Three-way group decisions based on multigranulation hesitant fuzzy decision-theoretic rough set over two universes

Abstract

Three-way decisions, deduced from decision-theoretic rough set (DTRS), provide useful approaches for uncertain decision-making problems from a new perspective. Considering situations where decision-makers hesitate among several evaluation values, the hesitant fuzzy set, an innovative extension of fuzzy set, is introduced into multigranulation DTRS under multi-criteria group decision-making (MCGDM) environment. More specifically, we incorporate DTRS with hesitant fuzzy information into MCGDM in this paper, and explore the related decision-making mechanism. Firstly, the variable precision multigranulation hesitant fuzzy DTRS over two universes is defined by utilizing hesitant relation and conditional probability of a hesitant fuzzy event, and the associated decision rules and properties are derived and carefully investigated. Accordingly, as two special types, optimistic and pessimistic multigranulation hesitant fuzzy DTRS over two universes are constructed similarly at the same time. Then, different from loss functions with fixed values in most of the existing DTRS models, the loss functions in the paper are not given directly but calculated from evaluation values expressed by hesitant fuzzy elements. With the aid of the distance measure of hesitant fuzzy elements, the calculation of loss functions is presented. Furthermore, the three-way group decision-making model based on multigranulation hesitant fuzzy DTRS over two universes is established to address MCGDM problems, and key steps of this model are designed. Finally, we elaborate the application of the model by an example of green supplier selection problem.

Keywords

Three-way decisions rough set over two universes hesitant fuzzy rough set multigranulation multi-criteria group decision-making green supplier selection

1 Introduction

As an innovative extension of fuzzy sets, the concept of hesitant fuzzy sets was originally proposed by Torra and Narukawa [23, 24] to address uncertain decision situations, where decision-makers are hesitant and irresolute among several possible evaluation values [38]. Ever since its introduction, hesitant fuzzy sets have been extensively studied and widely applied to various practical problems. For instance, Xia and Xu [27] proposed several aggregation operators for hesitant fuzzy information and offered their application in handling decision-making problems. Zhang [37] studied the hesitant fuzzy linguistic term set, which is fused into rough sets over two universes, and discussed its application to person-job fit, compared to classical method. Yu et al. [33] put forth a new approach to MCGDM issues with unbalanced hesitant fuzzy linguistic term set by taking the psychological behavior of decision makers into consideration. Li et al. [8] initially introduced the concept of hesitance degree, and with the aid of this developed a series of novel distances of hesitant fuzzy sets. Zhang et al. [42] explored the improvement of the additive consistency of a hesitant fuzzy preference relation by constructing several mixed 0-1 linear programming models. Zhang et al. [43] developed a novel method to determine a priority weight vector deduce from an incomplete hesitant fuzzy preference relation by utilizing the logarithmic least squares approach.

Another powerful mathematical approach to addressing uncertainty, imprecision and incompleteness in decision-making is Pawlak’s rough set [13], owing to its advantages in attributes selection and rules deduction. Decision-theoretic rough set (DTRS), a generalization of the classical rough set, was firstly proposed by Yao [29 , 32] in light of Bayesian risk decision and thus has been playing a key role in bridging the rough set theory and risk decision theory [10]. More specifically, DTRS offers a reasonable semantic interpretation from the perspective of determination of the thresholds values in probability rough set [30, 32]. In view of positive region, boundary region and negative region in rough set, DTRS captures the decision semantic explanation and accordingly deduces three-way decisions, i.e., the acceptance, the non-commitment and the rejection [10 , 31]. Presently, research on three-way decisions based on DTRS has formed many important issues and been studied by considerable scholars. Sun et al. [18] introduced the idea of three-way decisions based on DTRS into multiple attributes group decision under linguistic information environment. Hu [4] initially defined the three-way decision space and systematically explored the relative properties. Li and Zhou [7] took the decision-maker’s preferences into consideration and designed the corresponding three-way decision model based on DTRS. Jia and Liu [5] defined the concept of relative loss functions and further determined the calculation of loss functions in three-way decisions. Azam et al. utilized game theory idea to determine the evaluation function and decision conditions of three-way decisions with DTRS [1].

Group decision-making, a crucial branch of decision-making, aims to aggregate a group of decision-makers’ preferences to determine the optimal collective alternative solution to a decision issue [40]. A wide variety of issues has been explored, under group decision-making setting, such as venture investment [11], consensus efficiency [40], intuitionistic multiplicative preference relations [41], linguistic group decision-making model with hesitant fuzzy evaluations [26], heterogeneous preference structures [35], consensus reaching process with hesitant fuzzy linguistic term sets [36]. In particular, multigranulation rough set originated by Qian et al. [14, 15] is also a key approach to handling group decision-making problems. It extends the range of application of single granularity rough set, where a binary relation defined on universe is considered to be a granularity. When utilizing multigranulation rough set to solve group decision problems, each decision maker’s preference is regarded as an independent information system and its relative granular structure is deduced accordingly [38]. Thus, the group decision makers’ different preferences could be aggregated efficiently by multigranulation rough set. Furthermore, optimistic and pessimistic multigranulation rough sets model are two special cases, i.e., completely risk-preferring and risk-averse in decision-making practice [15, 19]. Presently, owing to the advantage in group decision-making, multigranulation rough set has attracted a wide range of studies theoretically and practically [17 , 38]. Research of the combination of multigranulation rough set and DTRS has been studied extensively [3 , 20], but less effort on the exploration of hesitant fuzzy MCGDM is made by combining multigranulation rough set and DTRS over two universes.

In view of the merit of multigranulation DTRS in aggregating decision-makers’ preferences and the flexibility of hesitant fuzzy set in conveying decision maker’s understandings and knowledge, exploring hesitant fuzzy group decision-making issues by utilizing multigranularity three-way decisions is meaningful. With the above discussions, in this paper, the variable precision multigranulation hesitant fuzzy DTRS over two universes is put forth and applied to a MCGDM problem, green supplier selection. Specifically, the issue of green supplier selection depends on judgements by a group of decision-makers regarding performance of green suppliers on a set of criteria. Uncertainty between the green suppliers and criteria, which belong to two different but relevant universes, is taken into consideration in the decision process. Meanwhile, the self-evaluation set, representing the actual requirement of a green supplier selector, is regarded as a hesitant fuzzy set on the criteria in the setting. In light of the established variable precision multigranulation hesitant fuzzy DTRS over two universes, an efficient three-way group decision-making approach to addressing the above green supplier selection can be developed.

The aim of this paper is to propose new risk decision mechanisms and design a new three-way decision model based on multigranulation DTRS under hesitant fuzzy information environment. This work makes the following main contribution: (1) By introducing the idea of hesitant fuzzy set into DTRS, the variable precision multigranulation hesitant fuzzy DTRS is established. The corresponding rules are induced and some relative properties are examined. As two special cases, pessimistic and optimistic multigranulation hesitant fuzzy DTRS over two universes are also given. (2) By the definition of distance between two hesitant fuzzy elements, the calculation method of loss functions that are not fixed but derived from hesitant fuzzy evaluation values, is determined. (3) A new group decision-making model based on hesitant fuzzy DTRS over two universes is developed.

The rest of this paper is organized as follows: Section 2 reviews some basic concept of hesitant fuzzy set and rough set. In Section 3, multigranulation hesitant fuzzy DTRS over two universes is constructed and discussed. In Section 4, the calculation of relative loss function and aggregated loss function is given. In Section 5, the three-way decision group decision-making model based on hesitant fuzzy DTRS over two universes is presented. Also, an example of green supplier selection under MCGDM is utilized to illustrate the model. Section 6 concludes the study and outlines the future work.

2 Preliminaries

In this section, basic concepts of hesitant fuzzy set and rough set are briefly reviewed.

Definition 2.1.[23, 24] Let U be a non-empty universe. A hesitant fuzzy set E is in terms of a function that when applied to U returns a subset of [0, 1]. For simplicity, Xia and Xu [27] proposed the following mathematical symbol of the hesitant fuzzyset $E = {< x, h_{E} (x) > | x \in U},$ where h_E (x) is a set of some different finite values in [0, 1], indicating the possible membership degrees of the element x ∈ U to the set U. h (x) = h_E (x) is called a hesitant fuzzy element.

We use HF (U) to denote the set of all hesitant fuzzy sets on U. For any two hesitant fuzzy sets A, B ∈ HF (U), some operations of the two hesitant fuzzy elements h_A (x) and h_B (x), where x ∈ U, are defined as follows by Torra [23, 24]:

∼h_A (x) = {1 - e₁ :∀ e₁ ∈ h_A (x)} ;

h_A (x)⊕ h_B (x) = {e₁ + e₂ - e₁e₂ : ∀ e₁ ∈ h_A (x) , ∀ e₂ ∈ h_B (x)} ;

h_A (x)⊗ h_B (x) = {e₁e₂ : ∀ e₁ ∈ h_A (x) , ∀ e₂ ∈ h_B (x)} ;

θh_A (x) = {1 - (1 - e₁) ^θ : ∀ e₁ ∈ h_A (x)} , θ is constant.

