Social network analysis based approach to group decision making problem with fuzzy preference relations

Abstract

There are plenty of researches on group decision making (GDM) problem and most of them assume that all the experts are independent. However, the social network connection is an important characteristic among experts, and should been taken into account in the GDM decision process. The social network analysis (SNA) is a rapidly developing technology to deal with the problem about social network connections. In this paper, we address GDM problem with fuzzy preference relations where the experts have directed social network connections. And we use in-degree and out-degree centrality indices in SNA to build the social characteristics of the experts. Then we utilize the importance induced ordered weighted averaging(I-IOWA) operator to aggregate all the individual fuzzy preference relations with social indices. Furthermore,we investigate the reciprocity, additive consistency and acceptable consistency properties about the obtained group fuzzy preference relations. A procedure of addressing GDM problem with fuzzy preference relations where the experts are within directed social network connections is developed. Moreover, we compare the results of the case where all the experts are independent and the case where all the experts are within directed social network connections. Some useful tips that affect final outranking result in the directed graphical social network connection are obtained. In the end, we summarize the main results of this paper and point out some research directions for future research.

Keywords

Social network analysis (SNA)fuzzy preference relation I-IOWA operator group decision making (GDM)

1 Introduction

Group decision making problems are usually characterized as that several experts provide his/her preferences depending on the nature of the alternatives in order to choose the alternative(s) which is most accepted by these specialists [12]. Preference relation is the most common representation of information used in decision making problem because it is a useful tool in modelling decision processes, above all when we want to aggregate experts preferences into group preferences [11 , 20]. And the fuzzy preference relation (FPR) [20] introduced by Tanino is one typical preference relation which has been attracting wide attention till now.

Many researchers have been applying FPRs to describe the individual experts’ judgment information under GDM environment [16 , 35]. And in most literatures the researchers suppose that all the individual experts are independent, that is, they are not within any social network connections among them. However, this assumption is irrational to some extent. Because in most cases experts involved in a group decision-making problem may have developed opinions about the reliability of other experts’ judgments, either because they have previous history of interaction with each other or because they have knowledge that informs them on the reliability of other colleagues in the group in solving decision-making problem in the past [18]. Therefore, we should take the social network connections among experts into consideration. In other words, we should address GDM problem in the framework of social network.

Social network is a theoretical construct useful in the social sciences to study relationships between individuals, groups, organizations, or even entire societies. The methods of social network analysis (SNA) are a relatively new and still developing subject that focuses on the analysis relationships among social entities, and on the patterns and implications of these relationships [24]. The approach could be particularly suitable for (but not limited to) consensus analysis, coalition formation, group problem solving and communities analysis [2]. There are numerous researchers having been investigating GDM problem within the framework of social network decision making [1 , 33]. Brunelli, Fedrizzi and Michele [3] addressed the problem of consensus evaluation by endogenously computing the importance of the decision makers in terms of their influence strength in the social network. Wu and Chiclana [25] developed a SNA trust–consensus based group decision making model with interval-valued fuzzy reciprocal preference relation, which is one of the first efforts in combining trust degree and consensus level into the field of GDM problem. Wu et al. [26] put forward a theoretical framework to consensus building within a networked social group. They investigated a trust based estimation and aggregation methods as part of a visual consensus model for multiple criteria GDM with incomplete linguistic information. Duong et al. [7] proposed a collaborative algorithm for semantic video annotation using consensus-based social network analysis (SNA). Pérez et al. [18] presented three new social network analysis based induced ordered weighted averaging (IOWA) operators that take advantage of the linguistic trustworthiness information gathered from the experts’ social network to aggregate the social group preferences within the framework of social network decision making. Maturo and Ventre [15] described a procedure and a suitable algorithm for obtaining consensus in a group decision context where the experts were constituted to take decisions about a subject of social interest.

However, there are few literatures studying GDM problem with fuzzy preference relations under framework of social network connections and analyzing the corresponding aggregated group preference relation.What’s more,few researchers have analyzedthe reasons on the social network connections’ influence towards outranking alternatives in above decision making environment. Therefore, in this paper we would like try to define the group preference relation with directed social network connections and study its properties. And we develop a feasible approach to address GDM problem with fuzzy preference relations under the framework of directed social network connections.

The rest of the paper is organized as follows. In Section 0, we review the fuzzy preference relation, induced ordered weighted averaging (IOWA) operator and some basic concepts about social network analysis(SNA). In Section 3, we define the group preference relation with directed social network connections and study its reciprocity, additive consistency and acceptable consistency properties. In Section 4, we develop a procedure of addressing GDM problem where the experts are within directed social network connections and point out the characteristics and advantages of our approach. In Section 5, we use a numerical example to illustrate the feasibility of the proposed approach. And we compare it with the case where all the experts are independent. In Section 6, we summarize this paper and explain the direction for future research.

2 Preliminaries

In this section, we review some basic definitions regarding the fuzzy preference relation, induced ordered weighted averaging (IOWA) operator and some basic concepts associated with social network analysis (SNA).

2.1 Fuzzy preference relation (FPR)

Let Y = { Y₁, …, Y_n } (n ≥ 2) be a finite set of alternatives, where Y_i denotes the ith alternative. The concept of fuzzy preference relation (FPR) is defined as follows.

Definition 1. [21] A fuzzy preference relation on Y is a fuzzy set on the product set Y × Y, which is characterized by a membership function μ_p : Y × Y → [0, 1], the preference relation is conveniently represented by the n × n matrix P = (p_ij), where μ_p (Y_i, Y_j) = p_ij and p_ij is interpreted as the preference degree of the alternative x_i over x_j: $p_{ij} = \frac{1}{2}$ indicates indifference between x_i and x_j, p_ij = 1 indicates that x_i is absolutely better than x_j, $p_{ij} > \frac{1}{2}$ indicates that x_i is preferred to x_j. In this case, the fuzzy preference relation is generally assumed to be an additive reciprocal verifying: p_ji + p_ij = 1, p_ii = 0.5, ∀i, j, k = 1, …, n.

Considering the consistency of a fuzzy preference relation, Herrera et al. [11] introduced the following concept of additive consistency.

Definition 2. [11] A reciprocal fuzzy preference relation P = (p_ij) _n×n is consistent if $p_{ij} + p_{jk} + p_{ki} = \frac{3}{2}, \forall i, j, k = 1, \dots, n$ (1)

It is noted that this consistency is called additive consistency, which distinguishes the multiplicative consistency of fuzzy preference relation based on multiplicative transitivity [20].

Ma et al. [13] developed one feasible approach for constructing a fuzzy preference relation with additive consistency based on a given fuzzy preference relation by the following theorem.

