Cosine-similarity based approach for weights determination under hesitant fuzzy environment and its extension to priority derivation

Abstract

The theory of hesitant fuzzy set is powerful in capturing fuzziness when a decision maker hesitates among some possible values in assessment. In this paper, a novel method is developed for weight determination based on the hesitant fuzzy cosine similarity. This method fully considers the impact of each criterion as well as their mutual relationship on decision analysis. Additionally, we extend this cosine similarity-based method for deriving priorities from a hesitant multiplicative preference relation. This method has a clearer modeling mechanism that the priority vector derived from a hesitant multiplicative preference relation should be as highly cosine-similarity correlated with its column vectors as possible. Then an optimization model can be sequentially established. After that, the proposed methods are demonstrated with two numerical examples and compared with other similar approaches to show the validity and superiority.

Keywords

Hesitant fuzzy set hesitant multiplicative preference relation cosine similarity weights determination multi-criteria decision making

1 Introduction

As a generalization of fuzzy sets, the hesitant fuzzy set (HFS) [1, 2] opens a new window in fuzzy decision making. HFSs are recognized as capable of depicting uncertain situations when two or more sources of vagueness emerge simultaneously and have been evolved into different structures, such that hesitant multiplicative set (HMS) using an unsymmetrical scale [3] or hesitant fuzzy topological spaces [4]. These concepts have already been applied in different fields, such as multiple criteria decision making (MCDM), fuzzy preference relations, to facilitate with imprecise information [5].

In MCDM, how to weigh criteria plays a key role in determining the final result [6]. Solving such a problem with complete weights follows a common resolution scheme [7]: 1) aggregation process and 2) ranking process. The aggregation process can be tackled in various ways and yields an overall evaluation. However, multiple criteria are often correlated or even in conflict with each other, so it is crucial to deal with correlation among criteria, especially in a hesitant fuzzy environment. This problem also exists in the derivation of priority weights from hesitant multiplicative preference relation (HMPR).

Similarity (or dissimilarity) measure, which can be utilized for gauging the closeness between two arguments and implying how well they move together [8], is an effective indicator in weight determinations. Several formats of similarity measures have been examined [9, 10], and among them one of the most commonly used is the cosine similarity, which is defined as the inner product of two vectors divided by the product of their lengths. Nevertheless, there is still little attention on weight determination as well as prioritization under hesitant fuzzy context from the perspective of cosine similarity.

With this gap in mind, this paper aims to present a new approach to overcome this weakness. The contribution of this paper can be summarized as below: (1) We develop an objective weight derivation method based on cosine similarity measure in hesitant fuzzy context, where the similarity degrees are calculated by eliminating each criterion from the overall assessment of alternatives; (2) An optimization model is presented to derive the priorities from an HMPR based on the cosine similarity and consistent property; (3) We apply the hesitant fuzzy cosine similarity-based method in decision making such as green supplier evaluation.

The remainder of this article is structured as below. In Section 2, brief reviews on weight derivation and cosine similarity are outlined. Section 3 gives some basic knowledge on HFS and HMS. In Section 4, we develop a method for weight determination based on cosine similarity under the hesitant fuzzy framework and extend this method for generating priorities from HMPRs. Two numerical examples are provided to illustrate the proposed methods in Section 5. Section 6 concludes this article.

2 Literature review

2.1 Review on weight derivation in hesitant fuzzy environment

A wealth of weighting methods have been studied, which can be roughly subdivided into two categories: the subjective methods and the objective methods [11, 12]. The previous ones reflect opinions of decision makers (DMs) in terms of knowledge, experience, or expertise, including the eigenvector method [13], the analytic hierarchy process [14], etc. The latter ones based on the adverse judgments of DMs’ evaluations without taking personal judgments into account, such as the entropy method [15], the criteria importance through inter-criteria correlation [16]. It is argued that objective weight determinations are preferable since subjective methods are not always reliable.

Up to now, more and more scholars’ attention has been attracted to weight determinations under a hesitant fuzzy environment. Xu and Xia [17] studied the entropy, cross-entropy, similarity measures of HFEs and their relationships, then established an exact model using hesitant fuzzy entropy for generating weights. Lin et al. [18] derived the relative weights of hesitant fuzzy criteria by building a maximizing differential model. Xue et al. [19] determine weights in accordance with the differences among alternatives with dual hesitant fuzzy elements, which are computed by a new distance measure. Tan et al. [20] developed an extended Choquet-based TODIM to handle hesitant fuzzy interactive MCDM, where the weights of interactive criteria are measured by Shapley values. Liao et al. [21] presented two methods, the improved maximum entropy method, and the minimum average deviation method, to obtain the dynamic weighting vector.

Sometimes, a decision maker may prefer to express preferences by comparing a pair of alternatives and establish a specific preference relation. Xia and Xu [3] introduced the concept of HMPR, a matrix consisting of hesitant multiplicative elements (HMEs) that allows a decision maker to hesitate about several values of the membership. A natural issue related to HMPR is how to derive an associated priority, or equivalently, how to assign relative weights to each alternative. Zhang et al. [22] used a hesitant goal programming model to derive priorities based on α-normalization principle. Zhu et al. [23] defined two notions measuring the degree of consistency and consensus of HMPR, then a new method of hesitant preference analysis is found to derive holistic priorities from a stochastic point of view. Lin et al. [24] investigated the least square deviation method from a HMPR and applied this method for selecting a logistics service provider. A convergent iterative algorithm was also presented for generating priority vector. Bashir et al. [25] defined the concept of HMPR with self-confidence, and derived a collective priority based on heterogeneous hesitant preference relations with self-confidence. In order to obtain hesitant fuzzy priority weights, Meng et al. [26] provided a new consistency concept for HMPRs and put forward a consistency probability-based method. However, no researches have focused on the issue of prioritization with HMPR based on the cosine similarity measure.

