Consensus Reaching Process in the Two-Rank Group Decision-Making with Heterogeneous Preference Information

Huali Tang; Shoufu Wan; Cong-Cong Li; Haiming Liang; Yucheng Dong

doi:10.2991/ijcis.d.210201.001

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Volume 14, Issue 1, 2021, Pages 758 - 768

Consensus Reaching Process in the Two-Rank Group Decision-Making with Heterogeneous Preference Information

Authors

Huali Tang¹, Shoufu Wan²^{, *}, Cong-Cong Li³, Haiming Liang¹, Yucheng Dong¹

¹Business School, Sichuan University, Sichuan, China

²School of Management, Shenzhen Institute of Information Technology, Shenzhen, China

³School of Economics and Management, Southwest Jiaotong University, Sichuan, China

^*Corresponding author. Email: wansoff@qq.com

Corresponding Author

Shoufu Wan

Received 20 October 2020, Accepted 15 January 2021, Available Online 10 February 2021.

DOI: 10.2991/ijcis.d.210201.001 How to use a DOI?
Keywords: Two-rank group decision-making; Consensus reaching; Heterogeneous preference information; Feedback adjustment rule
Abstract: This paper proposes a novel consensus reaching process (CRP) for the two-rank group decision-making (GDM) problems with heterogeneous preference information. The methods for deriving the individual and collective preference vector are provided. And the individual and collective two-rank vectors are obtained. Then, the feedback adjustment rules are proposed. Next, an algorithm is given to describe the two-rank CRP with heterogeneous preference information. Finally, we present a practical example to illustrate the feasibility of the proposed method.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Group decision-making (GDM) [1,2] as an important field in decision analysis has gained the increasing attentions since its first appearance. In real applications of GDM problems, due to the differences of the professional knowledge, cultural background, and social experience among different individuals, it is natural that different decision makers may utilize heterogeneous representation structures to express their preferences on the alternatives [3]. Recently, the study on consensus reaching process (CRP) has become an interesting and necessary area [4–7].

Over the last few decades, a mass of consensus models for the GDM with heterogeneous preference structures have been developed from different focuses: (i) Heterogeneous formats of expressions. For example, in the existing studies [8–14] it is assumed that different individuals will express the additive preference relations, multiplicative preference relations, preference orderings, and utility values, etc. (ii) Multi-granularity linguistic term information. For example, with the consideration of the individuals may use different scale of linguistic terms to express their preferences, various CRPs with the multi-granularity linguistic term information have been proposed [15–19]. (iii) The optimization-based models. The existing studies optimize some available standards such as cost, time, adjustments, and information losses to achieve a better solution to increase the agreements among individuals [20–28].

Despite the research already conducted on the CRP with heterogeneous preference structures, there exists a major challenge that contradicts the real applications. Most of the existing CRPs are based on the individual and group rankings of alternatives to build a consensus, while in some real-life GDM problems in which individuals don't need to rank alternatives, and instead hope to divide the alternatives into two categories [1,17] (i.e., two-rank GDM problems). For example:

Example 1.

In the problem of reviewing the projects approved by Natural science foundation of China, based on the novelty of topic, quality of writing, and previous performances, each expert will divide the candidate proposals into two categories, i.e., the approved proposals and the rejected proposals.

Example 2.

In the problem of bank loaning, with the consideration of some factors such as income, deposit, profession, and history records, the bank will divide the individuals who applied for a loan into two categories, i.e., arranging loans for some individuals and rejecting loans for other individuals.

To overcome the weaknesses in the existing CRP with heterogeneous preference structures, the aim of this paper is to develop a novel CRP for the two-rank GDM problems with heterogeneous preference information. The rest of this study is organized as follows: Section 2 introduces the preliminaries regarding four kinds of heterogeneous preference representation structures, and formulates the two-rank GDM problems. In Section 3, we develop a novel CRP for the two-rank GDM with heterogeneous preference representation structures. Next, a practical example under the background of selecting the subway construction plan is given to show the feasibility of the proposed two-rank CRP in Section 4. Finally, conclusions are summarized in Section 5.

2. PRELIMINARIES

In this section, the basic knowledge regarding heterogeneous preference information is firstly introduced. Then, the formulation of two-rank GDM problems is presented.

2.1. Heterogeneous Preference Information in GD M

Let X=x1,x2⋯,xn be an alternative set, where xi represents the ith alternative, i=1,2,…,n. Let D=d1,d2⋯,dm be the set of individuals, where dk denotes the kth individual, k=1,2,…,m.

In practice, due to the differences in the professional knowledge, cultural background, social experience, and personality among different individuals, sometimes these people use different representation structures to express the preferences on the alternatives.

Let DL, DU, DP, and DArepresent the subset of individuals who express their preferences using linguistic preference relations, utility functions, additive preference relations, and multiplicative preference relations, respectively, satisfying DL∪DU∪DP∪DA=D, Dk∩Dq=∅, k,q=L,U,P,A and k≠q. Let S=s0,s1,…,sT be a linguistic term set with odd granularities.

