Computer Modeling in Engineering & Sciences

DOI: 10.32604/cmes.2022.020598

ARTICLE

Group Decision-Making Method with Incomplete Intuitionistic Fuzzy Preference Relations Based on a Generalized Multiplicative Consistent Concept

Xiaoyun Lu1, Jiuying Dong2,3,*, Hecheng Li1 and Shuping Wan4

1School of Computer Science, Qinghai Normal University, Xining, 810008, China
2School of Science, Shanghai Institute of Technology, Shanghai, 201418, China
3School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China
4School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, 330013, China
∗Corresponding Author: Jiuying Dong. Email: jiuyingdong@126.com
Received: 02 December 2021; Accepted: 26 January 2022

1  Introduction

Multi-attribute decision-making (MADM) aims to identify the best alternative from a set of finite feasible alternatives by considering multiple attributes that are used extensively in numerous fields of modern life [1–3]. However, because of the many evaluation factors of complex decision-making problems, it is difficult for experts to provide a corresponding evaluation value for each factor, leading to the emergence of preference DMs (PDMs) [4,5]. The PDM is used to identify the best alternative from a set of finite feasible alternatives using preference relations (PRs) between alternatives offered by experts, which is simpler and more convenient than MADM. PR is a form of pairwise comparison between alternatives, and was initially reported by Saaty [6] in the analytic hierarchy process. Since PRs are highly consistent with people’s evaluation habits and experts can provide them easily, consequently, PDM has received widespread attention in recent years. Currently, there are two primary types of PRs, i.e., additive preference relations [7] and multiplicative preference relations [8]. With extensive research on PDM, many derivatives of the two PRs have emerged through the intervention of various fuzzy sets, such as interval fuzzy preference relations (IVFPRs) [9,10], intuitionistic fuzzy preference relations (IFPRs) [11], intuitionistic multiplicative preference relations (IMPRs) [12], complex intuitionistic fuzzy preference relations (CIFPRs) [13], hesitant fuzzy preference relations (HFPRs) [14], probabilistic hesitant fuzzy preference relations (PHFPRs) [15], etc. [16]. Among them, since IFPRs can express the pairwise comparison between alternatives from the following three dimensions: affirmative, negative, and hesitant, IFPRs are more popular in group decision-making (GDM) when expressing the pairwise comparison between alternatives with uncertain information. Therefore, GDM with IFPRs is an interesting topic. Generally, it is necessary to perform three processes for GDM with IFPRs. First, the consistency of IFPRs is an essential issue in GDM, the quality of which directly affects the reasonability and acceptability of the final DM result. Researches related to the consistency of IFPRs in GDM includes how to define, check, and repair the consistency. Second, the consensus among individual IFPRs is another crucial issue in GDM with IFPRs, the quality of which directly impacts the recognition and acceptance of the final DM result by all members in the group. Researches related to consensus in GDM with IFPRs includes checking and improving the consensus. In addition to the above two issues, the ranking process of alternatives is the third issue that must be solved in GDM with IFPRs. Research on the ranking process of alternatives primarily refers to the selection of a reasonable method for determining the priority weight vector of the alternatives. Although many achievements have been made in the research on GDM with IFPRs, these methods are imperfect for numerous reasons. Therefore, this study continued to study GDM with IFPRs.

The remainder of the paper is organized as follows. Section 2 briefly recalls the literature review of GDM with IFPRs. This section also resumes the research motivations, novelties and research methodology of this study. In Section 3, the concepts regarding IFPRs are reviewed. In Section 4 a generalized multiplicative consistency definition of IFPRs is presented. Section 5 constructs respectively two mathematically programming models to supplement the default values in incomplete IFPRs and improve the consistency level of the unacceptably consistent IFPRs. After that develops a new consistency improving method with incomplete IFPRs. Section 6 introduces a new method of generating the experts’ weights, and then designs a targeted CPR and develops a method determining priority weight vector of alternatives. Further, based on the above theoretical basics, a new GDM method with incomplete IFPRs is developed. In Section 7, an application example is supplied to verify the practicability of the method presented in this study, comparative and sensitivity analysis result shows the advantages of this method. Concluding remarks are given in Section 8.

