Influence model and doubly extended TOPSIS with TOPSIS based matrix of interpersonal influences

Abstract

The aim of this paper is to construct a matrix of interpersonal influences employing TOPSIS and then to apply the matrix in influence model and doubly extended TOPSIS. Entries of that matrix are obtained from coefficients of relative closeness. Such a systematically constructed matrix performs better than the direct influence matrix because of the consideration of alternatives under certain criteria/attributes. Implementation of such influence matrix improves an influence model and group decision process. In this paper, TOPSIS is used for individual as well as group decisions. Once the decisions are reached by individuals with the help of TOPSIS, then coefficients of relative closeness are obtained and matrix of interpersonal influences is constructed. This matrix is used in influence model and to construct the influenced decision matrices. These influenced decision matrices are aggregated to get the collective decision. This strategy is based on the fact that the decisions taken by individuals affect their collective decision in future.

Keywords

Group decision making social influence networks multi criteria decision making TOPSIS

1 Introduction

Decision making is one of the most important processes that human being go through in their daily life activities such as businesses, services and managements. In this process, certain weights are assigned to different criterion, whereas all the weights are obtained either from the group of experts or from intuitive ways. These weights are an essential part of any decision be it individual or collective and heavily dependant on an input data available to group of experts or individuals. Ambiguities and uncertainties in the data make decision making process even more complex. Fuzzy set theory [11] has played a vital role in decision making since its inception in 1965. Fuzzy sets constitutes a suitable framework to formulate decision making models given the fact that uncertainties and ambiguities are present in input data.

Aruldoss et al., [1] studied and compared different multi criteria decision making (MCDM) methods. It is observed that TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) plays a significant role in MCDM. This method is based on the fact that the solution should be near to the ideal situation and far from the worst situation. More specifically, relative closeness (RC) of alternatives to the ideal solution is calculated and an alternative for which coefficient of RC is maximum is chosen. Many researchers have employed TOPSIS to solve various decision making problems based on different theories such as fuzzy soft set, intuitionistic fuzzy soft set, hesitant fuzzy soft set and dual hesitant fuzzy set theories [2–6 , 17]. Some recent advancements of TOPSIS in high-order fuzzy extensions and its applications have been observed in [26, 27].

Kacprzak [23] presented a doubly extended TOPSIS method for GDM where TOPSIS is applied twice. At first stage, each expert suggests its own ranking of alternatives using TOPSIS and at the second step, it is applied on their combined data which is then aggregated by using the weights of experts obtained from their individual decisions.

Group decision making (GDM) is a procedure by which a group of experts collectively selects the best alternative out of a finite set of available alternatives. On the other hand, Social influence network theory presents a mathematical modal of the process of attitude changes which occur in a social network of interpersonal influences. Degroot [10] presented this process as an iterative process and studied its convergence. The goal of the social influence network theory (SINT) is to predict and explain the attitude changes in small groups [7]. It uses an individual’s initial attitude and attitudes of other members in a small group to predict the attitudes of group members as a result of their interactions. This idea originated from the French’s formal theory of social power (1956), which proposes a simple model of network of interpersonal influence entering into the process of opinion formation. Social network analysis has grown rapidly since 1970 and has several applications in Sociology, Anthropology and Business Managements. After 30 years of French’s theory, Friedkin [24] studied this theory with the relationship between social and cognitive structures. He presented a model in the form of N equations for N members of a group. Friedkin and Johnsen [9] presented a recursive definition for the influence process in a group of N members, where the interpersonal influences of N members were represented in the form of a matrix W = (w_ij) and w_ij represents the influence of jth member on the ith member. This matrix plays a vital role in influence models and group decision processes. Several researchers are investigating group decision making (GDM) problems employing the social network theory [8, 21]. Capuano et al. [21] presented fuzzy group decision making process with incomplete information guided by a social influence. Recently, Lu et al. [25] proposed a framework based on social network analysis.

Influence models contribute significantly in GDM. Khalid and Beg [8] proposed an influence model of evasive decision makers. Chu et al. [12] defined the group preference relation with directed social network connections. Preference relations express the likings of experts for each alternative over others. In order to express the preferences, a widely adopted way is by means of pairwise comparison. Many researchers have applied fuzzy preference relations (FPRs) to describe the experts’ judgment in GDM process [8 , 13–16].

Experts are usually allowed to interact freely with each other and exchange their opinions. To the best of our knowledge, influence models in the existing literature involve the matrix of interpersonal influences which is formed according to experts’ opinions.

In this paper, we present an influence model and a doubly extended TOPSIS involving a matrix of interpersonal influences which is formed by using the coefficients of RC. This matrix plays a significant role in influence model and doubly extended TOPSIS. Doubly extended TOPSIS with this matrix of interpersonal influences is an extension of the idea presented by Kacprzak [23]. According to him, the influence of one expert on all other experts remains the same whereas our method proposes pairwise influences of the experts. Each individual of a group of experts provide its ranking of alternatives. They may chose the same or different alternatives. Taking into consideration of these two cases, entries of matrix W are taken from the coefficients of RC. This is a systematically developed matrix rather than the experts^, free opinions.

