the squared error criterion E converges:(14)El=∑j=1k∑f˜il∈Cjldpvf˜il,m˜jl,where k is the number of clusters, Cjl is the jth cluster of the lth decision scheme, m˜jl is the cluster center of cluster Cjl, f˜il is the decision information of the lth decision scheme given by the ith decision-maker, and pvf˜il is the prospect value of f˜il:(15)pvf˜il=∑j=1nwp˜ijlvx˜ijl,where wp˜ijl and vx˜ijlare the weight function and the value function of the jth attribute in f˜il, respectively. x˜ijl is a TrIFN and the value function vx˜ijl is(16)vx˜ijl=dx˜ijl,r˜ij,x˜ijl≥r˜ij,−δdx˜ijl,r˜ijβ,x˜ijl<r˜ij,where r˜ij is the jth reference point given by the ith decision-maker.Before clustering the group decision information, the outliers should be eliminated. The threshold value of Hamming distance is denoted asdmax and the threshold value of exceed dmax is denoted as δ. If the count of dpvf˜il,pvf˜jl;j=1,2,…,n more than dmax exceeds δ, the ith decision information should be eliminated. The process of elimination is process of consensus analysis.The genetic variant of theK-means clustering algorithm involves the evolution of chromosomes, and the final result is obtained by genetic operations such as selection, crossover, and mutation. The steps involved in the genetic K-means clustering algorithm are as follows:(a)
Real-number encoding is adopted to transform the cluster centers into genesG1,G2,…,Gk on a chromosome, as shown in Figure 1. The encoding values are integers from 1 to n, where n is the number of decision-makers. The size of a chromosome varies with the number of clusters.(b)
The clustering results should satisfy tightness and separability requirements. Tightness means that the schemes within one cluster are as similar as possible, while separability means that the schemes in different clusters are as different as possible. Therefore, we define the fitness functionfitDl of the genetic K-means clustering algorithm as(17)fitDl=Gblb+aEl,where a and b are positive constant coefficients and Gbl is the sum of the distances between different clusters when dealing with the lth decision scheme, given as(18)Gbl=2kk−1∑i=1k∑j=i+1km˜i−m˜j2,which is used to quantify separability. The fitness function fitDl indicates that the clustering is better the pieces of decision information in the same cluster are closer to each other other (El has a smaller value) and the cluster centers of different clusters are farther from each other (Gbl has a larger value).Figure 1
Encoding of cluster centers as genes of a chromosome.Using genetic operations including roulette-based selection and single-point crossover (denoted asPc), we propose a new mutation operation that leads to the automatic learning of the optimal number of clusters k. The population chromosome with the largest fitness value is selected as the model chromosome of the optimal cluster number. Other chromosomes in the same population should learn from this model to achieve better fitness by decreasing or increasing the number of genes of chromosomes. The chromosomes in the initial population have the same length, a small number of initial clusters are set, and an increasing trend is assumed when the first mutation occurs. With the reoccurrence of mutation operation, the offspring chromosomes decrease or increase based on whether their lengths are longer or shorter than the model chromosome, respectively. On the one hand, decreasing the number of genes is achieved by eliminating the nearest genes to cluster centers of the model chromosome. On the other hand, the number of genes is increased by adding the farthest schemes to the cluster centers of the model chromosome.The decision information is partitioned intok clusters. The weight of decision information in each cluster is determined by considering two factors, namely, the tightness of decision information within a cluster and the count of decision-makers in each group. Let ct and Etl be the count of decision-makers and the squared error criterion of decision information in the tth cluster. Then, the weight μtl of the tth cluster of the lth decision scheme is(19)μtl=c¯t+E¯tl2,where c¯t and E¯tl are the normalized values of ct and Etl, respectively.(20)c¯t=ctn,(21)E¯tl=Etl∑t=1kEtl,where n is the count of decision-makers.The aggregation result of thelth decision scheme can be written as follows.(22)al=∑t=1t=TμtlCtl,where T is the count of clusters.This aggregation method considers both intercluster information aggregation and intracluster information aggregation, which can fully integrate decision information. Compared with other MAGDM based on clustering methods, the proposed MAGDM method can realize the learning of clustering numbers automatically.


### 3.2. Calculation Steps of the Proposed MAGDM Method

The calculation steps of the proposed MAGDM method are as follows.