2  Literature Review

As an extension of the conventional fuzzy set portrayed by Atanassov [17], the intuitionistic fuzzy set (IFS) theory uses three-dimensional (3D) degrees to characterize the experts’ positive, negative, and indeterminate cognitions. Owing to its robust advantages in describing the uncertainty and vagueness of human thinking, the IFS has received special attention in the field of DM. Therefore, many DM methods based on the IFS have also been developed. For instance, Liao et al. manifested the intuitionistic fuzzy Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) [18], Liao et al. [19] extended Analytic Network Process (ANP) into intuitionistic fuzzy environments and developed the intuitionistic ANP (IFANP) method, and Wan et al. [20] introduced the Best-worst method with IFPRs, and Efe [21] proposed IF modelling-based risk assessment approaches using IF numbers-based quality function deployment (QFD) and VIseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) technique. Stanujkić et al. [22] proposed a new extension of the WASPAS method for DM problems with intuitionistic fuzzy numbers. Moreover, Kushwaha et al. [23] presented an integrated framework based on Intuitionistic Fuzzy-Failure Mode Effect Analysis (IF-FMEA) and IF-Technique for Order Preference by Similarity to Ideal Solution (IF-TOPSIS) techniques to rank the listed failure causes, etc. Among the methods based on IFS for handling DM problems, the DM method based on IFPRs is one of the fundamental and efficient methods for modeling DM issues. In recent years, many achievements have been made regarding DM with IFPRs. Xu et al. [24] developed a consensus analysis method for GDM with IFPRs based on a defined similarity measure between IFSs. Xu et al. [25] first presented the multiplicative consistency definition and used it to estimate probable missing elements in an incomplete IFPR, after which they developed a GDM method with incomplete IFPRs. Based on the consistency definition [25], Xu et al. [26] developed an approach to enhance the consistency level of unacceptably consistent IFPRs and extended the classic analytic hierarchy process (AHP) into intuitionistic fuzzy AHP (IFAHP). Furthermore, Liao et al. [27] found that the multiplicative consistency condition proposed in [25] conflicts with Saaty’s multiplicative transitivity property because of stringent constraints and proposed a general multiplicative consistency definition of IFPRs and developed a novel approach to derive the priority weights from an IFPR. Based on the multiplicative consistency definition presented in [27], Liao et al. [28] devised an iterative algorithm to enhance the consistency level of IFPRs, and then presented an interesting consensus reaching process, and Xu et al. [29] constructed an optimization model to repair the consistency of the unacceptably consistent IFPRs and developed a GDM method with IFPRs by simultaneously considering the acceptable consensus and consistency. Jin et al. [30] developed a novel approach to derive the priority weights from an IFPR and proposed a GDM method with IFPRs. In addition, according to the isomorphism between IVFPRs and IFPRs, Gong et al. [31] introduced the definition of multiplicative consistent IFPR, based on which the intuitionistic fuzzy priority weight vector (IFPWV) was produced through some goal programming models. Moreover, Wu et al. [32] initiated multiplicative consistency of IFPRs by a proposed multiplicative consistency definition for IVFPRs, and designed a consistency and consensus improving method based on feedback mechanism. Based on the multiplicative consistency of IFPRs presented in [32], Wu et al. [33] constructed a multiplicative consistency-based multi-objective programming (MOP) model to derive the priority vector from an IFPR. More recently, Meng et al. [34] provided an innovative multiplicative consistency definition of IFPRs based on the quasi-interval-based transitivity equation of IVFPRs and constructed a consensus model to enhance consensus among experts. Based on Krejčí [35] multiplicative consistency definition of IVFPRs, Meng et al. [36] redefined the multiplicative consistency of IFPRs, and Wang [37] presented a novel multiplicative consistency definition of IFPRs and developed a representable uninorm based IFAHP.

2.1 Research Motivations

The literature reviews reveal that the above mentioned achievements are effective in dealing with real DM problems; however, there are some gaps in the following aspects, which can be summarized as follows:

(1) The first gap obtained from existing studies is in terms of consistency management in GDM with IFPRs. In [24], the multiplicative consistency of individual IFPRs was not considered, which could lead to an unreasonable DM result. In [37], Wang pointed out that these multiplicative consistency definitions for IFPRs presented in [25,27,32] are not intuitionistic fuzzy extensions of Tanino’s multiplicative consistency. Therefore, the DM outcomes obtained by GDM methods [28,29,32], which were respectively developed based on the consistency definitions in [25,27,32] may lack rationality and persuasiveness. Moreover, because the GDM method with IFPRs [34] is developed based on complete multiplicative consistency, and the DM result is produced at the cost of a large amount of original information loss (it can be seen from the comparative analysis in Section 6). For GDM methods [30,31], the consistency of the IFPRs has not been considered. Given the shortcomings of consistency management in GDM with IFPRs [25,27–29,32,34], it is essential to present a more reasonable consistency management framework for GDM with IFPRs, which is the first motivation of this study.