To compare hesitant fuzzy elements, the following criterion is introduced by Xia and Xu 27.

Definition 2.2. [27] Let U be a non-empty finite universe. For any hesitant fuzzy element h (x) , x ∈ U, the score function of h (x) is defined as $s (h_{A} (x)) = \frac{1}{l (h_{A} (x))} \sum_{e \in h_{A} (x)} e,$ where l (h_A (x)) is the length of l (h_A (x)), namely the number of element in h_A (x) . For any two hesitant fuzzy elements h_A (x) and h_B (x), if s (h_A (x))>s (h_B (x)), then h_A (x)> h_B (x) ; if s (h_A (x)) < s (h_B(x)), then h_A (x)< h_B (x) ; If s (h_A (x)) = s (h_B (x)), then h_A (x) = h_B (x) .

In some cases, for any two hesitant fuzzy elements, their lengths are not the same. To operate in computation, the shorter one needs to be implemented by adding the minimum value, the maximum value, or any other value in it until it has the same length as that of the longer one [28]. In this work, the extension is always performed by adding the maximum value to the shorter one, and this extension will be used in the following without further explanation.

Definition 2.3. [23, 24] For any two hesitant fuzzy sets A and B ∈ HF (U), if h_A (x) ⪯ h_B (x) holds for any x ∈ U such that $h_{A} (x) ⪯ h_{B} (x) \Leftrightarrow h_{A}^{j} (x) \leq h_{B}^{j} (x), j \in l_{x},$ then A is called a hesitant fuzzy subset of B, and it is denoted by A ⊆ B, where $h_{A}^{j}$ and $h_{B}^{j}$ are jth value in h_A (x) and h_B (x) respectively, and l_x = max {l (h_A (x)) , l (h_B (x))} .

In what follows, we briefly review the concept of rough set.

Definition 2.4. [13] Let U be a non-empty finite universe and R be an equivalence relation of U × U . The equivalence relation R induces a partition of U, which is denoted by U/R. Then (U, R) is the Pawlak approximation space. For any subset X ⊆ U, its lower and upper approximation with respect to (U, R) are defined respectively: $\begin{matrix} \underline{R} (X) = {x \in U | [x] \subseteq X}, \\ \bar{R} (X) = {x \in U | [x] \cap X \neq \emptyset} . \end{matrix}$ where [x] denotes the equivalence class containing x. Based on the approximations of X, the universe U can be divided into three disjoint regions: the positive region Pos (X), the boundary region Bnd (X) and negative region Neg (X) as follows: $\begin{matrix} Pos (X) = \underline{R} (X), Bnd (X) = \underline{R} (X) - \bar{R} (X) \\ Neg (X) = U - \bar{R} (X) \end{matrix}$

One can deduce certain rules from Pos (X) and Neg (X), i.e., an element x ∈ Pos (X) is certainly accepted as a member of X and an element x ∈ Neg (X) is certainly rejected as a member of X. For x ∈ Bnd (X), one can not conclude whether or not x is a member of X. The structure of these three regions leads to the idea of three-way decisions [31].

3 Multigranulation hesitant fuzzy DTRS over two universes

Definition 3.1. [39] Let U, V be two non-empty finite universes. A hesitant fuzzy relation R between U and V is a hesitant fuzzy subset of U × V, and R is defined by: $R = {< (x, y), h_{R} (x) (y) > | (x, y) \in U \times V},$ where h_R (x) (y) is a set of some different finite values in [0, 1], representing the possible membership degrees between x and y.

Definition 3.2. Let U, V be two non-empty finite universes, R ∈ U × V is a hesitant fuzzy relation between U and V. The binary hesitant fuzzy relation class is a hesitant fuzzy subset of universe V and expressed by $\begin{matrix} H_{R} (x) (y) & = \frac{h_{R} (x) (y_{1})}{y_{1}} + \frac{h_{R} (x) (y_{2})}{y_{2}} + \dots \\ + \frac{h_{R} (x) (y_{n})}{y_{n}}, n = | V | . \end{matrix}$

By the concepts above, we can show how to compute the conditional probability of a hesitant fuzzy event given the hesitant fuzzy description as follows.

Definition 3.3. Let U, V be two non-empty finite universes. Suppose R ∈ HF (U × V) is an arbitrary hesitant fuzzy relation between U and V. For any x ∈ U, y ∈ V, and A ∈ HF (V), P (A|H_R (x) (y)) is the conditional probability of A given the hesitant fuzzy description H_R (x) (y). Define $P (A | H_{R} (x) (y)) = \frac{\sum_{y \in V} (s (h_{A} (y)) \cdot s (h_{R} (x) (y)))}{\sum_{y \in V} s (h_{R} (x) (y))},$ and the following conditions hold:

0≤ P (A|H_R (x) (y)) ≤1 ;

If A, B ∈ HF (U), and A ⊆ B, then P (A|H_R (x) (y))≤ P (A|H_R (x) (y)) ;

P (A^c|H_R (x) (y)) =1 - P (A^c|H_R (x) (y)) .

In fact, ∀x ∈ U, $\begin{matrix} P (A^{c} | H_{R} (x) (y) \\ = \frac{\sum_{y \in V} (s (h_{A^{c}} (y)) \cdot s (h_{R} (x) (y)))}{\sum_{y \in V} s (h_{R} (x) (y))} \\ = \frac{\sum_{y \in V} (1 - s (h_{A} (y)) \cdot s (h_{R} (x) (y)))}{\sum_{y \in V} s (h_{R} (x) (y))} \\ = 1 - P (h_{A} | H_{R} (x) (y)) . \end{matrix}$

In what follows, in light of Bayesian decision process [29 , 32], the hesitant fuzzy DTRS over two universes is given.

Let U, V be two non-empty finite universes. Ω = {A, A^c} stands for the set of state, denoting that an object is in state A or not in state A, respectively. d = {d₁, d₂, d₃} stands for the set of decision actions in classifying an object x ∈ U, in which d₁, d₂ and d₃ indicate x ∈ Pos (A) , x ∈ Neg (A) and x ∈ Bnd (A), respectively. λ (d₁|A), λ (d₂|A) and λ (d₃|A) stand for the loss function about the risk or cost incurred for taking decision d₁, d₂ and d₃, respectively, when x belongs to A. Similarly, λ (d₁|A^c), λ (d₂|A^c) and λ (d₃|A^c) stand for the loss function about the risk or cost incurred for taking the same decision actions, when x dose not belong to A. For simplicity, let us denote $λ_{11} = λ (d_{1} | A), λ_{21} = λ (d_{2} | A), λ_{31} = λ (d_{3} | A),$ $λ_{12} = λ (d_{1} | A^{c}), λ_{22} = λ (d_{2} | A^{c}), λ_{32} = λ (d_{3} | A^{c}),$ as shown in Table 1. And P (A|H_R (x)) is the conditional probability of A for the object x given the hesitant fuzzy description H_R (x).

Table 1
Loss function

A A ^c

d ₁ λ ₁₁ λ ₁₂

d ₃ λ ₃₁ λ ₃₂

d ₂ λ ₂₁ λ ₂₂

	A	A ^c
d ₁	λ ₁₁	λ ₁₂
d ₃	λ ₃₁	λ ₃₂
d ₂	λ ₂₁	λ ₂₂

Thus, for any alternative with the hesitant fuzzy description H_R (x), the expected loss R (d_p|H_R (x)) of taking decision action d_p can be expressed as: $\begin{matrix} R (d_{p} | H_{R} (x)) = λ (d_{p} | A) P (A | H_{R} (x)) \\ + λ (d_{p} | A^{c}) P (A^{c} | H_{R} (x)), p = 1, 2, 3 . \end{matrix}$

The Bayesian decision process suggests the the minimum-risk decision rules (P), (N) and (B) in the following:

(P) Decide Pos (A) if R (d₁|H_R (x)) ≤ R (d₂|H_R (x)) and R (d₁|H_R (x)) ≤ R (d₃|H_R (x));

(N) Decide Neg (A) if R (d₂|H_R (x)) ≤ R (d₁|H_R (x)) and R (d₂|H_R (x)) ≤ R (d₃|H_R (x));

(B) Decide Bnd (A) if R (d₃|H_R (x)) ≤ R (d₁|H_R (x)) and R (d₃|H_R (x)) ≤ R (d₂|H_R (x)), where the above three decision rules (P)-(B) are three-way decisions with DTRS introduced by Yao [31], and d₁, d₂ and d₃ are assigned semantic explanations: acceptance decision, rejection decision, and non-commitment decision, respectively [29].