Specifically, the heterogeneous preference information is introduced in the below:

Case 1. dk∈DL. In this case, the individual dk will provide linguistic preference relations L(k)=lij(k)n×n on X, where lij(k)=sα∈S and lji(k)=sT−α [10,29–31].

Case 2. dk∈DU. In this case, the individual dk will give provide the utility vectors U(k)=u1(k),u2(k),⋯,un(k)T on X, where ui(k) represents the utility value of xi given by the individual dk, ui(k)∈0,1, for i=1,2,…,n [32].

Case 3. dk∈DP. In this case, the individual dk will give additive preference relations P(k)=pij(k)n×n on X, where pij(k)+pji(k)=1, pij(k)∈[0,1] [33].

Case 4. dk∈DA. In this case, the individual dk will provide multiplicative preference relations A(k)=aij(k)n×n on X, where aij(k)×aji(k)=1, aij(k)≥0 [34].

2.2. Formulation of the Two-Rank GDM Problems

The main notations of the two-rank GDM problem are listed as follows:

X=x1,x2⋯,xn: The set of alternatives;
D=d1,d2⋯,dm: The set of individuals;
DL=d1,d2⋯,dm1: The set of individuals expressing the linguistic preference relations;
DU=dm1+1,dm1+2⋯,dm2: The set of individuals expressing the utility vectors;
DP=dm2+1,dm2+2⋯,dm3: The set of individuals expressing the additive preference relations;
DA=dm3+1,dm3+2⋯,dm: The set of individuals expressing the multiplicative preference relations;
w(k)=w1(k),w2(k),⋯wn(k)T: Individual preference vector on X associated with dk;
w(k∗)=w1(k∗),w2(k∗),⋯wn(k∗)T: Standardized individual preference vector on X associated with dk;
w(c)=w1(c),w2(c),⋯wn(c)T: Collective preference vector on X;

The problem concerned in this paper is how to obtain the collective two-rank result with the acceptable consensus level based on the above notations and variables.

3. THE PROPOSED METHOD

In this section, a novel consensus reaching framework for the two-rank GDM with heterogeneous preference information is firstly developed. Then, the detailed two-rank CRPs which comprises of selection process and feedback adjustments are given.

3.1. Two-Rank Consensus Reaching Framework

Inspired by the idea of the classical CRP, we build a novel consensus reaching framework for the two-rank GDM with heterogeneous preference information, which is depicted in Figure 1.

As shown in Figure 1, the two-rank selection process and the feedback adjustments are included in the framework.

Two-rank selection process. Firstly, the heterogeneous preference information expressed by the individuals is transformed into the individual preference vectors. Then, the collective preference vector is obtained by aggregating the individual preference vectors. Furthermore,
Feedback adjustment. The consensus measure is further defined as the deviation between the individual preference vectors and collective preference vector. Once the consensus measure is not satisfied, it is necessary to use the feedback adjustment to improve the consensus level among different individuals. Based on the inherent characteristics of preference representation structures, the feedback adjustment rules are derived to provide the suggestions to adjust the opinions for each individual. The procedure of feedback adjustment is repeated until the consensus among individuals is achieved.

3.2. Two-Rank Selection Process

The procedure of two-rank selection process with heterogeneous preference information is given as follows:

Step 1: Obtaining the individual preference vectors

The individual preference vector w(k)=w1(k),w2(k),⋯wn(k)T(k∈M) is determined by considering the following four cases:

Case 1. dk∈DL

In Case 1, the individual dk provides his/her preference information on X using the linguistic preference relation L(k)=lij(k)n×n, where lij(k)∈S.

For computation convenience, by means of 2-tuple linguistic model, the linguistic term is usually transformed as a numerical scale. Thus, let B(k)=bij(k)n×n be a numerical preference relation associated with L(k)=lij(k)n×n, where

bij(k)=Δ−1lij(k)(1)

From Eq. (1), it is easier to obtain that: bii(k)=T2 and bij(k)+bji(k)=T.

Then, the preference on the alternative xi by the individual dk is calculated as follows:

wi(k)=1n∑j=1nbij(k)(2)

Case 2. dk∈DU

In Case 2, the individual dk uses utility function U(k)=u1(k),u2(k),⋯,un(k)T to provide his/her preference information on alternatives X. Since the larger value ui(k) implies that the larger preference value wi(k). Thus, the preference on the alternative xi by the individual dk is calculated as follows:

wi(k)=ui(k)(3)

Case 3. dk∈DP

In Case 3, the individual dk uses additive preference relationPk=pijkn×n to provide his/her preference information on X. Thus, the preference vector of the individual dk on the alternative xi is calculated by

wi(k)=1npi1(k)+pi2(k)+⋯+pin(k)(4)

Case 4. dk∈DA

In Case 4, the individual dk provides his/her preference information on X by means of the multiplicative preference relationA(k)=aij(k)n×n. Then, the row geometric mean method (RGMM) in AHP is used to derive the preference of the individual dk on the alternative xi.