2.2 Review on cosine similarity measure

Several formats of similarity measures have been examined [9, 10], such as Jaccard similarity, Kumar-Hassebrook similarity, Dice similarity and Cosine similarity. Among them, one of the most commonly used is cosine similarity, which defined as the inner product of two vectors divided by the product of their lengths. As the name suggests, its geometric meaning is to quantify the magnitude of the angle between two vectors. Salton and McGill [27] brought this concept into fuzzy sets theory, which fundamentally is a sort of coefficient or a quotient. For two fuzzy sets ${\tilde{F}}_{k} = {x, u_{{\tilde{F}}_{k}} (x) | x \in X}$ (k = 1, 2), the cosine similarity is, $C o s ({\tilde{F}}_{1}, {\tilde{F}}_{2}) = \frac{\sum_{x \in X} u_{{\tilde{F}}_{1}} (x) u_{{\tilde{F}}_{2}} (x)}{\sqrt{\sum_{x \in X} u_{{\tilde{F}}_{1}}^{2} (x)} \sqrt{\sum_{x \in X} u_{{\tilde{F}}_{2}}^{2} (x)}}$ (2.1)

Formula (2.1) indicates that the value getting larger when ${\tilde{F}}_{1}$ , ${\tilde{F}}_{2}$ evolving toward more similar, and approaches 1.0 when ${\tilde{F}}_{1} = {\tilde{F}}_{2}$ .

When uncertainty is present, it is important to consider the similarity or correlation between criteria in MCDM. Recently, Liao and Xu [28] presented a collection of cosine distance and similarity measures for hesitant fuzzy linguistic term set (HFLTSs) from the geometric point. Later, Liu et al. [29] combined the cosine similarity and the Euclidean distance together, which from both point views of algebra and geometry. Garg and Arora [30] offered some axioms of distance and similarity measures based on Hamming, Euclidean, and Hausdorff metrics with the dual hesitant fuzzy soft set. Moreover, different relations between them have also been provided. Under the context of probabilistic dual HFS (PDHFS), Garg and Kaur [31] further defined the informational energy and studied the covariance between two PDHFSs. Then a kind of weighted correlation coefficient was developed for handling MCDM problems. Dong et al. [32] advocate a linear programming model is constructed to objectively determine the criteria weights on the basis of adjusted cosine similarity.

As for other types of fuzzy sets, Yang et al. [33] studied the relationship between two HMSs by employing some correlation coefficient formulas. The weighted forms of the correlations and correlation coefficients are also developed. Liu et al. [29] suggested a cosine distance measure between neutrosophic HFLTSs according to the relationship between the similarity and distance measure. Wu et al. [34] introduced the definitions of similarity measure and correlation measure between probabilistic linguistic elements; afterwards they proposed a combined weight determination by integrating the subjective preference with the correlation coefficients.Additionally, Wang et al. [35] redefined the Frank operations of bipolar neutrosophic numbers and presented some operators that considering the interactions and interrelationships among criteria.

Based on the analysis of the state-of-the-art above, much research has been done on similarity measures using different fuzzy sets. Some scholars paid attention to the issue of weight determinations in MCDM, while others studied the problem of priority derivation in complicate preference relations. However, little research has yet considered how a criterion can be related to the overall assessment of alternatives. Particularly, the existing works on cosine similarity mainly focus on comparing different fuzzy information in algebra formulas and their properties. Few attention have been paid to its advantages in weight determination, especially from an overall view of mutual affinity among multi-criteria. This paper aims to develop an objective approach of weight determination for hesitant fuzzy MCDM and to derive priorities from a HMPR based on the cosine similarity measure. As can be seen, the weights information acquired by the proposed approach sounds more persuasive as they take mutual relevance among criteria into consideration.

3 Preliminaries

Definition 1. [2] Let X be a reference set, a HFS E on X is defined by $E = {〈 x, h_{E} (x) | x \in X 〉}$ (3.1) where h_E (x) is a set of some values in [0, 1], denoting the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu [36] called h_E a hesitant fuzzy element (HFE) and rearrange it in increasing order, i.e., $h_{E} = {h_{E}^{σ (t)} | t = 1, 2, \dots, l}$ , where $h_{E}^{σ (t)}$ be the t-th smallest value in h_E, and l is the length of h_E. Some operational laws of HFEs are described as below,

Definition 2. [36, 37] Given three HFEs h, h₁ and h₂, then

$h_{1} \oplus h_{2} = \cup_{h_{1}^{σ (t)} \in h_{1}, h_{2}^{σ (t)} \in h_{2}} {h_{1}^{σ (t)} + h_{2}^{σ (t)} - h_{1}^{σ (t)} h_{2}^{σ (t)}}$ ;

$h_{1} \otimes h_{2} = \cup_{h_{1}^{σ (t)} \in h_{1}, h_{2}^{σ (t)} \in h_{2}} {h_{1}^{σ (t)} h_{2}^{σ (t)}}$ ;

h₁ ⊖ h₂ = {ζ}, where $ζ = {\begin{matrix} \frac{h_{1}^{σ (t)} - h_{2}^{σ (t)}}{1 - h_{2}^{σ (t)}} if h_{1}^{σ (t)} ⩾ h_{2}^{σ (t)} and h_{2}^{σ (t)} \neq 1 \\ 0 otherwise \end{matrix}$

h₁ ø h₂ = {ζ}, where $ζ = {\begin{matrix} \frac{h_{1}^{σ (t)}}{h_{2}^{σ (t)}} if h_{1}^{σ (t)} ⩽ h_{2}^{σ (t)} and h_{2}^{σ (t)} \neq 0 \\ 1 otherwise \end{matrix}$

h^α = ∪ _h^σ(t)∈h {(h^σ(t)) ^α};

αh = ∪ _h^σ(t)∈h {1 - (1 - h^σ(t)) ^α}.