Lemma 1. Let P = (p_ij) _n×n be a fuzzy preference relation. A matrix G = (g_ij) _n×n can be constructed, where

$\begin{matrix} g_{ij} & = & 0.5 + \frac{1}{n} (\sum_{l = 1}^{n} p_{il} - \sum_{l = 1}^{n} p_{jl}), \\ \forall i, j \in {1, 2, \dots, n} \end{matrix}$ (2)

For g_ij, ∀i, j∈ { 1, 2, …, n }, there are two cases to be considered:

Case 1: g_ij ≥ 0, ∀i, j∈ { 1, 2, …, n }. In this case, G = (g_ij) _n×n is a fuzzy preference relation with additive consistency;

Case 2: There exists at least one element less than 0 in matrix G = (g_ij) _n×n. In this case, the matrix G = (g_ij) _n×n is transformed into $G^{'} = {(g_{i j}^{'})}_{n \times n}$ , the elements of are given by $g_{ij}^{'} = \frac{g_{ij} + a}{1 + 2 a}, \forall i, j \in {1, 2, \dots, n}$ (3) where a = max{ |g_ts||g_ts < 0, t, s ∈ { 1, 2, …, n } }. Therefore, matrix is a fuzzy pre-ference relation with additive consistency.

It is worth noting that, for simplicity, the matrix obtained by Equation (2) is always assumed to be a consistent fuzzy preference relation correspondingto P.

2.2 Induced ordered weighted averaging operator

The ordered weighted averaging (OWA) operator proposed by Yager [29] provides a unified framework for the traditional decision making methods under uncertainty. The OWA operator provides a new information aggregation technique and has already aroused considerable research interests. Yager and Filev introduced in [32] a more general type of OWA operator, which they named the induced ordered weighted averaging (IOWA) operator. And the IOWA operators induce their ordering by using an additional variable or criterion, called the order inducing variable.

Definition 3. (IOWA operator) An IOWA operator of dimension n is a function Φ_W : (R × R)ⁿ→ R, to which a set of weights or weighting vector is associated, W = (w₁, w₂, ⋯ , w_m), such that w_i ε [0, 1] and $\sum_{k = 1}^{m} w_{k} = 1$ , and it is defined to aggregate the set of second arguments of a list of n 2-tuples {〈 u₁, p₁ 〉 , ⋯ 〈 u_n, p_n 〉 } according to the following expression: $Φ (〈 u_{1}, p_{1} 〉, \dots 〈 u_{n}, p_{n} 〉) = \sum_{i = 1}^{n} w_{i} p_{σ (i)}$ (4) being σ: {1, ⋯ , n } → { 1, ⋯ , n } a permutation such that u_σ(i) ≥ u_σ(i+1), ∀i = 1, …, n - 1, i.e., 〈u_σ(i), p_σ(i)〉 is the 2-tuple with u_σ(i) the ith largest value in the set {u₁, … , u_n }.

Chiclana, Herrera-Viedma and Herrera [5] introduced a special IOWA operator named importance induced ordered weighted averaging (I-IOWA) operator where the expert is assigned an importance degree to him/her. In this case, each component in the aggregation consists of a triple (v_k, u_k, p_k): p_k is the argument value to aggregate, v_k is the importance weight value associated to p_k, and u_k is the order inducing value. In this case the aggregation is

$\begin{matrix} Φ_{I - IOWA} (〈 v_{1}, u_{1}, p_{1} 〉, 〈 v_{2}, u_{2}, p_{2} 〉, \dots, \\ 〈 v_{n}, u_{n}, p_{n} 〉) = \sum_{k = 1}^{n} w_{k} p_{σ (k)} \end{matrix}$ (5) with $w_{k} = Q (\frac{S (k)}{S (n)}) - Q (\frac{S (k - 1)}{S (n)})$ (6) where $S (k) = \sum_{l = 1}^{k} v_{σ (l)}$ and σ is the permutation such that v_σ(k) in (u_σ(k), p_σ(k)) is the kth largest value in the set {v₁, v₂, … , v_n }. The function Q is denoted as basic unit-interval monotonic (BUM) function [30].

2.3 Social network analysis

Social network analysis (SNA) studies the relationships between social entities such as members of a group, corporations, or nations and give us a background that allows us, among other things, to examine the structural and locational properties including centrality, prestige, and structural balance [18]. The main three elements in SNA are: the set of actors, the relations themselves, and the actor attributes (see Table 1). We can refer to important network concepts in a unify manner, using the three different and possible representation schemes:

Sociometric: relational data are often presented in two-ways matrices called sociomatrix or adjacency matrix.

Graph theoretical: the network is viewed as a graph, consisting of nodes joined by lines.

Algebraic: this notation allows to distinguish several distinct relations and represent combinations of relations.

Definition 4. (In-degree and out-degree centrality indices) Let G = (E, L) be a directed dichotomous graph, E ={ e₁, e₂, ⋯ e_m } be the set of nodes, and L ={ l₁, l₂, ⋯ l_q } be the set of directed lines, or arcs, between pairs of nodes.

The number of arcs originating at a node,d⁺(e_i) is known as the node out-degree centrality index,C_oD (e_i), i.e.,

C_{oD} (e_{i}) = d^{+} (e_{i})

(7)

The number of arcs terminating at a node,d^-(e_i) is known as the node in-degree centrality index,C_iD (e_i), i.e.,

C_{iD} (e_{i}) = d^{-} (e_{i})

(8)

As with the centrality index, both out-degree centrality and in-degree centrality indices depend on the cardinality of the set E, and therefore the corresponding standardized measures are $C_{oD}^{'} (e_{i}) = \frac{d^{+} (e_{i})}{m - 1}$ (9) $C_{iD}^{'} (e_{i}) = \frac{d^{-} (e_{i})}{m - 1}$ (10)

With directed social network connections, choices received are quite interesting to a network analyst. Thus, measures of centrality may not be of as much concern as measures of prestige. The simplest actor-level measure of prestige is the in-degree centrality index of each actor. The idea is that actors who are prestigious tend to receive many nominations or choice [24]. The larger this index is, the more prestigious is the actor.

3 Group fuzzy preference relation with directed social network connections

In group decision making model, we normally aggregate all the individual information into a collective one and then rank the alternatives based on it. In this section, we introduce group fuzzy preference relation without considering social network connections and group fuzzy preference relation based on directed social network connections under group decision making environment and study their reciprocity and additive consistency properties.

3.1 The group fuzzy preference relation without considering social network connections

Let D ={ d₁, d₂, …, d_m } be a group of experts and Λ = { λ₁, λ₂, …, λ_m } be the weight vector of experts, where λ_k > 0, $\sum_{k = 1}^{m} λ_{k} = 1$ for all k = 1, 2, …, m. It is worth noting that each expert contributes to the decision making despite that their degrees of importance are different. Thus we let λ_k > 0. In this section, we assume all the experts are independent, that is, they are no social network connections between each other. To express his/her opinions about alternatives, each decision maker d_k compares each pair of alternatives (Y_i, Y_j) with number q_ijk ∈ [0, 1] and we assume that the judgment matrix given by d_k is a fuzzy preference relation denoted by Q_k = (q_ijk) _n×n(k = 1, 2, …, m).