Let wi(k) be as defined before, then the RGMM can be described as follows [35]:

min ∑i=1n∑j>inlnaij(k)−lnwi(k)−lnwj(k)2s.t. ∑i=1nwi(k)=1wi(k)≥0(5)

Crawford and Williams [35] have shown that the optimal solution to model (5) is written as the geometric means of the rows of matrix A(k):

wi(k)=∏j=1naij(k)1/n∑i=1n∏j=1naij(k)1/n(6)

Step 2: Obtaining the collective preference vectors

Due to the scale differences in the individual preference vectors of different decision makers, we first standardize the individual preference vector into the standardized preference vector.

Let w(k∗)=w1(k∗),w2(k∗),⋯wn(k∗)T be the standardized individual preference vectors of dk, where

wi(k∗)=wi(k)∑i=1nwi(k)(7)

Then, let w(c)=w1(c),w2(c),⋯,wn(c)T be the collective preference vector, where

wi(c)=1mwi(1∗)+wi(2∗)+⋯+wi(m∗)(8)

Step 3: Obtaining the individual and collective two-rank vectors

Let O(k)=o1(k),o2(k),⋯,on(k)T be the individual ranking vector of dk, and let O(c)=o1(c),o2(c),⋯,on(c)T be collective ranking vector. The ranking vector O(k) and O(c) can be determined by the preference vectors w(k∗) and w(c), respectively. Meanwhile, the larger values wi(k∗) and wi(c) implies the smaller values for oi(k∗) and oi(c), respectively.

Then, let p be the required number in the first category. In practical two-rank GDM, p can be determined by the real demand of moderator. Let R(k)=r1(k),r2(k),⋯,rn(k)T be the individual two-rank vector of dk and let R(c)=r1(c),r2(c),⋯,rn(c)T be collective two-rank vector, where

ri(k)=1,oi(k)≤p0,otherwise(9a)

and

ri(c)=1,oi(c)≤p0,otherwise(9b)

3.3. Feedback Adjustments

Before presenting the feedback adjustment rules, the consensus measure is defined as follows:

Definition 1.

Let R(k)=r1(k),r2(k),⋯,rn(k)T and R(c)=(r1(c),r2(c),⋯,rn(c))T be as defined before. Then, the two-rank consensus level among all the individuals is given by

CMd1,d2,…,dm=1mp∑k=1m∑i=1nti(k)(10)

where ti(k) indicates whether the individual dk determines the same category on xi as the collective two-rank vector, i.e.,

ti(k)=1,ri(k)=1 and ri(c)=10,otherwise,(11)

If CMd1,d2,…,dm=1, then all the individuals have full consensus with the two-rank vectors. Otherwise, the larger CMd1,d2,…,dm indicates the higher two-rank consensus level among the individuals.

Let CM¯ be the predetermined threshold for judging the acceptable consensus level. If CMd1,d2,…,dm≥CM¯, then the acceptable two-rank consensus among the individuals is achieved.

Then, the aim of feedback adjustments is to provide individuals with adjustment suggestions for heterogeneous preference information, thus the consensus level among the individuals will be improved.

In this paper, we design a new feedback adjustment rules by considering the heterogeneous preference information. The feedback adjustment rules are composed of two key components: (i) Transforming the collective preference vector into the type of preference information expressed by each individual. For example, if the individual dk uses the linguistic preference information to express his/her preferences, then the collective preference vector should also be transformed into the linguistic preference relation. (ii) The individuals adjust their preferences based on the original preference information and transformed preference information.

Furthermore, with the consideration of the heterogeneous preference structures, the detailed feedback adjustment rules are introduced as follows:

Case 1. dk∈DL. In this case, the collective preference vector is transformed into a linguistic preference relation. Let Bc,k=bijc,kn×n be the transformed linguistic preference relation. According to 2-tuple linguistic representation model, we obtain the values of Δ−1bijc,k as follows:

Δ−1bij(c,k)=Twi(c∗)wi(c∗)+wj(c∗)(12)

Then, we encode the values of Δ−1bij(c,k) into bij(c,k) as follows:

bij(c,k)=ΔΔ−1bij(c,k)=lij(c,k),αij(c,k)(13)

Furthermore, let B¯(k)=b¯ij(k)n×n be the adjusted linguistic preference relation, where

b¯ij(k)∈minbij(k),bij(c,k),maxbij(k),bij(c,k),i<j,b¯ij(k)=sT/2,i=j,bij(k)¯=ΔT−Δ−1bji(k),i>j.(14)

Case 2. dk∈DU. In this case, the collective preference vector is transformed into a utility preference vector. Let U(c,k)=u1(c,k),u2(c,k),⋯,un(c,k)T be the transformed utility vector, where

ui(c,k)=wi(c∗)∑i=1nui(k)(15)