(2) The second gap obtained from existing studies is in terms of the method of determining the weights of experts. In these methods [24,25,28,30,31], the experts’ weights are assigned in advance, which makes it difficult to avoid subjective randomness. In [29], the experts’ weights were derived by simultaneously considering the consistency index and confidence degree. In [32], experts’ weights were derived by considering the consistency index. In [34], experts’ weights were derived by considering the degree of confidence. Although these methods determine experts’ weights in [29,32,34] effectively avoid the subjective randomness of the experts’ weights, they only consider one or two aspects affecting experts’ weights. In fact, some factors, such as the hesitancy degree of the IFPRs and the ability to retain the original information, have a greater impact on the experts’ weights. In view of this fact, it is essential to propose a method determining the experts’ weights that comprehensively considers these aspects, which is the second motivation of this research.

(3) The third gap obtained from existing studies is in terms of the consensus-reaching process (CRP) in GDM with IFPRs. In the methods [25,30,31], group consensus among experts was not considered. In [24,28,32,34], the information loss of the original individual IFPRs was ignored in CRP. In [29], although the information loss of the original individual IFPRs has gained attention, the adjustments for individual IFPRs were accomplished by modifying all individual IFPRs via an optimization model, which illustrates the adjustment without the target. Thus, the final adjusted individual IFPRs may not be optimal in terms of preserving the original information (it is evident from the comparative analysis in Section 6). In view of the shortcomings of the CRP proposed in [24,28–32,34], it is essential to develop a more reasonable consensus-improving method, which is the third motivation of this study.

(4) The fourth gap obtained from existing studies exist in terms of the method employed for obtaining the priority weights of alternatives. In the methods [25,28,29,32,34], the priority weights of alternatives have been primarily determined by some aggregation operator; however, for some situations, it may cause a number of information losses if this method is adopted. However, although the priority weights of alternatives are derived by the optimization model in [27,30,31], which effectively prevents the loss of information, the DM result may lack persuasiveness because the priority weights of alternatives are generated based on the group IFPR without checking the consistency. In view of the shortcomings of methods for obtaining the priority weights of alternatives in [25,27–32,34], it is necessary to develop a more reasonable method to obtain the priority weights of alternatives, which is the fourth motivation of this study.

2.2 Novelties of this Article

To bridge the above four research motivations, this study aims to develop a novel GDM method using IFPRs, the key novelties of which are listed below:

(1) A more generalized multiplicative consistency definition for IFPRs is presented, which compensates for the defects of the existing multiplicative consistency definitions for IFPRs. Furthermore, a more reasonable consistency management framework for GDM with IFPRs is presented, in which the acceptable multiplicative consistency of individual IFPRs is considered.

(2) Three reliable sources influencing experts’ weights were first identified by analyzing the factors affecting experts’ weights, and a method of determining experts’ weights was developed by simultaneously considering three reliable sources.

(3) A targeted consensus-improving method is developed by which the individual IFPRs obtained retain the original information to the greatest extent while achieving an acceptable consensus and consistency level.

(4) The priority weights of the alternatives were derived from the acceptably multiplicative consistent IVFPR via the optimization model, which prevents information loss and enhances the persuasiveness of the obtained DM result.

2.3 Research Methodology

To achieve the goal mentioned above, the research methodology for the present study is shown in Fig. 1.

Figure 1: Research methodology

Stage 1: Consistency improving phase: Based on the proposed multiplicative consistency definition for IFPRs, two mathematical programming models are respectively constructed to supplement the default values in incomplete IFPRs and improve the consistency level of unacceptably consistent IFPRs.

Stage 2: Consensus improving phase: This stage introduces a new method of generating experts’ weights, and then designs a targeted consensus improving method to enhance the consensus level of individual IFPRs.

Stage 3: Exploitation phase: Based on the proposed multiplicative consistency definition for IFPRs, a method determining the priority weights of alternatives is developed.

3  Preliminaries

This section mainly reviews some relevant concepts of IFSs and IFPRs. Let N={1,2,⋯,n} and Q={1,2,⋯,q} be the corresponding sets of indices.