#### 3.2.1. Construction of Decision Information Matrix

LetX˜l and P˜l be the attributes and weights of decision information matrices of the lth decision scheme:(23)X˜l=x˜11lx˜12l⋯x˜1mlx˜21lx˜22l⋯x˜2ml⋮⋮⋮⋮x˜n1lx˜n2l⋯x˜nml,(24)P˜l=p˜11lp˜12l⋯p˜1mlp˜21lp˜22l⋯p˜2ml⋮⋮⋮⋮p˜n1lp˜n2l⋯p˜nml,where x˜ijl and p˜ijl are the jth attribute and the weight of the lth decision scheme given by the ith decision-maker, n is the count of decision-makers, and m is the count of attributes. x˜il=x˜i1l,x˜i2l,…,x˜iml is called the lth decision scheme given by the ith decision-maker.


#### 3.2.2. Aggregation of Reference Points

The reference points matrixR˜ is given according to the subjective feelings of the decision-makers as(25)R˜=r˜11r˜12⋯r˜1mr˜21r˜22⋯r˜2m⋮⋮⋮⋮r˜n1r˜n2⋯r˜nm,where r˜ij is the jth reference point given by the ith decision-maker. The ordered weighted average (OWA) operator is applied to aggregate the reference point r˜ given by decision-makers. TrIFNs of each column in the matrix R˜ are sorted by comparing rules of TrIFNs (Section 2). The sorted R˜ is written as R˜s as follows:(26)R˜s=r˜11sr˜12s⋯r˜1msr˜21sr˜22s⋯r˜2ms⋮⋮⋮⋮r˜n1sr˜n2s⋯r˜nms.Then, the reference pointr˜ is obtained as(27)r˜=⊕i=1nw1r˜i1s,⊕i=1nw2r˜i2s,…,⊕i=1nwmr˜ims,where w=w1,w2,…,wm is the weighted vector relevant to OWA.


#### 3.2.3. Determination of the Gain or Loss Values

The gain or loss valuesdx˜ijl,r˜j of the jth attribute of the lth decision scheme given by the ith decision-maker are obtained by formula (13). Then, the decision information set F˜l=f˜1l,…,f˜il,…,f˜nl is obtained, and f˜il=dx˜i1l,r˜1,p˜i1l;dx˜i2l,r˜2,p˜i2l;…;dx˜iml,r˜m,p˜iml is the prospect used to express decision information of the lth decision scheme given by the ith decision-maker.


#### 3.2.4. Computation and Normalization of Prospect Values

The prospect valuepvlf˜i of f˜il is obtained by formula (15). Let pvlf˜i be written as follows:(28)pvlf˜i=pv11lf˜i,pv12lf˜i,pv13lf˜i,pv14lf˜i,pv21lf˜i,pv22lf˜i,pv23lf˜i,pv24lf˜i;μpvlf˜i,vpvlf˜i.The prospect valuepvlf˜i is normalized.(29)p¯vlf˜i=pv11lf˜i−γ11l−γ11l+−γ11l−,pv12lf˜i−γ12l−γ12l+−γ12l−,pv13lf˜i−γ13l−γ13l+−γ13l−,pv14lf˜i−γ14l−γ14l+−γ14l−,pv21lf˜i−ζ21l−ζ21l+−ζ21l−,pv22lf˜i−ζ22l−ζ21l+−ζ22l−,pv23lf˜i−ζ23l−ζ23l+−ζ23l−,pv24lf˜i−ζ24l−ζ24l+−ζ24l−;μpvlf˜i,vpvlf˜i,where p¯vlf˜iis the normalized value of pvlf˜i; γ1sl−,s=1,2,3,4 and γ2sl− are mini=1npvl1sf˜i and mini=1npv2slf˜i, respectively; and γ1sl+ and γ2sl+ are mini=1npvl1sf˜i and maxi=1npv2slf˜i, respectively.


#### 3.2.5. Consensus Analysis and Aggregation of Decision Information Given by all Decision-Makers

The population and the maximum number of iterations are denoted byP and TM, respectively. The consensus analysis and aggregation procedure are the same as those described in Section 3.1.


#### 3.2.6. Ranking of Decision Schemes

The count of decision schemes isL. All decision schemes perform Steps 1–5. All expected values EVal of all=1,2,…,L are computed according to formula (10). Then, score function Sal and accuracy function Aal values are computed. Ranking of all decision schemes is done according to the comparing rules of TrIFNs (Section 2).