Let U(k)¯=u1(k)¯,u2(k)¯⋯,un(k)¯T be the adjusted utility preference vector, where

ui(k)¯∈minui(k),ui(c,k),maxui(k),ui(c,k)(16)

Case 3. dk∈DP. In this case, the collective preference vector is transformed into an additive preference relation. Let P(c,k)=pij(c,k)n×n be a transformed additive preference relation, where

pij(c,k)=wi(c∗)wi(c∗)+wj(c∗)(17)

Let P¯(k)=p¯ij(k)n×n be the adjusted additive preference relation, where

pij(k)¯∈minpij(k),pij(c,k),maxpij(k),pij(c,k),i<j,pij(k)¯=0.5,i=j,pij(k)¯=1−pji(k)¯,i>j.(18)

Case 4. dk∈DA. In this case, the collective preference vector is transformed into a multiplicative preference relation. Let A(c,k)=aij(c,k)n×n be a transformed multiplicative preference relation, where

aij(c,k)=wi(c∗)wj(c∗)(19)

Let A¯(k)=a¯ij(k)n×n be the adjusted multiplicative preference relation, where

aij(k)¯∈minaij(k),aij(c,k),maxaij(k),aij(c,k),i<j,aij(k)¯=1,i=j,aij(k)¯=1aji(k)¯,i>j.(20)

3.4. Two-rank Consensus Reaching Algorithm

Based on the proposed feedback adjustment rules, we design a two-rank consensus reaching algorithm with heterogeneous preference information.

Let z be the iteration number. Then the algorithm is given in the below.

Algorithm 1:

Input: Maximum number of iterations τ, the required number p in the first category, a predetermined consensus threshold CL¯, and the original preference information {L(1),⋯,L(m1),U(m1+1),⋯,U(m2),P(m2+1),⋯,P(m3),A(m3+1),⋯,A(m)} by individuals.

Output: The adjusted individual preference information {L¯(1),⋯,L¯(m1),U¯(m1+1),⋯,U¯(m2),P¯(m2+1),⋯,P¯(m3),A¯(m3+1),⋯,A¯(m), the standardized collective preference vector w(c∗)¯, the collective two-rank vector R¯(c)=r¯1(c),r¯2(c),⋯,r¯n(c)T, and number of iterations.

Step 1: Let z=0, Lz(k)=lij,z(k)n×n=lij(k)n×nk=1,2,⋯,m1,Uz(k)=(u1,z(k),u2,z(k),⋯,un,z(k))T=(u1(k),u2(k),⋯,un(k))T(k=m1+1,m1+2,⋯,m2),Pz(k)=pij,z(k)n×n=pij(k)n×n(k=m2+1,m2+2,⋯,m3) andAz(k)=aij,z(k)n×n=aij(k)n×n(k=m3+1,m3+2,⋯,m).

Step 2: Based on the two-rank selection process in Section 3.2, four cases are considered to obtain individual preference vectors.

Case 1. dk∈DL(k=1,2,⋯,m1). wi(k,z)=1n∑j=1nΔ−1lij,z(k),αij,z(k).

Case 2. dk∈DU(k=m1+1,m1+2,⋯,m2). wi(k,z)=ui,z(k).

Case 3. In this case, dk∈DP(k=m2+1,m2+2,⋯,m3). wi(k,z)=1npi1,z(k)+pi2,z(k)+⋯+pin,z(k).

Case 4. In this case, dk∈DA(k=m3+1,m3+2,⋯,m). wi(k,z)=∏j=1naij,z(k)1/n∑i=1n∏j=1naij,z(k)1/n.

Step 3: Transform w(k,z)=w1(k,z),w2(k,z),⋯,wn(k,z)T into w(k∗,z)=w1(k∗,z),w2(k∗,z),⋯,wn(k∗,z)T by using Eq. (7). Then, using Eq. (8) derives the collective preference vector w(c,z)=w1(c,z),w2(c,z),⋯,wn(c,z)T.

Step 4: According to the preference vectors w(k∗,z)=w1(k∗,z),w2(k∗,z),⋯,wn(k∗,z)T and w(c,z)=w1(c,z),w2(c,z),⋯,wn(c,z)T, we obtain the ranking vectors Oz(k)=o1,z(k),o2,z(k),⋯,on,z(k)T and Oz(c)=o1,z(c),o2,z(c),⋯,on,z(c)T. Next, utilizing the value of p, the two-rank vectors Rz(k)=r1,z(k),r2,z(k),⋯,rn,z(k)T and Rz(c)=r1,z(c),r2,z(c),⋯,rn,z(c)T are determined.

Step 5: Using Eq. (10) calculates the two-rank consensus level CM{d1,d2,…,dm} among all the individuals. If CM{d1,d2,…,dm}≥CL¯, go to Step 8; otherwise, go to Step 6.

