Hesitant fuzzy multi-attribute group decision making method based on weighted power operators in social network and their application

Abstract

This paper aims to put forward a hesitant fuzzy multi-attribute group decision making (MAGDM) method based on the weighted power aggregation operators in social network. From the point of view of social network analysis, decision makers (DMs) are interconnected in the process of MAGDM. Furthermore, the dimension of the obtained hesitant fuzzy element (HFE) by original power operators will be greater with the increasing number of attributes and alternatives and DMs, which will lead to the problem of “intermediate expression swell". This paper combines the order operation laws with the power operators to redefine two novel hesitant fuzzy power aggregation operators to simplify the involved calculation and explore new operators’ properties. Meanwhile, when two given elements have different number of values, we use the strength of social ties and social influence to develop an algorithm for extending the HFEs objectively. On the other hand, the PageRank algorithm and the deviation method are used to determine DMs’ combined weights. The feasibility of the proposed hesitant fuzzy MAGDM method based on social network is illustrated by the application to the actual issue of decision making and the comparative analysis with the existing method.

Keywords

Group decision making hesitant fuzzy set social network power operator PageRank algorithm

1 Introduction

With the increasing complexity and uncertainty of the environment, multi-attribute group decision making (MAGDM) has attracted more attention [1], because a single decision maker (DM) can not comprehensively consider in the decision making process. The MAGDM is termed as the decision making situation in which numbers of DMs are employed to find a common solution on a multi-attribute problem among a set of feasible alternatives according to their preferences. However,concerning each attribute in the real world, it is much more difficult for DMs to give precise evaluation values of the alternatives. In order to further deal with the vagueness and hesitancy of decision making problems, the theory of fuzzy set originally introduced by Zadeh [2] has been developed into several different forms. One extension of the fuzzy set is called the hesitant fuzzy set (HFS), which was developed to handle hesitant and absonant problems [5 –7] by Torra and Narukawa [3] and Torra [4]. In HFSs, the element’s membership degree is permitted to a set of some possible real values between 0 and 1 [3, 4]. Under hesitant fuzzy environment, a large number of studies on methods and applications of MAGDM problems have gained a lot of attention [8 –15].

In many studies, the aggregation operator has always held a pivotal role as an important information fusion tool. A significant number of aggregation operators have been designed to fuse the fuzzy decision information in various situations over the past decades. Among them, the power average (PA) operator and the power geometric (PG) operator are two popular aggregation operators, both of which can eliminate the influence of extreme evaluation values from the bias DMs. The PA operator first proposed by Yager [16] was a very useful technique to aggregate the correlated data, and the latter one was proposed by Xu and Yager [17]. To accommodate the hesitant fuzzy information, both the PA operator and the PG operator have been extended to a series of hesitant fuzzy power aggregation operators by Zhang [18], which allowed the aggregated values to support and reinforce each other. However, these operators can substantially increase the dimensions of the derived hesitant fuzzy elements (HFEs), which usually result in higher calculations. Let h₁, h₂, h₃ and h₄ be four HFEs, and they contain 4, 5, 5 and 4 values, respectively. When one of the aggregation operators [18] is used for these four HFEs, the obtained element contains 400 values. In the meantime, with the increasing number of attributes and alternatives and the DMs, the dimensions of the obtained elements will be greater, and the amount of calculation will increase.

We notice that the length of HFE can not be ignored. They always have different numbers of values for different HFEs. In most cases, the shorter HFEs should be extended [6, 19]. Generally speaking, adding the maximum or minimum value based on the DMs’ risk preference is commonly used [6]. The optimist may add the maximum values, on the contrary the pessimist may add the minimum values. For example, if we have two HFEs h₁ = {0.5, 0.3, 0.2} and h₂ = {0.6, 0.4}, then the optimist may extend h₂ as h₂ = {0.6, 0.6, 0.4}, while the pessimist may extend it as h₂ = {0.6, 0.4, 0.4}. However, it is very hard to judge and determine DMs’ risk altitude and the degree of DMs’ risk preference.

At the same time, most group decision making methods generally assume that DMs do not interact with each other, that is, the social ties and social influence among DMs are not taken into consideration by these methods. Actually, Pérez and his coworkers [20] claimed that there were always some relationships between DMs. Social Network Analysis (SNA) [21, 22] just studied the relationship between all members of the social network. People always prefer to share their idea with others who have strong ties with them based on SNA; meanwhile, their stronger social ties have, more impact on each other [23]. A social evaluation approach to aid decision-making of individuals was proposed in 2014through online social network [23]. Quijano-Sánchez et al. [24] developed a group recommendation application according to social interaction information in online social networks. A novel model and and approximate final solution to group decision making problem based on social influence was presented [25]. Qian et al. [26] utilized the strength of social ties and social influence to complete the incomplete preference relations based on additive consistency. The HybridRank algorithm was used in online social networks to detect opinion leaders who had a significant influence, and related applications were discussed [27]. Akkuzu et al. [28] presented a decision framework focused on a fuzzy logic study of social networks and used data sensitivity values and trust values (confidence values) for group decision making. In [29], a framework based on SAS was suggested to effectively deal with multi-attribute social networks matching with the unknown relative weights of influencers. Therefore, social ties and social influence should also not be overlooked and play an important part in managing real-life decision making problems.

Consequently, the MAGDM problems in which the values of attributes are HFEs in the social network will be addressed. Two novel hesitant fuzzy power aggregation operators are developed to simplify the complexity of the computation in this paper. In the meantime, the properties and relationship of the proposed operators are discussed. Then, we employ social ties and social influence between DMs to construct an algorithm to extend the HFEs, which have fewer number of values. In reality, when a DM has a strong connection with another DM, the DM is more influenced by him or her. Based on this idea, we get the shorter HFE risk preference degree of and extend it until they have the same length. In addition, due to the DMs’ difference in individual preferences, knowledge structure and so on, DMs usually have various views and give different evaluation values for the same problem. So, we should not disregard the degree of significance of each DM. Inspired by the PageRank algorithm, we can create a directed adjacency matrix based on the DMs’ social network to deduce the first weight information of DMs. Simultaneously, the second weight information of the DMs can be obtained using the deviation method based on the hesitant fuzzy decision information themselves. Combine these two types of weight information in a linear way. The combination values are taken as the final weights of the DMs. Derived weight information is more comprehensive because it takes into account both the network structure of the DMs and the decision evaluation values. On the basis of the concepts above, we propose an approach to address hesitant fuzzy MAGDM problems in the social network.

The rest of this paper is structured as follows. Section 2 briefly reviews some basic definitions, concepts and theories regarding HFSs and social networks. We propose two novel hesitant fuzzy power aggregation operators and discuss their significant properties in Section 3. Section 4 consistes of two parts. Based on social ties and social influence, an algorithm is given to extend the shorter HFEs. An enhanced hesitant fuzzy MAGDM method based on the aforementioned operators and extension algorithm, is generated after deriving weights of DMs by the PageRank algorithm and deviation method. To validate the proposed method, an illustrative example is provided to demonstrate the applicability in Section 5. The paper finishes with some conclusions in Section 6.

2 Preliminaries

In this paper, we consider a set of DMs, denoted by E = {e₁, e₂, ⋯ , e_f} (f ≥ 2) and assume that the DMs’ social network is directed. They provide their alternative evaluation values with respect to the attributes using HFEs.

This section starts with some preliminaries on HFSs, including definitions, the operational laws, the comparison rule and so on. Next, we briefly represent and redefine some concepts in social network.

2.1 Hesitant fuzzy sets

Definition 1. [3, 4] X is a fixed set, a HFS on X is in terms of a function h that returns a subset of [0, 1] when applied to X. The HFS can be defined as follows: $E = {< x, h (x) > | x \in X},$ (2.1) where h (x) is a set of values in [0, 1] and it denotes the possible membership degrees of the element x ∈ X to the set E.

Generally, h (x) is called hesitant fuzzy element (HFE), denoted as h. l_h is the length of h, which is the number of values in h.

Subsequently, Xia and Xu [30] gave the following operations and comparison rule on HFEs.

Definition 2. [30] Give three HFEs h, h₁ and h₂, there are some operations as:

(1) h^λ = ∪_γ ∈ h {γ^λ};

(2) λh = ∪_γ ∈ h {1 - (1 - γ) ^λ};

(3) h₁ ⊕ h₂ = ∪_{γ₁ ∈ h₁, γ₂ ∈ h₂} {γ₁ + γ₂ - γ₁γ₂};

(4) h₁ ⊗ h₂ = ∪_{γ₁ ∈ h₁, γ₂ ∈ h₂} {γ₁γ₂} .

Definition 3. (Comparison rule)[30] For a HFE h, the score function of h is defined as s (h) = ∑_γ∈hγ/l_h, where l_h is the number of values in h. For two HFEs h₁ and h₂, if s (h₁) > s (h₂), then h₁ is superior to h₂ (denoted by h₁ ≻ h₂); if s (h₁) = s (h₂), we say h₁ is indifferent to h₂ (denoted by h₁ ∼ h₂).