Suppose the following reasonable conditions for loss function λ_pq (p = 1, 2, 3, q = 1, 2) are satisfied: $λ_{11} \leq λ_{31} \leq λ_{21}, λ_{22} \leq λ_{32} \leq λ_{12} .$ Then the following decision rules can be generated by P (A|H_R (x)) + P (A^c|H_R (x)) =1 :

(P) Decide x ∈ Pos (A), if P (A|H_R (x)) ≥ α and P (A|H_R (x)) ≥ γ;

(N) Decide x ∈ Neg (A), if P (A|H_R (x)) ≤ β and P (A|H_R (x))≤ γ ;

(B) Decide x ∈ Bnd (A), if P (A|H_R (x)) ≤ α and P (A|H_R (x)) ≥ β,

where β ≤ γ ≤ α and $\begin{matrix} α = \frac{λ_{12} - λ_{32}}{(λ_{12} - λ_{32}) + (λ_{31} - λ_{11})}, \\ β = \frac{λ_{32} - λ_{22}}{(λ_{32} - λ_{22}) + (λ_{21} - λ_{31})}, \\ γ = \frac{λ_{12} - λ_{22}}{(λ_{12} - λ_{22}) + (λ_{21} - λ_{11})} . \end{matrix}$

If β < α, then β < γ < α. The decision rules of hesitant fuzzy DTRS over two universes can be deduced as follows:

(P) If P (A|H_R (x)) ≥ α, decide x ∈ Pos (A);

(N) If P (A|H_R (x)) ≤ β, decide x ∈ Neg (A);

(B) If β < P (A|H_R (x)) < α, decide x ∈ Bnd (A).

Meantime, the hesitant fuzzy DTRS over two universes can be obtained by the above (P)-(B) decision rules: $\begin{matrix} {\underline{Apr}}_{P}^{α} (A) = {x \in U | P (A | H_{R} (x)) \geq α}; \\ {\bar{Apr}}_{P}^{β} (A) = {x \in U | P (A | H_{R} (x)) > β} . \end{matrix}$

If β = α, then β = γ = α. And α-probabilistic hesitant fuzzy rough set over two universes can be deduced.

In the following, we will propose a variable precision multigranulation hesitant fuzzy DTRS over two universes and examine some relevant properties.

Definition 3.4. Let U, V be two non-empty finite universes. Suppose that $R$ is a family of binary hesitant fuzzy relations between U and V, R_k ∈ HF (U × V) and $R_{k} \in R$ , k = 1, 2, ⋯ , l. P is a probability defined on the hesitant fuzzy subset classes of V for x ∈ U. Then $(U, V, R, P)$ is said to be a multigranulation hesitant fuzzy probabilistic approximation space over two universes U and V.

Definition 3.5. Let $(U, V, R, P)$ be a multigranulation hesitant fuzzy probabilistic approximation space over two universes. Given two threshold parameters, 0 ≤ β < α ≤ 1, and precision parameter 0 < δ ≤ 1, for a hesitant fuzzy set A ∈ HF (V), the δ-lower and δ-upper approximations of A with respect to $(U, V, R, P)$ , denoted as ${\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A)$ and ${\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A)$ , are defined in the following: $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A) = \\ {x | \frac{| {R_{k} | P (A | H_{R_{k}} (x) (y)) \geq α} |}{l} \geq δ, \\ x \in U, y \in V, k = 1, 2, \dots, l}, \\ {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A) = \\ U - {x | \frac{| {R_{k} | P (A | H_{R_{k}} (x) (y)) \leq β} |}{l} > 1 - δ, \\ x \in U, y \in V, k = 1, 2, \dots, l} . \\ = {x | \frac{| {R_{k} | P (A | H_{R_{k}} (x) (y)) > β} |}{l} > 1 - δ, \\ x \in U, y \in V, k = 1, 2, \dots, l} . \end{matrix}$

Remark 3.1. Definition 3.5 will generate into variable precision multigranulation fuzzy DTRS over two universes when the binary hesitant fuzzy relation degenerates into fuzzy relation, which was discussed in our previous work [20].

Proposition 3.1. Let $(U, V, R, P)$ be a multigranulation hesitant fuzzy approximation space over two universes. Given two threshold parameters, 0 ≤ β < α ≤ 1, and precision parameter 0 < δ ≤ 1, for hesitant fuzzy sets A₁, A₂, ⋯ , A_m ∈ HF (V) such that A₁ ⊆ A₂ ⊆ ⋯ ⊆ A_m, then we have $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A_{1}) \subseteq {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A_{2}) \\ \subseteq \dots \subseteq {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A_{m}), \\ {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A_{1}) \subseteq {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A_{2}) \\ \subseteq \dots \subseteq {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A_{m}) . \end{matrix}$ Proof. We just prove the first relation holds for k = 1, 2, and other cases can be proved similarly.

For any y ∈ V, if A₁ ⊆ A₂, then on the basis of Definition 2.3, we have $h_{A_{1}} (y) ⪯ h_{A_{2}} (y) \Leftrightarrow h_{A_{1}}^{j} (y) \leq h_{A_{2}}^{j} (y), j \in l_{y},$ where $h_{A_{1}}^{j}$ and $h_{A_{2}}^{j}$ are jth value in h_{A
₁} (y) and h_{A
₂} (y) respectively, and l_y = max {l (h_{A
₁} (y)) , l (h_{A
₂} (y))} . Thus by Definition 3.3, for any $x \in {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A_{1})$ , we have α ≤ P (A₁|H_{R
_k} (x) (y)) ≤ P (A₂|H_{R
_k} (x) (y)). It follows that $x \in {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A_{2})$ .

The second relation can be proved similarly.

In what follows, by use of δ-lower and δ-upper approximation, the positive region ${Pos}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ , negative region ${Neg}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ and boundary region ${Bnd}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ are defined respectively: $\begin{matrix} {Pos}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) = {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A), \\ {Neg}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) = U - {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A), \\ {Bnd}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) = \\ U - {Pos}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) - {Neg}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) . \end{matrix}$

Accordingly, the relevant decision rules of the proposed variable precision multigranulation hesitant fuzzy DTRS over two universes are deduced as follows:

(P^δ) If there exist t (t ≥ δl) binary hesitant fuzzy relations out of R₁, R₂, ⋯ , R_l ∈ HF (U × V) such that P (A|H_{R
_k} (x)) ≥ α with any x ∈ U, decide x ∈ Pos (A);

(N^δ) If there exist t (t > l - δl) binary hesitant fuzzy relations out of R₁, R₂, ⋯ , R_l ∈ HF (U × V) such that P (A|H_{R
_k} (x)) ≤ β with any x ∈ U, decide x ∈ Neg (A);

(B^δ) Other cases excluding (P^δ) and (N^δ), decide $x \in {Bnd}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A) .$

We know that R_k (d_p|H_{R
_k} (x) (y)) (k = 1, 2, ⋯ , l, p = 1, 2, 3) is the decision loss function for object x ∈ U of taking action d_p regarding the kth binary hesitant fuzzy relation R_k over U and V. Suppose that the opinion weight vector of the l relations is u = (u₁, u₂, ⋯ , u_l), where u_k ≥ 0, and $\sum_{k = 1}^{l} u_{k} = 1$ , then we can obtain the total loss function for all l binary hesitant fuzzy relations: $R_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (d_{p} ∣ x) = δ \sum_{k = 1}^{l} u_{k} R_{k} (d_{p} ∣ H_{R_{k}} (x)),$ where R_k (d_p ∣ H_{R
_k} (x) = λ (d_p ∣ A) P (A ∣ H_{R
_k}) + λ(d_p ∣ A^c (x)) P (A^c ∣ H_{R
_k} (x)) , p = 1, 2, 3 .

In what follows, by taking special values of the precision parameter δ, the optimistic and pessimistic multigranulation hesitant fuzzy DTRS over two universes can be induced.

Proposition 3.2. Let $(U, V, R, P)$ be a multigranulation hesitant fuzzy probabilistic approximation space over two universes. For ant two threshold parameters 0 ≤ β < α ≤ 1, and a hesitant fuzzy set A ∈ HF (V), let $δ = \frac{1}{l}$ , then $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A) = {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{O, α} (A), \\ {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A) = {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{O, β} (A), \end{matrix}$ and $({\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{O, α} (A), {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{O, β} (A))$ is called the optimistic multigranulation hesitant fuzzy DTRS over two universes, a special case of Definition 3.5. Furthermore, the decision rules of optimistic multigranulation hesitant fuzzy DTRS over two universes are clear:

(P^O) If there exists one binary hesitant fuzzy relation R_k ∈ HF (U × V) (k = 1, 2, ⋯ , l) such that P (A|H_{R_k(x)}) ≥ α, decide x ∈ Pos (A);

(N^O) If all binary hesitant fuzzy relations R₁, R₂, ⋯ , R_l ∈ HF (U × V) such that P (A|H_{R
_k} (x)) ≤ β, decide x ∈ Neg (A);

(B^O) Other cases excluding (P^O) and (N^O), decide $x \in {Bnd}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ .

Meantime, with respect to the optimistic multigranulation hesitant fuzzy DTRS over two universes, the total loss function for all l hesitant fuzzy relations between U and V is defined as: $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{O, α} (d_{p} | x) \\ = \sum_{k = 1}^{l} u_{k} R_{k} (d_{p} | H_{R_{k}} (x)), p = 1, 2, 3 . \end{matrix}$

Proposition 3.3. Let $(U, V, R, P)$ be a multigranulation hesitant fuzzy probabilistic approximation space over two universes. For ant two threshold parameters 0 ≤ β < α ≤ 1, and a hesitant fuzzy set A ∈ HF (V), let δ = 1, there are: $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, α} (A) = {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{P, α} (A), \\ {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{δ, β} (A) = {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{P, β} (A), \end{matrix}$ and $({\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{P, α} (A), {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{P, β} (A))$ is called the optimistic multigranulation hesitant fuzzy DTRS over two universes, a special case of Definition 3.5.

Similarly, the decision rules of optimistic multigranulation hesitant fuzzy DTRS over two universes are clear:

(P^P) If all binary hesitant fuzzy relations R₁, R₂, ⋯ , R_m ∈ HF (U × V) such that P (A|H_{R
_k} (x)) ≥ α, decide $x \in {Pos}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ ;

(N^P) If there exists one binary hesitant fuzzy relation R_k ∈ HF (U × V) (k = 1, 2, ⋯ , l) such that P (A|H_{R
_k} (x)) ≤ β, decide $x \in {Neg}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ ;

(B^P) Other cases excludes (P^P) and (N^P), decide $x \in {Bnd}_{\sum_{k = 1}^{l} R_{k}}^{α, β} (A)$ . Meantime, with respect to the pessimistic multigranulation hesitant fuzzy DTRS over two universes, the total loss function for all m hesitant fuzzy relations between U and V is determined as: $\begin{matrix} {\underline{R}}_{\sum_{k = 1}^{l} R_{k}}^{P, α} (d_{p} | x) \\ = \prod_{k = 1}^{l} u_{k} R_{k} (d_{p} | H_{R_{k}} (x)), p = 1, 2, 3 . \end{matrix}$

4 Calculation of loss functions derived from hesitant fuzzy evaluation values

In this paper, the loss function in decision-making process is not fixed as in original three-way decisions, but can be calculated by evaluation values expressed in the form of hesitant fuzzy element, with the assistance of distance measure of hesitant fuzzy sets introduced by Xu and Xia [28]. It should be noted that this calculation method is based on the idea of Jia and Liu [5].

Definition 4.1. Let U be a non-empty universe. For any A, B ∈ HF (U), and any x ∈ U, the distance between the two hesitant fuzzy elements h_A (x) and h_B (x) is defined as: $d_{h} (h_{A} (x), h_{B} (x)) = \frac{1}{l_{x}} \sum_{j = 1}^{l_{x}} | h_{A}^{j} (x) - h_{B}^{j} (x) |,$ where $h_{A}^{j} (x)$ and $h_{B}^{j} (x)$ are jth values in h_A (x) and h_B (x), respectively, l_x = max {l (h_A (x)) , l (h_B (x))} .

Suppose that the evaluation value of alternative x_i in criteria y_j with associated state C_j is denoted by h_ij, where $C_{j}^{c}$ denotes an alternative does not belong to C_j, and $\min {h_{ij}} = h_{\min}^{j}$ , $\max {h_{ij}} = h_{\max}^{j},$ which are expressed by hesitant fuzzy element. In light of distance measure of any two hesitant fuzzy elements in Definition 4.1, the relative loss functions derived from h_ij can be given in Table 2, where the so called relative loss function means λ₁₁ = 0 and λ₂₂ = 0 [5]. The parameter δ_j ∈ [0, 1] in the relative loss function expression shown in Table 2, introduced by Li and Zhou [7], implies that the higher the value of δ_j, the more hesitant the decision maker [5].

Table 2
The relative loss function λ_{y
_j} (x_i) of x_i

x _ij C _j $C_{j}^{c}$

d ₁ 0 $d_{h} (h_{\max}^{j}, h_{ij})$

d ₃ $δ_{j} d_{h} (h_{ij}, h_{\min}^{j})$ $δ_{j} d_{h} (h_{\max}^{j}, h_{ij})$

d ₂ $d_{h} (h_{ij}, h_{\min}^{j})$ 0

x _ij	C _j	$C_{j}^{c}$
d ₁	0	$d_{h} (h_{\max}^{j}, h_{ij})$
d ₃	$δ_{j} d_{h} (h_{ij}, h_{\min}^{j})$	$δ_{j} d_{h} (h_{\max}^{j}, h_{ij})$
d ₂	$d_{h} (h_{ij}, h_{\min}^{j})$	0

For clarity, we use an example to illustrate the calculation of the relative loss function shown inTable 2.

Example 1. Two alternatives x₁ and x₂ are evaluated in criterion y_j with associated state C_j by decision-makers in the form of hesitant fuzzy element, which implies that $h_{\min}^{j} = {0}$ , $h_{\max}^{j} = {1}$ . Let h_1j = {0.2, 0.4}, h_2j = {0.6}, and δ_j = 0.4. The relative loss function of x₁ and x₂ can be determined in Tables 3 and 4.

Table 3

The relative loss function λ_{y
_j} (x₁) of x₁

x ₁	C _j	${\tilde{C_{j}}}^{c}$
d ₁	0	0.7
d ₃	0.12	0.28
d ₂	0.3	0

Table 4

The relative loss function λ_{y
_j} (x₂) of x₂

x ₂	C _j	$C_{j}^{c}$
d ₁	0	0.6
d ₃	0.16	0.24
d ₂	0.4	0

Suppose the weight vector of criteria {y₁, y₂, ⋯ , y_m} is w = (w₁, w₂, ⋯ , w_n) , where w_j ≥ 0 and $\sum_{j = 1}^{n} w_{j} = 1$ , and c is the comprehensive criteria with state C, where C^c denotes an alternative does not belong to C. For alternative x_i ∈ U, its aggregated loss functions, taking all criteria {y₁, y₂, ⋯ , y_m} into consideration, is calculated by: $λ^{C} (x_{i}) = \sum_{j = 1}^{n} w_{j} λ_{y_{j}} (x_{i}), i = 1, 2, \dots, m,$ where λ_{y
_j} (x_i) denotes the relative loss function of x_i in criterion y_j given in Table 2. The calculation of aggregated relative loss function λ^C (x_i) is shown in Table 5.

Table 5

Aggregated relative loss function λ^C (x_i) of x_i

x _i	C	C ^c
d ₁	0	$d_{h} (h_{\max}^{j}, \sum_{j = 1}^{n} δ_{j} h_{ij})$
d ₃	$\sum_{j = 1}^{n} δ_{j} d_{h} (h_{ij}, h_{\min}^{j})$	$\sum_{j = 1}^{n} δ_{j} d_{h} (h_{\max}^{j}, h_{ij})$
a ₂	$d_{h} (h_{ij}, \sum_{j = 1}^{n} δ_{j} h_{\min}^{j})$	0

5 Three-way group decision-making model based on hesitant fuzzy DTRS over two universes and application in green supplier selection

5.1 Problem statement

The green revolution aimed at achieving sustainable development and environmental protection has been vey flourish and infiltrated into a wide variety of industries and domains, including supply chain management. Green supply chain management is a strategy that takes environmental factors into account throughout supply chain [9]. Essential to green supply chain management are the decision procedures of green supplier evaluation and selection [12], which can improve the companies’ performance in terms of environmental competencies. Therefore, this section establishes the three-way decision-making model based on hesitant fuzzy DTRS over two universes under the background of green supplier selection.

Let U = {x₁, x₂, ⋯ , x_m} be a green supplier set and V = {y₁, y₂, ⋯ , y_n} be a criteria set. Suppose that R₁, R₂, ⋯ , R_l are l invited decision-makers by the green supplier selector. If several values of the green supplier x_i ∈ U with respect to the criterion y_j ∈ V are given by the decision-maker R_k, these values can be regarded as a hesitant fuzzy element h_{R
_k} (x_i, y_j), i.e., R_k ∈ HF (U × V), where i ∈ {1, 2, ⋯ , m} , j ∈ {1, 2, ⋯ , n} and k ∈ {1, 2, ⋯ , l}. Meantime, let A ∈ HF (V) be a self-evaluation set, a hesitant fuzzy set of the criteria set V, denoting the actual requirement of a green supplier selector. Denote the opinion weight vector of decision-makers as u = {u₁, u₂, ⋯ , u_l}, where u_k ≥ 0 and $\sum_{k = 1}^{l} u_{k} = 1 .$ The action (i.e., deciding selection immediately, deciding not selection, and further information) need be made for every green supplier x_i ∈ U (i = 1, 2, ⋯ , m).

5.2 The key steps of the model

In this section, we model the procedure of group decision-making with the background of a green supplier selection problem by using the proposed variable precision multigranulation HFDTRS over two universes in Section 3. According to above descriptions, the key steps of the above decision-making method with hesitant fuzzy information are given as follows.

Step 1. For decision maker R_k, calculate the aggregated loss function of λ^{k
_C} with respect to comprehensive criteria c with state C.

