A consensus reaching process with fuzzy matrix of interpersonal influences

Abstract

Keywords: Fuzzy Inverse Matrix; Group Decision Making; Influence Model; Triangular Fuzzy Numbers

Introduction

Decision-making is an imperative element of our routine life whether we are at home or work and have become complicated due to the development of society. Organizations arrange group of members for their decision-making processes, which is known as group decision making (GDM). GDM models are usually concerned with two processes: the consensus reaching process (CRP) [1-3] and the selection process [4,5]. CRP is an iterative process leading to the final solution after discussions and interpersonal influences of experts. Yao and Gu [6] propose a consensus model based on *Corresponding author. *E-mail address: asmamahmood@gcuf.edu.pk This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey

an influence network for large-scale GDM. After the influence model was introduced by Friedkin and Johnsen [7], there have been presented its related concepts and applications by different authors [8-10]. Influence models contribute particularly in GDM, where initial opinions of n experts are revised due to their interpersonal and social influences [5,11]. These models are based on an iterative process where the matri 𝑊 = [𝑤𝑖𝑗]n×n, (𝑤𝑖𝑗 ∈ [0,1]), of interpersonal influences is a basic component. Weights 𝑤𝑖𝑗 are assumed to be satisfy the normalization property i.e. . Fuzzy set theory [12] is suggested to deal with uncertainty which is an important aspect of almost all

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decision-making processes. Fuzzy set have many generalizations with a large number of applications in optimization and other decision making [13-15]. The mathematical theory of interpersonal influences leads us to construct the influence models. Fuzzy preference relations (FPRs) also play an important role in such problems. FPRs are utilized to form the initial opinions of experts and through a mathematical model final opinions are obtained [16-18]. Fuzzy numbers have been introduced by Jain [19] and Dubois and Prade [20] to model the uncertain information. Ranking fuzzy numbers is a crucial step in investigating fuzzy information in decision making process. A lot of contibutions have been made in ranking of fuzzy numbers [21-24]. A fuzzy number gets its extension from the real number and can be considered as a function whose domain is a specified set. A fuzzy set satisfying some conditions, is called a fuzzy number. Fuzzy numbers and the extension principle are the basis for fuzzy arithmetic [25]. Lee [26] has also discussed fuzzy numbers with their basic operations ⊕, ⊖, ⊗ and ⊘, where these operations are conducted for triangular fuzzy numbers (TFNs), with the help of 𝛼-cuts. Such operations are not fuzzy, the numbers on which the operations are performed are fuzzy, so the results of these operations are also fuzzy. TFN (𝑎𝑙, 𝑎0, 𝑎𝑟) assigns an interval, called 𝛼-cut, for each value in the interval [0,1]. Fuzzy numbers have a wide utilization in decision sciences and engineering applications [27,28]. A fuzzy matrix has two different meanings in the literature: firstly, A = (aij)m×n is called a fuzzy matrix if 𝑎𝑖𝑗 ∈ [0,1], (𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛), on the other hand a matrix with entries of fuzzy numbers is called a fuzzy matrix too. Fuzzy matrices of second class is considered in this article. There is a strong connection between interval matrix and fuzzy matrix because each 𝛼-cut of a fuzzy matrix is an interval matrix that contains interval numbers. The interval matrix, its operations and singularity were discussed by Rohn [29,30]. Clustering with covariance matrix and mediative fuzzy relations which contain a large number of uncertainties, also took advantage of fuzzy matrices [31-34]. An approximate inverse of an uncertain matrix was introduced in Ghaoui [35]. A square matrix 𝐴 of order 𝑛 represents the co-efficient matrix in a system 𝐴𝑥 = 𝑏 of linear equations whose solution is uniquely determined by finding the inverse of 𝐴. In the case of a fuzzy matrix, the computation of its inverse is quite complicated. Dehghan et al. [36] presented some conditions for the invertibility of fuzzy matrices in terms of interval matrices. Farahani et.al. [37] presented a method to find the inverse of a fuzzy matrix by using eigen value method. Dequan and Guo [38] investigated a class of fuzzy linear matrix equation by using the embedding approach. Some recent developments are observed in neutrosophic fuzzy matrices [39,40]. Basaran [41] has suggested a method to calculate fuzzy inverse matrix by using a fuzzy linear equation

system. This method of obtaining the inverse fuzzy matrix is utilized in this paper for the matrix of triangular fuzzy numbers and to present the influence model where the matrix of interpersonal influences is a fuzzy matrix with triangular fuzzy numbers as its entries. The concepts of social influence network theory and opinion dynamics are not mathematical in nature, but formation of the matrices of opinions and influences prepares the fundamentals for mathematical procedures. This research aims to distend the insufficiency of mathematical techniques in opinion dynamics and social influences. Crisp numbers are replaced by fuzzy numbers in the CRP [7]. TFNs are utilized which are suitable for ranking and comparison of the influences of experts. TFNs give comprehensive results while inverse of the matrix is computed.