Wu and Xu [28] defined the group fuzzy preference relation in the group decision making environment where the individual judgment matrix is expressed in term of fuzzy preference relation.

Definition 5. Let {Q₁, Q₂, … , Q_m } be m fuzzy preference relations, then $Q = λ_{1} Q_{1} + λ_{2} Q_{2} + \dots λ_{m} Q_{m}$ (11) is called the group fuzzy preference relation, where Q = (q_ij) _n×n and $q_{ij} = \sum_{k = 1}^{m} λ_{k} q_{ijk}$ .

It is worth noting that the group fuzzy preference relation is reciprocal as well. Furthermore, Wu and Xu [28] defined the acceptable consistency of fuzzy preference relation and we change its expression in this paper as follows for understanding easily.

Definition 6. Let P = (p_ij) _n×n be a fuzzy preference relation. The corresponding consistent preference relation obtained using (2) is denoted by G. Given a threshold value α, if the consistency index satisfies the following: $CI (P) = 1 - \frac{1}{n (n - 2) / - 2} \sum_{i < j} | p_{ij} - g_{ij} | \geq α$ (12) then P is called a fuzzy preference relation with acceptable consistency.

Wu and Xu [28] gave the following theorem about relationship between the acceptable consistency of group fuzzy preference relation and individual ones.

Theorem 1.Let {Q₁, Q₂, … , Q_m } bemfuzzy preference relations,Qis the group fuzzy preference relation utilizing Equation (11). Then $CI (Q) \geq min_{k \in {1, 2, \dots, m}} {CI (Q_{k})}$ (13)

It is found that Theorem 1 shows that the group fuzzy preference relationcan maintain acceptable consistency based on the assumption that all the individual preference relations are with acceptable consistency.

3.2 An extension of group fuzzy preference relation corresponding to the directed social network connections

Suppose that we have a single set of experts, and a single, dichotomous directional relation shown in Fig. 1 measured on the pairs of experts. Due to the difference of professional knowledge, experience and research field, any individual expert play different role in decision making. We can impose different weights to different experts to show their intrinsic importance. Here we assume Λ = (λ₁, λ₂, …, λ_m) is the weight vector with respect to experts themselves. The nodein-degree centrality can reflect the expert’s importance when we consider the directional social network connections between experts. We let $C_{iD}^{'} = (C_{iD}^{'} (e_{1}), C_{iD}^{'} (e_{2}), \dots, C_{iD}^{'} (e_{m}))$ be the vector of nodein-degree centrality index, where $C_{iD}^{'} (e_{k})$ (k = 1, 2, …, m) is computed by Equation (10).

3.2.1 Group fuzzy preference relation corresponding to directed social network connections

Since we consider the directional social network connections between experts, we not only take the expert’s intrinsic weight into consideration but also the corresponding prestige should be considered when we aggregate the individual fuzzy preference relations. Moreover, we obtain the group fuzzy preference relation in this case by means of I-IOWA operator and we named it directed graph based group fuzzy preference relation (DG-GFPR). We let W = (w₁, w₂, …, w_m) be the weight vector derived by Equation (6) associated with I-IOWA operator. And we consider the nodein-degree centrality with respect to expert as the importance degree to him/her. Before aggregating the individual preference relations, we normalize the vector of nodein-degree centrality expert as follows.

$\begin{matrix} C_{iD}^{″} = (C_{iD}^{″} (e_{1}), C_{iD}^{″} (e_{2}), \dots, C_{iD}^{″} (e_{m})) \\ = (\frac{e^{C_{iD}^{'} (e_{1})}}{\sum_{i = 1}^{m} e^{C_{iD}^{'} (e_{i})}}, \frac{e^{C_{iD}^{'} (e_{2})}}{\sum_{i = 1}^{m} e^{C_{iD}^{'} (e_{i})}}, \dots, \frac{e^{C_{iD}^{'} (e_{m})}}{\sum_{i = 1}^{m} e^{C_{iD}^{'} (e_{i})}}) \end{matrix}$ (14)

Then, DG-GFPR Q_DG-GFPR is computed as follows:

$\begin{matrix} Q_{DG - GFPR} = Φ_{DG - GFPR} (〈 C_{iD}^{″} (e_{1}), λ_{1}, P_{1} 〉, \\ 〈 C_{iD}^{″} (e_{2}), λ_{2}, P_{2} 〉, \dots, 〈 C_{iD}^{″} (e_{m}), λ_{m}, P_{m} 〉) \\ = \sum_{i = 1}^{m} \frac{w_{i} λ_{σ^{- 1} (i)} C_{iD}^{″} (e_{σ (i)})}{\sum_{t = 1}^{m} w_{t} λ_{σ^{- 1} (t)} C_{iD}^{″} (e_{σ (t)})} P_{σ (i)} \end{matrix}$ (15) being σ: {1, ⋯ , n } → { 1, ⋯ , n } a permutation such that $C_{iD}^{″} (e_{σ (i)}) \geq C_{iD}^{″} (e_{σ (i + 1)})$ , ∀i = 1, …, n - 1, i.e., $〈 C_{iD}^{″} (e_{σ (i)}), λ_{σ (i)}, P_{σ (i)} 〉$ is the 3-tuple with $C_{iD}^{″} (e_{σ (i)})$ the ith largest value in the set ${C_{iD}^{″} (e_{1}), C_{iD}^{″} (e_{2}), \dots, C_{iD}^{″} (e_{m})}$ . And here σ^-1(i) denotes the value in the set initial set {1, ⋯ , n } corresponding to σ (i) in permutation set {σ (1) , σ (2) , … , σ (n) }.

In special environment, when we reordering the nodein-degree centrality indices, if there are two nodein-degree centrality indices are equal, then we consider their nodeout-degree centrality indices subsequently. It is worth noting that in this paper φ^-1(e_i) denotes the value in the permutation set {σ (1) , σ (2) , … , σ (n) } corresponding to e_i. For example, if $C_{iD}^{″} (e_{x}) = C_{iD}^{″} (e_{y})$ , then we investigate their nodeout-degree centrality indices $C_{oD}^{'} (e_{x})$ and $C_{oD}^{'} (e_{y})$ . If $C_{oD}^{'} (e_{x}) > C_{oD}^{'} (e_{y})$ , then φ^-1(e_x) > φ^-1(e_y); if $C_{oD}^{'} (e_{x}) < C_{oD}^{'} (e_{y})$ , then φ^-1(e_x) < φ^-1(e_y); if $C_{oD}^{'} (e_{x}) = C_{oD}^{'} (e_{y})$ , then |φ^-1(e_x) - φ^-1(e_y) | = 1 and we substitute $w_{φ - 1 (e_{x})}^{'} = w_{φ - 1 (e_{y})}^{'} = \frac{w_{φ^{- 1} (e_{x})} + w_{φ^{- 1} (e_{y})}}{2}$ for w_{φ^-1(e_x)} and w_{φ^-1(e_y)} respectively when we perform I-IOWA operator.