## 3.1. Aggregation of Decision Information by UsingK-Means Clustering Algorithm

TheK-means clustering algorithm is sensitive to the initial cluster centers, cannot determine the optimal number of clusters, and is easily trapped in local optima. Genetic algorithms can overcome the shortcomings of K-means clustering algorithms and realize the automatic learning of the cluster count. The decision information expressed by TrIFNs is clustered by a genetic K-means clustering algorithm. First, we review the conventional K-means clustering algorithm as follows. k pieces of decision information in F˜l are randomly selected as the initial cluster centers. Then, each piece of the remaining decision information is assigned to the cluster center of the nearest distance. The cluster centers are recalculated to minimize the squared error criterion E converges:(14)El=∑j=1k∑f˜il∈Cjldpvf˜il,m˜jl,where k is the number of clusters, Cjl is the jth cluster of the lth decision scheme, m˜jl is the cluster center of cluster Cjl, f˜il is the decision information of the lth decision scheme given by the ith decision-maker, and pvf˜il is the prospect value of f˜il:(15)pvf˜il=∑j=1nwp˜ijlvx˜ijl,where wp˜ijl and vx˜ijlare the weight function and the value function of the jth attribute in f˜il, respectively. x˜ijl is a TrIFN and the value function vx˜ijl is(16)vx˜ijl=dx˜ijl,r˜ij,x˜ijl≥r˜ij,−δdx˜ijl,r˜ijβ,x˜ijl<r˜ij,where r˜ij is the jth reference point given by the ith decision-maker.Before clustering the group decision information, the outliers should be eliminated. The threshold value of Hamming distance is denoted asdmax and the threshold value of exceed dmax is denoted as δ. If the count of dpvf˜il,pvf˜jl;j=1,2,…,n more than dmax exceeds δ, the ith decision information should be eliminated. The process of elimination is process of consensus analysis.The genetic variant of theK-means clustering algorithm involves the evolution of chromosomes, and the final result is obtained by genetic operations such as selection, crossover, and mutation. The steps involved in the genetic K-means clustering algorithm are as follows:(a)
Real-number encoding is adopted to transform the cluster centers into genesG1,G2,…,Gk on a chromosome, as shown in Figure 1. The encoding values are integers from 1 to n, where n is the number of decision-makers. The size of a chromosome varies with the number of clusters.(b)
The clustering results should satisfy tightness and separability requirements. Tightness means that the schemes within one cluster are as similar as possible, while separability means that the schemes in different clusters are as different as possible. Therefore, we define the fitness functionfitDl of the genetic K-means clustering algorithm as(17)fitDl=Gblb+aEl,where a and b are positive constant coefficients and Gbl is the sum of the distances between different clusters when dealing with the lth decision scheme, given as(18)Gbl=2kk−1∑i=1k∑j=i+1km˜i−m˜j2,which is used to quantify separability. The fitness function fitDl indicates that the clustering is better the pieces of decision information in the same cluster are closer to each other other (El has a smaller value) and the cluster centers of different clusters are farther from each other (Gbl has a larger value).Figure 1
Encoding of cluster centers as genes of a chromosome.Using genetic operations including roulette-based selection and single-point crossover (denoted asPc), we propose a new mutation operation that leads to the automatic learning of the optimal number of clusters k. The population chromosome with the largest fitness value is selected as the model chromosome of the optimal cluster number. Other chromosomes in the same population should learn from this model to achieve better fitness by decreasing or increasing the number of genes of chromosomes. The chromosomes in the initial population have the same length, a small number of initial clusters are set, and an increasing trend is assumed when the first mutation occurs. With the reoccurrence of mutation operation, the offspring chromosomes decrease or increase based on whether their lengths are longer or shorter than the model chromosome, respectively. On the one hand, decreasing the number of genes is achieved by eliminating the nearest genes to cluster centers of the model chromosome. On the other hand, the number of genes is increased by adding the farthest schemes to the cluster centers of the model chromosome.The decision information is partitioned intok clusters. The weight of decision information in each cluster is determined by considering two factors, namely, the tightness of decision information within a cluster and the count of decision-makers in each group. Let ct and Etl be the count of decision-makers and the squared error criterion of decision information in the tth cluster. Then, the weight μtl of the tth cluster of the lth decision scheme is(19)μtl=c¯t+E¯tl2,where c¯t and E¯tl are the normalized values of ct and Etl, respectively.(20)c¯t=ctn,(21)E¯tl=Etl∑t=1kEtl,where n is the count of decision-makers.The aggregation result of thelth decision scheme can be written as follows.(22)al=∑t=1t=TμtlCtl,where T is the count of clusters.This aggregation method considers both intercluster information aggregation and intracluster information aggregation, which can fully integrate decision information. Compared with other MAGDM based on clustering methods, the proposed MAGDM method can realize the learning of clustering numbers automatically.


## 3.2. Calculation Steps of the Proposed MAGDM Method

The calculation steps of the proposed MAGDM method are as follows.


### 3.2.1. Construction of Decision Information Matrix

LetX˜l and P˜l be the attributes and weights of decision information matrices of the lth decision scheme:(23)X˜l=x˜11lx˜12l⋯x˜1mlx˜21lx˜22l⋯x˜2ml⋮⋮⋮⋮x˜n1lx˜n2l⋯x˜nml,(24)P˜l=p˜11lp˜12l⋯p˜1mlp˜21lp˜22l⋯p˜2ml⋮⋮⋮⋮p˜n1lp˜n2l⋯p˜nml,where x˜ijl and p˜ijl are the jth attribute and the weight of the lth decision scheme given by the ith decision-maker, n is the count of decision-makers, and m is the count of attributes. x˜il=x˜i1l,x˜i2l,…,x˜iml is called the lth decision scheme given by the ith decision-maker.