2 Preliminaries

This section includes some preparatory definitions that are required in the sequel.

Definition 2.1. [18] Let X ={ x₁, x₂, . . . , x_m } be the set of alternatives. A fuzzy preference relation (FPR) P on X is defined by the membership function μ_P : X × X ⟶ [0, 1]. The membership function μ_P (x_i, x_j) = p_ij is interpreted as follows: The alternative x_i is absolutely preferred over the alternative x_j if p_ij = 1.

The alternative x_i is absolutely outranked by the alternative x_j if p_ji = 1.

The alternative x_i is preferred over the alternative x_j if p_ij ∈ (0.5, 1].

The alternative x_i is outranked by the alternative x_j if p_ji ∈ (0.5, 1].

There exists an indifference between the alternatives x_i and x_j if p_ij = 0.5 .

A FPR P can be represented by m × m matrix P = (p_ij) .

Once experts have expressed their preferences, n individual FPRs P₁, P₂, . . . , P_n are available. A matrix $P_{k} = (p_{ij}^{k})$ denotes the fuzzy preference relation of k- th expert, where k ∈ {1, 2, . . . , n} and i, j ∈ {1, 2, . . . , m}. To obtain a collective FPR P from a set of individual ones, the family of ordered weighted average (OWA) operators is often adopted [19, 20].

2.1 Social influence network theory

Social influence network theory deals with the problems related to the change of opinions with the passage of time. Interpersonal influences of experts play an important role in setting up their reconsidered opinions. These problems can be formulated as an iterative process.

Friedkin et al. [9] presented a simple recursive concept for the influence process in a group of N actors. This influence model [8] was investigated in terms of its convergence.

Let W = (w_ij) be the matrix of interpersonal influences among n experts with w_ij ∈ [0, 1] and ∑_j=1ⁿw_ij = 1. The diagonal matrix A = diag (a₁₁, a₂₂, . . . , a_nn) is obtained from the matrix W by calculating a_ii = 1 - w_ii, i = 1, 2, . . . , n. The matrix A is interpreted as the susceptibility of all experts to interpersonal influence. With the initial opinion y⁽¹⁾, the following iterative scheme is defined to find the revised and final opinion: $y^{(t)} = {AWy}^{(t - 1)} + (I - A) y^{(1)}$

If (I - AW) is non-singular, then this process reaches the following $y^{(\infty)} = {(I - AW)}^{- 1} (I - A) y^{(1)} .$

2.2 TOPSIS method

TOPSIS is a commonly used MCDM method [1] which was introduced by Hwang and Yoon [22]. The process is carried out as follows:

Step 1. Construction of a decision matrix D .

$D = \begin{matrix} \begin{matrix} c_{1} & c_{2} & \dots & c_{n} \end{matrix} \\ \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{i} \\ ⋮ \\ A_{m} \end{matrix} & (\begin{matrix} d_{11} & d_{12} & \dots & d_{1 n} \\ d_{21} & d_{22} & \dots & d_{2 n} \\ ⋮ & ⋮ & ⋮ \\ d_{i 1} & d_{i 2} & \dots & d_{in} \\ ⋮ & ⋮ & ⋮ \\ d_{m 1} & d_{m 2} & \dots & d_{mn} \end{matrix}) \end{matrix} = {[d_{ij}]}_{m \times n}$

here A_i (i ∈ {1, 2, . . . , m}) and c_j (j ∈ {1, 2, . . . , n}) denote the alternatives and criteria, respectively.

Step 2. Formation of a standard (normalized) decision matrix R .

$r_{ij} = \frac{d_{ij}}{\sqrt{{\sum^{m}}_{k = 1} d_{ij}^{2}}}$ ∀d_ij ≠ 0, ∀i ∈ {1, 2, . . . , m} and ∀j ∈ {1, 2, . . . , n}

$R = (\begin{matrix} r_{11} & r_{12} & \dots & r_{1 n} \\ r_{21} & r_{22} & \dots & r_{2 n} \\ ⋮ & ⋮ & ⋮ \\ r_{i 1} & r_{i 2} & \dots & r_{in} \\ ⋮ & ⋮ & ⋮ \\ r_{m 1} & r_{m 2} & \dots & r_{mn} \end{matrix}) = {[r_{ij}]}_{m \times n}$ .

If matrix D is already normalized, then skip this step.

Step 3. Obtain the weighted normalized decision matrix V .

V = [v_ij] _m×n = [w_jr_ij] _m×n, i ∈ {1, 2, . . . , m}, where $w_{j} = \frac{W_{j}}{{\sum^{n}}_{j = 1} W_{j}}$ , j = 1, 2, . . . , n so that ∑ⁿ_j=1w_j = 1, and W_j is the original weight given to the criteria c_j, j = 1, 2, . . . , n .

$V = (\begin{matrix} v_{11} & v_{12} & \dots & v_{1 n} \\ v_{21} & v_{22} & \dots & v_{2 n} \\ ⋮ & ⋮ & ⋮ \\ v_{i 1} & v_{i 2} & \dots & v_{in} \\ ⋮ & ⋮ & ⋮ \\ v_{m 1} & v_{m 2} & \dots & v_{mn} \end{matrix}) = {[v_{ij}]}_{m \times n}$

Step 4. Determination of positive ideal solution (PIS) A⁺ and negative ideal solution (NIS) A^-.