Definition 3.1. [17] Assume that Z={z1,z2,⋯,zn} is a fixed set, an IFS in Z is defined as T={<z,μT(z),υT(z)>|z∈Z}, where μT(z) and υT(z) denote the membership and non-membership degrees of ∀z∈Z to set T, respectively, such that μT(z),υT(z)∈[0,1] and μT(z)+υT(z)≤1 for ∀z∈Z. πT(z)=1−μT(z)−υT(z) is called the hesitancy degree of ∀z∈Z to set T.

In addition, Xu et al. [38] introduced the intuitionistic fuzzy number (IFN), which is generally denoted as a=(μa,υa), where 0≤μa,υa≤1 and μa+υa≤1.

Definition 3.2. [39] Assume that a1=(u1,ν1) and a2=(u2,ν2) are two IFNs, the distance between a1 and a2 is defined as Eq. (1):

d(a1,a2)=12(|u1−u2|+|ν1−ν2|)(1)

Definition 3.3. [11] Let X be a finite set of compared alternatives. An IFPR A on X is represented by a matrix A=(aij)n×n=(⟨μij,υij⟩)n×n, where μij and υij denote the membership and non-membership degrees to which xi is preferred to xj, respectively. Moreover, μij and υij satisfy: μij,υij∈[0,1], μij=υji, υij=μji, μij+υij≤1, μii=υii=1, for all i,j∈N.

4  A New Multiplicative Consistency Framework for IFPRs

In this section, based on the multiplicative consistency definition for IFPRs proposed by Meng et al. [36], a new consistency index for IFPRs is first defined, after which the corresponding acceptably multiplicative consistency definition for IFPRs is presented.

Definition 4.1. [36] Let A=(aij)n×n=(⟨μij,υij⟩)n×n be an IFPR. If for any preferred degree εij∈[μij,1−υij], there always exist two preferred degrees εik∈[μik,1−υik] and εkj∈[μkj,1−υkj], such that the three preferred degrees εij, εik and εkj satisfy Tanino’s multiplicative transitivity [40], i.e., Eq. (2):

εijεji=εikεkiεkjεjk(i,k,j∈N;i≠k≠j)(2)

Then A is regarded as multiplicative consistent.

Property 4.1. [36] Let A=(aij)n×n=(⟨μij,υij⟩)n×n be an IFPR. If A satisfies the following inequalities:

μij1−υji≥μik1−υkiμkj1−υjk,1−υijμji≤1−υikμki1−υkjμjk(i,k,j∈N;k≠i,j;i<j)(3)

Then A is regarded as a multiplicative consistent IFPR.

Since μji=υij and υji=μij, Eq. (3) can be reduced as:

1−υijυij≤1−υikυik1−υkjυkj(i,k,j∈N;i≠k≠j)(4)

By taking the logarithm on both sides of Eq. (4), we can obtain

log⁡(1−υij)−log⁡υij≤log⁡(1−υik)−log⁡υik+log⁡(1−υkj)−log⁡υkj(i,k,j∈N;i≠k≠j)(5)

It should be noted that Eq. (5) holds only when the IFPR A is completely consistent. However, in the real DM problems, it is not easy for an expert to offer a completely consistent IFPR, that is to say, it is difficult to find an IFPR meeting Eq. (5). Given the above facts, the smaller deviations are usually permitted. Thus, we expect to seek an IFPR to satisfy Eq. (5) in fuzzy sense, i.e., Eq. (6):

log⁡(1−υij)−log⁡υij−(log⁡(1−υik)−log⁡υik+log⁡(1−υkj)−log⁡υkj)≤~0(i,k,j∈N;i≠k≠j)(6)

where the symbol “≤~” is the fuzzy version for symbol “≤” in the real number and has the linguistic interpretation “essentially equal to”. Let

Raikj=log⁡(1−υij)−log⁡υij−(log⁡(1−υik)−log⁡υik+log⁡(1−υkj)−log⁡υkj)(i,k,j∈N;i≠k≠j)(7)

then Eq. (6) can be rewritten as Raikj ≤~0 (i,k,j∈N;i≠k≠j). Furthermore, we construct the linear membership function to represent the intensity of satisfaction for fuzzy constraints given in Eq. (7) as the following Eq. (8):

M(Raikj)={0Raikj>tikj1−Raikjtikj0≤Raikj≤tikj1Raikj<0(i,k,j∈N;i≠k≠j)(8)

where tikj(i,k,j∈N;i≠k≠j) are tolerance parameters.