### 3.2.2. Aggregation of Reference Points

The reference points matrixR˜ is given according to the subjective feelings of the decision-makers as(25)R˜=r˜11r˜12⋯r˜1mr˜21r˜22⋯r˜2m⋮⋮⋮⋮r˜n1r˜n2⋯r˜nm,where r˜ij is the jth reference point given by the ith decision-maker. The ordered weighted average (OWA) operator is applied to aggregate the reference point r˜ given by decision-makers. TrIFNs of each column in the matrix R˜ are sorted by comparing rules of TrIFNs (Section 2). The sorted R˜ is written as R˜s as follows:(26)R˜s=r˜11sr˜12s⋯r˜1msr˜21sr˜22s⋯r˜2ms⋮⋮⋮⋮r˜n1sr˜n2s⋯r˜nms.Then, the reference pointr˜ is obtained as(27)r˜=⊕i=1nw1r˜i1s,⊕i=1nw2r˜i2s,…,⊕i=1nwmr˜ims,where w=w1,w2,…,wm is the weighted vector relevant to OWA.


### 3.2.3. Determination of the Gain or Loss Values

The gain or loss valuesdx˜ijl,r˜j of the jth attribute of the lth decision scheme given by the ith decision-maker are obtained by formula (13). Then, the decision information set F˜l=f˜1l,…,f˜il,…,f˜nl is obtained, and f˜il=dx˜i1l,r˜1,p˜i1l;dx˜i2l,r˜2,p˜i2l;…;dx˜iml,r˜m,p˜iml is the prospect used to express decision information of the lth decision scheme given by the ith decision-maker.


### 3.2.4. Computation and Normalization of Prospect Values

The prospect valuepvlf˜i of f˜il is obtained by formula (15). Let pvlf˜i be written as follows:(28)pvlf˜i=pv11lf˜i,pv12lf˜i,pv13lf˜i,pv14lf˜i,pv21lf˜i,pv22lf˜i,pv23lf˜i,pv24lf˜i;μpvlf˜i,vpvlf˜i.The prospect valuepvlf˜i is normalized.(29)p¯vlf˜i=pv11lf˜i−γ11l−γ11l+−γ11l−,pv12lf˜i−γ12l−γ12l+−γ12l−,pv13lf˜i−γ13l−γ13l+−γ13l−,pv14lf˜i−γ14l−γ14l+−γ14l−,pv21lf˜i−ζ21l−ζ21l+−ζ21l−,pv22lf˜i−ζ22l−ζ21l+−ζ22l−,pv23lf˜i−ζ23l−ζ23l+−ζ23l−,pv24lf˜i−ζ24l−ζ24l+−ζ24l−;μpvlf˜i,vpvlf˜i,where p¯vlf˜iis the normalized value of pvlf˜i; γ1sl−,s=1,2,3,4 and γ2sl− are mini=1npvl1sf˜i and mini=1npv2slf˜i, respectively; and γ1sl+ and γ2sl+ are mini=1npvl1sf˜i and maxi=1npv2slf˜i, respectively.


### 3.2.5. Consensus Analysis and Aggregation of Decision Information Given by all Decision-Makers

The population and the maximum number of iterations are denoted byP and TM, respectively. The consensus analysis and aggregation procedure are the same as those described in Section 3.1.


### 3.2.6. Ranking of Decision Schemes

The count of decision schemes isL. All decision schemes perform Steps 1–5. All expected values EVal of all=1,2,…,L are computed according to formula (10). Then, score function Sal and accuracy function Aal values are computed. Ranking of all decision schemes is done according to the comparing rules of TrIFNs (Section 2).


## 3.2.1. Construction of Decision Information Matrix

LetX˜l and P˜l be the attributes and weights of decision information matrices of the lth decision scheme:(23)X˜l=x˜11lx˜12l⋯x˜1mlx˜21lx˜22l⋯x˜2ml⋮⋮⋮⋮x˜n1lx˜n2l⋯x˜nml,(24)P˜l=p˜11lp˜12l⋯p˜1mlp˜21lp˜22l⋯p˜2ml⋮⋮⋮⋮p˜n1lp˜n2l⋯p˜nml,where x˜ijl and p˜ijl are the jth attribute and the weight of the lth decision scheme given by the ith decision-maker, n is the count of decision-makers, and m is the count of attributes. x˜il=x˜i1l,x˜i2l,…,x˜iml is called the lth decision scheme given by the ith decision-maker.