Preliminaries

Triangular Fuzzy Numbers Definition 1 [27] A fuzzy number 𝐴 = (𝑎𝑙, 𝑎0, 𝑎𝑟) where 𝑎𝑙, 𝑎0, 𝑎𝑟 ∈ 𝑅 and 𝑎𝑙 ≤ 𝑎0 ≤ 𝑎𝑟, is a triangular fuzzy number if it is represented as the following membership function:

Operations on fuzzy numbers The arithmetic operations on fuzzy numbers can be defined by the extension principle. Definition 2 [43] Let 𝐴 and 𝐵 be two fuzzy numbers and × be an operation on 𝑅, such as +, −,∗,÷, ⋯. By extension principle, the extended operation ⊗ on fuzzy numbers can be defined by:

Suppose 𝐴 = (𝑎𝑙, 𝑎0, 𝑎𝑟) and B= (𝑏𝑙, 𝑏0, 𝑏𝑟) are two triangular fuzzy numbers. The formulae for the extended addition, subtraction, multiplication and division operations become [17,23]:

2. Subtraction

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4. Division

Ordering of triangular fuzzy numbers [43] The set of all fuzzy numbers is denoted by 𝑁𝐹. A total ordering ≼ on 𝑁𝑇 (the set of all triangular fuzzy numbers) may be defined as: (𝑎𝑙, 𝑎0, 𝑎𝑟) ≼ (𝑏𝑙, 𝑏, 𝑏𝑟) if and only if 1. 𝑎𝑙 < 𝑏𝑙, or 2. 𝑎𝑙 = 𝑏𝑙 but 𝑎0 < 𝑏0, or 3. 𝑎𝑙 = 𝑏𝑙, 𝑎0 = 𝑏0 but 𝑎𝑟 < 𝑏𝑟. Fuzzy Inverse Matrix

Definition 3 [44] A matrix 𝐴̃ = (𝑎̃𝑖𝑗) is called a fuzzy matrix, if each element of 𝐴̃ is a fuzzy number. let 𝐴 = (𝑎𝑙, 𝑎0, 𝑎𝑟) and B = (𝑏𝑙, 𝑏0, 𝑏𝑟) be two 𝑚 × 𝑛 and 𝑛 × 𝑝 fuzzy matrices. The size of the product of two fuzzy matrices is 𝑚 × 𝑝 and is written as follows: 𝐴̃ × 𝐵̃ = 𝐶̃ = (𝑐̃𝑖𝑗), where 𝑐̃𝑖𝑗 = ⨁(𝑎̃𝑖𝑘⨂𝑏̃𝑘𝑗) where ⨂ is the approximated multiplication. Before defining fuzzy inverse matrix, it is necessary to define fuzzy zero number and fuzzy one number to develop fuzzy identity matrix. Definition 4 [41] If the center value of a fuzzy number is 0 and the left and right spread values are 𝛼 and 𝛽 where 0 < 𝛼 < 𝛽 < 1, this fuzzy number is called fuzzy zero number and is denoted as 0̃ = (−𝛼, 0, 𝛽).

Definition 5 If the center value of a fuzzy number is 1̃ and the left and right spread values are 𝛿 and 𝜆 where 0 < 𝛿 < 𝜆 < 1, this

fuzzy number is called fuzzy one number and is denoted as 1̃ = (1 − 𝛿, 1,1 + 𝜆).

Definition 6 If the diagonal elements of a fuzzy matrix are fuzzy one numbers and the off-diagonal elements are fuzzy zero numbers, then this fuzzy matrix is called fuzzy identity matrix and is denoted by 𝐼̃.

Social Influence Network Theory Let 𝑊 = 𝑤𝑖𝑗 be the matrix of interpersonal influences among 𝑛 experts with 𝑤𝑖𝑗 ∈ [0,1] and . The diagonal matrix 𝐴 = 𝑑𝑖𝑎𝑔(𝑎11, 𝑎22, … , 𝑎𝑛𝑛) is obtained from the matrix 𝑊 computing 𝑎𝑖𝑖 = 1 − 𝑤𝑖𝑖, 𝑖 = 12, … , 𝑛. The matrix 𝐴 is clarified as the susceptibility of all experts to interpersonal influence [8]. With the first opinion 𝑔(1), the following iterative plan is suggested to find the revised and final opinion: (1) If 𝐼 − 𝐴𝑊 is non-singular, then this process reaches the following Figure 1 describes the general consensus reaching process with interpersonal influences of experts.