Recently, Liao [12] gave other operations with the following forms:

Definition 4. [12] Give three HFEs h, h₁ and h₂ with same length l_h = l_{h
₁} = l_{h
₂} = l, then some operational laws are as follows:

(1) h^λ = ∪_{γ^σ(t) ∈ h} {(γ^σ(t)) ^λ} = {(γ^σ(t)) ^λ, t = 1, 2, ⋯ , l} ;

(2) λh = ∪_{γ^σ(t) ∈ h} {1 - (1 - γ^σ(t)) ^λ} = {1 - (1 - γ^σ(t)) ^λ, t = 1, 2, ⋯ , l} ;

$(3) h_{1} \oplus h_{2} = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}}{\cup} {γ_{1}^{σ (t)} + γ_{2}^{σ (t)} - γ_{1}^{σ (t)} γ_{2}^{σ (t)}} = {γ_{1}^{σ (t)} + γ_{2}^{σ (t)} - γ_{1}^{σ (t)} γ_{2}^{σ (t)}, t = 1, 2, \dots, l};$

$(4) h_{1} \otimes h_{2} = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}}{\cup} {γ_{1}^{σ (t)} γ_{2}^{σ (t)}} = {γ_{1}^{σ (t)} γ_{2}^{σ (t)}, t = 1, 2, \dots, l},$ where $γ_{i}^{σ (t)}$ is the tth largest value in h_i, and λ > 0.

For two HFEs h₁ and h₂, they always have different numbers of values, namely l_{h
₁} ≠ l_{h
₂}. In most cases, the element with fewer number of values should be extended until they have the same length. An extension method for HFEs with parameter was proposed by Xu and Hang [19], which sonsiders all possible risk preferences of DMs by modifying or adjusting the parameters.

Definition 5. [19] For a HFE h = {r^σ(t)|t = 1, 2, ⋯ , l_h}, the extension value is defined as $h^{*} = λ h^{+} + (1 - λ) h^{-},$ where r^σ(t) (t = 1, 2, ⋯ , l_h) is the tth largest value in h, h⁺ and h^- denote the maximum and minimum values in the HFE h, respectively. λ (0 ≤ λ ≤ 1) is the parameter determined by the risk preferences of DMs.

For the sake of simplicity, we take l = max {l_{h
₁}, l_{h
₂}}. The hesitant Hamming distance developed by Xu and Xia [31] was defined as below, which will be used thereafter. $d (h_{1}, h_{2}) = \frac{1}{l} \sum_{i = 1}^{l} | γ_{1}^{σ (t)} - γ_{2}^{σ (t)} |,$ (2.2) where $γ_{1}^{σ (t)}$ and $γ_{2}^{σ (t)}$ are the tth largest values in h₁ and h₂, respectively.

In order to derive the weight information from hesitant fuzzy information, Zhan, et al. [32] gave the definition of the mean of a collection of the HFEs h_i (i = 1, 2, ⋯ , n) as follows:

Definition 6. Let $h_{i} = {γ_{i}^{σ (t)} | t = 1, 2, \dots, l} (i = 1, 2, \dots, n)$ be a collection of the HFEs with same length l, then the mean of these HFEs $\bar{h} = {{\bar{γ}}_{i}^{σ (t)} | t = 1, 2, \dots, l}$ is defined as follows: ${\bar{γ}}_{i}^{σ (t)} = \frac{1}{n} \sum_{i = 1}^{n} γ_{i}^{σ (t)},$ where $γ_{i}^{σ (t)}$ is the tth largest values in h_i.

2.2 Social network

The SNA method is an approach of quantitative analysis that integrate various disciplines such as complex network theory, graph theory and mathematical method. It analyzes the relationships between individuals or groups to identify the rules. With the development of network technology, social relationships will be formed among DMs in group decision making with certain characteristics such as trust relationships, influence relationships and cognition relationships. The social relationships between the DMs and the structural features of the social network in which they are situated would also affect the results of group decision making. Subsequently, this paper intends to construct an appropriate social network based on the social relationships between DMs for MAGDM problems.

Let G = (E, L) be a directed graph, E = {e₁, e₂, ⋯ , e_f} be the set of DMs and L = {l₁, l₂, ⋯ , l_n} be the set of directed arcs between pairs of DMs. Based on relationships between DMs, a directed adjacency matrix can be obtained, which is defined as follows: $AM = {\begin{matrix} a_{ij} = 1, & the edge exists from e_{i} to e_{j}; \\ a_{ij} = 0, & otherwise . \end{matrix}$ (2.3)

Note that if there is at least one path connection from e_i to e_j, we call e_i and e_j the information seeker and information provider, respectively.

The concept of “network” is often seen in several fields. In the subway rail transit diagram, the stations are considered to be nodes. If one station can reach another station, there is a path between the two stations, then a “rail transit network diagram” can be constructed. Similarly, considering the website as a node in the site group, the relationship between the sites can be used to construct a directed adjacency matrix and determine the importance degrees of these sites, i.e., PageRank (PR) algorithm. The PR algorithm was put forward by Larry Page and Sergey Brin [33] to rank websites. Based on directed adjacency matrix, we will obtain the degree of importance of each DM according to the PR algorithm that will be used afterward. The PR algorithm can be represented as below:

Definition 7. (PageRank algorithm) The PR value of node e_i at t + 1 step is ${PR}_{i} (t + 1) = \sum_{j = 1}^{f} a_{ji} \frac{{PR}_{j} (t)}{d^{+} (e_{j})} .$ (2.4)

Here d⁺ (e_j) denotes the out-degree centrality of e_j, which is described as follows $d^{+} (e_{j}) = \sum_{i = 1}^{f} a_{ji},$ (2.5)f is the total number of nodes.

Most group decision making methods always assume that DMs are independent, which implies that DMs do not interact with each other. Generally, there are some links between the DMs [20]. In order to study the relationships between DMs, we introduce and define some basic concepts in social network.

Definition 8. [21, 23] Social connection strength between e_i and e_j is given as follows: ${SC}_{ij} = \frac{n_{ij}}{(d^{+} (e_{i}) - 1) + (d^{+} (e_{j}) - 1) - n_{ij}} .$ (2.6)

Here d⁺ (e_i) and d⁺ (e_j) denote the out-degree centrality of e_i and e_j, respectively. n_ij is the number of common social connections (common acquaintances) between e_i and e_j.

Remark 2.1. The more common social connections of e_i and e_j are, the stronger the social tie between e_i and e_j is. It indicates that there is a stronger tie between two social units if most of their friends overlap.

Meanwhile, the stronger social relationships between DMs always lead to higher degree of opinion acceptance with each other. So we have the following definition.

Definition 9. Let Nei_r = {e_t|d (e_r, e_t) =1}, the direct degree of opinion acceptance from e_r to e_t is defined as: $DOA (e_{t} | e_{r}) = \frac{{SC}_{rt}}{\sum_{e_{l} \in {Nei}_{r}} {SC}_{rl}} .$ (2.7)

In social network, there is always at least one path connecting the information seeker and the information provider. If e_r goes to e_t along path q (i.e., $e_{r} = e_{r 1}^{q} \to e_{r 2}^{q} \to, \dots, \to e_{{rn}_{q}}^{q} = e_{t}$ , $e_{ri}^{q} \neq e_{rj}^{q} (i, j = 1, 2, \dots, n_{q}, i \neq j)$ ), the degree of opinion acceptance from e_r to e_t is related with the number of node n_q and the direct degree of opinion acceptance from $e_{ri}^{q}$ to $e_{r (i + 1)}^{q}$ . Meanwhile, the degree of opinion acceptance declines during the propagation process, when e_r reaches e_t through n_q nodes. Thus, we give the following definition about the degree of opinion acceptance from e_r to e_t based on the given path q.

Definition 10. Let Nei_r = {e_t|d (e_r, e_t) =1}, if e_r also goes to e_t along path q_k (k = 1, 2, ⋯ , m) (i.e., $e_{r} = e_{r 1}^{q_{k}} \to e_{e 2}^{q_{k}} \to, \dots, \to e_{{rn}_{q_{k}}}^{q_{k}} = e_{t}$ , $e_{ri}^{q_{k}} \neq e_{rj}^{q_{k}}, i, j = 1, 2, \dots, n_{q_{k}}, i \neq j$ ), the degree of opinion acceptance from e_r to e_t is defined as $OA (e_{t} | e_{r}) = \sum_{k = 1}^{m} {OA}^{q_{k}} (e_{t} | e_{r}),$ (2.8) where ${OA}^{q_{k}} (e_{t} | e_{r}) = \prod_{i = 1}^{n - 1} DOA (e_{r (i + 1)}^{q_{k}} | e_{ri}^{q_{k}})$ (2.9) is the degree of opinion acceptance from e_r to e_t along path q_k.

In social network, an information seeker e_r has more than one information providers. Due to different relationship with e_r, these information providers provide different level of information. That is to say, their importance degrees are different for e_r.