Remark 1. For two HFEs, their lengths may be different. Xia and Xu [36] suggested the short one should be extended until both of them have equal length. More detail can refer to [36].

Definition 3. [8] For a reference set X, the mean and variance of a HFE are given as, $\bar{h} = \frac{1}{l} \sum_{t = 1}^{l} h^{σ (t)}$ (3.2) $Va r (h) = \frac{1}{l} \sum_{t = 1}^{l} (h^{σ (t)} - \bar{h})$ (3.3)

Definition 4. [10] For two HFEs $h_{A} = {h_{A}^{σ (t)} | t = 1, 2, \dots, l_{A}}$ and $h_{B} = {h_{B}^{σ (t)} | t = 1, 2, \dots, l_{B}}$ on set X, the cosine similarity between h_A and h_B is defined as follows, $Cos (h_{A}, h_{B}) = \frac{\sum_{t = 1}^{l} h_{A}^{σ (t)} h_{B}^{σ (t)}}{\sqrt{\sum_{t = 1}^{l} (h_{A}^{σ (t)})^{2}} \sqrt{\sum_{t = 1}^{l} (h_{B}^{σ (t)})^{2}}}$ (3.4) where l = max {l_A, l_B}. In general, the larger the value of Cos (h_A, h_B), the bigger the similarity of them.

Similar to Definition 1, if we employ the Saaty’s 1∼9 ratio scale instead of the 0∼1 scale for denoting the membership degrees in HFS, we have,

Definition 5. [3] Let X be a reference set, a HMS is defined as $M = {〈 x, h_{m} (x) 〉 | x \in X}$ (3.5) where h_m (x) is a set of values in [1/9, 9] following the Saaty’s asymmetrical scale, implying possible membership degrees of the element x ∈ X to M. h_m (x) is called a hesitant multiplicative element (HME). Also, the concept of HMPR can be defined based on the HMEs.

Definition 6. [3] Let X ={ x₁, x₂, …, x_n } be a reference set, a HMPR is defined by a matrix Z = (z_ij) _n×n, where $z_{ij} = {γ_{ij}^{λ} | λ = 1, 2, \dots, | z_{ij} |}$ is a HME indicating all preference degrees of x_i over x_j, i, j = 1, 2, …, n, satisfies that $γ_{ij}^{λ} \times γ_{ji}^{λ} = 1, γ_{ii}^{λ} = {1}, | z_{ij} | = | z_{ji} |$ (3.6) where $γ_{ij}^{λ}$ is the λ-th smallest element of z_ij.

4 Methodology

4.1 Cosine similarity-based weights determination

Given alternative set A ={ A₁, A₂ , …, A_m } and criteria set X ={ x₁, x₂, …, x_n }, and A_j is judged on each of the n criteria and the judgment is described as an HFE, i.e., $h_{A_{j}} (x_{i}) = {h_{ji}^{σ (t)} | t = 1, 2, \dots, l}$ indicating l possible membership degrees of A_j under criterion x_i, j = 1, 2, …, m, i = 1, 2, …, n; For convenience, we denote h_{A
_j} (x_i) as h_ji for short. Notice that there exist two types of criteria, i.e., the benefit and the cost. As they differentiate only in normalization, we here just take benefit criteria for example. Thus, the hesitant fuzzy MCDM problem can be concisely written in the decision matrix as follows. $\begin{matrix} x_{1} x_{2} \dots x_{n} \\ H = \begin{matrix} A_{1} \\ A_{2} \\ ⋮ \\ A_{m} \end{matrix} (\begin{matrix} h_{11} & h_{12} & \dots & h_{n} \\ h_{21} & h_{22} & \dots & h_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ h_{m} & h_{m 2} & \dots & h_{mn} \end{matrix}) \\ W = (w_{1}, w_{2}, . . ., w_{n})^{T} \end{matrix}$ where W = (w₁, w₂ , …, w_n) ^T is the weight vector satisfying w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

In reality, due to the increasing complexity and uncertainty involved in practice, as well as the inherently subjective nature of human judgments, sometimes it is unrealistic and infeasible to acquire complete weight information of criteria. Several studies [12] summarized the structure forms of weights, which are roughly grouped into the following five basic forms: (1) A weak ranking: {w_{j
₁} ⩾ w_{j
₂} } , j₁ ≠ j₂; (2) A strict ranking: {w_{j
₁} - w_{j
₂} ⩾ ξ_j₁ j₂ } , j₁ ≠ j₂, where ξ_j₁ j₂ > 0 is a constant; (3) A ranking with multiples: {w_{j
₁} ⩾ ξ_j₁ j₂w_{j
₂} } , j₁ ≠ j₂, where 0 ⩽ ξ_j₁ j₂ ⩽ 1 is a constant; (4) An interval form: ${τ_{j}^{L} ⩽ w_{j} ⩽ τ_{j}^{U}}$ , where $0 ⩽ τ_{j}^{L} < τ_{j}^{U} ⩽ 1$ are constants; (5) A ranking of differences: {w_{j
₁} - w_{j
₂}⩾ w_{j
₃} - w_{j
₄} }, j₁ ≠ j₂ ≠ j₃ ≠ j₄. Without loss of generality, we assume the relative weights of criteria are complete unknown beforehand.