In what follows, based on the above linear membership functions for fuzzy constraint given in Eq. (7), a new multiplicative consistency index to quantify the multiplicative consistency level of IFPRs is presented.

Definition 4.2. Let A=(aij)n×n=(⟨μij,υij⟩)n×n be an IFPR. The multiplicative consistency index MCI(A) of the IFPR A is defined as the following Eq. (9):

MCI(A)=∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(Raikj))n×(n−1)×(n−2)(9)

where μ(M(Raikj))=min{max{M(Raikj),0},1}(i,k,j∈N;i≠k≠j) represent the linear memberships for fuzzy constraints given in Eq. (6).

Remark 4.1. It is apparent that 0≤MCI(A)≤1. The bigger the value of MCI(A), the better consistency of A. Specifically, if MCI(A)=1, then A is called a completely multiplicative consistent IFPR.

Definition 4.3. Let MCI0∈[0,1] be a consistent threshold. If MCI(A)≥MCI0, then A is regarded as an acceptably multiplicative consistent IFPR. Otherwise, it is called an unacceptably one.

However, due to the complexity and uncertainty of DM problems involved in real life, it is not an easy thing for experts to provide acceptably consistent IFPRs. For these unacceptably consistent IFPRs, generally, the judgments of IFPRs needs to be adjusted by establishing related optimization model to make the IFPRs reach an acceptable consistency level.

5  A New Method for Improving the Consistency of the Incomplete IFPRs

It is common to encounter information default in DM because of experts’ limited expertise and time pressure or the complex DM environment. Therefore, it is very necessary to present an effective method to deal with the DM problems with incomplete information under intuitionistic fuzzy context. To do this, we first need to supplement these missing values. In this section, based on the above multiplicative consistency measure concept, the missing values in an incomplete IFPR are estimated by constructing related optimization model.

5.1 Optimization Model for Estimating Missing Values in Incomplete IFPRs

Before constructing the optimization model to estimate missing values in incomplete IFPRs, the relevant concepts for incomplete IFPRs are first introduced in the following.

Definition 5.1. Let A=(aij)n×n=(⟨μij,υij⟩)n×n be an IFPR. If some membership degrees, non-membership degrees or both membership and non-membership degrees of its pairwise judgments are missing, then A is regarded as an incomplete IFPR.

In this paper, A~=(a~ij)n×n=(⟨μ~ij,υ~ij⟩)n×n is employed to denote the incomplete IFPR to distinguish the complete IFPR A. As to A~, let Nij={(i,j)|μ~ij is unknown i,j∈N} and Nμ={(i,k,j)|μ~ij∨μ~ik∨μ~kj is unknown, i,k,j∈N}.

Definition 5.2. Let A~=(a~ij)n×n be an incomplete IFPR. If there exist some values in [0,1] for all unknown values in Nμ satisfying Eq. (4), then the incomplete IFPR A~ is multiplicative consistent.

In fact, for an incomplete IFPR A~, we often cannot guarantee that Eq. (4) holds. That is to say, the membership value for fuzzy constraints given in Eq. (6) may be not equal to 1. Thus, we often seek some values in [0,1] for all unknown values in Nμ to make the membership value for fuzzy constraints given in Eq. (6) reach the maximum. Given the above facts, we construct the following mathematical programming model:

max∑(i,k,j)∈Nμμ(M(Ra~ikj))s.t.{min{max{M(Ra~ikj),0},1}=μ(M(Ra~ikj))μ~ij≥0,μ~ij≤1−υ~ij(i,j)∈Nijμ~ij≥0,μ~ji≥0,μ~ij+μ~ji≤1(i,j)∈Nij∧(j,i)∈Nji(10)

where M(Ra~ikj) are derived by Eq. (8). The first constrain comes from Definition 4.2, and the rest constrains ensure that the supplemented values for A~ meet the requirement of IFNs. By solving the Eq. (10), we obtain the optimal solutions, denoted by μ~ij(i,j)∈Nij. Furthermore, based on these optimal solutions, the complete IFPR A associated with A~ is derived.