Step 6: With the consideration of heterogeneous preference information, the feedback adjustment rules are proposed.

Case 1. dk∈DL(k=1,2,⋯,m1). Let Bz(c,k)=bij,z(c,k)n×n. Using Eq. (12) obtains the semantics Δ−1bij,z(c,k). Then, we encode the values of Δ−1bij,z(c,k) into bij(c,k)=lij,z(c,k),αij,z(c,k) by using Eq. (13). Next, the linguistic preference relations Lz+1(k)=lij,z+1(k),αij,z+1(k)n×n are derived as:

lij,z+1(k),αij,z+1(k)∈minlij,z(k),lij,z(c,k),maxlij,z(k),lij,z(c,k),i<j,lij,z+1(k)=sT/2,i=j,lij,z+1(k)=ΔT−Δ−1lji,z+1(k),i>j.

Case 2. dk∈DU(k=m1+1,m1+2,⋯,m2). Let Uz(c,k)=u1,z(c,k),u2,z(c,k),⋯,un,z(c,k)T. Using Eq. (15) obtains Uz(c,k). Next, the utility preference vectors Uz+1(c,k)=u1,z+1(c,k),u2,z+1(c,k),⋯,un,z+1(c,k)T are derived as:

ui,z+1(k)∈minui,z(k),ui,z(c,k),maxui,z(k),ui,z(c,k)

Case 3. dk∈DP(k=m2+1,m2+2,⋯,m3). Let Pz(c,k)=pij,z(c,k)n×n. Using Eq. (17) obtains Pz(c,k). Next, the additive preference relations Pz+1(k)=pij,z+1(k)n×n are derived as:

pij,z+1(k)∈minpij,z(k),pij,z(c,k),maxpij,z(k),pij,z(c,k),i<j,pij,z+1(k)=0.5,i=j,pij,z+1(k)=1−pji,z+1(k),i>j.

Case 4. dk∈DA(k=m3+1,m3+2,⋯,m). Let Az(c,k)=aij,z(c,k)n×n. Using Eq. (19) obtains Az(c,k). Next, the multiplicative preference relations Az+1(k)=aij,z+1(k)n×n are derived as:

aij,z+1(k)∈minaij,z(k),aij,z(c,k),maxaij,z(k),aij,z(c,k),i<j,aij,z+1(k)=1,i=j,aij,z+1(k)=1aji,z+1(k),i>j.

Step 7: If z≥τ, go to step 8, otherwise let z=z+1, then go to step 2.

Step 8: Let L(k)¯=Lz(k)(k=1,2,⋯,m1), U(k)¯=Uz(k)(k=m1+1,m1+2,⋯,m2), P(k)¯=Pz(k)(k=m2+1,m2+2,⋯,m3), A(k)¯=Az(k)(m3+1,m3+2,⋯,m), w(c∗)¯=w(c∗,z), R(k)¯=Rz(k) and R(c)¯=Rz(c).

Step 9: Output adjusted individual preference information {L¯(1),⋯,L¯(m1),U¯(m1+1),⋯,U¯(m2),P¯(m2+1),⋯,P¯(m3),A¯(m3+1),⋯,A¯(m)}, the standardized collective preference vector w(c∗)¯, the collective two-rank vector R¯(c)=r¯1(c),r¯2(c),⋯,r¯n(c)T, and the number of iterations z.

Note 1. Compared with the existing consensus reaching models, the proposed consensus reaching algorithms increase the agreements of individuals on the two categories of alternatives (i.e., two–rank). Meanwhile, the provided feedback adjustment rules facilitate the individuals adjust their own preference information in the premise of without transforming the preference representation structures.

4. NUMERICAL EXAMPLE

In this section, a practical example in the background of subway construction plan is given to show the feasibility of the proposed two-rank CRPs.

Nowadays, the subway has become one of the most popular transportation tools for people. Thus, in this context, many cities all over the world paid the consideration attention on the subway construction. B city is an important city in western China, which is also famous for its ancient culture. To ease the road congestion and contribute to better air quality, B city decides to construct a new subway. Due to the complicated terrain and heritage conservation, the government in B city obtains the subway construction plans of other cities via various legal channels. Then, some approved plans will provide the important references for determining the final subway construction plan.

Suppose B city obtains six subway construction plans X={x1,x2,⋯,x6} after preliminary investigations. B city government invites eight experts (i.e., decision makers) D={d1,d2,⋯,d8} to distinguish six plans into two categories: approved plans and disapproved plans. To facilitate the expressions of decision makers, B city government allows the decision makers can use their own means to express the preference information.