Definition 11. Let M = {e_r|r ∈ {1, 2, ⋯ , f}} be the set of DMs who are information seekers, and $D_{ij}^{r} = {e_{t} | t \in {1, 2, \dots, m, t \neq r}}$ be the set of DMs who are information providers for e_r. The importance degree of each DM $e_{t} \in D_{ij}^{r}$ to aid e_r is defined as follows $δ_{rt} = \frac{OA (e_{t} | e_{r})}{\sum_{e_{l} \in D_{ij}^{r}} OA (e_{l} | e_{r})} .$ (2.10)

3 Novel hesitant fuzzy power aggregation operators

How to aggregate evaluation information of DMs on alternatives is an essential part of the decision making process. One of the most useful tools to aggregate information is aggregation operator, which can combine DMs’ information and derive collective values for each alternative. In 2001, the power average (PA) operator, a nonlinear weighted average aggregation operator, was introduced by Yager [16], allowing exact arguments to support each other in the aggregation process. In 2008, Xu and Yager [17] developed the power geometric (PG) operator based on PA operator and geometric mean. It can be regarded as a nonlinear weighted geometric aggregation operator. Associated with the studies of the HFEs, Zhang developed the PA operator and PG operator into weighted hesitant fuzzy power average (WHFPA) operator and weighted hesitant fuzzy power geometric (WHFPG) operator [18], respectively. The definitions are as follows:

Definition 12. [18] For a given set of HFEs {h₁, h₂, ⋯ , h_n}, the standardized importance degree is ω = (ω₁, ω₂, ⋯ , ω_n) ^T. Then, the WHFPA operator and WHFPG operator are defined respectively: $\begin{matrix} WHFPA (h_{1}, h_{2}, \dots \dots, h_{n}) = {oplus}_{i = 1}^{n} \frac{ω_{i} (1 + T (h_{i})) h_{i}}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))} \\ = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots \dots, γ_{n} \in h_{n}}{\cup} {1 - \prod_{i = 1}^{n} (1 - γ_{i})^{\frac{ω_{i} (1 + T (h_{i}))}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))}}}, \end{matrix}$ (3.1)

$\begin{matrix} WHFPG (h_{1}, h_{2}, \dots \dots, h_{n}) = ⨂_{i = 1}^{n} \frac{ω_{i} (1 + T (h_{i})) h_{i}}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))} \\ = \underset{γ_{1} \in h_{1}, γ_{2} \in h_{2}, \dots \dots, γ_{n} \in h_{n}}{\cup} {\prod_{i = 1}^{n} (γ_{i})^{\frac{ω_{i} (1 + T (h_{i}))}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))}}}, \end{matrix}$ (3.2) where T (a_i) is given by $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} ω_{j} Sup (h_{i}, h_{j}), \sum_{i = 1}^{n} ω_{i} = 1,$ $ω_{i} \in [0, 1] (i = 1, 2, \dots, n),$ with Sup (h_i, h_j) =1 - d (h_i, h_j) being the support of h_i obtained from h_j, and d is a distance measure.

3.1 Two novel hesitant fuzzy power aggregation operators

Both the WHFPA operator and the WHFPG operator consider the relationship between evaluation values during the information fusion. However they will both increase the dimensions of the derived HFEs in the process of calculation. Motivated by operation laws given in Definition 4, the above aggregation operators can be adapted to the following forms.

Definition 13. Let h_i (i = 1, 2, ⋯ , n) be a collection of HFEs. Assume h_i (i = 1, 2, ⋯ , n) have the same length, the standardized importance degree is ω = (ω₁, ω₂, ⋯ , ω_n) ^T. We define the novel hesitant fuzzy power average (NHFPA) operator as follows: $\begin{matrix} NHFPA (h_{1}, h_{2}, \dots, h_{n}) = \frac{{oplus}_{i = 1}^{n} ω_{i} (1 + T (h_{i})) h_{i}}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))} \\ = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}, \dots \dots, γ_{n}^{σ (t)} \in h_{n}}{\cup} \\ {1 - \prod_{i = 1}^{n} (1 - γ_{i}^{σ (t)})^{\frac{ω_{i} (1 + T (h_{i}))}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))}}}, \end{matrix}$ (3.3) where $γ_{i}^{σ (t)}$ is the tth largest value in h_i and $\sum_{i = 1}^{n} ω_{i} = 1, ω_{i} \in [0, 1], i = 1, 2, \dots, n$ .

Similarly, we can define another new operator, that is, the novel hesitant fuzzy power geometric (NHFPG) operator.

Definition 14. Let h_i (i = 1, 2, ⋯ , n) be a collection of HFEs. Assume h_i (i = 1, 2, ⋯ , n) have the same length, the standardized importance degree is ω = (ω₁, ω₂, ⋯ , ω_n) ^T. Then, the NHFPG operator is defined as the following form: $\begin{matrix} NHFPG (h_{1}, h_{2}, \dots, h_{n}) = \frac{⨂_{i = 1}^{n} ω_{i} (1 + T (h_{i})) h_{i}}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))} \\ = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}, \dots \dots, γ_{n}^{σ (t)} \in h_{n}}{\cup} {\prod_{i = 1}^{n} (γ_{i}^{σ (t)})^{\frac{ω_{i} (1 + T (h_{i}))}{\sum_{i = 1}^{n} ω_{i} (1 + T (h_{i}))}}}, \end{matrix}$ (3.4) where $γ_{i}^{σ (t)}$ is the tth largest value in h_i and $\sum_{i = 1}^{n} ω_{i} = 1, ω_{i} \in [0, 1], i = 1, 2, \dots, n$ .

It is easily seen that the involved calculation will be simplified greatly when we adopt the above two new operators. Here let h_i (i = 1, 2, ⋯ , n) be a collection of HFEs, and denote the length of h_i as l_{h
_i}. If we adopt the WHFPA operator or WHFPG operator, the dimension of the derived HFEs is l_{h
₁} × l_{h
₂} × ⋯ × l_{h
_n} [12]. If we use the new operator given by Equation (3.3) or Equation (3.4) after extending elements which have fewer values [6], the dimension of obtained HFEs is max_i (l_{h
_i}), which is smaller than l_{h
₁} × l_{h
₂} × ⋯ × l_{h
_n} obviously.

Example 1. Let h₁ = {0.1, 0.2}, h₂ = {0.2, 0.4} and h₃ = {0.7, 0.9} be three HFEs with the same important degree. Suppose that d (h_i, h_j) (i, j = 1, 2, 3, i ≠ j) is the hesitant Hamming distance between h_i and h_j, we can get the following results from Equations (3.1)-(3.4): $WHFPA (h_{1}, h_{2}, h_{3}) = {0.3641, 0.4005, 0.4007,$ $0.4350, 0.5397, 0.5660, 0.5662, 0.5910},$ $WHFPG (h_{1}, h_{2}, h_{3}) = {0.2044, 0.2201, 0.2358,$ $0.2539, 0.2891, 0.3113, 0.3334, 0.3590},$ $NHFPA (h_{1}, h_{2}, h_{3}) = {0.3641, 0.5910},$ $NHFPG (h_{1}, h_{2}, h_{3}) = {0.2044, 0.3590} .$

3.2 Properties of two novel operators

The WHFPA operator and WHFPG operator are neither idempotent, bounded, nor monotonic with respect to the input arguments [18]. With respect to the NHFPA operator and the NHFPG operator, we have the following properties.

Theorem 1. (Idempotency) Let h_i (i = 1, 2, ⋯ , n) be a collection of HFEs with the same length l_{h
_i} = l. If h_i = h for all i, we have $NHFPA (h_{1}, h_{2}, \dots, h_{n}) = h .$ (3.5)

Proof. The proof of Theorem 1 is shown in Appendix A.

Theorem 2. (Boundedness) Let h_i (i = 1, 2, ⋯ , n) be a collection of HFEs with the same length l_{h
_i} = l, and $H^{-} = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}, \dots, γ_{n}^{σ (t)} \in h_{n}}{\cup} {min_{i} (γ_{i}^{σ (t)})},$ (3.6) $H^{+} = \underset{γ_{1}^{σ (t)} \in h_{1}, γ_{2}^{σ (t)} \in h_{2}, \dots, γ_{n}^{σ (t)} \in h_{n}}{\cup} {max_{i} (γ_{i}^{σ (t)})},$ (3.7) then we have $H^{-} \leq NHFPA (h_{1}, h_{2}, \dots, h_{n}) \leq H^{+} .$ (3.8)

Proof. The proof of Theorem 2 is shown in Appendix B.□

However, the NHFPA operator is not monotonic, as illustrated by the following example.

Example 2. Let h₁ = {0.8, 0.6}, h₂ = {0.7, 0.7}, h₃ = {0.9, 0.4}, h₄ = {0.7, 0.5} be four HFEs. We assume that the important degrees of arguments are equal, and d (h_i, h_j) (i, j = 1, 2, 3, 4, i ≠ j) is the hesitant Hamming distance between h_i and h_j. Based on the Definition 13, we have NHFPA (h₁, h₁, h₁) = {0.8, 0.6} ,

NHFPA (h₁, h₂, h₃) = {0.8174, 0.5851} ,

NHFPA (h₁, h₂, h₄) = {0.7379, 0.6085} .

By the Definition 3, one can calculate that s (h₁) = s (h₂) =0.7, s (h₃) =0.65, s (h₄) =0.6,

s (NHFPA (h₁, h₁, h₁)) =0.7, s (NHFPA (h₁, h₂, h₃)) =0.7012, s (NHFPA (h₁, h₂, h₄)) =0.6727.