5.2 Optimization Model for Improving Consistency Level of Complete IFPRs

The consistency level of PR has a crucial impact on the final DM result. It happens too often that an unreasonable DM result may be generated from a PR with poor consistency. In fact, a complete IFPR A given by expert (or derived by solving Eq. (10)) is generally unacceptably consistent, namely, MCI(A)<MCI0. To acquire an acceptably consistent IFPR A¯=(a¯ij)n×n=(⟨μ¯ij,υ¯ij⟩)n×n associated with A, a mathematical programming model is constructed as follows:

∇∗=min1n(n−1)∑i=1n−1∑j=i+1n(|μij−μ¯ij|+|υij−υ¯ij|)s.t.{min{max{M(Ra¯ikj),0},1}=μ(M(Ra¯ikj))MCI(A¯)=∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(Ra¯ikj))n×(n−1)×(n−2)≥MCI00≤μ¯ij,υ¯ij≤1,μ¯ij+υ¯ij≤1,μ¯ij=υ¯ji,υ¯ij=μ¯ji(11)

where M(Ra¯ikj) are derived by Eq. (8). The first two constrains come from Definition 4.2, which ensure that the obtained IFPR is acceptably consistent. The third constrain ensures that the obtained matrix is an IFPR.

To solve Eq. (11) conveniently, let ϕij+=(|μij−μ¯ij|+(μij−μ¯ij))2, ϕij−=(|μij−μ¯ij|−(μij−μ¯ij))2, φij+=(|υij−υ¯ij|+(υij−υ¯ij))2, φij−=(|υij−υ¯ij|−(υij−υ¯ij))2, then Eq. (11) can be converted into a liner program model as:

∇∗=min1n(n−1)∑i=1n−1∑j=i+1n(ϕij++ϕij−+φij++φij−)s.t.{μij−μ¯ij=ϕij+−ϕij−,υij−υ¯ij=φij+−φij−ϕij+≥0,ϕij−≥0,φij+≥0,φij−≥0i,j∈N;i<jother constraints are the same as those of model (11)(12)

By solving Eq. (12), we can obtain the optimal solutions μ¯ij and υ¯ij. Furthermore, according to Definition 3.3, the modified acceptably multiplicative consistent IFPR A¯ associated with A is obtained.

5.3 Algorithm Description for Obtaining the Acceptably Multiplicative Consistent IFPRs

Based on the above theory, a new method for obtaining the acceptably multiplicative consistent IFPRs is brought forward as follows:

6  GDM with Incomplete IFPRs

It is well known that in real life, when facing a complex DM problem, experts or scholars from related fields are usually invited to make decisions collectively, namely, GDM. This section focuses on introducing a new GDM method with incomplete IFPRs. To this end, two parts need to be completed: derivation of experts’ weights and consensus improvement. Without loss of generality, assume that X={x1,x2,⋯,xn} is a set of alternatives that are required for comparison by q experts E={e1,e2,⋯,eq}; A~h corresponds to the incomplete individual IFPRs given by the expert eh(h∈Q); Ah denotes the complete individual IFPRs associated with A~h obtained by solving Eq. (10); A¯h indicates the acceptably multiplicative consistent IFPR associated with Ah obtained by solving Eqs. (11) and (12).

6.1 Derivation of Experts’ Weights

After obtaining the individual acceptably multiplicative consistent IFPR A¯h=(a¯ij)n×n=(⟨μ¯ij,υ¯ij⟩)n×n(h∈Q) via Algorithm 1, the next step involves generating the group acceptably multiplicative consistent IFPR A¯C by fusing all individual acceptably multiplicative consistent ones. To achieve this, we must determine the weights of the experts in advance. In the following section, three reliable sources influencing the experts’ weights are identified, after which the experts’ weights are determined by simultaneously considering the three reliable sources.

Generally, the degree of confidence reflects the degree of support from other experts. An expert with a higher confidence degree should be given a larger weight, whereas an expert with a lower confidence degree should be allocated less weight. Moreover, the confidence index CFh of expert eh was calculated using Eq. (13):

CFh=1q−1∑l=1,l≠hq(1−D(A¯h,A¯l))(h∈Q)(13)

The similarity between the original complete IFPR obtained by solving Eq. (10) (or given by experts) and the acceptably multiplicative consistent IFPR obtained by solving Eq. (11) reflects the degree to which the adjusted IFPR retains original information. More specifically, the larger the deviation, the lesser weight given to the expert, whereas the smaller the deviation degree, the more weight given to the expert. In general, the similarity index between A¯h and Ah was calculated using Eq. (14):

SIh=1−D(A¯h,Ah)(h∈Q)(14)