$\begin{matrix} A^{+} = {v_{1}^{+}, . . ., v_{j}^{+}, . . ., v_{n}^{+}} = {(max v_{ij} ∣ j \in J_{1}), \\ (min v_{ij} ∣ j \in J_{2}), i \in {1, 2, . . ., m}} \end{matrix}$

and $\begin{matrix} A^{-} = {v_{1}^{-}, . . ., v_{j}^{-}, . . ., v_{n}^{-}} = {(min v_{ij} ∣ j \in J_{1}), \\ (max v_{ij} ∣ j \in J_{2}), i \in {1, 2, . . ., m}} \end{matrix}$ where J₁ and J₂ are associated with the benefit and cost attribute sets, respectively.

Step 5. Calculation of separation measurements of positive ideal $(S_{i}^{+})$ and the negative ideal $(S_{i}^{-})$ solutions. $S_{i}^{+} = \frac{\underset{j = 1}{\sum^{n}} | v_{ij} - v_{j}^{+} |}{n}$ and $S_{i}^{-} = \frac{\underset{j = 1}{\sum^{n}} | v_{ij} - v_{j}^{-} |}{n}$

∀i ∈ {1, 2, . . . , m}

Step 6. Evaluation of RC of alternatives to the ideal solution, that is, $C_{i} = \frac{S_{i}^{-}}{S_{i}^{+} + S_{i}^{-}},$ 0 ≤ C_i ≤ 1, ∀i ∈ {1, 2, . . . , m} .

Step 7. Set up the preference order.

3 TOPSIS based matrix of interpersonal influences

TOPSIS is a widely utilized MCDM method in which an alternative is chosen on the basis of preference order. This method is a process of obtaining the coefficients of RC to rank the alternatives under consideration of some attributes and starts with the construction of decision matrices by the experts. Coefficients of RC for each alternative are calculated through distance measures between the entries of decision matrix and positive and negative ideal solutions. An alternative is chosen for which the coefficient is maximum without consideration of how much greater is this than the others. Coefficients of RC calculated by the experts are considered as their influence weights for other experts. In this way, n² influence weights for n experts are evaluated and a matrix of order n × n is formed. Taking into account of the preference order of alternatives as well as the amount of coefficients, the following definition suggests how the matrix W = (w_ij) _p×p of interpersonal influences of p experts is developed.

Definition 3.1. Let {x₁, x₂, . . . , x_m} be a set of alternatives and {c₁, c₂, . . . , c_n} a set of criterion/attributes. Let {e₁, e₂, . . . , e_p} be a set of experts with weight vectors W_k = { w_k1, w_k2, . . . , w_kn } , k = 1, 2, . . . , p with ∑ⁿ_j=1w_kj = 1, where w_kj is the weight for jth criteria given by kth expert. Let $x_{σ (1)}^{k} ⪯ x_{σ (2)}^{k} ⪯ . . . ⪯ x_{σ (m)}^{k}$ be the preference order of alternatives concluded by the expert e_k, (k = 1, 2, . . . , p) individually where σ : {1, 2, . . . , m} ⟶ {1, 2, . . . , m} is a permutation function such that $x_{σ (i)}^{k} ⪯ x_{σ (i + 1)}^{k}$ for all i = 1, 2, . . . , m - 1. If $RC (x_{i}^{k})$ is the coefficient of RC of alternative x_i determined by the kth expert, then the matrix W of interpersonal influences of p experts is constructed according to two different situations: (1) experts choose the same alternative in their individual decisions, then

W = [a_lk] _p×p with

$a_{lk} = RC (x_{σ (m)}^{k})$ where l = k or e_l and e_k choose the same alternative,

(2) experts chose different alternatives in their individual decisions, then

W = [a_lk] _p×p with

$a_{lk} = | RC (x_{σ (m)}^{k}) - RC (x_{σ (n)}^{l}) |$ where e_l and e_k choose different alternatives, σ (n) ≤ σ (m) such that $x_{σ (m)}^{k}$ and $x_{σ (n)}^{l}$ represent the same alternative.

Entries a_lk of the matrix W represent the amount of influence of k^th expert upon l^th expert. When two experts choose the same alternative and RC (x_σ(m)) is maximum for k^th expert, then this value is taken as a_lk due to its characteristic of being maximum. When two experts choose different alternatives then the influence of k^th expert is the amount of difference between $RC (x_{σ (m)}^{k})$ and $RC (x_{σ (n)}^{l})$ .

Now we present influence model and doubly extended TOPSIS utilizing the matrix W. These two approaches consists of three significant steps (Fig 1):

Individual decisions of experts.

Construction of the matrix W.

Combine decision of experts by using W.

Fig. 1

Graphical representation of the procedure.