Thus, we obtain s (NHFPA (h₁, h₁, h₁)) < s (NHFPA (h₁, h₂, h₃)) ,

s (NHFPA (h₁, h₁, h₁)) > s (NHFPA (h₁, h₂, h₄)), and NHFPA (h₁, h₁, h₁) ≺ NHFPA (h₁, h₂, h₃),

NHFPA (h₁, h₁, h₁) ≻ NHFPA (h₁, h₂, h₄).

Therefore, it is concluded that the NHFPA operator is not monotonic.

Performing the similar analysis for the NHFPG operator, one can prove that the NHFPG operator is idempotent and bounded, but it is not monotonic. Meanwhile, there is also a relationship between these two new operators.

Theorem 3. Suppose that h_i (i = 1, 2, ⋯ , n) is a collection of HFEs with the same length l_{h
_i} = l. Then we have $NHFPG (h_{1}, h_{2}, \dots, h_{n}) \leq NHFPA (h_{1}, h_{2}, \dots, h_{n}) .$

Proof. The proof of Theorem 3 is shown in Appendix C.□

4 Hesitant fuzzy MAGDM in social network

This section introduces a new method based on social network to solve the MAGDM problem under hesitant fuzzy environment. The method not only uses the novel aggregation operators to minimize the computational complexity, but also considers the relationship between DMs in the decision making process. At first, we give the following description of the hesitant fuzzy MAGDM problem in the social network.

Let E = {e₁, e₂, ⋯ , e_f} (f ≥ 2) be a set of DMs that have weight vector ω = (ω₁, ω₂, ⋯ , ω_f) ^T with ω_k ∈ [0, 1] (k = 1, 2, ⋯ , f) and $\sum_{k = 1}^{f} ω_{k} = 1$ in social network. In the graph, each DM is denoted as a node. Suppose A = {A₁, A₂, ⋯ , A_m} is the set of m alternatives, and C = {C₁, C₂, ⋯ , C_n} is the set of attributes that have the weight vector w = (w₁, w₂, ⋯ , w_n) ^T, with w_j ∈ [0, 1] (j = 1, 2, ⋯ , n) and $\sum_{j = 1}^{n} w_{j} = 1$ . The hesitant fuzzy decision matrix $H^{(k)} = (h_{ij}^{(k)})_{m \times n}$ is given by the DM e_k ∈ E, where $h_{ij}^{(k)} (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ is the hesitant fuzzy evaluation value of the alternative A_i with respect to the criterion C_j.

4.1 Extend hesitant fuzzy sets

There are often different number of values in HFEs in most situations. The common method for making these HFEs have the same length is to add the values into the shorter HFEs. One widely useful method is to add the maximum or minimum value based on the DMs’ risk preference [19, 31]. According to Definition 5, we will have different λ and add various values to the HFEs. However, it emphasizes the subjectivity of DMs and makes HFEs lose original information and characteristic. In addition, it is difficult to determine the degree of risk preference λ.

In this section, we will consider the relationships between DMs to decide the added values. Let $M = {e_{r} | l_{h_{ij}^{r}} < l}$ (r ∈ {1, 2, ⋯ , f}) be the set of DMs who are information seeker, $D_{ij}^{r} = {e_{t} | l_{h_{ij}^{t}} = l}$ (t ∈ {1, 2, ⋯ , f, t ≠ r}) be the set of DMs who are information providers for e_r to extend $h_{ij}^{r}$ , where $l = \max_{i, j, k} {l_{h_{ij}^{k}}}$ . Let $ω^{1} = (ω_{1}^{1}, ω_{2}^{1}, \dots, ω_{f}^{1})^{T}$ be the weight vector of DMs obtained by the PR algorithm, and $ω_{k}^{1} \in [0, 1] (k = 1, 2, \dots, f)$ , $\sum_{k = 1}^{f} ω_{k}^{1} = 1$ .

According to SNA, we can obtain the social influence of each DM in social network. Generally speaking, the more social relationship a DM has built with others, the more influential power the DM has. So we will use the risk preference of information providers to determine the risk preference of the seeker, which can be used to derive the extension value. See more information below for details.

Extension algorithm

Case 1: If $1 = l_{h_{ij}^{r}} < l$ , add the values which equal to the only one value.

Case 2: If $1 < l_{h_{ij}^{r}} < l$ , then

Step 1: Calculate the importance degree of each DM $e_{t} \in D_{ij}^{r}$ to extend the HFE $h_{ij}^{r}$ for e_r.

If DOA (e_t|e_r) =0, $\forall e_{t} \in D_{ij}^{r}$ , then $δ_{rt} = \frac{ω_{t}^{1}}{\sum_{e_{l} \in D_{ij}^{r}} ω_{l}^{1}},$ (4.1) where $ω_{t}^{1}$ is the weight of e_t.

Otherwise, using Definition 11 to calculate the importance degree of e_t, $δ_{rt} = \frac{OA (e_{t} | e_{r})}{\sum_{e_{l} \in D_{ij}^{r}} OA (e_{l} | e_{r})} .$ (4.2)

Step 2: Let $h_{ij}^{t} = {h_{ij 1}^{t}, h_{ij 2}^{t}, \dots, h_{ijl}^{t}}$ , according to Definition 5, we obtain $λ_{ijx}^{t}$ from $h_{ijx}^{t} = λ_{ijx}^{t} h_{ij 1}^{t} + (1 - λ_{ijx}^{t}) h_{ijl}^{t}$ , (x ∈ {2, 3, ⋯ , l - 1}).

The mean degree of risk preference of $e_{t} \in D_{ij}^{r}$ can be calculated by following approach $λ_{ij}^{t} = \frac{1}{l - 2} \sum_{x = 2}^{l - 1} λ_{ijx}^{t} .$ (4.3)

Step 3: Obtain the risk preference $λ_{ij}^{r}$ of $h_{ij}^{r}$ for e_r which is associated with providers e_t. Using providers’ important degree δ_rt obtained in Step 1, we have the risk preference $λ_{ij}^{r}$ of $h_{ij}^{r}$ :

$λ_{ij}^{r} = \sum_{t = 1}^{{sn}_{ij}^{r}} δ_{rt} λ_{ij}^{t},$ where ${sn}_{ij}^{r}$ is the number of information providers for e_r to extend $h_{ij}^{r} .$

Step 4: Recalling from Definition 5, we get the extension value $h_{ij *}^{r} = λ_{ij}^{r} h_{ij 1}^{r} + (1 - λ_{ij}^{r}) h_{ijl}^{r} .$ (4.4) Then the hesitant fuzzy matrix can be adjusted which all HFEs have same length.

4.2 The approach to hesitant fuzzy MAGDM

In this section, the PR algorithm and the deviation method are used to derive the weight vector of DMs, respectively. Based on the linear combination of these two kinds of weight information, the combined weights of the DMs are determined. In addition, the novel aggregation operators and the extension algorithm are employed to construct an improved method for dealing with the hesitant fuzzy MAGDM problem. The method is depicted in Fig. 1 and made up of seven steps as follows.

Fig. 1

General framework of the proposed method.

The MAGDM approach

Step 1: Collect related information of the MAGDM problem, including the set of alternatives A, attributes C and the relationship between DMs. Then give hesitant fuzzy decision matrix $H^{(k)} = (h_{ij}^{(k)})_{m \times n}$ of DM e_k, k ∈ {1, 2, ⋯ , f}.

Step 2: Determine the weight $ω_{k}^{1}$ of each DM e_k. We consider the DM as a node and use their relationships to construct a directed adjacency matrix. Next, the PR algorithm is used to get weight vector $ω^{1} = (ω_{1}^{1}, ω_{2}^{1}, \dots, ω_{f}^{1})$ of the DMs.

Step 3: Extend the hesitant fuzzy sets, see Section 4.1 for more details. In this step, we get DMs’ risk preferences, then calculate the relevant values to extend the HFEs which have shorter length. Then new hesitant fuzzy decision matrixes that all elements have same length are obtained.

Step 4: Compute the mean matrix of the new hesitant fuzzy decision matrixes obtained in step 3 by Definition 6. $\bar{H} = ({\bar{h}}_{ij})_{m \times n} = (\frac{1}{f} \sum_{k = 1}^{f} γ_{kij}^{σ (t)})_{m \times n},$ (4.5) where $γ_{kij}^{σ (t)}$ is the tth largest value in $h_{ij}^{k}$ .

Step 5: Derive the weights ω² and ω of each DM. $ω_{k} = α ω_{k}^{1} + (1 - α) ω_{k}^{2},$ (4.6) where α ∈ [0, 1] is the preference degree given by DMs. Here, $ω_{k}^{2} = \frac{d (H^{k}, \bar{H})}{\sum_{k = 1}^{f} d (H^{k}, \bar{H})} .$ (4.7)

Step 6: Derive the collective hesitant fuzzy decision matrix H = (h_ij) _m×n by aggregating all of the individual hesitant fuzzy decision matrices $H^{(k)} = (h_{ij}^{(k)})_{m \times n}, k \in {1, 2, \dots, f}$ .