## 3.2.2. Aggregation of Reference Points

The reference points matrixR˜ is given according to the subjective feelings of the decision-makers as(25)R˜=r˜11r˜12⋯r˜1mr˜21r˜22⋯r˜2m⋮⋮⋮⋮r˜n1r˜n2⋯r˜nm,where r˜ij is the jth reference point given by the ith decision-maker. The ordered weighted average (OWA) operator is applied to aggregate the reference point r˜ given by decision-makers. TrIFNs of each column in the matrix R˜ are sorted by comparing rules of TrIFNs (Section 2). The sorted R˜ is written as R˜s as follows:(26)R˜s=r˜11sr˜12s⋯r˜1msr˜21sr˜22s⋯r˜2ms⋮⋮⋮⋮r˜n1sr˜n2s⋯r˜nms.Then, the reference pointr˜ is obtained as(27)r˜=⊕i=1nw1r˜i1s,⊕i=1nw2r˜i2s,…,⊕i=1nwmr˜ims,where w=w1,w2,…,wm is the weighted vector relevant to OWA.


## 3.2.3. Determination of the Gain or Loss Values

The gain or loss valuesdx˜ijl,r˜j of the jth attribute of the lth decision scheme given by the ith decision-maker are obtained by formula (13). Then, the decision information set F˜l=f˜1l,…,f˜il,…,f˜nl is obtained, and f˜il=dx˜i1l,r˜1,p˜i1l;dx˜i2l,r˜2,p˜i2l;…;dx˜iml,r˜m,p˜iml is the prospect used to express decision information of the lth decision scheme given by the ith decision-maker.


## 3.2.4. Computation and Normalization of Prospect Values

The prospect valuepvlf˜i of f˜il is obtained by formula (15). Let pvlf˜i be written as follows:(28)pvlf˜i=pv11lf˜i,pv12lf˜i,pv13lf˜i,pv14lf˜i,pv21lf˜i,pv22lf˜i,pv23lf˜i,pv24lf˜i;μpvlf˜i,vpvlf˜i.The prospect valuepvlf˜i is normalized.(29)p¯vlf˜i=pv11lf˜i−γ11l−γ11l+−γ11l−,pv12lf˜i−γ12l−γ12l+−γ12l−,pv13lf˜i−γ13l−γ13l+−γ13l−,pv14lf˜i−γ14l−γ14l+−γ14l−,pv21lf˜i−ζ21l−ζ21l+−ζ21l−,pv22lf˜i−ζ22l−ζ21l+−ζ22l−,pv23lf˜i−ζ23l−ζ23l+−ζ23l−,pv24lf˜i−ζ24l−ζ24l+−ζ24l−;μpvlf˜i,vpvlf˜i,where p¯vlf˜iis the normalized value of pvlf˜i; γ1sl−,s=1,2,3,4 and γ2sl− are mini=1npvl1sf˜i and mini=1npv2slf˜i, respectively; and γ1sl+ and γ2sl+ are mini=1npvl1sf˜i and maxi=1npv2slf˜i, respectively.


## 3.2.5. Consensus Analysis and Aggregation of Decision Information Given by all Decision-Makers

The population and the maximum number of iterations are denoted byP and TM, respectively. The consensus analysis and aggregation procedure are the same as those described in Section 3.1.


## 3.2.6. Ranking of Decision Schemes

The count of decision schemes isL. All decision schemes perform Steps 1–5. All expected values EVal of all=1,2,…,L are computed according to formula (10). Then, score function Sal and accuracy function Aal values are computed. Ranking of all decision schemes is done according to the comparing rules of TrIFNs (Section 2).