In some real life situations, a person faces the problem of selecting an alternative, it follows the decisions taken previously by some experts individually. But after some time, there may arise some influences among those experts. Taking into account of those influences, a decision making technique is applied again. Influence model presented by Friedkin et al. [9] and doubly extended TOPSIS presented by Kacprzak [23] are two different techniques. Influence model is a group decision technique and an iterative procedure while TOPSIS is utilized both for individual and group decisions and a non-iterative procedure. The purpose of presenting the extended ideas together in both techniques is to show the vast utilization of the matrix of interpersonal influences.

3.1 Influence model with TOPSIS based matrix of interpersonal Influences

Influence model represented in section 2.1 involves a matrix W = (w_ij) _n×n of interpersonal influences of n experts. The values of w_ij are not chosen by a mathematical formula or technique in the existing comparable literature, and experts are usually independent to interact with each other. The values of w_ij are chosen by direct observation such that w_ij ∈ [0, 1] and ∑_j=1ⁿw_ij = 1 for i = {1, 2, . . . , n}.

To solve a practical problem, influence model based on the above defined matrix W consists of the following steps:

Step 1: The two types of matrices, one is of fuzzy preference relations P_k over the set of alternatives X = {x₁, x₂, . . . , x_m}

$P_{k} = (\begin{matrix} p_{11}^{k} & p_{12}^{k} & \dots & p_{1 m}^{k} \\ p_{21}^{k} & p_{22}^{k} & \dots & p_{2 m}^{k} \\ ⋮ & ⋮ & ⋮ \\ p_{m 1}^{k} & p_{m 2}^{k} & \dots & p_{mm}^{k} \end{matrix}),$ and the other is of decision matrices D_k over the set of alternatives X = {x₁, x₂, . . . , x_m} and the set C = {c₁, c₂, . . . , c_n} of criterion

$D_{k} = (\begin{matrix} d_{11}^{k} & d_{12}^{k} & \dots & d_{1 n}^{k} \\ d_{21}^{k} & d_{22}^{k} & \dots & d_{2 n}^{k} \\ ⋮ & ⋮ & ⋮ \\ d_{m 1}^{k} & d_{m 2}^{k} & \dots & d_{mn}^{k} \end{matrix}),$ are provided by k experts, k = {1, 2, . . . , p}.

Also weight vectors W_k = {w_k1, w_k2, . . . , w_kn} with ∑ⁿ_j=1w_kj = 1, w_kj ∈ [0, 1] are provided for n criterion/attributes by p experts.

Step 2: Calculate the preference order of alternatives for experts e₁, e₂, . . . , e_p with different weight vectors by using TOPSIS as represented in section 2.2. Preference orders provided by the different experts may be same. Different preference orders occur due to the different weight vectors and different decision matrices provided by the experts.

Step 3: Develop the matrix of interpersonal influences W as defined in definition 3.1 over the set of experts E = {e₁, e₂, . . . , e_p} by using the coefficients of relative closeness obtained in step 2. And then find the matrix A of susceptibilities of each expert as described in section 2.1.

Step 4: Find the column sum of the jth column in P_k as ∑^m_l=1p_lj and then dividing each value by its corresponding column sum.

$(\begin{matrix} \frac{P_{11}^{k}}{{\sum^{m}}_{l = 1} p_{l 1}} & \dots & \frac{P_{1 m}^{k}}{{\sum^{m}}_{l = 1} p_{lm}} \\ ⋮ & ⋮ \\ \frac{P_{m 1}^{k}}{{\sum^{m}}_{l = 1} p_{l 1}} & \dots & \frac{P_{mm}^{k}}{{\sum^{m}}_{l = 1} p_{lm}} \end{matrix})$

and get the matrix $(u_{ij}^{k}) m \times m$ .

Get the corresponding priority vectors

$U^{k} = (\begin{matrix} u_{1}^{k} \\ ⋮ \\ u_{m}^{k} \end{matrix})$

of P_k where $u_{i}^{k} = \frac{{\sum^{m}}_{l = 1} u_{il}^{k}}{m}$ , represents that alternative x_i is preferred over other alternatives by this value.

Now find $y_{x_{i}}^{(1)}$ for i = 1, 2, . . . , m as follows:

$y_{x_{i}}^{(1)} = (\begin{matrix} u_{i}^{1} \\ ⋮ \\ u_{i}^{p} \end{matrix})$

$y_{x_{i}}^{(1)}$ separates the information relevant to the alternative x_i.

Step 5: Using the matrix W and A as developed in step 3, apply the formula described in section 2.1 to find the final opinions

$y_{x_{i}}^{(\infty)} = (\begin{matrix} u_{i}^{1 \infty} \\ ⋮ \\ u_{i}^{p \infty} \end{matrix})$

on each alternative x_i.

Find the column vector that represents the final preference of each expert over the set of alternatives X .

$U^{(k \infty)} = (\begin{matrix} u_{1}^{k \infty} \\ ⋮ \\ u_{m}^{k \infty} \end{matrix})$

Rank the alternatives by using

$φ (u_{i}^{1 \infty}, . . ., u_{i}^{p \infty}) = {\sum_{k = 1}}^{p} u_{i}^{k \infty}$

Choose the alternative x_i for which ${\sum_{k = 1}}^{p} u_{i}^{k \infty}$ is maximum.

An illustrative example is given to explain the method.