In Equation (15), for simplicity, we let $v_{i} = \frac{w_{i} λ_{σ^{- 1} (i)} C_{iD}^{″} (e_{σ (i)})}{\sum_{t = 1}^{m} w_{t} λ_{σ^{- 1} (t)} C_{iD}^{″} (e_{σ (t)})}$ . We can get $\sum_{i = 1}^{m} v_{i} = 1$ . Then, we have

$\begin{matrix} Φ_{DG - GFPR} (〈 C_{iD}^{″} (e_{1}), λ_{1}, P_{1} 〉, \\ 〈 C_{iD}^{″} (e_{2}), λ_{2}, P_{2} 〉, \dots, 〈 C_{iD}^{″} (e_{m}), λ_{m}, P_{m} 〉) \\ = \sum_{i = 1}^{m} v_{i} P_{σ (i)} \end{matrix}$ (16) Thus, the group preference $Q_{DG - GFPR} = {(q_{ij}^{DG - GFPR})}_{n \times n}$ of alternative Y_i over alternative Y_j can be computed by the following equation. $q_{ij}^{DG - GFPR} = \sum_{k = 1}^{m} v_{i} q_{ij σ (k)}$ (17)

Remark 1. We transfer the vector of nodein-degree centrality indices into a normalized one utilizing the exponential function in Equation (14). The especial nodes whose in-degree centrality indices are equal to 0 (includes the isolated ones) should also be taken into consideration when we aggregate the individual judgment matrices into DG-GFPR. That is the reason why we have done. If we do not transfer it like this, the corresponding weighted part associated with the special nodes mentioned above in Equation (15) will be equal to 0. And that neglects the contribution(s) of the special node(s), which is irrational in real-world decision making environment. Since the expert is invited to make a decision, we should not ignore his/her contribution even though he/she never known other experts.

In this paper, we assume that the higher the prestige of an expert the higher the weighting value of that expert in the aggregation utilizing IOWA operator, i.e., in Equation (15)

$\begin{matrix} C_{iD}^{″} (e_{σ (1)}) > C_{iD}^{″} (e_{σ (2)}) > \dots > C_{iD}^{″} (e_{σ (m)}) \\ \Rightarrow w_{1} > w_{2} > \dots > w_{m} \end{matrix}$ (18) Chiclana, Herrera-Viedma and Herrera [5] pointed out that the following BUM function Q (r) satisfies the above requirement. $Q (r) = r^{a}, a \in [0, 1]$ (19)

For simplicity, in this paper we let $a = \frac{1}{2}$ , i.e., $Q (r) = r^{1 / - 2}$ (20)

3.2.2 The properties about directed graph based group fuzzy preference

Although all the individual preference relations Q_k = (q_ijk) _n×n(k = 1, 2, …, m) are reciprocal, considering the directed social network connections between individual experts, we should investigate the reciprocity and consistency properties of DG-GFPR. Due to that group fuzzy preference relation Q aggregated by Equation (11) is reciprocal, so we would like to know if the social network connections impact the reciprocity and consistency properties of group fuzzy preference relation where we take the directed social network connections between individual experts into consideration.

It is clear that $\begin{matrix} q_{ij}^{DG - GFPR} = \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} = \sum_{k = 1}^{m} v_{k} (1 - q_{ji σ (k)}) \\ = \sum_{k = 1}^{m} v_{k} - \sum_{k = 1}^{m} v_{k} q_{ji σ (k)} = 1 - q_{ji}^{DG - GFPR} \end{matrix}$ (21)

If all the individual fuzzy preference relations are additively consistent, i.e., $q_{ijk} + q_{jlk} + q_{lik} = \frac{3}{2} \forall i, j, l \in {1, \dots, n} k \in {1, \dots, m}$ (22) And $q_{ij}^{DG - GFPR} = \sum_{k = 1}^{m} v_{k} q_{ij σ (k)}$ , then we get

$\begin{matrix} q_{ij}^{DG - GFPR} + q_{jl}^{DG - GFPR} + q_{li}^{DG - GFPR} \\ = \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} + \sum_{k = 1}^{m} v_{k} q_{jl σ (k)} + \sum_{k = 1}^{m} v_{k} q_{li σ (k)} \\ = \sum_{k = 1}^{m} v_{k} (q_{ij σ (k)} + q_{jl σ (k)} + q_{li σ (k)}) = \frac{3}{2} \sum_{k = 1}^{m} v_{k} = \frac{3}{2} \end{matrix}$ (23) i.e. DG-GFPR verifies reciprocity property. Thus, we get the following useful theorem.

Theorem 2.Let {Q₁, Q₂, … , Q_m } bemfuzzy preference relations, DG-GFPR $Q_{DG - GFPR} = {(q_{ij}^{DG - GFPR})}_{n \times n}$ is constructed by Equation (15). Then, DG-GFPR verifies reciprocity and additive consistency properties.

It is worth noting that the above proof of reciprocity and consistency of the collective fuzzy preference relation is based on the assumption that the order inducing values are unchanged.

However, the completely additive consistency do not always achieve in the real-world environment. That the preference relation satisfies the acceptable consistency expressed as Equation (12) is the most common case. In what follows, we investigate whether the DG-GFPR can maintain the acceptable consistency when all the individual preference relations with acceptable consistency.

Let are constructed by Equation (2) associated with DG-GFPR $Q_{DG - GFPR} = {(q_{ij}^{DG - GFPR})}_{n \times n}$ . Then, we have $\begin{matrix} | q_{ij}^{DG - GFPR} - g_{ij}^{DG - GFPR} | \\ = | \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} - \frac{1}{n} (\sum_{l = 1}^{n} q_{il}^{DG - GFPR} - \sum_{l = 1}^{n} q_{jl}^{DG - GFPR}) - 0.5 | \\ = | \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} - \frac{1}{n} (\sum_{l = 1}^{n} \sum_{s = 1}^{m} v_{s} q_{il σ (s)} - \sum_{l = 1}^{n} \sum_{t = 1}^{m} v_{t} q_{jl σ (t)}) - 0.5 | \\ = | \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} - \frac{1}{n} (\sum_{s = 1}^{m} v_{s} \sum_{l = 1}^{n} q_{il σ (s)} - \sum_{t = 1}^{m} v_{t} \sum_{l = 1}^{n} q_{jl σ (t)}) - 0.5 | \\ = | \sum_{k = 1}^{m} v_{k} q_{ij σ (k)} - \frac{1}{n} \sum_{z = 1}^{m} v_{z} (\sum_{l = 1}^{n} (q_{il σ (s)} + q_{jl σ (t)})) - 0.5 | \\ = | \sum_{k = 1}^{m} v_{k} (q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5) | \\ \leq \sum_{k = 1}^{m} v_{k} | q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5 | \end{matrix}$ (24)