6.1 We first use Equation (2.2) to get distance $d (h_{ij}^{(p)}, h_{ij}^{(q)})$ , and calculate the supports $\begin{matrix} Sup (h_{ij}^{(p)}, h_{ij}^{(q)}) = 1 - d (h_{ij}^{(p)}, h_{ij}^{(q)}), \\ (p, q = 1, 2, \dots, f; i = 1, 2, \dots, m; \\ j = 1, 2, \dots, n) . \end{matrix}$ (4.8)

6.2 Based on the weight ω_p (p = 1, 2, ⋯ , f) of the DM e_p obtained in Step 5, the weighted support $T (h_{ij}^{(p)})$ of HFE $h_{ij}^{(p)}$ can be obtained by the other HFEs $h_{ij}^{(q)} (q = 1, 2, \dots, f; q \neq p)$ : $T (h_{ij}^{(p)}) = \sum_{p = 1, p \neq q}^{l} ω_{q} Sup (h_{ij}^{(p)}, h_{ij}^{(q)}) .$

6.3 In this case, H = (h_ij) _m×n is get by utilizing the NHFPA or NHFPG operator.

Step 7: Determine the collective overall preference value h_i (i = 1, 2, ⋯ , m) of alternative A_i (i = 1, 2, ⋯ , m). The process of this step is similar with the process in Step 6.

7.1 At first, calculate the supports based on distance d (h_ij, h_it), $\begin{matrix} Sup (h_{ij}, h_{it}) = 1 - d (h_{ij}, h_{it}), \\ (i = 1, 2, \dots, m; j, t = 1, 2, \dots, n) . \end{matrix}$ (4.9)

7.2 Utilize the weight w_j (j = 1, 2, ⋯ , n) of the attribute C_j (j = 1, 2, ⋯ , n), the weighted support T (h_ij) of HFE h_ij can be obtained $T (h_{ij}) = \sum_{t = 1, t \neq j}^{n} w_{p} Sup (h_{ij}, h_{it}), i = 1, 2, \dots \dots, m .$

7.3 By using the weight w_j and the weighted support T (h_ij) obtained in Step 7.2, we adopt the NHFPA or NHFPG operator to aggregate all values h_ij (j = 1, 2, ⋯ , n) to get the value h_i of the alternative A_i (i = 1, 2, ⋯ , m), respectively.

Step 8: Compute the score value s (A_i) of A_i (i = 1, 2, ⋯ , m) according to Definition 3.

Step 9: Rank the alternatives and select the best one. If s (A_i) > s (A_j), then A_i ≻ A_j.

5 Illustrative example

In order to clearly demonstrate the feasibility, effectiveness and practicality of the proposed method, a practical example is chosen only with 13 DMs. This method can be generalized to the cases with more attributes and DMs. It can be seen that the improved method not only simplifies the calculations involved, but also makes the decision-making results more reasonable and practical.

5.1 Background

Thirteen members with interactive relationships are expressed as E = {e₁, e₂, ⋯ , e₁₃}. They are organized to select the best choice for a vacation trip. In the graph, there are 13 nodes, namely e₁, e₂, , ⋯ , e₁₃. To visualize the relationships among these DMs, Fig. 2 gives the social graph of these 13 DMs. Directed arcs between pairs of nodes describe the relationships between DMs. For example, the link from node e₁ to node e₃ implies that e₁ is information seeker and e₃ is information provider. Suppose that there are four alternatives A = {A₁, A₂, A₃, A₄} and four attributes C = {C₁, C₂, C₃, C₄} are considered: C₁ = cost; C₂ = attraction; C₃ = traffic; C₄ = accommodation. The weight vector of C_j is w = (0.25, 0.25, 0.25, 0.25) ^T .

Fig. 2

The relationships between DMs.

All DMs evaluate these four alternatives by means of HFEs and 13 hesitant fuzzy decision matrixes are obtained as follows:

5.2 The decision-making process

According to the proposed method given in Subsection 4.2, the decision-making steps for the above MAGDM problem are described as follows:

Step 1: Calculate the weight $ω_{k}^{1}$ of the DM e_k (k = 1, 2, ⋯ , 13). According to the social graph Fig. 2, we get the adjacency matrix AM as follows $AM = (\begin{matrix} 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \end{matrix})$

Then, using the PR algorithm, the weight vector ω¹ of the DM e_k (k = 1, 2, ⋯ , 13) is obtained as: $\begin{matrix} ω^{1} = (0.0451, 0.1097, 0.0694, 0.1181, 0.0115, \\ 0.0474, 0.043, 0.1786, 0.0314, 0.0857, 0.0842, \\ 0.1393, 0.0366)^{T} . \end{matrix}$

Step 2: Complete hesitant fuzzy decision matrixes. Based on the relationships between DMs in Fig. 2, use Equation (2.6) to derive the social connection strength between DMs in Table 14.

Table 1
The decision matrix H⁽¹⁾ given by e₁

C ₁ C ₂ C ₃ C ₄

A ₁ {0.7, 0.4, 0.3} {0.9, 0.2} {0.8, 0.4, 0.2} {0.6}

A ₂ {0.5, 0.4} {0.9, 0.7, 0.6} {0.8, 0.1} {0.7, 0.5, 0.4}

A ₃ {0.8, 0.7, 0.4} {0.7, 0.4, 0.2} {0.7, 0.5, 0.3} {0.8, 0.2}

A ₄ {0.9, 0.7, 0.6} {0.8, 0.7, 0.6} {0.9, 0.6} {0.8, 0.7, 0.6}

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.7, 0.4, 0.3}	{0.9, 0.2}	{0.8, 0.4, 0.2}	{0.6}
A ₂	{0.5, 0.4}	{0.9, 0.7, 0.6}	{0.8, 0.1}	{0.7, 0.5, 0.4}
A ₃	{0.8, 0.7, 0.4}	{0.7, 0.4, 0.2}	{0.7, 0.5, 0.3}	{0.8, 0.2}
A ₄	{0.9, 0.7, 0.6}	{0.8, 0.7, 0.6}	{0.9, 0.6}	{0.8, 0.7, 0.6}

Table 2

The decision matrix H⁽²⁾ given by e₂

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.6, 0.5, 0.4}	{0.7, 0.6, 0.4}	{0.8, 0.7, 0.5}	{0.7, 0.6, 0.4}
A ₂	{0.6, 0.5, 0.3}	{0.6, 0.4, 0.3}	{0.9, 0.7, 0.6}	{0.8, 0.7, 0.6}
A ₃	{0.7, 0.4, 0.3}	{0.8, 0.5, 0.4}	{0.7, 0.5, 0.3}	{0.8, 0.6, 0.5}
A ₄	{0.8, 0.3, 0.2}	{0.7, 0.5, 0.3}	{0.7, 0.6, 0.5}	{0.9, 0.7, 0.5}

Table 3

The decision matrix H⁽³⁾ given by e₃

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.9, 0.7, 0.1}	{0.5, 0.4, 0.2}	{0.9, 0.6, 0.5}	{0.7, 0.6, 0.3}
A ₂	{0.9, 0.7, 0.6}	{0.8, 0.6, 0.5}	{0.8, 0.4, 0.3}	{0.8, 0.7, 0.6}
A ₃	{0.9, 0.7, 0.6}	{0.7, 0.5, 0.3}	{0.6, 0.5, 0.4}	{0.9, 0.8, 0.7}
A ₄	{0.7, 0.4, 0.2}	{0.8, 0.5, 0.1}	{0.9, 0.8, 0.6}	{0.8, 0.6, 0.3}

Table 4

The decision matrix H⁽⁴⁾ given by e₄

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.7, 0.3}	{0.8, 0.3}	{0.7, 0.4}	{0.6, 0.5, 0.4}
A ₂	{0.9, 0.7}	{0.8, 0.6, 0.5}	{0.4, 0.3, 0.2}	{0.5, 0.4, 0.3}
A ₃	{0.7, 0.6, 0.5}	{0.8, 0.7, 0.5}	{0.4, 0.3, 0.2}	{0.4, 0.3, 0.2}
A ₄	{0.8, 0.2}	{0.6, 0.5, 0.3}	{0.3, 0.2, 0.1}	{0.4}

Table 5

The decision matrix H⁽⁵⁾ given by e₅

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.6, 0.2, 0.1}	{0.5, 0.1}	{0.8, 0.4, 0.3}	{0.9, 0.5.0.3}
A ₂	{0.5, 0.2}	{0.3, 0.2}	{0.8, 0.3}	{0.8, 0.7, 0.5}
A ₃	{0.5, 0.3, 0.2}	{0.5, 0.3, 0.2}	{0.7, 0.5, 0.3}	{0.9, 0.5}
A ₄	{0.6, 0.2}	{0.8, 0.7, 0.6}	{0.7, 0.3}	{0.6, 0.5, 0.4}

Table 6

The decision matrix H⁽⁶⁾ given by e₆

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.5}	{0.7, 0.6, 0.4}	{0.7, 0.6}	{0.7, 0.6, 0.5}
A ₂	{0.7, 0.6, 0.5}	{0.8, 0.7, 0.5}	{0.7, 0.5, 0.4}	{0.5, 0.2}
A ₃	{0.9, 0.5, 0.2}	{0.5, 0.3}	{0.9, 0.8}	{0.8, 0.7, 0.6}
A ₄	{0.7, 0.1}	{0.8, 0.7}	{0.9, 0.6, 0.5}	{0.8, 0.3}