Example 3.1. Let us suppose that we have a set of four alternatives X = { x₁, x₂, x₃, x₄ } , a set of three experts E ={ e₁, e₂, e₃ } and a set of three criterion C ={ c₁, c₂, c₃ }.

Step 1: Preference relations over the set X provided by the experts are as follows:

$P_{1} = (\begin{matrix} 0.5 & 0.3 & 0.4 & 0.1 \\ 0.6 & 0.5 & 0.4 & 0.5 \\ 0.7 & 0.4 & 0.5 & 0.6 \\ 0.9 & 0.3 & 0.4 & 0.5 \end{matrix}),$ $P_{2} = (\begin{matrix} 0.5 & 0.5 & 0.9 & 0.6 \\ 0.1 & 0.5 & 0.2 & 0.4 \\ 0.9 & 0.7 & 0.5 & 0.3 \\ 0.7 & 0.8 & 0.6 & 0.5 \end{matrix}),$ $P_{3} = (\begin{matrix} 0.5 & 0.9 & 0.6 & 0.4 \\ 0.1 & 0.5 & 0.4 & 0.5 \\ 0.6 & 0.7 & 0.5 & 0.4 \\ 0.5 & 0.3 & 0.4 & 0.5 \end{matrix}) .$

Decision matrices provided by three experts are as follows:

$D_{1} = (\begin{matrix} 0.2 & 0.5 & 0.6 \\ 0.5 & 0.4 & 0.7 \\ 0.1 & 0.5 & 0.6 \\ 0.2 & 0.9 & 0.7 \end{matrix}),$ $D_{2} = (\begin{matrix} 0.5 & 0.4 & 0.6 \\ 0.3 & 0.4 & 0.7 \\ 0.2 & 0.5 & 0.6 \\ 0.9 & 0.4 & 0.7 \end{matrix}),$ $D_{3} = (\begin{matrix} 0.1 & 0.9 & 0.4 \\ 0.5 & 0.6 & 0.4 \\ 0.6 & 0.7 & 0.8 \\ 0.6 & 0.8 & 0.9 \end{matrix}) .$

And $W_{1} = (\begin{matrix} 0.3 & 0.4 & 0.3 \end{matrix}),$ $W_{2} = (\begin{matrix} 0.2 & 0.5 & 0.3 \end{matrix}),$ $W_{3} = (\begin{matrix} 0.5 & 0.4 & 0.1 \end{matrix})$ are the weight vectors for three experts.

Step 2: Calculate the preference order of alternatives for experts e₁, e₂, . . . , e_p with different weight vectors by using TOPSIS as represented in section 2.2.

For expert e₁ : C₁ = 0.2939, C₂ = 0.3430, C₃ = 0.1998, C₄ = 0.6569, note that

C₃ < C₁ < C₂ < C₄ .

For expert e₂ : C₁ = 0.4092, C₂ = 0.0901, C₃ = 0.3634, C₄ = 0.6366, obviously,

C₂ < C₃ < C₁ < C₄ .

For expert e₃ : C₁ = 0.4045, C₂ = 0.5954, C₃ = 0.7142, C₄ = 0.7857, clearly

C₁ < C₂ < C₃ < C₄ .

Step 3: Since three experts select the same alternative, therefore matrix of interpersonal influences based on coefficients of RC is as follows:

$W = (\begin{matrix} 0.6569 & 0.6366 & 0.7857 \\ 0.6569 & 0.6366 & 0.7857 \\ 0.6569 & 0.6366 & 0.7857 \end{matrix}),$

and normalized matrix W = $(\begin{matrix} 0.3159 & 0.3061 & 0.3778 \\ 0.3159 & 0.3061 & 0.3778 \\ 0.3159 & 0.3061 & 0.3778 \end{matrix}) .$

Entries of the matrix W are also explained in Fig. 2. The matrix A of susceptibilities of each expert is as follows

$A = (\begin{matrix} 0.6839 & 0 & 0 \\ 0 & 0.6937 & 0 \\ 0 & 0 & 0.6220 \end{matrix})$

Fig. 2

Graphical representation of W.

Step 4: Now let us convert the preference relations P₁, P₂ and P₃ into priority vectors.

$U^{1} = (\begin{matrix} 0.1698 \\ 0.2712 \\ 0.2932 \\ 0.2656 \end{matrix}),$ $U^{2} = (\begin{matrix} 0.2924 \\ 0.1396 \\ 0.2707 \\ 0.2972 \end{matrix}),$ $U^{3} = (\begin{matrix} 0.3018 \\ 0.1888 \\ 0.2825 \\ 0.2268 \end{matrix})$

The following matrices represent the available information according to each alternative.

$y_{x_{1}}^{(1)} = (\begin{matrix} 0.1698 \\ 0.2924 \\ 0.3018 \end{matrix}),$ $y_{x_{2}}^{(1)} = (\begin{matrix} 0.2712 \\ 0.1396 \\ 0.1888 \end{matrix}),$ $y_{x_{3}}^{(1)} = (\begin{matrix} 0.2932 \\ 0.2707 \\ 0.2825 \end{matrix}),$ $y_{x_{4}}^{(1)} = (\begin{matrix} 0.2656 \\ 0.2972 \\ 0.2268 \end{matrix}) .$

Step 5: Calculate (I - AW) ^-1 (I - A) to find the final opinion.