Assume the decision makers follow the axiom of expected utility theory. Since the decision matrix contains HFEs, we firstly introduce the hesitant fuzzy weighted averaging (HFWA) operator [36], which is defined as,

$\begin{matrix} HFWA (h_{1}, h_{2}, \dots, h_{n}) = \oplus_{i = 1}^{n} (w_{i} h_{i}) \\ = {1 - \prod_{i = 1}^{n} (1 - h_{i}^{σ (t)})^{w_{i}} | t = 1, 2, \dots, l} \end{matrix}$ (4.1) where l = max {l₁, l₂ , …, l_n}. The result fused by the operator (4.1) is still a HFE. Let g_j be the overall assessment value of A_j. Depended on the HFWA operator, we have,

$\begin{matrix} g_{j} & = \oplus_{i = 1}^{n} (w_{i} h_{j i}) = {1 - \prod_{i = 1}^{n} (1 - h_{ji}^{σ (t)})^{w_{i}} | t = 1, 2, \dots, l}, \\ j = 1, 2, \dots, m \end{matrix}$ (4.2)

According to expected utility theory, greater overall assessment with g_j implies a better alternative. In other words, the optimal alternative should be the one owning the greatest overall value.

In order to assess the importance of a criterion among the whole criteria set, i.e., the relative weight of x_i(i = 1, 2, …, n), in what follows, we intentionally drop out x_i from set X one at a time and see how its impact would be on the overall evaluation of an alternative. When x_i is taken away, the overall evaluation of A_j is redefined as,

$\begin{matrix} g_{j}^{(i)} & = \oplus_{k = 1, k \neq i}^{n} (w_{k} h_{jk}) \\ = {1 - \prod_{k = 1, k \neq i}^{n} (1 - h_{jk}^{σ (t)})^{w_{k}} | t = 1, 2, \dots, l}, \\ j = 1, 2, \dots, m \end{matrix}$ (4.3)

By Definition 4, the hesitant fuzzy cosine similarity between x_i and $g_{j}^{(i)}$ is computed as, $\begin{matrix} C_{i} = Cos (h_{ji}, g_{j}^{(i)}) = \frac{\sum_{j = 1}^{n} h_{ji} g_{j}^{(i)}}{\sqrt{\sum_{j = 1}^{n} (h_{ji})^{2}} \sqrt{\sum_{j = 1}^{n} (g_{j}^{(i)})^{2}}} \\ = \frac{\sum_{j = 1}^{n} (\sum_{t = 1}^{l_{i}} h_{ji}^{σ (t)} g_{j}^{(i) σ (t)})}{\sqrt{\sum_{j = 1}^{n} (\sum_{t = 1}^{l_{i}} {[h_{ji}^{σ (t)}]}^{2})} \sqrt{\sum_{j = 1}^{n} (\sum_{t = 1}^{l_{i}} {[g_{j}^{(i) σ (t)}]}^{2})}} \end{matrix}$ (4.4) where $l_{i} = max {l (h_{j i}), l (g_{j}^{(i)})}$ . Motivated by the implication of cosine similarity, one can infer that when C_i closes to 1, then the affinity between x_i and $g_{j}^{(i)}$ is nearly the same. Under this case, the exclusion of x_i has little impact on decision making, which illustrates that x_i is fairly unimportant comparing to the other criteria. Therefore, x_i should be assigned a relatively small weight. On the contrary, if C_i is small enough and closes to 0, it indicates that removal of x_i would have a big impact on alternative ranking, so a relatively large weight should be given. It is worth pointing out that this idea is agreed with the classical MCDM theory [38], namely, if a criterion has equal utilities for all alternatives, then it can be removed from the set of criteria without any effect on a final decision. From the above analysis, we suggest a mathematical form for assessing relative weights of the n criteria, $w_{i} = \frac{σ_{i} \sqrt{1 - C_{i}}}{\sum_{k = 1}^{n} σ_{k} \sqrt{1 - C_{k}}}, i = 1, 2, \dots, n$ (4.5) where σ_i is the standard deviation of x_i. According to Definition 3, σ_i can be calculated as follows,

$\begin{matrix} σ_{i} & = \sqrt{\frac{1}{l} \sum_{j = 1}^{m} \sum_{t = 1}^{l} (h_{ji}^{σ (t)} - {\bar{h}}_{ji})^{2}} \\ = \sqrt{\frac{1}{l} \sum_{j = 1}^{m} \sum_{t = 1}^{l} [h_{ji}^{σ (t)} - (\frac{1}{l} \sum_{t = 1}^{l} h_{ji}^{σ (t)})]^{2}} \\ i = 1, 2, \dots, n \end{matrix}$ (4.6)

Remark 2. The square-root imposed on (1 - C_i) is not compulsory, while it just to shorten the variance among different weights. If we get rid of this square-root, the variance between the biggest and the smallest weights will be greater.

Obviously, formula (4.5) is a system of n nonlinear formulas. Correspondingly, we need to calculate n time for weights determination. To overcome this drawback, we introduce the following deviationvariables ɛ_i, i = 1, 2, …, n, $ɛ_{i} = w_{i} - \frac{σ_{i} \sqrt{1 - C_{i}}}{\sum_{k = 1}^{n} σ_{k} \sqrt{1 - C_{k}}}, i = 1, 2, \dots, n$ (4.7)

Smaller values of ɛ_i(i = 1, 2, …, n) are always preferable. Based on such a modeling idea, we construct the nonlinear optimization model as follows, $\begin{matrix} min J = \sum_{i = 1}^{n} ɛ_{i}^{2} \\ s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i} = 1 \\ 0 ⩽ w_{i} ⩽ 1, i = 1, 2, \dots, n \end{matrix} \end{matrix}$ (4.8)

By solving model (4.8), the optimal weights of each criterion $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})^{T}$ can be captured. Then the overall evaluations of alternatives and their ranking can be obtained through an aggregated process. Since it is not the main focus in our discussion, we do not intend to clarify them anymore.