Table 7

The decision matrix H⁽⁷⁾ given by e₇

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.5, 0.4, 0.3}	{0.6, 0.5, 0.4}	{0.6, 0.5, 0.3}	{0.6, 0.3}
A ₂	{0.7, 0.6, 0.5}	{0.7, 0.4}	{0.6, 0.5, 0.4}	{0.6, 0.4, 0.3}
A ₃	{0.8, 0.6, 0.3}	{0.7, 0.6, 0.5}	{0.7, 0.2}	{0.7, 0.6, 0.5}
A ₄	{0.8, 0.2}	{0.5, 0.4, 0.1}	{0.7, 0.6, 0.4}	{0.8, 0.4}

Table 8

The decision matrix H⁽⁸⁾ given by e₈

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.8, 0.3, 0.2}	{0.9, 0.7, 0.6}	{0.5, 0.2}	{0.6, 0.5, 0.3}
A ₂	{0.7, 0.6, 0.5}	{0.7, 0.6, 0.4}	{0.5, 0.3, 0.2}	{0.6, 0.1}
A ₃	{0.5, 0.3}	{0.6, 0.1}	{0.9, 0.8, 0.7}	{0.9, 0.8, 0.7}
A ₄	{0.4, 0.2, 0.1}	{0.7, 0.2}	{0.5, 0.4, 0.3}	{0.9, 0.8, 0.3}

Table 9

The decision matrix H⁽⁹⁾ given by e₉

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.7, 0.6, 0.3}	{0.5, }	{0.9, 0.8, 0.7}	{0.4, 0.3, 0.2}
A ₂	{0.7, 0.5, 0.4}	{0.7, 0.6}	{0.5, 0.3}	{0.5, 0.3}
A ₃	{0.6, 0.4}	{0.9, 0.6, 0.4}	{0.6, 0.5, 0.3}	{0.6, 0.5, 0.4}
A ₄	{0.7, 0.6, 0.5}	{0.8, 0.3}	{0.7, 0.6, 0.5}	{0.8, 0.7, 0.6}

Table 10

The decision matrix H⁽¹⁰⁾ given by e₁₀

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.9, 0.7}	{0.5}	{0.8, 0.7, 0.5}	{0.6, 0.5}
A ₂	{0.8, 0.7, 0.6}	{0.6, 0.5, 0.3}	{0.7, 0.5, 0.3}	{0.5}
A ₃	{0.6, 0.5, 0.4}	{0.6, 0.3, 0.2}	{0.9, 0.6}	{0.5, 0.4, 0.3}
A ₄	{0.6, 0.3}	{0.8, 0.6, 0.5}	{0.7, 0.6}	{0.5, 0.2}

Table 11

The decision matrix H⁽¹¹⁾ given by e₁₁

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.8, 0.4, 0.3}	{0.7, 0.5, 0.4}	{0.8, 0.7, 0.6}	{0.6, 0.5, 0.4}
A ₂	{0.6, 0.4, 0.3}	{0.6, 0.5, 0.4}	{0.7, 0.6, 0.5}	{0.5, 0.4, 0.3}
A ₃	{0.8, 0.7, 0.6}	{0.7, 0.5, 0.4}	{0.9, 0.8, 0.7}	{0.6, 0.5, 0.4}
A ₄	{0.7, 0.4, 0.2}	{0.6, 0.4, 0.3}	{0.9, 0.7, 0.6}	{0.7, 0.6, 0.5}

Table 12

The decision matrix H⁽¹²⁾ given by e₁₂

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.7, 0.6, 0.5}	{0.7, 0.6}	{0.6, 0.4, 0.3}	{0.4, 0.3, 0.2}
A ₂	{0.9, 0.7, 0.4}	{0.8, 0.7, 0.6}	{0.7, 0.6, 0.5}	{0.8, 0.7, 0.5}
A ₃	{0.7, 0.5, 0.4}	{0.3, 0.2, 0.1}	{0.6, 0.4}	{0.9, 0.8}
A ₄	{0.9, 0.8, 0.7}	{0.8, 0.5, 0.4}	{0.8, 0.7, 0.6}	{0.8, 0.7, 0.4}

Table 13

The decision matrix H⁽¹³⁾ given by e₁₃

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.5, 0.4, 0.2}	{0.7, 0.6, 0.4}	{0.6, 0.4}	{0.9, 0.7}
A ₂	{0.7, 0.5}	{0.7}	{0.7, 0.6, 0.4}	{0.7, 0.6, 0.5}
A ₃	{0.8, 0.6}	{0.9, 0.6}	{0.8, 0.7, 0.6}	{0.5, 0.4}
A ₄	{0.5, 0.3, 0.2}	{0.8, 0.7, 0.3}	{0.7, 0.2}	{0.8, 0.7, 0.6}

Table 14

The strength of social ties between DMs.

Information searcher	Information supporters	The strength of social ties	Information searcher	Information supporters	The strength of social ties
e ₁	e ₂	0.3824	e ₄	e ₇	0.2390
	e ₄	0.5275		e ₈	0.9247
	e ₆	0.2315		e ₁₂	0.1080
	e ₇	0.2792		e ₁₃	0.2252
	e ₈	0.6200
	e ₉	0.1815
e ₅	e ₁	0.3361	e ₆	e ₂	0.5882
	e ₂	0.6282		e ₉	0.4341
	e ₃	0.7158
	e ₁₀	0.7179
e ₇	e ₈	1.0000	e ₈	e ₂	0.2177
				e ₄	0.8174
e ₉	e ₁₀	1.0000	e ₁₀	e ₃	0.2914
				e ₄	0.7103
				e ₈	0.7470
e ₁₂	e ₈	1.0000	e ₁₃	e ₈	0.8620
				e ₁₀	0.5628

Subsequently, we use the extension algorithm in Subsection 4.1 to get the results below

Shorter values	Decision supporters	Preference degree	Added values	Shorter values	Decision supporters	Preference degree	Added values
$h_{12}^{1}$	e₂, e₆, e₇, e₈	0.4289	0.5003	$h_{11}^{4}$	e₇, e₈, e₁₂, e₁₃	0.3192	0.4277
$h_{14}^{1}$	/	/	0.6	$h_{12}^{4}$	e₇, e₈, e₁₃	0.4161	0.5080
$h_{21}^{1}$	e₂, e₄, e₆, e₇, e₈, e₉	0.4821	0.5929	$h_{13}^{4}$	e₇, e₁₂	0.5629	0.5689
$h_{23}^{1}$	e₂, e₄, e₆, e₇, e₈	0.4193	0.3935	$h_{21}^{4}$	e₇, e₈	0.5	0.8
$h_{34}^{1}$	e₂, e₄, e₆, e₇, e₈, e₉	0.4713	0.828	$h_{41}^{4}$	e₈, e₁₂, e₁₃	0.3476	0.4086
$h_{43}^{1}$	e₂, e₄, e₆, e₇, e₈, e₉	0.4929	0.5464	$h_{44}^{4}$	/	/	0.4
$h_{12}^{5}$	e₂, e₃	0.6667	0.36667	$h_{11}^{6}$	/	/	0.5
$h_{21}^{5}$	e₂, e₃, e₁₀	0.4421	0.4211	$h_{13}^{6}$	e₂, e₉	0.5959	0.6596
$h_{22}^{5}$	e₁, e₂, e₃, e₁₀	3333	0.3333	$h_{24}^{6}$	e ₂	0.5	0.35
$h_{23}^{5}$	e₂, e₃, e₁₀	0.3451	0.4725	$h_{32}^{6}$	e₂, e₉	0.3137	0.3627
$h_{34}^{5}$	e₂, e₃	0.4221	0.6688	$h_{33}^{6}$	e₂, e₉	0.5708	0.6854
$h_{41}^{5}$	e₁, e₂, e₃	0.2994	0.3198	$h_{41}^{6}$	e₂, e₉	0.3082	0.2849
$h_{43}^{5}$	e₂, e₃	5888	0.5355	$h_{42}^{6}$	e ₂	0.5	05
/	/	/	/	$h_{43}^{6}$	e₂, e₉	0.5	0.55
$h_{14}^{7}$	e ₈	0.6667	0.5	$h_{13}^{8}$	e ₂	0.6667	0.4
$h_{22}^{7}$	e ₈	0.6667	0.6	$h_{24}^{8}$	e₂, e₄	0.5	0.35
$h_{33}^{7}$	e ₈	0.5	0.45	$h_{31}^{8}$	e₂, e₄	0.4474	0.3895
$h_{41}^{7}$	e ₈	0.3333	0.4	$h_{32}^{8}$	e₂, e₄	0.5790	0.3895
$h_{44}^{7}$	e ₈	0.8333	0.7333	$h_{42}^{8}$	e₂, e₄	0.6316	0.5158
$h_{12}^{9}$	/	/	0.5	$h_{11}^{10}$	e₃, e₈	0.1901	0.7380
$h_{21}^{9}$	e ₁₀	0.3333	0.6333	$h_{12}^{10}$	/	/	0.5
$h_{22}^{9}$	e ₁₀	0.5	0.4	$h_{14}^{10}$	e₃, e₄, e₈	0.6129	0.5613
$h_{23}^{9}$	e ₁₀	0.3333	0.3667	$h_{24}^{10}$	/	/	0.5
$h_{34}^{9}$	e ₁₀	0.75	0.55	$h_{33}^{10}$	e₃, e₄, e₈	0.5	0.75
$h_{41}^{9}$	e ₁₀	0.5	0.55	$h_{41}^{10}$	e₃, e₈	0.3520	0.4056
/	/	/	/	$h_{43}^{10}$	e₃, e₄, e₈	0.5278	0.6528
/	/	/	/	$h_{44}^{10}$	e₃, e₈	0.7678	0.4304
$h_{12}^{12}$	e ₈	0.3333	0.6333	$h_{13}^{13}$	e ₁₀	0.6667	0.5333
$h_{33}^{12}$	e ₈	0.5	0.5	$h_{14}^{13}$	e ₈	0.6667	0.8333
$h_{34}^{12}$	e ₈	0.5	0.85	$h_{21}^{13}$	e₈, e₁₀	0.5	0.6
/	/	/	/	$h_{22}^{13}$	/	/	0.7
/	/	/	/	$h_{31}^{13}$	e ₁₀	0.5	0.7
/	/	/	/	$h_{32}^{13}$	, e₁₀	0.25	0.675
/	/	/	/	$h_{34}^{13}$	e₈, e₁₀	0.4013	0.4401
/	/	/	/	$h_{43}^{13}$	e ₈	0.5	0.6