Suppose the decision makers d1 and d2 use the linguistic preference relations L(1) and L(2) to express preference information on X. Here, the term set used by the decision maker is S = {s₀= extremely poor, s₁= very poor, s₂= poor, s₃= moder ate, s₄= good, s₅= very good, s₆= extremely good}. Then, L(1) and L(2) are given as follows:

L(1)=s3s4s6s5s3s6s2s3s5s6s3s5s0s1s3s3s1s4s1s0s3s3s1s4s3s3s5s5s3s5s0s1s2s2s1s3,L(2)=s3s3s5s6s2s5s3s3s5s6s2s5s1s1s3s3s1s3s0s0s3s3s0s3s4s4s5s6s3s6s1s1s3s3s0s3

The decision makers d3 and d4 use the utility functions U(3) and U(4) to express preference information on X, i.e.,

U(3)=(0.7,0.8,0.2,0.4,0.7,0.2)T,U(4)=(0.5,0.9,0.6,0.2,0.7,0.1)T..

The decision makers d5 and d6 use the additive preference relation P(5) and P(6) to express preference information on X, i.e.,

P(5)=0.50.540.830.890.550.920.460.50.80.820.580.910.170.20.50.550.160.730.110.180.450.50.120.750.450.420.840.880.50.860.080.090.270.250.140.5,P(6)=0.50.450.850.850.530.870.550.50.930.90.650.960.150.070.50.490.150.520.150.10.510.50.160.550.470.350.850.840.50.840.130.040.480.450.160.5

The decision makers d7 and d8 use the multiplicative preference relation A(7) and A(8) to express preference information on X, i.e.,

A(7)=11/373253195481/71/911/61/51/41/31/5611/231/21/452131/51/841/31/31,A(8)=1561/41/471/5151/51/671/61/511/61/724561394671/3181/71/71/21/91/81

In the following, we use the CRP in Section 3 to determine the approved plans.

For d1,d2∈DL, we determine the standardized individual preference vectors by using Eqs. (1) and (2) and (7), i.e.,

w(1∗)=(0.2500,0.2222,0.1111,0.1111,0.2222,0.0834)T,w(2∗)=(0.2222,0.2222,0.1111,0.0833,0.2593,0.1019)T.

For d3,d4∈DU, we determine the standardized individual preference vectors by using Eqs. (3) and (7), i.e.,

w(3∗)=(0.2333,0.2667,0.0667,0.1333,0.2333,0.0667)T,w(4∗)=(0.1667,0.3000,0.2000,0.0667,0.2333,0.0333)T.

For d5,d6∈DP, we determine the standardized individual preference vectors by using Eqs. (4) and (7), i.e.,

w(5∗)=(0.2350,0.2261,0.1283,0.1172,0.2194,0.0740)T,w(6∗)=(0.2249,0.2493,0.1044,0.1094,0.2143,0.0977)T.

For d7,d8∈DA, we determine the standardized individual preference vectors by using Eqs. (6) and (7), i.e.,

w(7∗)=(0.2274,0.4520,0.0253,0.1028,0.1396,0.0529)T,w(8∗)=(0.1616,0.0826,0.0359,0.4048,0.2911,0.0240)T.

Then, based on Eq. (8), the collective standardized preference vectors are determined by

w(c∗)=(0.2151,0.2526,0.0979,0.1411,0.2266,0.0667)T.

And the individual ranking vectors and collective ranking vectors are obtained as

o(1)=(1,2,4,5,3,6)T,o(2)=(2,3,4,6,1,5)T,o(3)=(2,1,5,4,3,6)T,o(4)=(4,1,3,5,2,6)T,o(5)=(1,2,4,5,3,6)T,o(6)=(2,1,5,4,3,6)T,o(7)=(2,1,6,4,3,5)T,o(8)=(3,4,5,1,2,6)T,o(c)=(3,1,5,4,2,6)T.

Suppose the B city government wants to choose three approved plans, i.e., p=3. Based on Eqs. (9) and (10), we obtain the individual and collective two-rank vectors, i.e.,

R(1)=(1,1,0,0,1,0)T,R(2)=(1,1,0,0,1,0)T,R(3)=(1,1,0,0,1,0)T,R(4)=(0,1,1,0,1,0)T,R(5)=(1,1,0,0,1,0)T,R(6)=(1,1,0,0,1,0)T,R(7)=(1,1,0,0,1,0)T,R(8)=(1,0,0,1,1,0)T,R(c)=(1,1,0,0,1,0)T.

Next, the two-rank consensus level is calculated as CMd1,d2,…,dm=1112. Suppose B city government establishes the threshold as: CM¯=0.95. Obviously, CM{d1,d2,…,dm}<CM¯., i.e., the acceptable consensus is not reached. Thus, it is necessary to carry out the feedback adjustment stage.

In the feedback adjustment stage, the collective preference vector into the type of preference information expressed by each individual.