## 4. Simulations and Analyses of Results

### 4.1. Simulations

We assume that stock investors decide whether to buy a stock based on gain or loss of stock. The stock investors select one of the three stocksA=A1,A2,A3. The attributes of stocks are denoted as X˜=x˜1,x˜2,x˜3,x˜4,x˜5, where x˜1 means big gain, x˜2 means small gain, x˜3 means basically unchanged, x˜4 means small loss, and x˜5 means big loss. Ten decision-makers D=D1,D2,…,D10 are invited to make decisions. The values of attributes S evaluated by decision-makers D are listed in Table 1. The values of weights P˜=p˜1,p˜2,p˜3,p˜4,p˜5 and the reference points R˜=r˜1,r˜2,…,r˜10 provided by decision-makers D are listed in Tables 2 and 3, respectively. w1,w2,…,w10=0.1. dmax is 0.85 and δ is 2 (values of dmax and δ should be given according to a number of comparative experiments based on other MAGDM methods). The parameters of the genetic clustering algorithm are as follows: the population P is 5, the maximum number of iterations TM is 8, the genetic crossover Pc is 0.8, the initial value of k is 2, and the positive coefficients a and b are 2 and 1.2, respectively. The genetic K-means algorithm is executed 20 times independently (Tables 1–3). We compare our proposed method with the methods proposed by [20, 22, 23] and [10]. The parameters of the other three methods are taken from literature [10, 20, 22, 23].Table 1
Attributes evaluated by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.5, 0.7), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
A
3
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
2
A
1
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
3
A
1
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
D
4
A
1
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
A
2
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
D
5
A
1
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
A
3
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
D
6
A
1
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
A
3
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
7
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
8
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
D
9
A
1
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
D
10
A
1
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>Table 2
Weights evaluated by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
D
2
A
1
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
D
3
A
1
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
D
4
A
1
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
2
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
5
A
1
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
A
3
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
D
6
A
1
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
2
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
7
A
1
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
D
8
A
1
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
9
A
1
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
3
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
D
10
A
1
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
3
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>Table 3
Reference points given by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
D
2
A
1
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
A
2
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
3
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
D
3
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
2
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
3
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
D
4
A
1
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
5
A
1
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
D
6
A
1
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
2
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
D
7
A
1
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
3
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
8
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
D
9
A
1
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
D
10
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
2
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>


### 4.2. Analyses of Results

Figures2–4 present the curves of varying k values (in the experiment with the best clustering result Gbl) in aggregation procedures of schemes A1, A2, and A3, respectively, and they indicate that the values of k in the genetic K-means clustering algorithm realize automatic learning. Table 4 presents the ranking and computational time of four methods.Figure 2
Curve of varyingk value in the aggregation procedure of scheme A1.Figure 3
Curve of varyingk value in the aggregation procedure of scheme A2.Figure 4
Curve of varyingk value in the aggregation procedure of scheme A3.Table 4
Ranking of schemes and computational time.
Method
Ranking of schemes
Computational time (s)
Our proposed method
A
2
≻
A
1
≻
A
3
18.029
Zhao [23]
A
2
≻
A
1
≻
A
3
74.332
Liu & Chen [22]
A
1
≻
A
3
≻
A
2
5.956
Yuan & Li [20]
A
2
≻
A
3
≻
A
1
19.353
[10]
A
2
≻
A
3
≻
A
1
33.251Zhao [23] uses interval-valued intuitionistic fuzzy numbers to express the decision information, so we adopted the parameters and operational details of Zhao’s K-means clustering algorithm to conduct the experiments in the present study. Although the ranking of schemes obtained by this method is the same as our proposed method, there are two shortcomings of Zhao’s method: first, the decision information is expressed in the form of interval-valued intuitionistic fuzzy numbers, and, second, the computational time is longer than that in our proposed method because the k values are determined by using the traversing method.Liu and Chen [22] modify the clustering results on the basis of the gradient descent method. After computation, the ranking of schemes is different from that of our proposed method. The values of Gb1, Gb2, and Gb3 obtained by Liu and Chen’s method are 0.155, 0.163, and 0.146, respectively, whereas those obtained by our proposed method are 0.326, 0.320, and 0.402, respectively, clearly indicating that our proposed method yields higher values. The results indicate that, in Liu and Chen’s method, the k value is determined before the simulation, which leads to the lower performance of this clustering method compared with our proposed method.Yuan and Li [20] use the framework of the PT and TrIFN to solve the MAGDM problem. The preferred scheme is A2, which is the same as our proposed method, but the alternatives are different from those obtained by our proposed method. The difference between our proposed method and Yuan and Li’s method is the aggregation method used. As a representative MAGDM method, Yuan and Li’s method comprehensively considers the factors in the MAGDM problem. The decision results of all MAGDM methods are neither absolutely good nor absolutely poor and the quality of MAGDM methods depends on whether the factors in the decision-making process are fully considered. Therefore, we cannot just compare whether the results of Yuan and Li’s method and our proposed method are absolutely good or poor. The simulation results of these two methods can be used for mutual reference.Wu and Cao [10] use the TrIFN weighted geometric operator and hybrid geometric operator to obtain the collective overall values. The characteristic of the aggregation is on the basis of attribute independence and decision-makers’ risk preference. The preferred scheme is scheme A2, which is the same as our proposed method, but the alternatives are different from those obtained by our proposed method. The computational time of this method is medium among the five algorithms.The performance of the proposed method is better than the other methods analysed in this study on the basis of the clustering concept. In addition, the preferred scheme by our method is the same as that by Yuan and Li’s method. Furthermore, the proposed method is apprehensible and feasible to solve the MAGDM problems. It enriches the theory of solving MAGDM problems based on clustering algorithms.