Thus, the HFEs in all decision matrixes all have same length.

Step 3: Determine the weight vector ω of DMs.

(3.1) Based on the adjusted matrixes, we utilize Equation (4.5) to obtain the following mean of matrix shown in Table 15:

Table 15

The mean matrix $\bar{H}$

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.6846, 0.4743, 0.3231}	{0.6692, 0.5314, 0.3846}	{0.7308, 0.5663, 0.4231}	{0.6385, 0.5304, 0.3923}
A ₂	{0.7538, 0.5934, 0.4538}	{0.7154, 0.5744, 0.4615}	{0.6769, 0.4743, 0.3462}	{0.6385, 0.5128, 0.3923}
A ₃	{0.7154, 0.5492, 0.4000}	{0.6692, 0.4636, 0.3231}	{0.7231, 0.5758, 0.4154}	{0.7154, 0.5878, 0.4769}
A ₄	{0.7000, 0.4245, 0.2846}	{0.7308, 0.5435, 0.3231}	{0.7231, 0.5796, 0.4385}	{0.7385, 0.6241, 0.4231}

(3.2) By the Equation (4.7), get the weight vector ω² of DMs. $\begin{matrix} ω^{2} = (0.0767, 0.0789, 0.0766, 0.0748, 0.0750, \\ 0.0789, 0.0804, 0.0741, 0.0779, 0.0765, 0.0784, \\ 0.0746, 0.0770)^{T} . \end{matrix}$

(3.3) Let α = 0.5, from Equation (4.6), the final weight vector ω of DMs is as follows: $\begin{matrix} ω = (0.0609, 0.0943, 0.0730, 0.0965, 0.0432, \\ 0.0631, 0.0617, 0.1264, 0.0546, 0.0811, 0.0813, \\ 0.1070, 0.0568)^{T} . \end{matrix}$

Step 4: After calculating the support $Sup (h_{ij}^{k}, h_{ij}^{t})$ , we use the NHFPA operator to get the collective decision matrix H (see Table 16) by aggregating all individual decision matrixes {H^(k)| (k = 1, 2, ⋯ , 13)}.

Table 16

The collective decision matrix H

	C ₁	C ₂	C ₃	C ₄
A ₁	{0.7317, 0.4992, 0.3564}	{0.7260, 0.5576, 0.4281}	{0.7445, 0.5818, 0.4321}	{0.6517, 0.5400, 0.3985}
A ₂	{0.7848, 0.6216, 0.4803}	{0.7336, 0.5911, 0.4750}	{0.7011, 0.4894, 0.3660}	{0.6621, 0.5328, 0.4039}
A ₃	{0.7367, 0.5582, 0.4168}	{0.6989, 0.4774, 0.3259}	{0.7716, 0.6157, 0.4631}	{0.7775, 0.6444, 0.5373}
A ₄	{0.7352, 0.4569, 0.3157}	{0.7423, 0.5417, 0.3297}	{0.7657, 0.6038, 0.4693}	{0.7884, 0.6568, 0.4256}

Step 5: The collective overall preference value h_i of the alternative A_i is derived as follows: $\begin{matrix} h_{1} = {0.7156, 0.5457, 0.4046}, h_{2} = \\ {0.7242, 0.5617, 0.4334}, \\ h_{3} = {0.7483, 0.5789, 0.4412}, h_{4} = \\ {0.7587, 0.5706, 0.3887} . \end{matrix}$

Step 6: Use Definition 3 to compute the score value s (A_i) of the alternative A_i (i = 1, 2, 3, 4), which read: $\begin{matrix} A_{1} = 0.5553, A_{2} = 0.5731, A_{3} = 0.5895, \\ A_{4} = 0.5726 . \end{matrix}$

Step 7: Rank the alternative A_i (i = 1, 2, 3, 4) according to the score value s (A_i) (i = 1, 2, 3, 4): A₃ ≻ A₂ ≻ A₄ ≻ A₁, and thus the most desirable alternative is A₃.

5.3 Discussion and comparison

This section discusses and compares the results when adopt the proposed method and the method from Xia and Xu [30]. The analyses are shown in Tables 17 and 18 in details.

Table 17
Ranking order with different methods

Methods Added values Order of alternatives

The proposed method Values from the extension algorithm A₃ ≻ A₂ ≻ A₄ ≻ A₁

The method from Maximum value A₃ ≻ A₄ ≻ A₂ ≻ A₁

Xia and Xu [30] Minimum value A₃ ≻ A₂ ≻ A₄ ≻ A₁

Methods	Added values	Order of alternatives
The proposed method	Values from the extension algorithm	A₃ ≻ A₂ ≻ A₄ ≻ A₁
The method from	Maximum value	A₃ ≻ A₄ ≻ A₂ ≻ A₁
Xia and Xu [30]	Minimum value	A₃ ≻ A₂ ≻ A₄ ≻ A₁

Table 18

Dimension of h_i of the alternative A_i (i = 1, 2, 3, 4)

	A ₁	A ₂	A ₃	A ₄
The proposed method	3	3	3	3
The method from Xia and Xu [30]	1594323⁴	1594323⁴	1594323⁴	1594323⁴

It can be seen from Table 17 that the alternative A₃ is the best choice obtained from the proposed method, which is in agreement with the one obtained in Xia and Xu [30]. However, there is still little difference between these results. The ranking orders are the same by using the proposed method and the approach of Xia and Xu’s [30] with the addition the minimum value. That is, A₃ is ranked the first, then A₂ and A₄, respectively, and A₁ the last. In the shorter HFEs, when adding the maximum value according to Xia and Xu’s [30] approach, A₁ and A₃ are ranked the same as the proposed method, but the ranking orders of the other alternatives are changed: A₂ drops from the second to the third, while A₄ moves up from the third to the second. Note that the ranking results are significantly different when different values are applied to the shorter HFEs, even though the same method is used because applying different values to shorter HFEs results in the slightly different ranking results. However, we cannot assume that the DMs are optimistic or pessimistic objectively. In the proposed method, the risk preference of the DM is obtained by considering the degrees of opinion acceptance of other DMs, and then get corresponding extension values. In other words, the extension values obtained by the proposed method are relatively feasible, making the ranking results more reasonable.

In addition, the proposed method adopts the novel operator developed in this paper, which can simplify the calculations involved greatly. Table 18 shows that the dimension (length) of h_i of alternative A_i (i = 1, 2, 3, 4) are all 3 by the proposed method, while the dimension of each h_i (i = 1, 2, 3, 4) is 1594323⁴ by utilizing the WHFPA operator or WHFPG operator. If the number of attributes or alternatives or DMs in the example is much larger, the dimension of each h_i (i = 1, 2, 3, 4) by using WHFPA operator or WHFPG operator will be greater. That is to say, the proposed method is rather effective.

In the MAGDM method proposed in this paper, the weight vector of the DMs is composed of two parts, one is obtained based on the social relationship between the DMs, and the other is determined based on the evaluation matrixes given by DMs. DMs can choose different parameter α to interpret their preferences, which can provide more flexible and reasonable choices to DMs. The weight information also has an impact on the decision result. When different α is chosen, the decision result will be changed accordingly. Figure 3 provides an intuitive sense of the influence of the parameter α on the decision result. As shown in Fig. 3, scores of A₁, A₂ and A₃ increase with the increase of α, on the contrary the score of A₄ decreases with the increase of α. In addition, when α is selected from {0, 0.1, 0.2, 0.3, 0.4}, the ranking order is A₃ ≻ A₄ ≻ A₂ ≻ A₁. Other cases are all A₃ ≻ A₂ ≻ A₄ ≻ A₁.

Fig. 3

Ranking result based on different parameter α.