(I - AW) ^-1 (I - A) = $[\begin{matrix} 0.518 97 & 0.190 48 & 0.290 14 \\ 0.205 78 & 0.499 51 & 0.294 29 \\ 0.184 51 & 0.173 24 & 0.641 88 \end{matrix}]$

Final opinions about each alternative are as follows:

$y_{x_{1}}^{(\infty)} = (\begin{matrix} 0.231 38 \\ 0.269 81 \\ 0.275 7 \end{matrix}),$ $y_{x_{2}}^{(\infty)} = (\begin{matrix} 0.222 11 \\ 0.181 1 \\ 0.195 41 \end{matrix}),$ $y_{x_{3}}^{(\infty)} = (\begin{matrix} 0.285 69 \\ 0.278 69 \\ 0.282 33 \end{matrix}),$ $y_{x_{4}}^{(\infty)} = (\begin{matrix} 0.260 25 \\ 0.269 85 \\ 0.246 07 \end{matrix}) .$

Accordingly, the final opinions of three experts are as follows.

$U^{(1 \infty)} = (\begin{matrix} 0.23138 \\ 0.22211 \\ 0.28569 \\ 0.26025 \end{matrix}),$ $U^{(2 \infty)} = (\begin{matrix} 0.26981 \\ 0.1811 \\ 0.27869 \\ 0.26985 \end{matrix}),$ $U^{(3 \infty)} = (\begin{matrix} 0.2757 \\ 0.19541 \\ 0.28233 \\ 0.24607 \end{matrix})$

Now find the values of $φ (u_{i}^{1 \infty}, . . ., u_{i}^{p \infty})$ for each alternative, as follows

$φ (u_{1}^{1 \infty}, u_{1}^{2 \infty}, u_{1}^{3 \infty}) = 0.77689$

$φ (u_{2}^{1 \infty}, u_{2}^{2 \infty}, u_{2}^{3 \infty}) = 0.59862$

$φ (u_{3}^{1 \infty}, u_{3}^{2 \infty}, u_{3}^{3 \infty}) = 0.84671$

$φ (u_{4}^{1 \infty}, u_{4}^{2 \infty}, u_{4}^{3 \infty}) = 0.77617$

which implies x₃ ≽ x₁ ≽ x₄ ≽ x₂.

3.2 Doubly extended TOPSIS with matrix of interpersonal influences

In this method TOPSIS is applied twice in a group of k experts. Each expert gives a decision independently and then by using the coefficients of RC, matrix of interpersonal influences is developed as defined in Definition 3.1. Employing this matrix, influenced decision matrices are developed and then TOPSIS is applied to calculate a combined decision.

Stepwise description of this method is as follows:

Step 1: Choose the decision matrices D_k over the set of alternatives X = {x₁, x₂, . . . , x_m} and the set of criterion C = {c₁, c₂, . . . , c_n}

k = {1, 2, . . . , p}.

For k = {1, 2, . . . , p} , select the weight vectors W_k = {w_k1, w_k2, . . . , w_kn} , with ∑ⁿ_j=1w_kj = 1 and w_kj ∈ [0, 1].

Step 2: Develop the matrix W of interpersonal influences and A as developed in section 3.1.

Step 3: Develop the influenced decision matrices $D_{k}^{'} = [d_{ij}^{' k}]$ as:

$d_{ij}^{'} = [\begin{matrix} d_{ij}^{' 1} \\ ⋮ \\ d_{ij}^{' p} \end{matrix}] = {(I - AW)}^{- 1} (I - A) d_{ij}$

$= {(I - AW)}^{- 1} (I - A) [\begin{matrix} d_{ij}^{1} \\ ⋮ \\ d_{ij}^{p} \end{matrix}]$

Step 4: Now again apply TOPSIS for the decision matrices $D_{k}^{'}$ . First find the collective decision matrix D′ by taking the average values of corresponding entries from $D_{k}^{'}$ , k = 1, 2, . . . , p, then positive and negative ideal solutions and finally the coefficients of RC are obtained.

Step 5: Rank the alternatives.

To illustrate the method, consider the following example:

Example 3.2. As given in example 3.1, consider a set of four alternatives X = { x₁, x₂, x₃, x₄ } , a set of three experts E ={ e₁, e₂, e₃ } and a set of three criteria C ={ c₁, c₂, c₃ } where c₁ and c₂are benefit criteria and c₃ is cost criteria.

Step 1: Consider D₁, D₂ and D₃ as described in example 3.1 with weight vectors $W_{1} = (\begin{matrix} 0.4 & 0.2 & 0.4 \end{matrix}),$ $W_{2} = (\begin{matrix} 0.4 & 0.5 & 0.1 \end{matrix}),$ and $W_{3} = (\begin{matrix} 0.2 & 0.5 & 0.3 \end{matrix})$ respectively.

Step 2: Calculate the preference order of alternatives for experts e₁, e₂, . . . , e_p with different weight vectors by using TOPSIS as represented in section 2.2.