For d1,d2∈DL, we utilize Eq. (12) to obtain the transformed linguistic preference relations:

B(c,1)=B(c,2)=(s3,0.00)(s3,0.16)(s4,−0.04)(s4,−0.06)(s3,0.28)(s4,0.36)(s3,−0.16)(s3,0.00)(s4,−0.19)(s4,−0.21)(s3,0.13)(s4,0.23)(s2,0.04)(s2,0.19)(s3,0.00)(s3,−0.02)(s2,0.31)(s3,0.47)(s2,0.06)(s2,0.21)(s3,0.02)(s3,0.00)(s2,0.33)(s3,0.49)(s3,−0.28)(s3,−0.13)(s4,−0.31)(s4,−0.33)(s3,0.00)(s4,0.12)(s2,−0.36)(s2,−0.23)(s3,−0.47)(s3,−0.49)(s2,−0.12)(s3,0.00).

For d3,d4∈DU, we utilize Eq. (15) to obtain the transformed utility preference vectors:

U(c,3)=(0.7241,0.6515,0.3742,0.3793,0.5985,0.2725)T,U(c,4)=(0.7241,0.6515,0.3742,0.3793,0.5985,0.2725)T.

For d5,d6∈DP, we utilize Eq. (17) to obtain the transformed additive preference relation:

P(c,5)=P(c,6)=0.50000.52700.65780.65720.54970.72580.47300.50000.63310.63250.52290.70380.34220.36690.50000.49940.38840.57930.34280.36750.50060.50000.38900.57990.45030.47710.61160.61100.50000.68430.27420.29620.42070.42010.31570.5000.

For d7,d8∈DA, we utilize Eq. (19) to obtain the transformed multiplicative preference relation:

A(c,7)=A(c,8)=1.00001.11411.92221.91731.22092.64670.89761.00001.72541.72101.09592.37570.52020.57961.00000.99750.63521.37690.52160.58111.00251.00000.63681.38050.81910.91251.57441.57041.00002.16790.37780.42090.72620.72440.46131.0000.

Then, we obtain the adjusted preference information of each individual as follows:

For d1,d2∈DL, we utilize Eq. (14) to derive their adjusted preference information as

L(1)¯=(s3,0.00)(s4,−0.17)(s6,−0.41)(s5,−0.21)(s3,0.23)(s6,−0.33)(s2,0.17)(s3,0.00)(s5,−0.24)(s6,−0.44)(s3,0.10)(s5,−0.15)(s0,0.41)(s1,0.24)(s3,0.00)(s3,0.00)(s2,0.05)(s4,−0.11)(s1,0.21)(s0,0.44)(s3,0.00)(s3,0.00)(s2,0.06)(s4,−0.10)(s3,−0.23)(s3,−0.10)(s4,−0.05)(s4,−0.06)(s3,0.00)(s5,−0.18)(s0,0.33)(s1,0.15)(s2,0.11)(s2,0.10)(s1,0.18)(s3,0.00),

L(2)¯=(s3,0.00)(s3,0.13)(s5,−0.21)(s6,−0.41)(s3,0.03)(s5,−0.13)(s3,−0.13)(s3,0.00)(s5,−0.24)(s6,−0.44)(s3,−0.10)(s5,−0.15)(s1,0.21)(s1,0.24)(s3,0.00)(s3,0.00)(s2,0.05)(s3,0.38)(s0,0.41)(s0,0.44)(s3,0.00)(s3,0.00)(s2,−0.14)(s3,0.39)(s3,−0.03)(s3,0.10)(s4,−0.05)(s4,0.14)(s3,0.00)(s6,−0.38)(s1,0.13)(s1,0.15)(s3,−0.38)(s3,−0.39)(s0,0.38)(s3,0.00).

For d3,d4∈DU, we utilize Eq. (16) to obtain derive their adjusted preference information as

U(3)¯=(0.7202,0.7702,0.3419,0.3957,0.6788,0.2592)T,U(4)¯=(0.6802,0.8502,0.5555,0.3426,0.6788,0.2392)T.

For d5,d6∈DP, we utilize Eq. (18) to obtain derive their adjusted preference information as

P5¯=0.50000.53740.79560.84340.54990.88120.46260.50000.76660.78250.56860.86880.20440.23340.50000.53990.34280.69990.15660.21750.46010.50000.33520.71600.45010.43140.65720.66480.50000.82490.11880.13120.30010.28400.17510.5000,

P6¯=0.50000.51160.81160.81140.54580.84120.48840.50000.87060.84650.62460.90880.18840.12940.50000.49750.34080.56740.18860.15350.50250.50000.34320.57390.45420.37540.65920.65680.50000.81690.15880.09120.43260.42610.18310.5000.

For d7,d8∈DA, we utilize Eq. (20) to obtain derive their adjusted preference information as

A7¯=1132231132241/31/311/41/211/21/2411/221/21/222131/31/411/21/31,A8¯=1231141/2121131/31/211/31/221131131121131/41/31/21/31/31.