## 4.1. Simulations

We assume that stock investors decide whether to buy a stock based on gain or loss of stock. The stock investors select one of the three stocksA=A1,A2,A3. The attributes of stocks are denoted as X˜=x˜1,x˜2,x˜3,x˜4,x˜5, where x˜1 means big gain, x˜2 means small gain, x˜3 means basically unchanged, x˜4 means small loss, and x˜5 means big loss. Ten decision-makers D=D1,D2,…,D10 are invited to make decisions. The values of attributes S evaluated by decision-makers D are listed in Table 1. The values of weights P˜=p˜1,p˜2,p˜3,p˜4,p˜5 and the reference points R˜=r˜1,r˜2,…,r˜10 provided by decision-makers D are listed in Tables 2 and 3, respectively. w1,w2,…,w10=0.1. dmax is 0.85 and δ is 2 (values of dmax and δ should be given according to a number of comparative experiments based on other MAGDM methods). The parameters of the genetic clustering algorithm are as follows: the population P is 5, the maximum number of iterations TM is 8, the genetic crossover Pc is 0.8, the initial value of k is 2, and the positive coefficients a and b are 2 and 1.2, respectively. The genetic K-means algorithm is executed 20 times independently (Tables 1–3). We compare our proposed method with the methods proposed by [20, 22, 23] and [10]. The parameters of the other three methods are taken from literature [10, 20, 22, 23].Table 1
Attributes evaluated by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.5, 0.7), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
A
3
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
2
A
1
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
3
A
1
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
D
4
A
1
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
A
2
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
D
5
A
1
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
A
3
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
D
6
A
1
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
A
3
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
7
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
D
8
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
D
9
A
1
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
D
10
A
1
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>Table 2
Weights evaluated by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
D
2
A
1
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
D
3
A
1
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
D
4
A
1
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
2
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
5
A
1
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
A
3
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
D
6
A
1
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
2
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
7
A
1
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
3
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
D
8
A
1
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
3
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
D
9
A
1
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
3
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
D
10
A
1
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
3
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>Table 3
Reference points given by decision-makers.
D
A
x
˜
1
x
˜
2
x
˜
3
x
˜
4
x
˜
5
D
1
A
1
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
D
2
A
1
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
A
2
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
3
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
D
3
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
2
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
A
3
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
D
4
A
1
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
A
2
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
5
A
1
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
A
2
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
A
3
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
D
6
A
1
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
A
2
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
D
7
A
1
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
2
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
A
3
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
D
8
A
1
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
<(0.2, 0.4, 0.7, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
A
2
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.3, 0.4, 0.6, 0.7), (0.3, 0.4, 0.6, 0.7); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
D
9
A
1
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
A
2
<(0.1, 0.2, 0.3, 0.4), (0.1, 0.2, 0.3, 0.4); 0.7, 0.2>
<(0.1, 0.2, 0.4, 0.5), (0.1, 0.2, 0.4, 0.5); 0.7, 0.2>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.9), (0.4, 0.5, 0.7, 0.9); 0.5, 0.4>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
A
3
<(0.5, 0.6, 0.7, 0.8), (0.5, 0.6, 0.7, 0.8); 0.4, 0.3>
<(0.5, 0.6, 0.8, 0.9), (0.5, 0.6, 0.8, 0.9); 0.4, 0.3>
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.3, 0.4, 0.5, 0.6), (0.3, 0.4, 0.5, 0.6); 0.7, 0.2>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
D
10
A
1
<(0.2, 0.3, 0.6, 0.7), (0.2, 0.3, 0.6, 0.7); 0.5, 0.4>
<(0.2, 0.4, 07, 0.9), (0.2, 0.4, 0.7, 0.9); 0.4, 0.3>
<(0.2, 0.4, 0.6, 0.8), (0.2, 0.4, 0.6, 0.8); 0.4, 0.3>
<(0.3, 0.4, 0.7, 0.8), (0.3, 0.4, 0.7, 0.8); 0.7, 0.2>
<(0.2, 0.3, 0.5, 0.6), (0.2, 0.3, 0.5, 0.6); 0.5, 0.4>
A
2
<(0.4, 0.5, 0.6, 0.7), (0.4, 0.5, 0.6, 0.7); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.2, 0.5, 0.6), (0.1, 0.2, 0.5, 0.6); 0.7, 0.2>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.3, 0.5, 0.7, 0.9), (0.3, 0.5, 0.7, 0.9); 0.6, 0.3>
A
3
<(0.6, 0.7, 0.8, 0.9), (0.6, 0.7, 0.8, 0.9); 0.4, 0.3>
<(0.4, 0.5, 0.7, 0.8), (0.4, 0.5, 0.7, 0.8); 0.5, 0.4>
<(0.1, 0.3, 0.5, 0.7), (0.1, 0.3, 0.5, 0.7); 0.6, 0.3>
<(0.1, 0.3, 0.6, 0.8), (0.1, 0.3, 0.6, 0.8); 0.6, 0.3>
<(0.2, 0.3, 0.4, 0.5), (0.2, 0.3, 0.4, 0.5); 0.5, 0.4>