Remark 3. Model (4.8) can be extended if a DM wants to manipulate the weights. For example, if he hopes the weights satisfy w_i ⩽ w_i+1 (i = 2, 3), then this condition can be integrated into model (4.8) as, $\begin{matrix} min J = \sum_{i = 1}^{n} ɛ_{i}^{2} \\ s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i} = 1 \\ w_{i} ⩽ w_{i + 1}, i = 2, 3 \\ 0 ⩽ w_{i} ⩽ 1, i = 1, 2, \dots, n \end{matrix} \end{matrix}$

That is to say, a subjective preference can be involved as a constraint. In such a case, it turns to be an integrated subjective and objective method.

4.2 Cosine similarity-based priority derivation

According to [14], a crisp MPR A = (a_ij) _n×n is said to be consistent if a_ij = a_ika_kj holds for any i, j, k = 1, 2, …, n, which can be characterized by a priority W = (w₁, w₂, …, w_n) ^T with w_i ⩾ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ , such that a_ij = w_i/ - w_j, i, j = 1, 2, …, n. Similarity, for a HMPR Z = (z_ij) _n×n, since each element signifies a possible membership degree for pairwise comparison of x_i over x_j, a HMPR is called consistent if it satisfies, $w_{i} / - w_{j} = γ_{ij}^{1} or γ_{ij}^{2}, \dots, or γ_{ij}^{| z_{ij} |}, i, j = 1, 2, \dots, n$ (4.9)

Denoted $R (z_{ij}) = γ_{ij}^{1}$ or $γ_{ij}^{2}, \dots,$ or $γ_{ij}^{| z_{ij} |}$ , (4.9) can therefore be simplified as, $R (z_{ij}) = w_{i} / - w_{j}, i, j = 1, 2, \dots, n$ (4.10)

Let C_k be the cosine similarity between vector W = (w₁, w₂, …, w_n) ^T and the kth column of consistent HMPR, i.e., R (z_jk)(j = 1, 2, …, n), then we have,

$\begin{matrix} C_{k} & = Cos (R (z_{jk}), W) \\ = \frac{\sum_{j = 1}^{n} R (z_{jk}) w_{j}}{\sqrt{\sum_{j = 1}^{n} R (z_{jk})^{2}} \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}}, k = 1, 2, \dots, n \end{matrix}$ (4.11)

By (4.10), it is clear that C_k = 1; Otherwise, the HMPR Z is not consistent when C_k < 1, then (4.10) does not hold anymore. To produce a reliable priority, naturally, it expects W can be cosine correlated with R (z_jk) as much as possible. Therefore, C_k should be maximally approaching to 1.0; and (4.11) can be relaxed to the following optimization model,

$\begin{matrix} max C = \sum_{k = 1}^{n} C_{k} = \sum_{k = 1}^{n} \frac{\sum_{j = 1}^{n} R (z_{jk}) w_{j}}{\sqrt{\sum_{j = 1}^{n} R (z_{jk})^{2}} \sqrt{\sum_{j = 1}^{n} w_{j}^{2}}} \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} w_{j} = 1 \\ 0 ⩽ w_{j} ⩽ 1, j = 1, 2, \dots n \end{matrix} \end{matrix}$ (4.12)

To elicit an optimal analytical solution $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots, w_{n}^{*})^{T}$ , let

$\begin{matrix} R^{'} (z_{jk}) & = \frac{R (z_{jk})}{\sqrt{\sum_{i = 1}^{n} R (z_{ik})^{2}}} \\ = \frac{γ_{jk}^{1} or γ_{jk}^{2}, \dots, or γ_{jk}^{| z_{jk} |}}{\sqrt{\sum_{i = 1}^{n} {(γ_{ik}^{1} or γ_{ik}^{2}, \dots, or γ_{ik}^{| z_{ik} |})}^{2}}}, \\ j, k = 1, 2, \dots, n \end{matrix}$ (4.13) and define n new relative weights as, $w_{j}^{'} = \frac{w_{j}}{\sqrt{\sum_{k = 1}^{n} w_{k}^{2}}}, j = 1, 2, \dots, n$ (4.14)

Nonlinear model (4.12) can be rewritten as below, $\begin{matrix} max C = \sum_{j = 1}^{n} w_{j}^{'} (\sum_{k = 1}^{n} R^{'} (z_{jk})) \\ s . t . {\begin{matrix} \sum_{j = 1}^{n} (w_{j}^{'})^{2} = 1 \\ 0 ⩽ w_{j}^{'} ⩽ 1, j = 1, 2, \dots, n \end{matrix} \end{matrix}$ (4.15)

To solve the above model, we utilize the Lagrange multiplier method.