Given δ_j ∈ [0, 1], based on Table 2, calculate $λ_{y_{j}}^{k_{C}} (x_{i})$ of x_i in criteria y_j by decision-maker R_k, where x_i ∈ U, i ∈ {1, 2, ⋯ , m} , y_j ∈ V, j ∈ {1, 2, ⋯ , n}, and k ∈ {1, 2, ⋯ , l}. Further, given criteria weight vector w = (w₁, w₂, ⋯ , w_n) , the aggregated loss function λ^{k
_C} (x_i) of x_i in comprehensive criteria c by decision-maker R_k can be determined according to Table 5. Finally, for decision-maker R_k, we compute the aggregated loss function λ^{k
_C} by taking the average value of λ^{k
_C} (x_i) of each alternative x_i ∈ U .

Step 2. Suppose that the opinion weight vector of all l decision-makers is u = (u₁, u₂, ⋯ , u_l) , then in light of Bayesian risk decision-making method, compute the comprehensive threshold parameter α and β by all l decision-makers. Let $λ_{11}^{k_{C}} = λ^{k} (d_{1} | C), λ_{21}^{k_{C}} = λ^{k} (d_{2} | C), λ_{31}^{k_{C}} = λ^{k} (d_{3} | C),$ $λ_{12}^{k_{C}} = λ^{k} (d_{1} | C^{c}), λ_{22}^{k_{C}} = λ^{k} (d_{2} | C^{c}), λ_{32}^{k_{C}} = λ^{k} (d_{3} | C^{c})$ be the specific components of the aggregated loss function λ^{k
_C} with state C by decision-maker R_k. Further, let $λ_{pq}^{k_{A}} = λ_{pq}^{k_{C}} \cdot P (A | C) (p = 1, 2, 3, q = 1, 2),$ and let A^c denote the green supplier being not in state A, then the aggregated loss function of R_k with state A can be obtained. Here, for the green supplier selection, the acceptance decision d₁ means a good green supplier, the rejection decision d₂ means a bad one, and the noncommitment decision d₃ means the decision-makers need further information.

As for the loss function $λ_{pq}^{k_{A}} (i = 1, 2, 3, j = 1, 2)$ , it is obvious that the following relations aresatisfied: $λ_{11}^{k_{A}} \leq λ_{31}^{k_{A}} \leq λ_{21}^{k_{A}}, λ_{12}^{k_{A}} \geq λ_{32}^{k_{A}} \geq λ_{22}^{k_{A}},$ then the threshold parameter values α^k and β^k, during the green supplier selection process by decision-maker R_k, can be determined by: $\begin{matrix} α^{k} = \frac{λ_{12}^{k_{A}} - λ_{32}^{k_{A}}}{(λ_{12}^{k_{A}} - λ_{32}^{k_{A}}) + (λ_{31}^{k_{A}} - λ_{11}^{k_{A}})}, \\ β^{k} = \frac{λ_{32}^{k_{A}} - λ_{22}^{k_{A}}}{(λ_{32}^{k_{A}} - λ_{22}^{k_{A}}) + (λ_{21}^{k_{A}} - λ_{31}^{k_{A}})} . \end{matrix}$ By means of the decision-maker weight vector u = (u₁, u₂, ⋯ , u_m), where u_k > 0 and $\sum_{k = 1}^{m} u_{k} = 1$ , we can compute the comprehensive threshold parameters α and β by all l decision-makers as follows: $α = \sum_{i = 1}^{m} u_{k} α^{k}, β = \sum_{i = 1}^{m} u_{k} β^{k} .$

Step 3. Take the precision parameter δ = 0.67 as Sun suggested in Ref. [20].

Step 4. By use of definition 3.3, calculate conditional probability P (A|H_{R
_k} (x_i, y)), and according to definition 3.4 obtain the positive region Pos (A), negative region Neg (A) and boundary region Bnd (A) of multigranulation hesitant fuzzy probabilistic approximation over two universes $(U, V, R, P)$ . Then, develop the corresponding decision rules (P^δ) - (N^δ) in Section 3, and determine the decision of each green supplier x_i ∈ U.

5.3 A numerical example

In this section, we provide a group decision-making problem of green supplier to illustrate the decision method given in Section 5.2.

Let U = {x₁, x₂, x₃, x₄} be the set of possible green suppliers and V = {y₁, y₂, y₃, y₄} be the set of four criteria, where y₁ stands for quality, y₂ stands for service, y₃ stands for delivery and y₄ stands for price, which are adopted in Ref. [22]. Suppose that the green supplier selector invites four experts to provide evaluation and the opinion weight vector of the four experts is u = (u₁, u₂, u₃, u₄) = (0.1, 0.6, 0.2, 0.1) . During the green supplier selection process, the experts evaluate the four possible green supplier under four criteria based on their own experience and knowledge, and the evaluation values are represented as hesitant fuzzy element. By use of Definition 3.3, a hesitant fuzzy relation R_k ∈ HF (U × V), where k = 1, 2, 3, 4, is provided by each expert, and the evaluation information are shown in Table 9, respectively.

Table 6
The relation between green suppliers and criteria by R₁

R ₁ y ₁ y ₂ y ₃ y ₄

x ₁ {0.1, 0.2} {0.3} {0.2, 0.3} {0.6}

x ₂ {0.1, 0.3} {0.5} {0.6, 0.7} {0.8}

x ₃ {0.7} {0.3, 0.4} {0.5, 0.6} {0.7, 0.8}

x ₄ {0.1} {0.4, 0.6} {0.8} {0.2, 0.3}

R ₁	y ₁	y ₂	y ₃	y ₄
x ₁	{0.1, 0.2}	{0.3}	{0.2, 0.3}	{0.6}
x ₂	{0.1, 0.3}	{0.5}	{0.6, 0.7}	{0.8}
x ₃	{0.7}	{0.3, 0.4}	{0.5, 0.6}	{0.7, 0.8}
x ₄	{0.1}	{0.4, 0.6}	{0.8}	{0.2, 0.3}

Table 7

The relation between green suppliers and criteria by R₂

R ₂	y ₁	y ₂	y ₃	y ₄
x ₁	{0.2, 0.3}	{0.3, 0.4}	{0.1}	{0.5, 0.6}
x ₂	{0.3}	{0.2, 0.4}	{0.1}	{0.6, 0.7}
x ₃	{0.5}	{0.1, 0.2}	{0.3, 0.5}	{0.7, 0.8}
x ₄	{0.2, 0.3}	{0.6}	{0.4, 0.5}	{0.3}

Table 8

The relation between green suppliers and criteria by R₃

R ₃	y ₁	y ₂	y ₃	y ₄
x ₁	{0.1, 0.2}	{0.6, 0.7}	{0.5, 0.7}	{0.8}
x ₂	{0.6}	{0.7, 0.8}	{0.3, 0.6}	{0.5, 0.6}
x ₃	{0.8, 1}	{0.4}	{0.6, 0.7}	{0.4, 0.5}
x ₄	{0.2, 0.3}	{0.5}	{0.4, 0.6}	{0.7}

Table 9

The relation between green suppliers and criteria by R₄

R ₄	y ₁	y ₂	y ₃	y ₄
x ₁	{0.9, 1}	{0.7, 0.8}	{0.6}	{0.5, 0.6}
x ₂	{0.3, 0.4}	{0.6, 0.7}	{0.8}	{0.7, 0.9}
x ₃	{0.2, 0.4}	{0.4}	{0.7, 0.8}	{0.9}
x ₄	{0.6, 0.8}	{0.8, 0.9}	{0.5, 0.6}	{1}

At the beginning, given δ₁ = 0.35, δ₂ = 0.4, δ₃ = 0.4 and δ₄ = 0.5, calculate the relative loss function $λ_{y_{j}}^{k_{C}} (x_{i})$ by expert R_k, where k = 1, 2, 3, 4, and thus obtain Table 13. Then by criteria weight vector w = (0.4, 0.2, 0.1, 0.3), the aggregated loss function λ^{k
_C} (x_i) by R_k is computed as shown in Table 14. So, the aggregated loss function λ^{k
_C} associated with state C can be determined, and the results are listed in Table 15.