## 4.2. Analyses of Results

Figures2–4 present the curves of varying k values (in the experiment with the best clustering result Gbl) in aggregation procedures of schemes A1, A2, and A3, respectively, and they indicate that the values of k in the genetic K-means clustering algorithm realize automatic learning. Table 4 presents the ranking and computational time of four methods.Figure 2
Curve of varyingk value in the aggregation procedure of scheme A1.Figure 3
Curve of varyingk value in the aggregation procedure of scheme A2.Figure 4
Curve of varyingk value in the aggregation procedure of scheme A3.Table 4
Ranking of schemes and computational time.
Method
Ranking of schemes
Computational time (s)
Our proposed method
A
2
≻
A
1
≻
A
3
18.029
Zhao [23]
A
2
≻
A
1
≻
A
3
74.332
Liu & Chen [22]
A
1
≻
A
3
≻
A
2
5.956
Yuan & Li [20]
A
2
≻
A
3
≻
A
1
19.353
[10]
A
2
≻
A
3
≻
A
1
33.251Zhao [23] uses interval-valued intuitionistic fuzzy numbers to express the decision information, so we adopted the parameters and operational details of Zhao’s K-means clustering algorithm to conduct the experiments in the present study. Although the ranking of schemes obtained by this method is the same as our proposed method, there are two shortcomings of Zhao’s method: first, the decision information is expressed in the form of interval-valued intuitionistic fuzzy numbers, and, second, the computational time is longer than that in our proposed method because the k values are determined by using the traversing method.Liu and Chen [22] modify the clustering results on the basis of the gradient descent method. After computation, the ranking of schemes is different from that of our proposed method. The values of Gb1, Gb2, and Gb3 obtained by Liu and Chen’s method are 0.155, 0.163, and 0.146, respectively, whereas those obtained by our proposed method are 0.326, 0.320, and 0.402, respectively, clearly indicating that our proposed method yields higher values. The results indicate that, in Liu and Chen’s method, the k value is determined before the simulation, which leads to the lower performance of this clustering method compared with our proposed method.Yuan and Li [20] use the framework of the PT and TrIFN to solve the MAGDM problem. The preferred scheme is A2, which is the same as our proposed method, but the alternatives are different from those obtained by our proposed method. The difference between our proposed method and Yuan and Li’s method is the aggregation method used. As a representative MAGDM method, Yuan and Li’s method comprehensively considers the factors in the MAGDM problem. The decision results of all MAGDM methods are neither absolutely good nor absolutely poor and the quality of MAGDM methods depends on whether the factors in the decision-making process are fully considered. Therefore, we cannot just compare whether the results of Yuan and Li’s method and our proposed method are absolutely good or poor. The simulation results of these two methods can be used for mutual reference.Wu and Cao [10] use the TrIFN weighted geometric operator and hybrid geometric operator to obtain the collective overall values. The characteristic of the aggregation is on the basis of attribute independence and decision-makers’ risk preference. The preferred scheme is scheme A2, which is the same as our proposed method, but the alternatives are different from those obtained by our proposed method. The computational time of this method is medium among the five algorithms.The performance of the proposed method is better than the other methods analysed in this study on the basis of the clustering concept. In addition, the preferred scheme by our method is the same as that by Yuan and Li’s method. Furthermore, the proposed method is apprehensible and feasible to solve the MAGDM problems. It enriches the theory of solving MAGDM problems based on clustering algorithms.


## 5. Conclusion and Further Research

In this study, we proposed a MAGDM method using a geneticK-means clustering algorithm. The simulation results demonstrate that our proposed method is apprehensible and feasible to solve MAGDM problems. In the future, we will use the proposed MAGDM method to deal with other types of decision information, such as (a) trapezoidal fuzzy numbers in consensus models for MAGDM under trapezoidal fuzzy numbers environment; (b) hesitant fuzzy linguistic information in additiveconsensus of hesitant fuzzy linguistic preference relation with a new expansion principle for hesitant fuzzy linguistic term sets local, and hesitant fuzzy linguistic information in feedback mechanism based on consensus-derived for consensus building in MAGDM with hesitant fuzzy linguistic preference relations; and (c) linguistic fuzzy information in algorithms for improving additive consensus of linguistic preference relations with an integer optimization model.



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*Source: 8313892-2020-05-06.xml*