6 Conclusions

HFS is a powerful tool to express uncertain and hesitant information in order to determine alternatives in the process of MAGDM. Moreover, in the MAGDM process, social ties and social influence among DMs are seldom considered in the existing research. This paper proposes a MAGDM method based on social network in the hesitant fuzzy environment. The following questions were resolved:

(1) Propose two novel hesitant fuzzy weighted power aggregation operators, namely the NHFPA operator and NHFPG operator. A variety of different forms of hesitant fuzzy aggregation operators were have been proposed to to aggregate evaluation information in decision making. Among them, the WHFPA operator and the WHFPG operator have been widely used. However, the dimensions of the derived HFEs may increase significantly when the addition or multiplication operation is performed. To overcome the constraint, we have developed two novel power aggregation operators. These novel operators will reduce the complexity of computation greatly.

Next, we analyze some fundamental properties and explore their relationship of the NHFPA operator and NHFPG operator, including their idempotency, boundedness and monotonicity.

(2) By referring to the theoretical knowledge of SNA, we consider the relationships between DMs and construct a directed adjacency matrix. Based on the adjacency matrix, we use the PR algorithm to obtain the DMs’ weight vector ω¹. In addition, the DMs’ weight vector ω² is obtained according to decision matrixes themselves given by DMs. By adjusting the parameter α, the proportion and role of ω¹ and ω² can be determined in the DMs’ final weights. It makes the DMs’ weight information more flexible.

(3) Meanwhile, we propose an extension algorithm to deal with the case, which HFEs have different length. The lengths of the two HFEs are not always same. A more common approach is to calcute the added value based on the DM’s preference, which is difficult to determine. At the same time, if the preferences are set artificially, the decision-making results will be subjective. This paper uses the strength of social ties and social influence in social network to derive the degree of risk preference and obtain the added values. A hesitant fuzzy decision matrix can be adjusted where all HFEs have the same number of values. These extension values are relatively objective.

Obviously, there are still some drawbacks to the method proposed in this paper. On the one hand, the added values acquired by the extension algorithm are relatively objective, but there is still a difference between these added values and the reasoning of DMs. On the other hand, the calculation process of determining will become very difficult in a very large social network. So, we will further concentrate on the method to deal with hesitant fuzzy decision making problems without adding any values and apply it to other related fuzzy sets. In the meantime, complex network, cluster analysis and other theories will be used to mine the relationships of DMs and determine the classification of small groups in social network. In big social network, we will use the principle of “dividing into parts - integrating into zero” to construct a decision-making model and apply it to practical problems such as personalized recommendations, product design and supply chain planning.

Footnotes

Appendices

Acknowledgments

This work was supported by the General Project of Philosophy and Social Science Research in Universities of Jiangsu Province(Nos. 2020SJA1594).

References

Wang

and Xu

Z.S.

, Multi-groups decision making using intuitionistic-valued hesitant fuzzy information, International Journal of Computational Intelligence Systems 9(3) (2016), 468–482.

Zadeh

L.A.

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

Torra

and Narukawa

, On hesitant fuzzy sets and decision, IEEE International Conference on Fuzzy Systems Jeju Island, Korea, (2009), 1378–1382.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

Liu

X.D.

, Wang

Z.W.

and Zhang

S.T.

, A modification on the hesitant fuzzy set lexicographical ranking method, Symmetry 8 (2016), 153.

Z.S.

and Xia

M.M.

, On distance and correlation measures of hesitant fuzzy information, International Journal of Intelligent Systems 26 (2011), 410–425.

Farhadinia

, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Information Sciences 204 (2013), 129–144.

D.J.

, Group decision making under multiplicative hesitant fuzzy environment, International Journal of Fuzzy Systems 16(2) (2014), 233–241.

Xia

M.M.

, Xu

Z.S.

and Chen

, Some hesitant fuzzy aggregation operators with their application in group decision making, Group Decision and Negotiation 22 (2013), 259–279.

10.

Zhou

X.Q.

and Li

Q.G.

, Generalized hesitant fuzzy prioritized Einstein aggregation operators and their application in group decision making, International Journal of Fuzzy Systems, 16(3) (2016), 303–316.

11.

Zhao

, Xu

Z.S.

and Cui

, Generalized hesitant fuzzy harmonic mean operators and their application in group decision making, International Journal of Fuzzy Systems 814 (2016), 685–696.

12.

Liao

H.C.

, Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, International Journal of Information Technology and Decision Making 13(1) (2014), 47–76.

13.

Zhang

Z.M.

, Hesitant fuzzy multi-criteria group decision making with unknown weight informaion, International Journal of Fuzzy Systems 19(3) (2017), 615–636.

14.

Liu

Z.M.

, Li

, Wang

X.Y.

and Liu

P.D.

, A novel multiple-attribute decision making method based on power Muirhead mean operator under normal wiggly hesitant fuzzy environment, Journal of Intelligent & Fuzzy Systems 37 (2019), 7003–7023.

15.

Lin

, Wang

Y.M.

and Yan

L.Z.

, Cosine-similarity based approach for weights determination under hesitant fuzzy environment and its extension to priority derivation, Journal of Intelligent & Fuzzy Systems 38 (2020), 2261–2271.

16.

Yager

R.R.

, The power average operator, IEEE Transactions on Systems Man and Cybernetics-Part A 31 (2001), 724–731.

17.

Z.S.

and Yager

R.R.

, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning 48(1) (2008), 246–262.

18.

Zhang

Z.M.

, Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making, Information Sciences 234 (2013), 150–181.

19.

Z.S.

and Hang

X.L.

, Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information, Knowledge-Based Systems 52 (2013), 53–64.

20.

Pérez

L.G.

, Mata

and Chiclana

, Social network decision making with linguistic trustworthiness-based induced OWA operators, International Journal of Intelligent Systems 29 (2014), 1117–1137.

21.

Wasserman

and Faust

, Social network analysis: methods and applications, Cambridge University Press New York (1994).

22.

Scott

J.P.

and Carrington

P.J.

, The SAGE handbook of social network analysis, SAGE, London (2011).

23.

Y.M.

and Lai

C.Y.

, A social appraisal mechanism for online purchase decision support in the micro-blogosphere, Decision Support Systems 59 (2014), 190–205.

24.

Quijano-Sánchez

, Díaz-Agudo

and Recio-García

J.A.

, Development of a group recommender application in a Social Network, Knowledge-Based Systems 71 (2014), 72–85.

25.

Pérez

L.G.

, Mata

, Chiclana

, Kou

and Herrera-Viedma

, Modelling influence in group decision making, Soft Computing 20 (2016), 1653–1665.

26.

Liang

, Liao

X.W.

and Liu

J.P.

, A social ties-based approach for group decision-making problems with incomplete additive preference relations, Knowledge-Based Systems 119(2017), 68–86.

27.

Qiu

L.Q.

, Dai

J.L.

, Liu

H.Y.

and Wen

, Detecting opinion leaders in online social networks using HybridRank algorithm, Journal of Intelligent & Fuzzy Systems 35 (2018), 513–522.

28.

Akkuzu

, Aziz

and Adda

, A fuzzy modeling approach for group decision making in social networks, 22nd International Conference on Business Information Systems (BIS) 354 (2019), 74–85.

29.

Lua

Y.L.

, Xua

Y.J.

, Qu

S.J.

, Xu

Z.S.

, Ma

and Li

Z.W.

, Multiattribute social network matching with unknown weight and different risk preference, Journal of Intelligent & Fuzzy Systems 38 (2020), 4031–4048.

30.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52 (2011), 395–407.

31.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181 (2011), 2128–2138.

32.

, Xu

Z.S.

, Zhao

and Liu

, Distribution-based approaches to deriving weights from dual hesitant fuzzy information, Symmetry 11(1) (2019), 85.

33.

Brin

and Page

, The anatomy of a large-scale hypertextual web search engine, Computer Networks and ISDN Systems 33 (1998), 107–117.

34.

Torra

and Narukawa

, Modeling eecisions: information fusion and aggregation operators, Springer (2007).

Hesitant fuzzy multi-attribute group decision making method based on weighted power operators in social network and their application

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy sets

4.1 Extend hesitant fuzzy sets

5.1 Background

Table 1 The decision matrix H(1) given by e1 C 1 C 2 C 3 C 4 A 1 {0.7, 0.4, 0.3} {0.9, 0.2} {0.8, 0.4, 0.2} {0.6} A 2 {0.5, 0.4} {0.9, 0.7, 0.6} {0.8, 0.1} {0.7, 0.5, 0.4} A 3 {0.8, 0.7, 0.4} {0.7, 0.4, 0.2} {0.7, 0.5, 0.3} {0.8, 0.2} A 4 {0.9, 0.7, 0.6} {0.8, 0.7, 0.6} {0.9, 0.6} {0.8, 0.7, 0.6}

Table 17 Ranking order with different methods Methods Added values Order of alternatives The proposed method Values from the extension algorithm A3 ≻ A2 ≻ A4 ≻ A1 The method from Maximum value A3 ≻ A4 ≻ A2 ≻ A1 Xia and Xu [30] Minimum value A3 ≻ A2 ≻ A4 ≻ A1

Footnotes

Appendices

Acknowledgments

References