Furthermore, the hesitancy degree of the IFPR provided by an expert was considered an expert feature. The smaller the hesitancy of the IFPR, the more the weight given to the expert, whereas the bigger the hesitancy of the IFPR, the lesser the weight given to the expert. In general, the hesitancy index of IFPR A¯h is calculated using Eq. (15):

HI(A¯h)=∑j=in−1∑j=i+1n(1−(μ¯ij+υ¯ij))n(n−1)(h∈Q)(15)

Based on the above three reliable sources influencing experts’ weights, a comprehensive index of expert eh is generated as Eq. (16):

CXh=(1−HI(A¯h))+SIh+CFh3(h∈Q)(16)

Let λ=(λ1,λ2,⋯,λq)T be the experts’ weight vector derived from the comprehensive index. Then, λh(h∈Q) can be calculated as Eq. (17):

λh=CXh∑h=1qCXh(h∈Q)(17)

Consensus measures applied to quantify the recognition degree of individual opinions among a group of experts are divided into two types [41]: (1) based on the distance to the group PR [42] and (2) based on the distance between experts’ PRs [43]. Without loss of generality, the first type is employed in this study to measure the consensus level among individual IFPRs. Prior to presenting the consensus measure concept, we first introduce the calculation of the group IFPRs.

Theorem 6.1. [44] Let Ah=(aijh)n×n=(⟨μijh,υijh⟩)n×n(h∈Q) be any q complete IFPRs given by experts eh(h∈Q) or obtained by solving Eq. (10). The group IFPR AC=(aijC)n×n aggregated by Ah(h∈Q) is determined as Eq. (18):

aijC=⟨μijC,υijC⟩=⟨∏h=1q(μijh)λh∏h=1q(μijh)λh+∏h=1q(1−μijh)λh,∏h=1q(υijh)λh∏h=1q(υijh)λh+∏h=1q(1−υijh)λh⟩(i,j∈N)(18)

where λh∈[0,1](h∈Q) is the weight of expert eh(h∈Q), obtained by Eqs. (13)–(17).

Theorem 6.2. Let Ah=(aijh)n×n=(⟨μijh,υijh⟩)n×n(h∈Q) be q complete IFPRs. If all q IFPRs are acceptably multiplicative consistent, then the group IFPR RC=(rijC)n×n aggregated by Eq. (18) is also acceptably multiplicative consistent.

Proof: As per Eqs. (9) and (18), we have

RaikjC=log⁡(1−υijC)−log⁡υijC−(log⁡(1−υikC)−log⁡υikC+log⁡(1−υkjC)−log⁡υkjC)=log⁡(∏h=1q(1−υijh)λh)−log⁡(∏h=1q(υijh)λh)−(log⁡(∏h=1q(1−υikh)λh)−log⁡(∏h=1q(υikh)λh)+log⁡(∏h=1q(1−υkjh)λh)−log⁡(∏h=1q(υkjh)λh))=∑h=1q(λh(log⁡(1−υijh)−log⁡υijh−(log⁡(1−υikh)−log⁡υikh+log⁡(1−υkjh)−log⁡υkjh)))=∑h=1qλhRaikjh

MCI(AC)=∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(RaikjC))n×(n−1)×(n−2)=∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(∑h=1qλhRaikjh))n×(n−1)×(n−2)=∑h=1qλh∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(Raikjh))n×(n−1)×(n−2)=∑h=1qλhMCI(Ah)

Since Ah(h∈Q) are acceptably multiplicative consistent, one has

MCI(Ah)=∑i=1n∑j=1,j≠in∑k=1,k≠i,jnμ(M(Raikjh))n×(n−1)×(n−2)≥MCI0.

Therefore, MCI(AC)=∑h=1qλhMCI(Ah)≥∑h=1qλhMCI0=MCI0. According to Definition 4.3, AC is acceptably multiplicative consistent.

Remark 6.1. Theorem 6.2 shows that the prerequisite for a group IFPR to be acceptably multiplicative consistent is that all of individual IFPRs used to aggregate into it by Eq. (18) must be acceptably multiplicative consistent. Furthermore, according to Theorem 6. 2, we can also obtain that if all of individual IFPRs are completely multiplicative consistent, then the group IFPR aggregated by these individual IFPRs is completely multiplicative consistent.

6.2 Consensus Improving Process for GDM with IFPRs

In what follows, based on the group IFPR, we first give the concept of consensus measure of the individual IFPRs. Then, a mathematical programming model is constructed to improve the consensus level of individual IFPRs which do not meet the requirement of consensus.