Influence Model With Fuzzy Matrices

Fuzzy numbers generalize real numbers and are very useful to represent data corresponding to uncertain situations. In this section, an influence model is presented which is similar to the model presented in section 2.3 but the ordinary numbers are replaced by fuzzy numbers. TFNs consider only 3 data points and two linear functions so suitable to represent the influences and opinions. Final opinions are

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obtained when initial opinions and IIs of experts are given. Stepwise procedure is as follows:

be a matrix with entries of triangular fuzzy numbers , 𝑖, 𝑗 = 1,2,3. and represent left and right spreads of 𝑥̃𝑖𝑗 respectively. Let

is taken as a matrix of interpersonal influences, where 𝑤̃𝑖𝑗 are assumed to be triangular fuzzy numbers and 𝑤̃𝑖1 ⊕ 𝑤̃𝑖2 ⊕ ⋯ ⊕ 𝑤̃𝑖𝑛 = 1̃ for 𝑖 = 1,2, ⋯ , 𝑛. And

be the inverse of 𝑋̃ where , 𝑖, 𝑗 and represent left and right spreads of 𝑧̃𝑖𝑗 = 1,2,3. respectively. Then 𝑋̃ ⊗ 𝑍̃ = 𝐼̃. Step 2: We will solve the following system of equations:

is a matrix of sensitivities of experts to the interpersonal influences, 𝑎̃𝑖𝑖 = 1̃ − 𝑤̃𝑖𝑖.

is taken where its entries are the initial opinions of n experts. Step 3: Compute 𝐴̃𝑊̃

Step 3: Following system is solved for center part 𝑧𝑖𝑗 of the inverse matrix:

then 𝐼 ̃ − 𝐴̃𝑊̃ and 𝐼 ̃ − 𝐴̃ by using 𝐼̃ as defined in Definition 4. Step 4: Find (𝐼 ̃ − 𝐴̃𝑊̃ )−1 by the procedure described in section 3.1 and then final opinions 𝑦∞ by using the following equation:

Fuzzy Inverse Matrix Following is a step-wise procedure to find fuzzy inverse of a fuzzy matrix with the entries represented as triangular fuzzy numbers: Step 1:

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In this step identity matrix is considered while in the Step 4: Form the matrix 𝑍̃ by substituting the values of 𝑧𝑖𝑗,

of 𝑋̃ and 𝑍̃ respectively are considered, then 𝑋𝑍 = 𝐼 or 𝑍𝑋 =

The final opinion obtained in this way is not unique since the uniqueness of fuzzy inverse matrix is not guarenteed. Example Consider the initial opinion of 3 experts:

be the matrix of interpersonal influences of these experts. Computation of final opinions is required. Solution Step 1: Normalized matrix of interpersonal influences

be the inverse of 𝐼 ̃ − 𝐴̃𝑊̃, described above. Then to find the

is obtained by using operations defined in section 2.1.1 (see appendix):

To find the left spread part, following system of equations is solved:

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Conclusion

To find the right spread part, following system of equations is solved:

In this paper, CRP is remodeled with fuzzy numbers, instead of crisp numbers. There are some limitations in the example given for illustration. In step 1, diagonal entriesentries 𝑎̃𝑖𝑖, 𝑖 = 1,2,3 of the matrix 𝐴̃ are equal to 1̃ − 𝑤̃𝑖𝑖. 1̃ is chosen according to the entries of i.e. are the respective fuzzy one numbers for three rows. In step 2, fuzzy identity matrix 𝐼̃ consists of three equal fuzzy one numbers and six equal fuzzy zero numbers. This matrix can also be chosen with different fuzzy one and fuzzy zero numbers (means 𝛿 and 𝛼 are not necessarily same for each entry). As well as decision-making problems are capable to model with fuzzy numbers, interpersonal influences and opinion dynamics have also the potential to process with fuzzy numbers and fuzzy matrices in more general forms. Moreover some other types and generalizations of fuzzy numbers would contribute to the proceedings of influences and opinions in future. This work can facilitate some other consensus-reaching processes with fuzzy numbers and prepare for advancements in fuzzy inverse matrices.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw

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data that support the finding of this study are available from the corresponding author, upon reasonable request.

Mishra VN. Data envelopment analysis in the context of spherical fuzzy inputs and outputs. Eur J Pure Appl Math 2022;15:1158–1179. [CrossRef] Sharma MK, Sadhna, Bhargava AK, Kumar S, Rathour L, Mishra LN, Pandey S. A Fermatean fuzzy ranking function in optimization of intuitionistic fuzzy transportation problems. Adv Math Models Appl 2022;7:191–204. Pakhira R, Ghosh U, Sarkar S, Mishra LN. Study of memory effect in an EOQ model with fractional polynomial demand rate under fuzzy environment. Discontinuity Nonlinearity Complex 2022;11:583– 598. [CrossRef] Khalid A, Beg I. Influence model of evasive decision makers. J Intell Fuzzy Syst 2019;37:2539–2548.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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