Therefore, we get $\begin{matrix} CI (Q_{DG - GFPR}) \\ = 1 - \frac{1}{n (n - 2) / - 2} \sum_{i < j} | q_{ij}^{DG - GFPR} - g_{ij}^{DG - GFPR} | \\ = 1 - \frac{1}{n (n - 2) / - 2} \\ \sum_{i < j} | \sum_{k = 1}^{m} v_{k} (q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5) | \\ \geq 1 - \frac{1}{n (n - 2) / - 2} \\ \sum_{i < j} \sum_{k = 1}^{m} v_{k} | q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5 | \\ = 1 - \sum_{k = 1}^{m} v_{k} \frac{1}{n (n - 2) / - 2} \sum_{i < j} | q_{ij σ (k)} \\ - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5 | \\ \geq 1 - \sum_{k = 1}^{m} v_{k} max_{k} \\ {\frac{1}{n (n - 2) / - 2} \sum_{i < j} | q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5 |} \\ = 1 - max_{k} \\ {\frac{1}{n (n - 2) / - 2} \sum_{i < j} | q_{ij σ (k)} - \frac{1}{n} \sum_{l = 1}^{n} (q_{il σ (k)} + q_{jl σ (k)}) - 0.5 |} \\ = min_{k} {CI (Q_{k})} \end{matrix}$ (25) To summarize what we have discussed above, we get the following theorem.

Theorem 3.Let {Q₁, Q₂, … , Q_m } bemfuzzy preference relations, DG-GFPRQ_DG-GFPR = ${(q_{ij}^{DG - GFPR})}_{n \times n}$ is constructed by Equation (16). Then, $CI (Q_{DG - GFPR}) \geq min_{k \in {1, 2, \dots, m}} {CI (Q_{k})}$ (26)

The Theorem 3 shows that DG-GFPR Q_DG-GFPR $= {(q_{ij}^{DG - GFPR})}_{n \times n}$ maintains the acceptable consistency based on the assumption that all the individual preference relations satisfy the constraint of acceptable consistency. Thus, form the analysis above, we conclude that DG-GFPR verifies reciprocity and additive consistency properties. What’s more, DG-GFPR can maintain the acceptable consistency, which is an important issue to address decision making problem.

4 The solution process of GDM problem with directed social network connections

In this section, we develop a procedure of addressing GDM problem with fuzzy preference relations under framework of directed social network connections. And we use a numerical example to show its applicability and feasibility. Furthermore, we compare the numerical results with the case without considering social network connections and arrive at some useful conclusions.

4.1 The procedure of GDM problem with directed social network connections

The purpose of addressing GDM problem is to select the best alternative(s) or rank the alternatives. For simplicity in this paper, we utilize a simple and effective method introduced by Fedrizzi and Brunelli [8] to derive priority weight vector from additive fuzzy preference relation. Given H ={ h₁, h₂, …, h_n } is the priority weight vector corresponding to additive fuzzy preference relation P = (p_ij) _n×n, then priority weight vector can be computed as $h_{i} = \frac{2}{n} \sum_{j = 1}^{n} p_{ij}$ (27)

According to general process of addressing decision making problem, we can design the flow diagram in Fig. 2 with respect to the proposed addressing GDM problem with fuzzy preference relations under the framework of social network. In what follows, we show the calculation procedure corresponding to the flow diagram in Fig. 2 in detail.

Step 1. For a GDM problem with fuzzy preference relations under the framework of social network. Every individual expert d_k(k = 1, 2, …, m) construct his/her judgment matrix Q_k = (q_ijk) _n×n(k = 1, 2, …, m) denoting as fuzzy preference relation. The individual experts are with intrinsic weight vector Λ = (λ₁, λ₂, …, λ_m).

Step 2. Calculate the acceptable consistency indices CI (Q_k) by Equation (12) corresponding to individual fuzzy preference relation Q_k(k = 1, 2, …, m). Given a threshold value α, adjust the fuzzy preference relation that don’t achieve acceptable consistency. For simplicity, we assume the expressions about adjusted fuzzy preference relations don’t change.

Step 3. Obtain in-degree and out-degree centrality indices by Equations (10) and (9) associated with each expert. And all the indices can be denoted as $C_{iD}^{'} = (C_{iD}^{'} (e_{1}), C_{iD}^{'} (e_{2}), \dots, C_{iD}^{'} (e_{m}))$ , $C_{oD}^{'} = (C_{oD}^{'} (e_{1}), C_{oD}^{'} (e_{2}), \dots, C_{oD}^{'} (e_{m}))$ respectively. Then normalize them by Equation (14) gives rise to $C_{iD}^{″} = (C_{iD}^{″} (e_{1}), C_{iD}^{″} (e_{2}), \dots, C_{iD}^{″} (e_{m}))$ .

Step 4. Based on the $C_{iD}^{″}$ and $C_{oD}^{'}$ in Step 3, get the initial weight vector W = (w₁, w₂, …, w_m) associated with I-IOWA operator by Equations (6) and (20). Then we adjust W based on $C_{oD}^{'}$ to get the final weight vector associated with I-IOWA operator.

Step 5. Aggregate all the individual fuzzy preference relations Q_k = (q_ijk) _n×n (k = 1, 2, …, m) to obtain DG-GFPR Q_DG-GFPR by Equation (15).

Step 6. Make use of Equation (27) to rank all the alternatives.

Step 7. The end.

4.2 The advantages and characteristics of the proposed approach to GDM problem with directed social network connections

We take the directed social network connections among experts into consideration when we address GDM problem with preference relations under the framework of directed social network connections. Compared with the existing work, we can show the advantages of the proposed approach as follows.

Wu et al. [26] developed a social trust based consensus model for social network in a 2-tuple linguistic context for multiple criteria group decision making. A method based on trust propagation was proposed to obtain collective 2- tuple linguistic decision matrix. Subsequently, Wu et al. [27] developed a trust propagation based approach to build indirect trust relationship to determine experts’ weights. The information provided by individual experts are in terms of decision matrix in both literatures. In this paper, we use in-degree and out-degree centrality indices in SNA to build the social characteristics of the experts. Then we aggregate all the individual fuzzy preference relations with social indices to obtain a collective one. We address the group decision making with fuzzy preference relations where the experts are within directed social network connections. Wu and Chiclana [25] developed one fuzzy SNA methodology compute the trust degree of each expert, but didn’t investigate the reciprocity and additive consistency properties of the collective preference relation. Moreover, the acceptable consistency of the collective preference relation was not studied either by Wu and Chiclana [25]. Pérez et al. [18] presented three new social network analysis based IOWA operators that take advantage of the linguistic trustworthiness information gathered from the experts’ social network to aggregate the social group preferences. However, Pérez et al. [18] didn’t study the reciprocity and additive consistency properties and the acceptable consistency either. These properties are very important for collective preference relation. If the collective preference relation don’t satisfy the reciprocity and additive consistency properties, or the acceptable consistency, it is inappropriate to make decision based on it. In this paper, we prove that the collective preference relation meet the reciprocity and additive consistency properties and maintain acceptable consistency.