Table 10

The relative loss function $λ_{y_{j}}^{1_{C}} (x_{i})$ of R₁

		y ₁		y ₂		y ₃		y ₄
R ₁		C ₁	$C_{1}^{c}$	C ₂	$C_{2}^{c}$	C ₃	$C_{3}^{c}$	C ₄	$C_{4}^{c}$
x ₁	d ₁	0	0.85	0	0.7	0	0.75	0	0.4
	d ₃	0.045	0.255	0.12	0.28	0.1	0.3	0.3	0.2
	d ₂	0.15	0	0.3	0	0.25	0	0.6	0
x ₂	d ₁	0	0.8	0	0.5	0	0.35	0	0.2
	d ₃	0.06	0.24	0.2	0.2	0.26	0.14	0.4	0.1
	d ₂	0.2	0	0.5	0	0.65	0	0.8	0
x ₃	d ₁	0	0.3	0	0.65	0	0.45	0	0.25
	d ₃	0.21	0.09	0.14	0.26	0.22	0.18	0.375	0.125
	d ₂	0.7	0	0.35	0	0.55	0	0.75	0
x ₄	d ₁	0	0.9	0	0.5	0	0.2	0	0.75
	d ₃	0.03	0.27	0.2	0.2	0.32	0.08	0.125	0.375
	d ₂	0.1	0	0.5	0	0.8	0	0.25	0

Table 11

The relative loss function $λ_{y_{j}}^{2_{C}} (x_{i})$ of R₂

		y ₁		y ₂		y ₃		y ₄
R ₂		C ₁	$C_{1}^{c}$	C ₂	$C_{2}^{c}$	C ₃	$C_{3}^{c}$	C ₄	$C_{4}^{c}$
x ₁	d ₁	0	0.75	0	0.65	0	0.9	0	0.45
	d ₃	0.075	0.225	0.14	0.26	0.04	0.36	0.275	0.225
	d ₂	0.25	0	0.35	0	0.1	0	0.55	0
x ₂	d ₁	0	0.7	0	0.7	0	0.9	0	0.35
	d ₃	0.09	0.21	0.12	0.28	0.04	0.36	0.325	0.175
	d ₂	0.3	0	0.3	0	0.1	0	0.65	0
x ₃	d ₁	0	0.5	0	0.85	0	0.6	0	0.25
	d ₃	0.15	0.15	0.06	0.34	0.16	0.24	0.375	0.125
	d ₂	0.5	0	0.15	0	0.4	0	0.75	0
x ₄	d ₁	0	0.75	0	0.4	0	0.55	0	0.7
	d ₃	0.075	0.225	0.24	0.16	0.18	0.22	0.15	0.35
	d ₂	0.25	0	0.6	0	0.45	0	0.3	0

Table 12

The relative loss function $λ_{y_{j}}^{3_{C}} (x_{i})$ of R₃

		y ₁		y ₂		y ₃		y ₄
R ₃		C ₁	$C_{1}^{c}$	C ₂	$C_{2}^{c}$	C ₃	$C_{3}^{c}$	C ₄	$C_{4}^{c}$
x ₁	d ₁	0	0.85	0	0.35	0	0.4	0	0.2
	d ₃	0.045	0.255	0.26	0.14	0.24	0.16	0.4	0.1
	d ₂	0.15	0	0.65	0	0.6	0	0.8	0
x ₂	d ₁	0	0.4	0	0.25	0	0.55	0	0.45
	d ₃	0.18	0.12	0.3	0.1	0.18	0.22	0.275	0.225
	d ₂	0.6	0	0.75	0	0.45	0	0.55	0
x ₃	d ₁	0	0.1	0	0.6	0	0.35	0	0.55
	d ₃	0.27	0.03	0.16	0.24	0.26	0.14	0.225	0.275
	d ₂	0.9	0	0.4	0	0.65	0	0.45	0
x ₄	d ₁	0	0.75	0	0.5	0	0.5	0	0.3
	d ₃	0.075	0.225	0.2	0.2	0.2	0.2	0.35	0.15
	d ₂	0.25	0	0.5	0	0.5	0	0.7	0

Table 13

The relative loss function $λ_{y_{j}}^{4_{C}} (x_{i})$ of R₄

		y ₁		y ₂		y ₃		y ₄
R ₄		C ₁	$C_{1}^{c}$	C ₂	$C_{2}^{c}$	C ₃	$C_{3}^{c}$	C ₄	$C_{4}^{c}$
x ₁	d ₁	0	0.05	0	0.25	0	0.4	0	0.45
	d ₃	0.285	0.115	0.3	0.1	0.24	0.16	0.275	0.225
	d ₂	0.95	0	0.75	0	0.6	0	0.55	0
x ₂	d ₁	0	0.65	0	0.35	0	0.2	0	0.2
	d ₃	0.105	0.195	0.26	0.14	0.32	0.08	0.4	0.1
	d ₂	0.35	0	0.65	0	0.8	0	0.8	0
x ₃	d ₁	0	0.7	0	0.6	0	0.25	0	0.1
	d ₃	0.09	0.21	0.16	0.24	0.3	0.1	0.45	0.05
	d ₂	0.3	0	0.4	0	0.75	0	0.9	0
x ₄	d ₁	0	0.3	0	0.15	0	0.45	0	0
	d ₃	0.21	0.09	0.34	0.06	0.22	0.18	0.5	0
	d ₂	0.7	0	0.85	0	0.55	0	1	0

Table 14

The aggregated loss function λ^k
_C (x_i) of R_k, k = 1, 2, 3, 4

		R ₁		R ₂		R ₃		R ₄
x ₁	d ₁	0	0.675	0	0.655	0	0.51	0	0.245
	d ₃	0.142	0.248	0.1445	0.2455	0.214	0.176	0.2805	0.1095
	d ₂	0.325	0	0.345	0	0.49	0	0.755	0
x ₂	d ₁	0	0.515	0	0.615	0	0.4	0	0.41
	d ₃	0.21	0.18	0.1615	0.2285	0.2325	0.1575	0.233	0.157
	d ₂	0.485	0	0.385	0	0.6	0	0.59	0
x ₃	d ₁	0	0.37	0	0.505	0	0.36	0	0.455
	d ₃	0.2465	0.1435	0.2005	0.1895	0.2335	0.1565	0.233	0.157
	d ₂	0.63	0	0.495	0	0.64	0	0.545	0
x ₄	d ₁	0	0.705	0	0.645	0	0.54	0	0.195
	d ₃	0.1215	0.2685	0.141	0.249	0.195	0.195	0.324	0.066
	d ₂	0.295	0	0.355	0	0.46	0	0.805	0

Furthermore, the green supplier selector conducts a self-evaluation A ∈ HF (V), which is expressed by: $A = \frac{{0.4, 0.6}}{y_{1}} + \frac{{0, 7, 0.8}}{y_{2}} + \frac{{0.5}}{y_{3}} + \frac{{0.6, 0.9}}{y_{4}} .$

Since C is the standard state of comprehensive criteria c, we have $C = \frac{{1}}{y_{1}} + \frac{{1}}{y_{2}} + \frac{{1}}{y_{3}} + \frac{{1}}{y_{4}} .$ Then, based on Definition 3.3, the conditional probability P (A|C) can be computed as: $P (A | C) = \frac{\sum_{y \in V} s (h_{A} (y)) \cdot s (h_{C} (y))}{\sum_{y \in V} s (h_{C} (y))} = 0.625 .$ By $λ_{pq}^{k_{A}} = λ_{pq}^{k_{C}} P (A | C)$ and Table 15, the aggregated loss function λ^{k
_A} of R_k, where k = 1, 2, 3, 4, can be calculated as shown in Table 16. Meantime, by Definition 3.3, compute the conditional probability results for each alternative x_i of hesitant fuzzy event A, given the hesitant fuzzy description H_{R
_k} (x, y), and thus obtain Table 17.

Table 15

The aggregated loss function λ^k
_C of R_k, k = 1, 2, 3, 4

	$λ_{11}^{k_{C}}$	$λ_{31}^{k_{C}}$	$λ_{21}^{k_{C}}$	$λ_{12}^{k_{C}}$	$λ_{32}^{k_{C}}$	$λ_{22}^{k_{C}}$
R ₁	0	0.180	0.434	0.566	0.21	0
R ₂	0	0.162	0.395	0.605	0.228	0
R ₃	0	0.219	0.548	0.453	0.171	0
R ₄	0	0.268	0.674	0.326	0.122	0

Table 16

The aggregated loss function λ^k
_A of R_k, k = 1, 2, 3, 4

	$λ_{11}^{k_{A}}$	$λ_{31}^{k_{A}}$	$λ_{21}^{k_{A}}$	$λ_{12}^{k_{A}}$	$λ_{32}^{k_{A}}$	$λ_{22}^{k_{A}}$
R ₁	0	0.113	0.271	0.354	0.131	0
R ₂	0	0.1013	0.247	0.378	0.143	0
R ₃	0	0.137	0.342	0.283	0.107	0
R ₄	0	0.168	0.421	0.204	0.076	0

Table 17

The conditional probability of A

R _k	R ₁	R ₂	R ₃	R ₄
P (A\|H_{R _k} (x₁))	0.673	0.651	0.617	0.614
P (A\|H_{R _k} (x₂))	0.680	0.676	0.625	0.641
P (A\|H_{R _k} (x₃))	0.665	0.638	0.589	0.654
P (A\|H_{R _k} (x₄))	0.614	0.639	0.638	0.649