Next, based on the adjusted preference information, we obtain the corresponding standardized individual preference vectors as

w1(1∗)=(0.2419,0.2170,0.1260,0.1261,0.1976,0.0914)T,w1(2∗)=(0.2261,0.2217,0.1287,0.1121,0.2106,0.1009)T,w1(3∗)=(0.2275,0.2433,0.1080,0.1250,0.2144,0.0819)T,w1(4∗)=(0.2033,0.2541,0.1660,0.1024,0.2028,0.0715)T,w1(5∗)=(0.2282,0.2194,0.1400,0.1325,0.1960,0.0839)T,w1(6∗)=(0.2234,0.2355,0.1235,0.1257,0.1924,0.0995)T,w1(7∗)=(0.2409,0.2836,0.0919,0.1181,0.1875,0.0780)T,w1(8∗)=(0.2856,0.1799,0.1111,0.1603,0.2019,0.0612)T,w1(c∗)=(0.2346,0.2318,0.1244,0.1253,0.2004,0.0835)T.

And the individual ranking vectors and collective ranking vectors are obtained as

o¯1(1)=(1,2,5,4,3,6)T,o¯1(2)=(1,2,4,5,3,6)T,o¯1(3)=(2,1,5,4,3,6)T,o¯1(4)=(2,1,4,5,3,6)T,o¯1(5)=(1,2,4,5,3,6)T,o¯1(6)=(2,1,5,4,3,6)T,o¯1(7)=(2,1,5,4,3,6)T,o¯1(8)=(1,3,5,4,2,6)T,o¯1(c)=(1,2,5,4,3,6)T.

Furthermore, we obtain the individual and collective two-rank vectors, i.e.,

R¯(1)=(1,1,0,0,1,0)T,R¯(2)=(1,1,0,0,1,0)T,R¯(3)=(1,1,0,0,1,0)T,R¯(4)=(1,1,0,0,1,0)T,R¯(5)=(1,1,0,0,1,0)T,R¯(6)=(1,1,0,0,1,0)T,R¯(7)=(1,1,0,0,1,0)T,R¯(8)=(1,1,0,1,0,0)T,R¯(c)=(1,1,0,0,1,0)T.

Finally, the two-rank consensus level is calculated as CMd1,d2,…,dm=1>0.95. Thus, an acceptable consensus has been reached. From the obtained collective two-rank result, B city government determines three subway construction plans x1, x2, and x4 as the approved subway construction plans, and the subway construction plan x1is considered with the highest priority.

5. CONCLUSION

This paper develops a novel consensus reaching method for two-rank GDM with heterogeneous preference information. The main contribution of this paper is presented as follows:

With the consideration of the heterogeneous preference information, we propose the method for deriving the individual preferences vectors. Then, the collective preference vector is obtained by aggregating the individual preferences vectors. Furthermore, the individual and collective two-rank vectors are obtained by the corresponding preference vectors.
We design a new consensus measure for two-rank GDM. Then, the feedback adjustment rules are proposed to improve the consensus level among the individuals. The collective preference vector is transformed into the type of preference information expressed by each individual. On this basis, the suggestions for adjusting the preference information are provided.
We design an algorithm to describe the two-rank CRP with heterogeneous preference information. Then, we present a practical example to illustrate the feasibility of the proposed method.

In the future, we argue three interesting research topics:

In some practical GDM problems, the individuals are always in a social network [36–38]. This will strongly impact the communication among different individuals. Thus, it would be an interesting topic for extending the proposed method into a social network context.
It is assumed that small number of individuals participated in two-rank GDM. However, in many scenarios, the large number of individuals will be included in two-rank GDM [39–42]. It is also necessary to investigate the large-scale two-rank GDM.
Minimum adjustment which implies to optimize the adjusted cost for each individual, has been regarded as an important criterion for evaluating the performances of consensus reaching. Thus, it is important to develop a new consensus reaching model with minimum adjustments for two-rank GDM.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest regarding the publication for the paper.

AUTHORS' CONTRIBUTIONS

Huali Tang: Conceptualization, Methodology, Writing - original draft. Shoufu Wan: Conceptualization, Methodology, Writing - review & editing. Cong-Cong Li: Conceptualization, Methodology, Writing - review & editing. Haiming Liang: Conceptualization, Methodology, Writing - original draft. Yucheng Dong: Conceptualization, Methodology, Funding acquisition, Writing - review & editing.

ACKNOWLEDGMENTS

This work was supported by the grants (Nos. 71901182 and 71971149) from NSF of China.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 758 - 768
Publication Date: 2021/02/10
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.210201.001 How to use a DOI?
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TY  - JOUR
AU  - Huali Tang
AU  - Shoufu Wan
AU  - Cong-Cong Li
AU  - Haiming Liang
AU  - Yucheng Dong
PY  - 2021
DA  - 2021/02/10
TI  - Consensus Reaching Process in the Two-Rank Group Decision-Making with Heterogeneous Preference Information
JO  - International Journal of Computational Intelligence Systems
SP  - 758
EP  - 768
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210201.001
DO  - 10.2991/ijcis.d.210201.001
ID  - Tang2021
ER  -

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