Furthermore, we can get the following characteristics about proposed approach to GDM problem with directed social network connections asfollows.

From the analysis in Section 3.1 and Section 3.2, we can arrive at twointeresting conclusions. (a) The social network connections do not impact the reciprocity and additive consistency properties associated with group fuzzy preference relations. (b) The social network connections do not change acceptable consistency property compared with the group fuzzy preference relations without considering social network connections.

When we utilize I-IOWA operator to obtain the group preference relation, we impose a special weight vector associated with I-IOWA operator to the experts based on their in-degree and out-degree centrality indices. It may change the obtained corresponding group preference relation compared with the one without considering social network connections. Therefore, that may probably obtain different outranking about alternatives. However, there is no enough evidence to show that the directed social network connections must change the final ranking result compared with the case without considering social network connections. Because the ranking result is with some special sensitivity. In special cases, if the directed social network connections cannot change the robustness condition with the corresponding ranking result, then the ranking result will remain unchanged. However the robustness/sensitivity condition is complex and various in different cases, so we cannot conclude it as an unified mathematicalform.

Different directed social network connections may probably achieve different ranking results for address group decision making problem with fuzzy preference relations. Because we may probably get different weight vector associated with I-IOWA operator according to different social network connections. That may obtain different group preferencerelations.

According to common sense, the expert whose prestige (in-degree centrality index) is higher should be imposed bigger weight associated with I-IOWA operator to him/her. In this paper, based on this idea, we enlarge the influence of expert with higher prestige(in-degree centrality index) towards the final ranking results compared with the case without considering social network connections. It is worth noting that we don’t consider the interpersonal influences among experts. Pérez et al. proposed a feasible approach to model influence in group decision making [17]. In [17], an influence model was applied to estimate the evolution of the experts’ opinions with regards to the other experts’ opinions based on Social Influence Network Theory [9]. This model took into account the relationship among experts to create a social network. Experts were supposed to discuss their opinion among them, and depending on the influence between them, their opinions are modified. The evolution of the experts’ opinions and predicting the solution of the group decision making problem can be inferred and studied through thismodel.

5 Numerical examples and comparative analysis

In this section, we use a numerical example to illustrate the feasibility and application of the proposed approach to address group decision making problems with fuzzy preference relations under the framework of directed social networks. We compare it with the case where all the experts are independent and arrive some useful conclusions.

5.1 Numerical example corresponding to the proposed approach to GDM problem with directed social network connections

In what follows, we apply a simplenumerical example to show the procedure of proposed approach addressing GDM problem with directed social network connections. Suppose that there are a committee of eight experts e₁, e₂, e₃, e₄, e₅, e₆, e₇ and e₈ intend to rank six alternatives Y₁, Y₂, Y₃, Y₄, Y₅ and Y₆. And the eight experts’ directed social network connections are shown in Fig. 3.

Step 1.Each expert provide his/her judgment information by fuzzypreference relations Q₁, Q₂, Q₃,Q₄, Q₅, Q₆, Q₇ and Q₈ as follows.

$\begin{matrix} Q_{1} = (\begin{matrix} 0.5 & 0.4 & 0.2 & 0.6 & 0.7 & 0.8 \\ 0.6 & 0.5 & 0.1 & 0.6 & 0.9 & 0.7 \\ 0.8 & 0.9 & 0.5 & 0.3 & 0.1 & 0.1 \\ 0.4 & 0.4 & 0.7 & 0.5 & 0.5 & 0.2 \\ 0.3 & 0.1 & 0.9 & 0.5 & 0.5 & 0.7 \\ 0.2 & 0.3 & 0.9 & 0.8 & 0.3 & 0.5 \end{matrix}) Q_{2} = (\begin{matrix} 0.5 & 0.3 & 0.3 & 0.5 & 0.6 & 0.6 \\ 0.7 & 0.5 & 0.4 & 0.7 & 0.2 & 0.3 \\ 0.7 & 0.6 & 0.5 & 0.5 & 0.4 & 0.2 \\ 0.5 & 0.3 & 0.5 & 0.5 & 0.6 & 0.7 \\ 0.4 & 0.8 & 0.6 & 0.4 & 0.5 & 0.4 \\ 0.4 & 0.7 & 0.8 & 0.3 & 0.6 & 0.5 \end{matrix}) \\ Q_{3} = (\begin{matrix} 0.5 & 0.6 & 0.6 & 0.6 & 0.1 & 0.4 \\ 0.4 & 0.5 & 0.3 & 0.6 & 0.3 & 0.6 \\ 0.4 & 0.7 & 0.5 & 0.6 & 0.1 & 0.6 \\ 0.4 & 0.4 & 0.4 & 0.5 & 0.7 & 0.6 \\ 0.9 & 0.7 & 0.9 & 0.3 & 0.5 & 0.2 \\ 0.6 & 0.4 & 0.4 & 0.4 & 0.8 & 0.5 \end{matrix}) Q_{4} = (\begin{matrix} 0.5 & 0.2 & 0.1 & 0.5 & 0.8 & 0.8 \\ 0.8 & 0.5 & 0.2 & 0.9 & 0.2 & 0.4 \\ 0.9 & 0.8 & 0.5 & 0.8 & 0.1 & 0.1 \\ 0.5 & 0.1 & 0.2 & 0.5 & 1.0 & 0.8 \\ 0.2 & 0.8 & 0.9 & 0 & 0.5 & 0.6 \\ 0.2 & 0.6 & 0.9 & 0.2 & 0.4 & 0.5 \end{matrix}) \\ Q_{5} = (\begin{matrix} 0.5 & 0.6 & 0.3 & 0.6 & 0.6 & 0.7 \\ 0.4 & 0.5 & 0.1 & 0.7 & 0.8 & 0.4 \\ 0.7 & 0.7 & 0.5 & 0.3 & 0.3 & 0.2 \\ 0.4 & 0.3 & 0.7 & 0.5 & 0.5 & 0.2 \\ 0.4 & 0.2 & 0.7 & 0.5 & 0.5 & 0.7 \\ 0.3 & 0.6 & 0.8 & 0.8 & 0.3 & 0.5 \end{matrix}) Q_{6} = (\begin{matrix} 0.5 & 0.3 & 0.1 & 0.5 & 0.7 & 0.6 \\ 0.7 & 0.5 & 0.4 & 0.7 & 0.2 & 0.4 \\ 0.9 & 0.6 & 0.5 & 0.5 & 0.4 & 0.2 \\ 0.5 & 0.3 & 0.5 & 0.5 & 0.6 & 0.7 \\ 0.3 & 0.8 & 0.6 & 0.4 & 0.5 & 0.4 \\ 0.4 & 0.6 & 0.8 & 0.3 & 0.6 & 0.5 \end{matrix}) \\ Q_{7} = (\begin{matrix} 0.5 & 0.7 & 0.4 & 0.6 & 0.2 & 0.6 \\ 0.3 & 0.5 & 0.3 & 0.7 & 0.3 & 0.8 \\ 0.6 & 0.7 & 0.5 & 0.6 & 0.1 & 0.6 \\ 0.4 & 0.7 & 0.4 & 0.5 & 0.7 & 0.6 \\ 0.8 & 0.7 & 0.9 & 0.3 & 0.5 & 0.2 \\ 0.2 & 0.4 & 0.4 & 0.4 & 0.8 & 0.5 \end{matrix}) Q_{8} = (\begin{matrix} 0.5 & 0.4 & 0.3 & 0.3 & 0.6 & 0.7 \\ 0.6 & 0.5 & 0.2 & 0.9 & 0.2 & 0.4 \\ 0.7 & 0.8 & 0.5 & 0.8 & 0.1 & 0.1 \\ 0.7 & 0.1 & 0.2 & 0.5 & 1.0 & 0.8 \\ 0.4 & 0.8 & 0.9 & 0.0 & 0.5 & 0.6 \\ 0.3 & 0.6 & 0.9 & 0.2 & 0.4 & 0.5 \end{matrix}) \end{matrix}$ For simplicity, we directly assume that the weight vector of eight experts is given by Λ = (λ₁, λ₂, ⋯ , λ₈) = (0.124, 0.123, 0.126, 0.125, 0.127, 0.122, 0.125, 0.128) .