Definition 6.3. Let A¯h=(a¯ijh)n×n=(⟨μ¯ijh,υ¯ijh⟩)n×n(h∈Q) be q acceptably multiplicative consistent IFPRs, and let A¯C=(a¯ijC)n×n=(⟨μ¯ijC,υ¯ijC⟩)n×n be the group IFPR aggregated by A¯h (h∈Q) using Eq. (18). Then the consensus index GCM of A¯h is definition as Eq. (19):

GCM(A¯h)=1−1n(n−1)∑i=1n−1∑j=i+1n(|μ¯ijh−μ¯ijC|+|υ¯ijh−υ¯ijC|)(h∈Q)(19)

Clearly, it is 0≤GCM(A¯h)≤1, the larger GCM(A¯h), the higher the consensus level of A¯h. Let CM0∈[0,1] be the consensus threshold. If GCM(A¯t)≤CM0 for some t=1,2,⋯,s∧s≤q, then the consensus level of A¯t(t=1,2,⋯,s∧s≤q) need to be adjusted. Without loss of generality, the adjusted IFPR ⌢At=(⌢aijt)n×n=(⟨⌢μijt,⌢υijt⟩)n×n corresponding to A¯t (t=1,2,⋯,s∧s≤q) is determined as:

{⌢μijt=(μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(1−μ¯ijt)θij(1−μ¯ijC)(1−θij)⌢υijt=(υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(1−υ¯ijt)θij(1−υ¯ijC)(1−θij)θij=θji,θij∈[0,1](i,j∈N∧i<j)(20)

Property 6.1. Let ⌢At=(⌢aijt)n×n=(⟨⌢μijt,⌢υijt⟩)n×n be a matrix obtained by Eq. (20). Then the IFPR ⌢At is acceptably multiplicative consistent.

Proof: First, we proof ⌢At is an IFPR. Since μ¯jit,υ¯ijt,μ¯ijC,υ¯ijC∈[0,1] (i,j∈N;t=1,2,⋯,s∧s≤q), one has (μ¯ijt)θij(μ¯ijC)(1−θij), (υ¯ijt)θij(υ¯ijC)(1−θij), (1−μ¯ijt)θij(1−μ¯ijC)(1−θij), (1−υ¯ijt)θij(1−υ¯ijC)(1−θij), and (μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(1−μ¯ijt)θij(1−μ¯ijC)(1−θij), (υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(1−υ¯ijt)θij(1−υ¯ijC)(1−θij)∈[0,1].

In addition, since μ¯ijt+υ¯ijt≤1 and μ¯ijC+υ¯ijC≤1(i,j∈N,t∈Q), one has (μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(1−μ¯ijt)θij(1−μ¯ijC)(1−θij)≤(μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(υ¯ijt)θij(υ¯ijC)(1−θij) and (υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(1−υ¯ijt)θij(1−υ¯ijC)(1−θij)≤(υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(μ¯ijt)θij(μ¯ijC)(1−θij), i.e., (μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(1−μ¯ijt)θij(1−μ¯ijC)(1−θij)+(υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(1−υ¯ijt)θij(1−υ¯ijC)(1−θij)≤1.

Furthermore, as per μ¯jit=υ¯ijt, υ¯jit=μ¯ijt, μ¯jiC=υ¯ijC and υ¯jiC=μ¯ijC(i,j∈N,t∈Q), one has

⌢μijt=(μ¯ijt)θij(μ¯ijC)(1−θij)(μ¯ijt)θij(μ¯ijC)(1−θij)+(1−μ¯ijt)θij(1−μ¯ijC)(1−θij)=(υ¯jit)θji(υ¯jiC)(1−θji)(υ¯jit)θji(υ¯jiC)(1−θji)+(1−υ¯jit)θji(1−υ¯jiC)(1−θji)=⌢υjit

⌢υijt=(υ¯ijt)θij(υ¯ijC)(1−θij)(υ¯ijt)θij(υ¯ijC)(1−θij)+(1−υ¯ijt)θij(1−υ¯ijC)(1−θij)=(μ¯jit)θji(μ¯jiC)(1−θji)(μ¯jit)θji(μ¯jiC)(1−θji)+(1−μ¯jit)θji(1−μ¯jiC)(1−θji)=⌢μjit