$\begin{matrix} L (C, φ) = & \sum_{j = 1}^{n} (\sum_{k = 1}^{n} R^{'} (z_{jk})) w^{'} \\ + φ (\sum_{j = 1}^{n} (w_{j}^{'})^{2} - 1) \end{matrix}$ (4.16) where φ is a real number, signifying the Lagrange multiplier variable. The partial derivatives of L (C, φ) can be calculated as follows,

${\begin{matrix} \frac{\partial L}{\partial w_{j}^{'}} = \sum_{k = 1}^{n} R^{'} (z_{j k}) + 2 φ w_{j}^{'} = 0 \\ \frac{\partial L}{\partial φ} = \sum_{j = 1}^{n} (w_{j}^{'})^{2} - 1 = 0 \end{matrix}, j = 1, 2, \dots, n$ (4.17)

It can be derived from (4.17) that

$\begin{matrix} w_{j}^{'} & = \frac{- \sum_{k = 1}^{n} R^{'} (z_{jk})}{2 φ} \\ = \frac{- \sum_{k = 1}^{n} (γ_{ij}^{1} or γ_{ij}^{2}, \dots, or γ_{ij}^{| z_{ij} |})}{2 φ \sqrt{\sum_{k = 1}^{n} {(γ_{ij}^{1} or γ_{ij}^{2}, \dots, or γ_{ij}^{| z_{ij} |})}^{2}}} \end{matrix}$ (4.18)

Due to the fact $\sum_{j = 1}^{n} (w_{j}^{'})^{2} = 1$ , $w_{j}^{'} ⩾ 0$ , we have, $φ = - \frac{1}{2} \sqrt{\sum_{j = 1}^{n} (\sum_{k = 1}^{n} R^{'} (z_{jk}))^{2}}$ (4.19)

Putting this formula into (4.18), it yields $(w_{j}^{'})^{*} = \frac{\sum_{k = 1}^{n} R^{'} (z_{jk})}{\sqrt{\sum_{i = 1}^{n} (\sum_{k = 1}^{n} R^{'} (z_{jk}))^{2}}}, j = 1, 2, \dots, n$ (4.20)

According to (4.14), by summing both sides of this equation, it follows, $\sum_{j = 1}^{n} [(w_{j}^{'})^{*} \sqrt{\sum_{k = 1}^{n} (w_{k}^{*})^{2}}] = 1$ (4.21)

Thus, $w_{j}^{*} = \frac{(w_{j}^{'})^{*}}{\sum_{k = 1}^{n} (w_{k}^{'})^{*}}, j = 1, 2, \dots, n$ (4.22)

To sum up, the determination process of the priority vector involves the following steps:

Step 1. Normalize the HMPR Z = (z_ij) _n×n according to (4.13);

Step 2. Establish the cosine similarity-based optimization model (4.15);

Step 3. Formalize the transformed priorities $w_{j}^{'}$ (j = 1, 2, …, n) by (4.20);

Step 4. Determine the numerical value of $w_{j}^{'}$ (j = 1, 2, …, n) by solving model (4.15);

Step 5. Obtain the priority weighting vector W = (w₁, w₂, …, w_n) ^T by (4.22).

Note that if Z = (z_ij) _n×n is degenerated to a MPR, then the (4.22) reduces to $w_{j}^{*} = 1 / \sum_{i = 1}^{n} z_{ij}$ , which is the same in Kou’s method [39]. Based on the above analysis, our method can manage both HMPRs and MPRs, whereas the method in [39] just is a special case of our method that handles crisp preference relation.

5 Numerical example

In this section, two numerical examples are provided to illustrate the proposed approach.

Example 1. Suppose there is a decision problem involving the evaluation of four departments A = {A₁, A₂, A₃, A₄} in a university. After a detailed survey, the expert group offers the following hesitant fuzzy decision matrix, as shown in Table 1 (adapted from [17]).

Table 1
Hesitant fuzzy decision matrix

x ₁ x ₂ x ₃ x ₄

A ₁ (0.2, 0.4, 0.7) (0.1, 0.2, 0.5, 0.7) (0.2, 0.3, 0.5, 0.7, 0.8) (0.1, 0.4, 0.6)

A ₂ (0.4, 0.6, 0.7) (0.1, 0.2, 0.4, 0.6) (0.3, 0.4, 0.6, 0.8, 0.9) (0.1, 0.2, 0.4)

A ₃ (0.2, 0.3, 0.6) (0.3, 0.4, 0.5, 0.9) (0.2, 0.4, 0.6, 0.7, 0.8) (0.3, 0.4, 0.8)

A ₄ (0.2, 0.3, 0.5) (0.2, 0.3, 0.5, 0.7) (0.4, 0.6, 0.7, 0.8, 0.9) (0.1, 0.2, 0.7)

	x ₁	x ₂	x ₃	x ₄
A ₁	(0.2, 0.4, 0.7)	(0.1, 0.2, 0.5, 0.7)	(0.2, 0.3, 0.5, 0.7, 0.8)	(0.1, 0.4, 0.6)
A ₂	(0.4, 0.6, 0.7)	(0.1, 0.2, 0.4, 0.6)	(0.3, 0.4, 0.6, 0.8, 0.9)	(0.1, 0.2, 0.4)
A ₃	(0.2, 0.3, 0.6)	(0.3, 0.4, 0.5, 0.9)	(0.2, 0.4, 0.6, 0.7, 0.8)	(0.3, 0.4, 0.8)
A ₄	(0.2, 0.3, 0.5)	(0.2, 0.3, 0.5, 0.7)	(0.4, 0.6, 0.7, 0.8, 0.9)	(0.1, 0.2, 0.7)

Assume that the relative weight w_i of each criterion x_i(i = 1, 2, 3, 4) are completely unknown. Table 2 shows the normalized hesitant fuzzy matrix whose elements are with equal length.