Last, according to Table 16,

compute the threshold parameters α_k and β_k by

decision-maker R_k as shown in Table 18, then by the opinion weight vector u = (0.1, 0.6, 0.2, 0.1) of the four decision-makers, obtain the comprehensive parameter α and β as follows: $α = \sum_{k = 1}^{4} u_{k} α^{k} = 0.641, β = \sum_{k = 1}^{4} u_{k} β^{k} = 0.434 .$ Let δ = 0.67 as Sun suggested in [20], then we have $\begin{matrix} {Pos}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) = {\underline{R}}_{\sum_{i = 1}^{4} R_{k}}^{0.67, 0.641} (A) = \\ {x \in U | \frac{| {R_{k} | P (A | H_{R_{k}} (x) (y)) \geq 0.641)} |}{4} \\ \geq 0.67, y \in V, k = 1, 2, 3, 4}, \\ {Neg}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) = U - {\bar{R}}_{\sum_{k = 1}^{l} R_{k}}^{0.67, 0.434} (A) = \\ {x \in U | \frac{| {R_{k} | P (A | H_{R_{k}} (x) (y)) \leq 0.434)} |}{4} \\ > 0.33, y \in V, k = 1, 2, 3, 4}, \\ {Bnd}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) \\ = U - {Pos}_{\sum_{k = 1}^{l} R_{k}}^{0.641, 0.434} (A) - {Neg}_{\sum_{k = 1}^{l} R_{k}}^{0.641, 0.434} (A) . \end{matrix}$ Further, $\begin{matrix} {Pos}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) = {x_{2}}, \\ {Neg}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) = \emptyset, \\ {Bng}_{\sum_{k = 1}^{4} R_{k}}^{0.641, 0.434} (A) = {x_{1}, x_{3}, x_{4}} . \end{matrix}$

Table 18

The threshold parameters α_k and β_k

R _k	R ₁	R ₂	R ₃	R ₄
α ^k	0.664	0.699	0.562	0.432
β ^k	0.453	0.495	0.342	0.231

From the above results, the green supplier selector can find that the green supplier x₂ should be selected, and x₁, x₃, x₄ are not confirmed, namely, further investigation is needed.

5.4 Validity test of the presented model

The validity of the proposed MCGDM model of green supplier selection is tested by means of the following two main test criteria constructed by [6, 25]. In test criterion 1, replace a non-optimal alternative with another worse alternative, without changing the weight vector of criteria, then the determination of the optimal alternative of the changed problem should be identical to that of the original problem. In test criterion 2, decompose a MCDM question into smaller questions, and utilize the proposed MCDM method to address smaller problems and gain the rankings of the alternatives, then the combined ranking of the alternatives should be the same as that the of un-decomposed problem.

Considering test criteria 1, we replace a non-optimal alternative x₄ by {0} , {0.3, 0.5} , {0.7} , {0.1, 0.2} in Table 6, {0.1, 0.2} , {0.5} , {0.3, 0.4} , {0.2} in Table 7, {0.1, 0.2} , {0.4} , {0.3, 0.5} , {0.6} in Table 8, and {0.5, 0.7} , {0.7, 0.8} , {0.4, 0.5} , {0.9} in Table 9, respectively. Then the proposed decision-making model is used to tackle the new hesitant fuzzy relations between criteria and green supplier candidates. By the proposed decision-making model, the results of the original MCGDM question and the changed MCGDM question are shown in Table 19 and Table 20, respectively. With this change, the alternatives are ranked as x₂ > x₁ > x₃ = x₄ according to the total loss function. Since the best green supplier is again x₂ which is identical to the original MCGDM question, it means that the proposed decision-making model is valid under test criterion 1 constructed by [25].

Table 19
The decision results of the original problem

x ₁ x ₂ x ₃ x ₄

$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{1} ∣ x)$ 0.0806 0.0760 0.0833 0.0825

$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{3} ∣ x)$ 0.0801 0.0800 0.0805 0.0807

$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{2} ∣ x)$ 0.1226 0.1262 0.1209 0.1221

Decision rule B ^δ P ^δ B ^δ B ^δ

cost 0.0801 0.0760 0.0805 0.0807

	x ₁	x ₂	x ₃	x ₄
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{1} ∣ x)$	0.0806	0.0760	0.0833	0.0825
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{3} ∣ x)$	0.0801	0.0800	0.0805	0.0807
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{2} ∣ x)$	0.1226	0.1262	0.1209	0.1221
Decision rule	B ^δ	P ^δ	B ^δ	B ^δ
cost	0.0801	0.0760	0.0805	0.0807

Table 20

The decision results of the changed problem

	x ₁	x ₂	x ₃	x ₄
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{1} ∣ x)$	0.0848	0.0800	0.0876	0.0848
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{3} ∣ x)$	0.0797	0.0793	0.0802	0.0802
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{2} ∣ x)$	0.1150	0.1185	0.1135	0.1164
Decision rule	B ^δ	B ^δ	B ^δ	B ^δ
cost	0.0797	0.0793	0.0802	0.0802

To test the validity of the proposed decision-making model under test criteria 2, we decompose the original green supplier selection problem into two smaller three-way group decision-making problems {x₁, x₂, x₄} and {x₁, x₃, x₄}. Similarly, the decision results of the un-decomposed MCGDM question and the decomposed MCGDM question can be computed and listed in Tables 21 and 22, respectively. Then we can gain the rankings of alternatives x₂ > x₁ > x₄ and x₁ > x₃ > x₄. With the above two rankings, the combined result is the same as the un-decomposed green supplier selection question. Therefore, the validity of test criteria 2 constructed by [25] is demonstrated.

Table 21

The decision results of the decomposed problem, x₁, x₂ and x₄

	x ₁	x ₂	x ₄
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{1} ∣ x)$	0.0850	0.0799	0.0869
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{3} ∣ x)$	0.0788	0.0784	0.0796
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{2} ∣ x)$	0.1147	0.1182	0.1146
Decision rule	B ^δ	B ^δ	B ^δ
cost	0.0788	0.0784	0.0796

Table 22

The decision results of the decomposed problem, x₁, x₃ and x₄

	x ₁	x ₃	x ₄
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{1} ∣ x)$	0.0806	0.0834	0.0826
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{3} ∣ x)$	0.0800	0.0803	0.0806
$R_{\sum_{k = 1}^{4} R_{k}}^{δ, α} (d_{2} ∣ x)$	0.1223	0.1208	0.1220
Decision rule	B ^δ	B ^δ	B ^δ
cost	0.0800	0.0803	0.0806

5.5 Comparison and discussion

Motivated by the merit of multigranulation DTRS in aggregating decision-makers’ preferences and the flexibility of hesitant fuzzy set in conveying information, the variable precision multigranulation hesitant fuzzy DTRS is discussed, then a novel three-way group decision-making model for green supplier selection is designed. Among the existing papers, our work is most related to [10 , 27].

Different from traditional method of MCGDM, such as the hesitant fuzzy averaging operator and hesitant fuzzy geometric operator defined in [27], our presented approach not only incorporates the idea of risk decision-making into the MCGDM process by DTRS, that is to say, the optimal alternative is gained by minimum risk, but also offers a new aggregation approach to aggregate risk preferences of decision-makers by multigranulation. In view of this, the decision results of the presented model are more reasonable and efficient in light of the merit of three-way decisions.

Compared with the construction of DTRS [10, 20], where the loss functions were both given straightly, in our presented decision-making model, loss functions are not fixed but generated from hesitant fuzzy evaluation values. Particularly, loss functions could be adjusted in accordance with the risk attitudes of decision-makers, so the presented model is more applicable in real-life group decision-making situations. In addition, in light of the essentiality of exploring hesitant fuzzy information system, the presented three-way group decision-making model based on variable multigranulation DTRS in hesitant fuzzy setting expands the scope of multigranularity three-way decisions theory.

6 Conclusions

In this paper, we have discussed a new method of the three-way group decision-making under hesitant fuzzy information environment. By introducing the theory of hesitant fuzzy set into DTRS over two universes, we have constructed three types ofmultigranulation hesitant fuzzy DTRS over two universes and induced the corresponding decision rules. Then according to the loss functions being calculated by hesitant fuzzy evaluation value, the new three-way group decision-making model has been established with the aid of the proposed multigranulation hesitant fuzzy DTRS over two universes. In the future, we intend to do the extension of the proposed model in incomplete hesitant fuzzy information system. It is also meaningful to investigate the attribute reduction or determination of the weight of decision-makers associated with the proposed model.

Footnotes

Acknowledgments

The work was partly supported by the National Natural Science Foundation of China (71571090, 61772019), the National Science Foundation of Shaanxi Province of China (2017JM7022), the Interdisciplinary Foundation of Humanities and Information (RW180167), the Fundamental Research Funds for the Central Universities(22120190116).

References

Azam

, Zhang

, Yao

J.T.

, Evaluation functions and decision conditions of three-way decisions with game-theoretic rough sets, European Journal of Operational Research 261(2) (2017), 704–714.

Farhadinia

, A novel method of ranking hesitant fuzzy values for multiple attribute decision making problems, International Journal of Intelligent Systems 28(8) (2013), 752–767.

Gong

Z.T.

, Chai

R.L.

, Covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and applications, Journal of Intelligent and Fuzzy Systems 31 (2016), 1621–1633.

B.Q.

, Three-way decisions space and three-way decisions, Information Sciences 281(281) (2014), 21–52.