For expert e₁ : C₁ = 0.3330, C₂ = 0.5330, C₃ = 0.2, C₄ = 0.4665,

C₃ < C₁ < C₄ < C₂ .

For expert e₂ : C₁ = 0.3822, C₂ = 0.1174, C₃ = 0.1765, C₄ = 0.8234,

C₂ < C₃ < C₁ < C₄ .

For expert e₃ : C₁ = 0.7502, C₂ = 0.5751, C₃ = 0.4501, C₄ = 0.5,

C₃ < C₄ < C₂ < C₁ .

Since three experts select different alternatives, therefore matrix of interpersonal influences based on coefficients of RC is as follows.

$W = (\begin{matrix} 0.5330 & 0.3569 & 0.4172 \\ 0.4156 & 0.8234 & 0.3680 \\ 0.0421 & 0.3234 & 0.7502 \end{matrix}),$

whose entries are explained in Fig 3,

Fig. 3

Graphical representation of W

and normalized matrix $W = (\begin{matrix} 0.4077 & 0.2730 & 0.3191 \\ 0.2586 & 0.5123 & 0.2289 \\ 0.0377 & 0.2898 & 0.6724 \end{matrix})$

The matrix A of susceptibilities of each expert is as follows

$A = (\begin{matrix} 0.5923 & 0 & 0 \\ 0 & 0.4877 & 0 \\ 0 & 0 & 0.3276 \end{matrix})$

Calculate (I - AW) ^-1 (I - A) to find the influenced decision matrices.

(I - AW) ^-1 (I - A) =

$[\begin{matrix} 0.563 55 & 0.177 58 & 0.258 66 \\ 9 . 784 8 \times 10^{- 2} & 0.726 36 & 0.175 61 \\ 0.020 84 & 9 . 125 5 \times 10^{- 2} & 0.887 84 \end{matrix}]$

Step 3: By using the matrix W of interpersonal influences and the matrix A, calculate the influenced decision matrices.

$D_{1}^{'} = (\begin{matrix} 0.22737 & 0.5856 & 0.54814 \\ 0.46438 & 0.45165 & 0.62226 \\ 0.24707 & 0.55163 & 0.65161 \\ 0.42773 & 0.78516 & 0.75159 \end{matrix}),$

$D_{2}^{'} = (\begin{matrix} 0.40031 & 0.49752 & 0.56477 \\ 0.35464 & 0.43505 & 0.64719 \\ 0.26042 & 0.53503 & 0.63501 \\ 0.77866 & 0.51910 & 0.73499 \end{matrix}),$

$D_{3}^{'} = (\begin{matrix} 0.13858 & 0.84598 & 0.42239 \\ 0.48172 & 0.57754 & 0.4336 \\ 0.55304 & 0.67754 & 0.77753 \\ 0.619 & 0.76553 & 0.87752 \end{matrix}) .$

Step 4: Collective decision matrix D′ is obtained by calculating the average values of corresponding entries of $D_{1}^{'},$ $D_{2}^{'}$ and $D_{3}^{'},$

$D^{'} = (\begin{matrix} 0.25542 & 0.64303 & 0.51176 \\ 0.43358 & 0.48808 & 0.56768 \\ 0.35351 & 0.58806 & 0.68805 \\ 0.60846 & 0.68993 & 0.78803 \end{matrix}) .$

Positive and negative ideal solutions are:

$A^{+} = (\begin{matrix} 0.60846 & 0.68993 & 0.51176 \end{matrix})$

$A^{-} = (\begin{matrix} 0.25542 & 0.48808 & 0.78803 \end{matrix})$

and the distances are: $S_{1}^{+} = 0.13331$

$S_{2}^{+} = 0.14421$

$S_{3}^{+} = 0.12481$

$S_{4}^{+} = 0.09209$

$S_{1}^{-} = 0.14374$

$S_{2}^{-} = 0.13283$

$S_{3}^{-} = 0.09935$

$S_{4}^{-} = 0.18496$

Coefficients of RC are:

C₁ = 0.51882

C₂ = 0.47946

C₃ = 0.44321

C₄ = 0.66760

Step 5: Preference order of alternatives is x₄ ≽ x₁ ≽ x₂ ≽ x₃.

4 Discussion

The method presented in section 3.1 combines the influence model and TOPSIS. It works together with preference relations over the set of alternatives and the matrices consisting of values of alternatives under consideration of multi criteria. TOPSIS is a MCDM method which is applied on the decision matrices and results in the coefficients of RC. These coefficients lead to develop the matrix of interpersonal influences. This matrix plays an important role in the influence model. In example 3.1, three experts choose the same alternative x₄ by using TOPSIS but finally they select x₃ after applying the influence based decision making method which includes the preference relations. In this way, we can recognize the role of preference relations in the selection of alternatives.

Doubly extended TOPSIS was presented by Kacprzak [23] without consideration of interpersonal influences. He calculated the weights by using coefficients of RC obtained from expert’s individual decisions and then by using those weights, he developed an aggregated collective matrix. Moreover, individual decisions of experts were not considered in the sense, whether they take same or different decisions (whether they choose the same or different alternatives).