Table 2

Normalized hesitant fuzzy decision matrix

	x ₁	x ₂	x ₃	x ₄
A ₁	(0.2, 0.2, 0.2, 0.4, 0.7)	(0.1, 0.1, 0.2, 0.5, 0.7)	(0.2, 0.3, 0.5, 0.7, 0.8)	(0.1, 0.1, 0.1, 0.4, 0.6)
A ₂	(0.4, 0.4, 0.4, 0.6, 0.7)	(0.1, 0.1, 0.2, 0.4, 0.6)	(0.3, 0.4, 0.6, 0.8, 0.9)	(0.1, 0.1, 0.1, 0.2, 0.4)
A ₃	(0.2, 0.2, 0.2, 0.3, 0.6)	(0.3, 0.3, 0.4, 0.5, 0.9)	(0.2, 0.4, 0.6, 0.7, 0.8)	(0.3, 0.3, 0.3, 0.4, 0.8)
A ₄	(0.2, 0.2, 0.2, 0.3, 0.5)	(0.2, 0.2, 0.3, 0.5, 0.7)	(0.4, 0.6, 0.7, 0.8, 0.9)	(0.1, 0.1, 0.1, 0.2, 0.7)

As per (4.3), we aggregate the hesitant fuzzy overall evaluation $g_{j}^{(i)}$ for each alternative A_j that neglects x_i(i, j = 1, 2, 3, 4). Some of these results are written as, $\begin{matrix} g_{1}^{(1)} = {1 - (0.9 w_{2} \times 0.8 w_{3} \times 0.9 w_{4}), \\ 1 - (0.9 w_{2} \times 0.7 w_{3} \times 0.9 w_{4}), \\ 1 - (0.8 w_{2} \times 0.5 w_{3} \times 0.9 w_{4}), \\ 1 - (0.5 w_{2} \times 0.3 w_{3} \times 0.6 w_{4}), \\ 1 - (0.3 w_{2} \times 0.2 w_{3} \times 0.4 w_{4})}; \\ ⋮ \\ g_{4}^{(4)} = {1 - (0.8 w_{1} \times 0.6 w_{2} \times 0.8 w_{3}), \\ 1 - (0.8 w_{1} \times 0.4 w_{2} \times 0.8 w_{3}), \\ 1 - (0.8 w_{1} \times 0.3 w_{2} \times 0.7 w_{3}), \\ 1 - (0.7 w_{1} \times 0.2 w_{2} \times 0.5 w_{3}), \\ 1 - (0.5 w_{1} \times 0.1 w_{2} \times 0.3 w_{3})}; \end{matrix}$

By (4.4), the hesitant fuzzy cosine similarities can be calculated. Afterwards, the standard deviations of each criterion according to (4.6) are obtained as σ₁ = 0.3033, σ₂ = 0.4271, σ₃ = 0.42433 and σ₄ = 0.3847. Plugging C_i and σ_i(i = 1, 2, 3, 4) into model (4.8), we obtain the priority: $W^{*} = (0.2083, 0.1821, 0.2852, 0.3238) .$

Below we compare our approach with the method in [17]. Xu and Xia [17] derived the relative weights via both the entropy method and the maximizing deviation method. Solving these models n times, each time for one attribute’s weight, which resulted in W ′ = (0.1957, 0.2013, 0.1611, 0.4419), and W ″ = (0.2629, 0.2229, 0.2057, 0.3086), respectively. To visualize the comparison results, the relative weights that acquired in [17] and this paper are plotted into Fig. 1. It is straightforward to see that w₄ is assigned the largest weight by these three approaches while other weights distributions are a bit different, and a remarkable one is w₃ > w₂ in our method while it is reversed in the other two ways.

Fig.1

Result comparison for Example 2.

To find the weight differences between x₂ and x₃, let us consider the means and variances of them. Since their criterion values on alternatives are in the form of HFS, the means and variances of x₂ and x₃ can be computed by (3.2) and (3.3), which listed in Table 3. Generally, apart from A₁ the variances of x₃ on the alternative set are much greater; thus a bigger weight should be given to it as discussed in Section 3. This result is in accordance with the outcome of our method.

Table 3

Means and variance of x₂ and x₃

	A ₁	A ₂	A ₃	A ₄
S₂	0.375	0.325	0.525	0.425
\|V₂\|	0.175	0.0113	0.1683	0.0717
S₃	0.50	0.60	0.54	0.68
\|V₃\|	0.025	0.375	0.0317	0.7283

Let us look into the differences between w₂ and w₃ in term of the entropy measure. Table 4 is a part of entropy matrix taken from [17].

Table 4

Entropy matrix of x₂ and x₃

	A ₁	A ₂	A ₃	A ₄
x ₂	0.9350	0.8750	0.9750	0.9750
x ₃	0.9920	0.9600	0.9880	0.8680

Clearly, except for the entropy measure of A₄ (i.e., 0.8680 < 0.9750), all of the entropies on thealternatives of x₃ are greater than those of x₂. According to [40], a higher entropy implies greater randomness of an attribute. From the analysis in Section 3, we can infer that a criterion with disorder distribution among alternatives should be assigned a bigger weight. This also verifies the validity and robustness of the proposed method.

In what follows, we conduct a sensitivity analysis based on the method in Wang and Luo’s [11] to see in what range of weights can keep the ranking order of criteria unchanged. Such an analysis offers a decision reference for a person to make a sensible decision. Similar to the formulas of (20) and (21) in [11], we obtain the ratio parameter $η_{1}^{*} = 0.8523$ , and have weight intervals of each one as,

w₁ ∈ [0.0915, 0.4238], w₂ ∈ [0.0856, 0.3987],

w₃ ∈ [0.1172, 0.3515], w₄ ∈ [0.2360, 0.4088],

Namely, when the weights of multi-criteria vary under the intervals above, the ranking order of the four departments is stable. Analogously, according to the formulas (27) and (21) in [11], we have $η_{1}^{*} = 0.9745$ , and weight intervals as,

w₁ ∈ [0.0902, 0.4366], w₂ ∈ [0.0827, 0.4012],

w₃ ∈ [0.1065, 0.3729], w₄ ∈ [0.2058, 0.4280],

So when the weights fluctuate within the above intervals, A₄ remains to be the optimal one.