Jia

, Liu

P.D.

, A novel three-way decision model under multiple-criteria environment, Information Science 471 (2019), 29–51.

Joshi

, Kumar

, Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making, European Journal of Operational Research 248 (2016), 183–191.

H.X.

, Zhou

X.Z.

, Risk decision making based on decisiontheoretic rough set: A three-way view decision model, International Journal of Computational Intelligence Systems 4(1) (2011), 1–11.

D.Q.

, Zeng

W.Y.

, Li

J.H.

, New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making, Engineering Applications of Artificial Intelligence 40 (2015), 11–16.

, Wang

J.Q.

, An extended qualiflex method under probability hesitant fuzzy environment for selecting green suppliers, International Journal of Fuzzy Systems 9(6) (2017), 1866–1879.

10.

Liang

D.C.

, Xu

Z.S.

, Liu

, A new aggregation method-based error analysis for decision-theoretic rough sets and its application in hesitant fuzzy information systems, IEEE Transactions on Fuzzy Systems 25(6) (2017), 1685–1697.

11.

Liu

Y.T.

, Zhang

H.J.

, Wu

Y.Z.

, et al., Ranking range based approach to MADM under incomplete context and its application in venture investment evaluation, Technological and Economic Development of Economy. DOI: 10.3846/tede.2019.10296

12.

Park

, Shin

, Chang

T.W.

, et al., An integrative framework for supplier relationship management, Industrial Management and Data Systems 110(4) (2010), 495–515.

13.

Pawlak

, Rough sets, International Journal of Cmputer and Information Sciences 11(5) (1982), 341–356.

14.

Qian

, Liang

, Dang

, Incomplete multigranulation rough set, IEEE Transactions on Systems Man and Cybernetics Systems 20 (2010), 420–443.

15.

Qian

, Liang

, Yao

, Dang

, MGRS: A multi-granulation rough set, Information Sciences 180 (2010), 949–970.

16.

Qian

Y.H.

, Zhang

Y.H.

, Sang

Y.L.

, et al., Multigranulation decision-theoretic rough sets, International Journal of Approximate Reasoning 55(1) (2104), 225–237.

17.

Sun

B.Z.

, Ma

W.M.

, Multigranulation rough set theory over two universes, Journal of Intelligent and Fuzzy Systems 28(3) (2015), 1251–1269.

18.

Sun

B.Z.

, Ma

W.M.

, Li

B.J.

, Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set, International Journal of Approximation Reasoning 93 (2018), 424–442.

19.

Sun

B.Z.

, Ma

W.M.

, Qian

Y.H.

, Multigranulation fuzzy rough set over two universes and its application to decision making, Knowledge-Based Systems 123 (2017), 61–74.

20.

Sun

B.Z.

, Ma

W.M.

, Xiao

, Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes, International Journal of Approximate Reasoning 81 (2017), 87–102.

21.

Sun

B.Z.

, Ma

W.M.

, Zhao

H.Y.

, Decision-theoretic rough fuzzy set model and application, Information Sciences 283(5) (2014), 180–196.

22.

Tang

S.L.

, Green supplier selection model with hesitant fuzzy information, Journal of Intelligent and Fuzzy Systems 32 (2017), 189–195.

23.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25(6) (2010), 529–539.

24.

Torra

, Narukawa

, On hesitant fuzzy sets and decision, In: The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382.

25.

Wang

X.T.

, Triantaphyllou

, Ranking irregularities when evaluating alternatives by using some ELECTRE methods, Omega 36 (2008), 45–63.

26.

Y.Z.

, Li

C.C.

, Chen

, et al., Group decision making based on linguistic distributions and hesitant assessments: Maximizing the support degree with an accuracy constraint, Information Fusion 41 (2018), 151–160.

27.

Xia

, Xu

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning 52(3) (2011), 395–407.

28.

Z.S.

, Xia

M.M.

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

29.

Yao

Y.Y.

, Three-way decisions with probabilistic rough sets, Information Science 180(3) (2010), 341–353.

30.

Y.Y.

, Probabilistic rough set approximations, International Journal of Approximate Reasoning 49(2) (2008), 255–271.

31.

Yao

Y.Y.

, Three-way decision: An interpretation of rules in rough set theory, International Conference on Rough Sets and Knowledge Technology, 2009, pp. 642–649.

32.

Yao

Y.Y.

, Wong

S.K.M.

, A decision theoretic framework for approximating concepts, International Journal of Man-Machine Studies 37(6) (1992), 793–809.

33.

W.Y.

, Zhang

, Zhong

Q.Y.

, et al., Extended TODIM for multi-criteria group decision making based on unbalanced hesitant fuzzy linguistic term sets, Computers and Industrial Engineering 114 (2017), 316–328.

34.

Zhan

J.M.

, Sun

B.Z.

, Alcantud

J.R.

, Covering based multi-granulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences 476 (2019), 290–318.

35.

Zhang

B.W.

, Dong

Y.C.

, Herrera-Viedma

, Group decision making with heterogeneous preference structures: An automatic mechanism to support consensus reaching, Group Decision and Negotiation 28 (2019), 585–617.

36.

Zhang

B.W.

, Liang

H.M.

, Zhang

G.Q.

, Reaching a consensus with minimum adjustment in MAGDM with hesitant fuzzy linguistic term sets, Information Fusion 42 (2018), 12–23.

37.

Zhang

, Li

D.Y.

, Liang

J.Y.

, Hesitant fuzzy linguistic rough set over two universes model and its applications, International Journal of Machine Learning and Cybernetics 9 (2018), 577–588.

38.

Zhang

, Li

D.Y.

, Zhai

Y.H.

, et al., Multigranulation rough set model in hesitant fuzzy information systems and its application in personjob fit, International Journal of Machine Learning and Cybernetics 10 (2019), 717–729.

39.

Zhang

H.D.

, Shu

, Liao

S.L.

, Hesitant fuzzy rough set over two universes and its application in decision making, Soft Computing 21 (2015), 1–14.

40.

Zhang

H.J.

, Dong

Y.C.

, Chiclana

, et al., Consensus efficiency in group decision making: A comprehensive comparative study and its optimal design, European Journal of Operational Research 275 (2019), 580–598.

41.

Zhang

, Guo

C.H.

, Deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings, Journal of the Operational Research Society 68 (2017), 1582–1599.

42.

Zhang

, Kou

X.Y.

, Dong

Q.X.

, Additive consistency analysis and improvement for hesitant fuzzy preference relations, Expert Systems with Applications 98 (2018), 118–128.

43.

Zhang

, Kou

X.Y.

, Yu

W.Y.

, et al., On priority weights and consistency for incomplete hesitant fuzzy preference relations, Knowledge-Based Systems 143 (2018), 115–126.

Three-way group decisions based on multigranulation hesitant fuzzy decision-theoretic rough set over two universes

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Multigranulation hesitant fuzzy DTRS over two universes

Table 1 Loss function A A c d 1 λ 11 λ 12 d 3 λ 31 λ 32 d 2 λ 21 λ 22

Table 2 The relative loss function λ y j (x i ) of x i x ij C j C j c d 1 0 d h ( h max j , h ij ) d 3 δ j d h ( h ij , h min j ) δ j d h ( h max j , h ij ) d 2 d h ( h ij , h min j ) 0

5.1 Problem statement

5.2 The key steps of the model

5.3 A numerical example

Table 6 The relation between green suppliers and criteria by R1 R 1 y 1 y 2 y 3 y 4 x 1 {0.1, 0.2} {0.3} {0.2, 0.3} {0.6} x 2 {0.1, 0.3} {0.5} {0.6, 0.7} {0.8} x 3 {0.7} {0.3, 0.4} {0.5, 0.6} {0.7, 0.8} x 4 {0.1} {0.4, 0.6} {0.8} {0.2, 0.3}

6 Conclusions

Footnotes

Acknowledgments

References

Table 1
Loss function

A A ^c

d ₁ λ ₁₁ λ ₁₂

d ₃ λ ₃₁ λ ₃₂

d ₂ λ ₂₁ λ ₂₂

Table 2
The relative loss function λ_{y
_j} (x_i) of x_i

x _ij C _j $C_{j}^{c}$

d ₁ 0 $d_{h} (h_{\max}^{j}, h_{ij})$

d ₃ $δ_{j} d_{h} (h_{ij}, h_{\min}^{j})$ $δ_{j} d_{h} (h_{\max}^{j}, h_{ij})$

d ₂ $d_{h} (h_{ij}, h_{\min}^{j})$ 0

Table 6
The relation between green suppliers and criteria by R₁

R ₁ y ₁ y ₂ y ₃ y ₄

x ₁ {0.1, 0.2} {0.3} {0.2, 0.3} {0.6}

x ₂ {0.1, 0.3} {0.5} {0.6, 0.7} {0.8}

x ₃ {0.7} {0.3, 0.4} {0.5, 0.6} {0.7, 0.8}

x ₄ {0.1} {0.4, 0.6} {0.8} {0.2, 0.3}