Step 2. Using Equation (12), the initial acceptable consistency indices for each expert are computed as

CI (Q₁) = 0.9400, CI (Q₂) = 0.8533, CI (Q₃) = 0.8311, CI (Q₄) = 0.7089, CI (Q₅) = 0.8289, CI (Q₆) = 0.8400, CI (Q₇) = 0.8422, CI (Q₈) = 0.7511

The minimum threshold value α = 0.8 being fixed in advance, it can be seen that expert e₄ and e₈ need to revise their preferences. Using the method(Algorithm 1) developed in [28], controlled parameter being equal to 0.4, then the modified preference relations are determined denoted by and as follows, respectively. $\begin{matrix} Q_{4}^{′} = (\begin{matrix} 0.5000 & 0.3133 & 0.2400 & 0.4867 & 0.6733 & 0.6867 \\ 0.6867 & 0.5000 & 0.3067 & 0.7333 & 0.3200 & 0.4533 \\ 0.7600 & 0.6933 & 0.5000 & 0.6867 & 0.2700 & 0.2867 \\ 0.5133 & 0.2667 & 0.3133 & 0.5000 & 0.8067 & 0.7000 \\ 0.3267 & 0.6800 & 0.7267 & 0.1933 & 0.5000 & 0.5733 \\ 0.3133 & 0.5467 & 0.7133 & 0.3000 & 0.4267 & 0.5000 \end{matrix}) \\ Q_{8}^{′} = (\begin{matrix} 0.5000 & 0.4400 & 0.3667 & 0.3467 & 0.5333 & 0.6133 \\ 0.5600 & 0.5000 & 0.3067 & 0.7067 & 0.2933 & 0.4333 \\ 0.6333 & 0.6933 & 0.5000 & 0.6600 & 0.2467 & 0.2667 \\ 0.6533 & 0.2933 & 0.3400 & 0.5000 & 0.8067 & 0.7067 \\ 0.4667 & 0.7067 & 0.7533 & 0.1933 & 0.5000 & 0.5800 \\ 0.3867 & 0.5667 & 0.7333 & 0.2933 & 0.4200 & 0.5000 \end{matrix}) \end{matrix}$ The corresponding consistency indices are CI (Q₄) = 0.8253, and CI (Q₈) = 0.8507. Therefore, all the fuzzy preference relations achieve acceptable consistency.

Step 3. The in-degree and out-degree centrality indices with respect to the eight experts are shown as follows. $\begin{matrix} C_{iD} (e_{1}) = 2, C_{iD} (e_{2}) = 2, C_{iD} (e_{3}) = 0, C_{iD} (e_{4}) = 4, \\ C_{iD} (e_{5}) = 1, C_{iD} (e_{6}) = 3, C_{iD} (e_{7}) = 1, C_{iD} (e_{8}) = 2; \\ C_{oD} (e_{1}) = 2, C_{oD} (e_{2}) = 1, C_{oD} (e_{3}) = 0, C_{oD} (e_{4}) = 1, \\ C_{oD} (e_{5}) = 3, C_{oD} (e_{6}) = 1, C_{oD} (e_{7}) = 3, C_{oD} (e_{8}) = 3 . \end{matrix}$

Using Equation (14), we get $\begin{matrix} C_{iD}^{″} = (0.1255, 0.1255, 0.0943, 0.1670, 0.1088, \\ 0.1447, 0.1088, 0.1255) \end{matrix}$

Step 4. Then we can obtain the initial weight vector W = (w₁, w₂, ⋯ , w₈) associated with I- IOWA operator by Equations (6) and (20), i.e. $\begin{matrix} w_{k} = {(\frac{\sum_{i = 1}^{k} C_{iD}^{″} (e_{σ (i)})}{\sum_{i = 1}^{n} C_{iD}^{″} (e_{σ (i)})})}^{\frac{1}{2}} - {(\frac{\sum_{i = 1}^{k - 1} C_{iD}^{″} (e_{σ (i)})}{\sum_{i = 1}^{n} C_{iD}^{″} (e_{σ (i)})})}^{\frac{1}{2}} \\ = {(\sum_{i = 1}^{k} C_{iD}^{″} (e_{σ (i)}))}^{\frac{1}{2}} - {(\sum_{i = 1}^{k - 1} C_{iD}^{″} (e_{σ (i)}))}^{\frac{1}{2}}, \\ k = 1, 2, \dots, 8 \end{matrix}$

Therefore, we get $\begin{matrix} w_{1} & = & 0.4086, w_{2} = 0.1497, w_{3} = 0.1029, \\ w_{4} = 0.0889, \\ w_{5} & = & 0.0794, w_{6} = 0.0632, w_{7} = 0.0590, \\ w_{8} = 0.0483 \end{matrix}$ Because C_iD (e₅) = C_iD (e₇) = 1, C_oD (e₅) =C_oD (e₇) = 3, in addition φ^-1(e₅) = 6, φ^-1(e₇) = 7 or φ^-1(e₅) = 7, φ^-1(e₇) = 6, so we might let $w_{6}^{'} = w_{7}^{'} = \frac{w_{6} + w_{7}}{2} = 0.0611$ . Finally, the weight vector associated with I-IOWA operator are $W^{'} = (0.4086, 0.1497, 0.1029, 0.0889, 0.0794, 0.0611, 0.0611, 0.0483)$