Example 2. Economic and financial globalization is increasing substantially and is creating new opportunities for enterprises. Efficient green supply chain management turns out to be critical for a corporation to improve its core competence in the global market. This example studies a green supplier selection problem, where four suppliers X = {x₁, x₂, x₃, x₄} are invited as candidates. An expert committee compares each pair of candidates and provides the following HMPR (adapted from [3]):

$Z = {\begin{matrix} {1} & {\frac{1}{6}, \frac{1}{4}, \frac{1}{2}, 5} & {1, \frac{3}{4}, 3} & {\frac{1}{6}, \frac{5}{7}, \frac{5}{9}} \\ {6, 4, 2, \frac{1}{5}} & {1} & {1, 2} & {\frac{4}{5}, 3, 7} \\ {1, \frac{4}{3}, \frac{1}{3}} & {1, \frac{1}{2}} & {1} & {\frac{1}{3}, \frac{5}{7}, 3, 4} \\ {6, \frac{7}{5}, \frac{9}{5}} & {\frac{5}{4}, \frac{1}{3}, \frac{1}{7}} & {3, \frac{7}{5}, \frac{1}{3}, \frac{1}{4}} & {1} \end{matrix}}$

We use the cosine similarity-based priority method to solve this problem. First, the HMPR Z is normalized by (4.13) to generate the R′ (z_ij) _4×4; Then we integrate the obtained R′ (z_ij) _4×4 and build the optimization model by (4.15), and then solve this model using Lingo 11.0 to determine the weights W ′ = (0.4536, 0.5522, 0.4786, 05102) ^T. According to (4.22), the optimal weighting vector turns out to be $\begin{matrix} W & = (w_{1}, w_{2}, w_{3}, w_{4})^{T} \\ = (0.2274, 0.2769, 0.2400, 0.2558)^{T}, \end{matrix}$ which gives the ranking of w₂ > w₄ > w₃ > w₁. Correspondingly, the green supplier x₂ is the best one.

To validate the feasibility of the proposed model, we compare the result with the ones obtained by the methods in [22, 23]. Zhang and Wu [22] suggested a hesitant multiplicative optimization model and a convex combination method, which based on the principles of α-normalization and β-normalization, respectively, to derive the priorities from HMPRs. The former method can generate a crisp priority vector which reduces a HMPR to a corresponding MPR. Meanwhile, Zhu et al. [23] utilized the Monte Carlo simulation to produce the priority vector from a stochastic point of view, where the weights information involves interpretations of probability. Via the developed formula of holistic evaluation, the priorities of objectives can be obtained. The result comparisons of the two methods are listed in Table 5.

Table 5

Performance comparison

Approach	Priority vector	Ranking order
Zhang and Wu	(0.2231, 0.2605, 0.2457, 0.2689)	x₄ ≻ x₂ ≻ x₃ ≻ x₁
Zhu et al.	(0.2319, 0.2812, 0.2362, 0.2507)	x₂ ≻ x₄ ≻ x₃ ≻ x₁

From Table 5, it is observed that the priorities of the two methods have a minor difference, where the ranking order by Zhu et al. is same as our method does, yet the alternative x₄ is superior to x₂ in Zhang and Wu’s method. To check the result, let us look back on the original data. From the HMPR Z, we find HMEs of x₂ (the second row) respectively are ${6, 4, 2, \frac{1}{5}}$ , {1}, {1, 2}, ${\frac{4}{5}, 3, 7}$ . Most of these values are greater than or equal to 1.0, which illustrates green supplier x₂ outweighs the other candidates in this case. This indicates the outcome produced by our approach is more persuasive. Our method considers not only the hesitant multiplicative information but also the correlation between column vectors in HMPR and the corresponding priority vectors. On the other hand, the modelling mechanism of the proposed cosine similarity-based method is clearer and more time-efficient when compared with Zhu et al.’s method since there is no need to consider the probable stochastic error by the estimation of the Monte Carlo method.

6 Conclusion

Owning to fuzzy and hesitant naturally inherent in human perception. Hesitant fuzzy (or multiplicative) sets show clear advantages over traditional fuzzy sets in describing imprecise information since the memberships are featured by a set of possible elements. As an important and pragmatic index for relationship measure, the cosine similarity has been broadly applied in decision making. In this paper, we proposed an approach based on hesitant fuzzy cosine similarity for determining relative weights of criteria in MCDM. Furthermore, this method is extended for the prioritization from HMPRs. Specifically, the cosine similarities are determined by getting rid of each criterion from the overall estimation of alternatives. When cosine similarity for a criterion proves to be small and approaches zero, then the removal of this criterion will have an effect on MCDM, so a relatively large weight must be assigned. Besides, an optimization model is established with a HMPR by maximizing the cosine similarity between the priority vector and each column of the given HMPR, and then the priority can be obtained by solving this model. Both the proposed approaches have been illustrated with numerical examples. Compared with the existing methods, the weights or priorities produced by our methods are more objective and reasonable.

Note that there are still some limitations in this paper. For instance, we mainly focus on the process of priority derivation with a HMPR but not considering its consistent measure. As linguistic decision making is essential and common in group decision analyses. We want to integrate the proposed approach in group decision making with linguistic information [41, 42] in the near future. Additionally, how to extend or modify the cosine similarity-based method in practical decision problems, such as the evaluation of container port [43] or matching of investment projects [44], is a quite interesting and challenging topic that worthy of future study.

Conflicts of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Footnotes

Acknowledgments

The authors are very grateful to the five anonymous referees and the editor for their constructive comments that have led to an improved version of this paper. The work was supported by the National Social Science Foundation of China (19BGL092).

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