It is worth mentioning that the matrix W in this paper is developed by considering the decisions of experts with two aspects, whether they agree with each other or not. Entries of W are the weights which represent the influence of each expert on the others. By using this matrix, we formed the decision matrices for individuals instead of a collective matrix. Due to the construction of individuals’ decision matrices, there is an advantage of obtaining both individuals or collective decisions.

Note that the utilization of influence model described in section 3.1 depends upon the availability of preference relations, whereas the doubly extended TOPSIS described in section 3.2 does not require the preference relations.

References

Aruldoss

, Lakshmi

T.M.

and Venkatesan

V.P.

, A survey on multicriteria decision making methods and its applications, American Journal of Information Systems 1(1) (2013), 31–43.

Joshi

and Kumar

, Intuitionistic fuzzy entropy and distance measure based TOPSIS method for multi-criteria decision making, Egyptian Informatics Journal 15 (2014), 97–104.

Eraslan

and Cagman

, A decision making method by combining TOPSIS and grey relation method under fuzzy soft sets, Sigma Journal of Engineering and Natural Sciences 8(1) (2017), 53–64.

Eraslan

and Karaaslan

, A group decision making method based on TOPSIS under fuzzy soft environment, Journal of New Theory 3 (2015), 30–40.

Wang

, Wang

, Xu

and Ni

, Distance and similarity measures of dual hesitant fuzzy sets with their applications to multiple attribute decision making, IEEE International Conference on Progress in Informatics and Computing, (2014).

Peng

and Yang

, Interval-valued hesitant fuzzy soft sets and their application in Decision Making, Fundamenta Informaticae 141 (2015), 71–93.

Breiger

, Carley

and Pattison

, Dynamic social network modeling and analysis: workshop summary and papers, The National Academies Press Washington, (2003).

Khalid

and Beg

, Inuence model of evasive decision makers, Journal of Intelligent and Fuzzy Systems 37(2) (2019), 2539–2548.

Friedkin

and Johnson

, Social inuence networks and opinion change, Advances in Group Processes 16(1) (1999), 1–29.

10.

Degroot

M.H.

, Reaching a consensus, Journal of the American Statistical Association 69(345) (1974), 118–121.

11.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

12.

Chu

, Liu

and Wang

, Social network analysis based approach to group decision making problem with fuzzy preference relations, Journal of Intelligent & Fuzzy Systems 31 (2016), 1271–1285.

13.

Meng

and Chen

, A new method for group decision making with incomplete fuzzy preference relations, Knowledge Based Systems 73 (2015), 111–123.

14.

Tapia Garća

J.M.

, del Moral

M.J.

, Martinez

M.A.

and Herrera-Viedma

, A consensus model for group decision making problems with linguistic interval fuzzy preference relations, Expert Systems with Applications 39 (2012), 10022–10030.

15.

Yan

H.B.

and Ma

T.J.

, A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority, European Journal of Operational Research 241 (2015), 815–829.

16.

Zhu

and Xu

, A fuzzy linear programming method for group decision making with additive reciprocal fuzzy preference relations, Fuzzy Sets and Systems 246 (2014), 19–33.

17.

Eraslan

and Cagman

, A decision making method by combining TOPSIS and grey relation method under fuzzy soft sets, Sigma Journal of Engineering and Natural Sciences 8(1) (2017), 53–64.

18.

Zadeh

L.A.

, A computational approach to fuzzy quantifiers in natural languages, Computers and Mathematics with Applications 9(1) (1983), 149–184.

19.

Yager

R.R.

, Quantifier guided aggregation using OWA operators, International Journal of Intelligent Systems 11(1) (1996), 49–73.

20.

Yager

R.R.

, Quantifiers in the formulation of multiple objective decision function, Information Sciences 31(2) (1983), 107–139.

21.

Capuano

, Chiclana

, Fujita

, Herrera-Veidma

and Loia

, Fuzzy group decision making with incomplete information guided by social innuence, IEEE Transactions on Fuzzy Systems (2017), DOI 10.1109/TFUZZ.2017.2744605

22.

Hwang

C.L.

and Yoon

, Multuple attribute decision making, Methods and Applications (1981), New York, Springer-Verlag.

23.

Kacprzak

, A doubly extende TOPSIS method for group decision making based on ordered fuzzy numbers, Expert Systems with Applications 116 (2019), 243–254.

24.

Friedkin

N.E.

, A formal theory of social power, Journal of Mathematical Psychology 12(2) (1986), 103–126.

25.

, Qu

, Xu

, Ma

and Li

, Multiattribute social network matching with unknown weight and different risk preference, Journal of Intelligent and Fuzzy Systems DOI: 10.3233/JIFS-182535

26.

Chen

Z.S.

, Yang

, Wang

X.J.

, Chin

K.S.

and Tsui

K.L.

, Fostering linguistic decision making under uncertainty: A proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization problems, Information Sciences 500 (2019), 229–258.

27.

Chen

Z.S.

, Li

, Kong

W.T.

and Chin

K.S.

, Evaluation and selection of HazMat transportation alternatives: A PHFLTS and TOPSIS integrated multi-perspective approach, Environmental Research and Public Health 16(21) (2019).