Step 5. We can aggregate the individual fuzzy preference relations by Equation (15) to obtain DG-GFPR Q as follows. $Q = (\begin{matrix} 0.5000 & 0.3702 & 0.2540 & 0.4998 & 0.6169 & 0.6570 \\ 0.6298 & 0.5000 & 0.3006 & 0.7056 & 0.3579 & 0.4706 \\ 0.7508 & 0.6899 & 0.5000 & 0.5878 & 0.2725 & 0.2720 \\ 0.5002 & 0.3132 & 0.4122 & 0.5000 & 0.7144 & 0.6297 \\ 0.3831 & 0.6421 & 0.7275 & 0.2856 & 0.5000 & 0.5223 \\ 0.3336 & 0.5388 & 0.7280 & 0.3703 & 0.4777 & 0.5000 \end{matrix})$ Using Equation (12), the consistency index corresponding to group fuzzy preference relation can be computed as CI (Q) = 0.8587 > 0.8.

Step 6. We can get priority weight vector H ={ h₁, h₂, …, h_n } computed by Equation (27) with h₁ = 0.9659, h₂ = 0.9882, h₃ = 1.0227, h₄ = 1.0232, h₅ = 1.0202 and h₆ = 0.9828.

Therefore, the six alternatives are ranked as Y₄ ≻ Y₃ ≻ Y₅ ≻ Y₂ ≻ Y₆ ≻ Y₁.

5.2 Comparative analysis of the ranking alternatives about the proposed approach

We would like to further investigate whether the social network connections can impact the ranking results of alternatives. Therefore, in this subsection we compare two cases: the one considering directed graph based group fuzzy preference, the other one is the case without considering social network connections.

In order to do it, we assume all the eight experts are independent completely. Then we can obtain the group fuzzy preference relation by Equation (11) directly as follows. $\begin{matrix} Q = 0.124 \times Q_{1} + 0.123 \times Q_{2} + 0.126 \times Q_{3} + 0.125 \times Q_{4}^{'} \\ + 0.127 \times Q_{5} + 0.122 \times Q_{6} + 0.125 \times Q_{7} + 0.128 \times Q_{8}^{'} \end{matrix}$ We get $Q = (\begin{matrix} 0.5000 & 0.4579 & 0.3145 & 0.5289 & 0.5122 & 0.6248 \\ 0.5421 & 0.5000 & 0.2760 & 0.6800 & 0.4150 & 0.5110 \\ 0.6855 & 0.6986 & 0.5000 & 0.5187 & 0.2393 & 0.3074 \\ 0.4711 & 0.3700 & 0.4813 & 0.5000 & 0.6523 & 0.5503 \\ 0.4878 & 0.5850 & 0.7607 & 0.3477 & 0.5000 & 0.4698 \\ 0.3502 & 0.5140 & 0.6926 & 0.4497 & 0.5302 & 0.5000 \end{matrix})$

Using Equation (12), the consistency index corresponding to group fuzzy preference relation can be computed as CI (Q) = 0.8907 > 0.8. Then, we can get priority weight vector H ={ h₁, h₂, …, h_n } computed by Equation (27) with h₁ = 0.9795, h₂ = 0.9747, h₃ = 0.9832, h₄ = 1.0083, h₅ = 1.0503 and h₆ = 1.0122.

Therefore, the six alternatives are ranked as $Y_{5} ≻ Y_{6} ≻ Y_{4} ≻ Y_{3} ≻ Y_{1} ≻ Y_{2}$

We show the corresponding compared results in Table 2. And in order to further show the directed social network connections’ influence towards ranking result, we consider other social network connections shown as the third row and first column in Table 2. This social network connections are changed partially compared with the social network connections in Section 5. Without loss of generality, we change it in the following three areas. First, the direction of social network connection between e₄ and e₇ changes. Second, we add one directed graphical social network connection from e₁ to e₃. Third, we remove the directed graphical social network connection from e₅ to e₆.

Combining the corresponding comparative results in Table 2, the computational procedures in Section 5.1 and Section 5.2 and the discussion in Section 4.2, we can arrive at the following points.

The directed social network connections can impact the final ranking result about alternatives with fuzzy preference relations. However, we try to perform different social network connections compared with the case in Fig. 3, then we find that the ranking results usually but not always change in most cases. When we take numerous different social network connections into consideration, we find that in most cases the directed social network connections can change the final ranking result to some extent.

Analyzing the Table 2, we can find all the corresponding group fuzzy preference relations are different. Therefore, we may usually obtain different ranking results based them. However all the corresponding group fuzzy preference relations can maintain reciprocity property, acceptable consistency completely, which verifies the Theorem 1 and the Theorem 3.

Form analyzing the computational procedure in Section 5 and Section 5.2, we find that the directed social network connections can impact the experts’ weights/importances when we aggregate the individual fuzzy preference relations utilizing I-IOWA operator.

Furthermore, we take undirected social network connections into consideration in practice, which can be treated a special social network connections with in-degree centrality index being equal to 0. We find that this type social network connections can impact the outranking alternatives as well and the expert with bigger centrality index will further impact the final outranking alternatives.

All in all, the directed social network connections can usually but not always change the ranking result compared with the case without considering social network connections.

6 Conclusion and future research

In this paper, social network analysis based approach to group decision making problem with fuzzy preference relations has been investigated. We have defined the group fuzzy preference relation corresponding to the group decision making (GDM) problem under the framework of directed social network connections, which is called directed graph based group fuzzy preference (DG-GFPR). Furthermore, we have proved the reciprocity, additive consistency and acceptable consistency properties associated with directed graph based group fuzzy preference (DG-GFPR). We have developed a procedure for addressing group decision making (GDM) problem under the framework of directed social network connections with fuzzy preference relations and we use a numerical example to show its applicability and feasibility. Specially, we have compared the representative results about two cases with numerical examples: the case considering directed social network connections, the other case without considering any social network connections. We have analyzed the numerical comparison results and given some useful tips and have further discussedthem.

In further research, we would like to develop a similar work based on the use of the multiplicative consistency as characterized in [4]. We will intend to address group decision making (GDM) problem under directed valued social network connections with fuzzy preference relations, and we would like to investigate the consensus issue associated with those problem.

Furthermore, we expect to develop one feasible approach based on evolutionary game theory to achieve acceptable consensus index considering directed social network connections.

Footnotes

Acknowledgments

We would like to appreciate constructive comments and valuable suggestions from the anonymous referees, which have helped us efficiently improve the quality of this paper. This work was supported by the Scientific Research and Innovation Project for College Graduates of Jiangsu Province (KYLX-0209) and National Natural Science Foundation of China (NSFC) (71171048 and 71371049).

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