An approach to multi-attribute group decision making based on multigranulation probabilistic fuzzy rough set and Multimoora method

Abstract

This paper proposes a new approach to uncertainty multiple attribute group decision problem with linguistic preference relation based on multigranulation probabilistic rough set and the Multimoora method. According to the classical Pawlak rough set and the neighborhood rough set, we present a multigranulation probabilistic fuzzy rough set based on neighborhood relation with linguistic preference information. We investigate the rough approximation of a crisp decision-making object and a fuzzy decision-making object under the framework of multigranulation rough set theory with linguistic preference inforamtion, respectively. That is, a multigranulation probabilistic rough set model and a multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation are established, respectively. Meanwhile, the proposed multigranulation probabilistic fuzzy rough set model is compared with the existing model and illustrate the superiority of the new model established. Furthermore, by combining the multigranulation probabilistic fuzzy rough set and the Multimoora method, a new method for multiple attribute group decision making with linguistic preference information is proposed. The decision-making procedure and the algorithm of the proposed method are also given. Finally, the effectiveness and validity of the proposed method is verified by investigating a multiple attribute group decision problem with the selection of suppliers in the context of e-commerce.

Keywords

Neighborhood relationship multigranulation probability rough set multiple attribute group decision-making Multimoora linguistic preference information

1 Introduction

Because of the complexity and uncertainty of decision-making problems, the incompleteness of the obtained information, and the limitation of human understanding, it is difficult to deal with the decision-making problems in real life only with a single decision maker. Therefore, it is necessary for multiple decision makers to make decisions on different attributes of the same problem, which is called multi-attribute group decision making problem. A group decision-making problem is defined as a decision problem in which multiple decision-makers provide their judgment or evaluation of a set of alternatives, choose the appropriate method, obtain the optimal solution or sorting result through calculation and analysis. In recent years, many scholars have established several new methodologies and decision theories for multi-attribute group decision making. Generally, evaluation given by decision-makers on alternatives of decision-making problems are mainly in the form of qualitative and quantitative. When decision makers consider the evaluation of the choice of green supply chain implementation [3], the selection of talents [5], the choice of suppliers [20], the strategic decision [23], and the emergency plans [27], they tend to describe preference information by linguistic information in the form of linguistic variables. Though there are many theories and methodologies of multiple attribute group decision-making under uncertainty with linguistic information, most of the previous literature have limitations in practical applications. The existing literatures are mainly focus on and the uncertainty decision making [13], evaluation of the effect of multi-attribute group decision making [18], the discussion of consensus measurement for GDM with linguistic preference information [26] and the use of aggregation operators [29], less effort on the discussion of multiple attribute fuzzy group decision-making problem is solved by the combination of multigranulation rough sets and other decision making methods. In fact, due to the instability of the aggregation operator, the approach of information fusion should not be limited to the use of aggregation operators. Therefore, it is necessary to improve the limitations of the previous methods as much as possible, and to explore new theories and methods to deal with the multiple attribute group decision-making problem with linguistic information under uncertainty.

The rough set theory [15] was firstly proposed by Pawlak as a mathematical tool for analyzing and dealing with incomplete information such as inaccuracy, uncertainty, and incompletely. Rough set theory does not require any prior knowledge, and through the effective analysis and reasoning of the current data, it discovers the implicit knowledge and finally reveals the underlying law of the data [16]. The rough set theory is mainly concerned with the approximation of sets described by a single binary relation on the universe. At present, the theory is widely used in the fields of conflict analysis [20], machine learning [6], medical diagnosis [12], decision support and analysis [11]. However, the classical Pawlak rough set has some limitations in the application of practical problems. In the past, several extensions of the rough set model have been proposed in terms of various requirements of practical applications, In order to improve the shortcomings of the classical Pawlak rough set in practical application, some extensions of the rough set model are proposed such as the rough set model based on neighborhood relationship [31], the variable precision rough set (VPRS) model [35], the fuzzy rough set model [34] and the rough fuzzy set model [5], the Bayesian rough set model [24], probabilistic rough set model [33]. Nowadays, the classical Pawlak rough set and its extension have become important and effective theories and tools for dealing with various decision problems under uncertainty. Based on the classical Pawlak rough set and its extensions, many scholars have proposed a variety of models and methods for uncertain decision-making problems [25]. The neighborhood rough set [32] and the multi-granular rough set [21] are useful extensions of the classical rough set theory.

From the perspective of the granular computing, an equivalence relation of universe can be regarded as a granulation, and a partition on the universe can be regarded as a granulation space. In order to solve the existing practical problems, on the basis of the classical Pawlak rough set model, Qian et al. [21] combined the granular computing and the rough set theory to propose a multigranulation rough set model (MGRS). This theory consists of two basic models: two models of optimistic multigranulation rough set and pessimistic multigranulation rough set are defined [21].

Pawlak’s rough set theory is applicable to discrete data systems, but it cannot be effectively applied to the processing of continuous data. By extending the equivalence relation in the traditional rough set theory, Lin [14] replaced the equivalence relation with the neighborhood relationship and presented the neighborhood-based rough set. Later, Yao [33] studied the neighborhood information system and analyzed the characteristic of the neighborhood approximate space. Since the neighborhood rough set model can directly process numerical data, the neighborhood has wider applicability when dealing with problems. At the same time, neighborhood relations are more general than equivalence relations. Ref. [33] discusses the related properties of approximate spaces.

Multimoora (MultiObjective Optimization by Ratio Analysis) method [2], as a multiple attribute group decision making method, becomes the most robust system of multiple objectives optimization. Compared with other methods, this method is simple and effective, and it is easy to sort and optimize the scheme. In order to improve the robustness of multi-attribute decision-making, Brauers et al. [1] proposed the MOORA method, which uses the ratio method (RS) and the reference point method (RP) to obtain the scheme of basic comprehensive decision information. Then, according to the advantage theory, the final program ranking result is obtained. The Multimoora method improves the MOORA method by introducing a full multiplication model (FMF) to process comprehensive decision information, obtaining program ordering. The Multimoora method integrating the three sorting methods is more accurate than the MOORA method. Also, the results obtained by the Multimoora method are more reasonable and scientific due to the fusion of the results of the three sorting methods. Therefore, Multimoora method is more stable than other existing MCDM methods(such as TOPSIS method, VIKOR method).

Based on the above analysis of the existing methods of group decision making and the concept of multigranulation rough set theory, the purpose of this paper is to present a new decision-making method for multiple attribute group decision-making problem with linguistic information under uncertainty. First, after processing the linguistic variables, the definition of the distance measure for any two alternatives with linguistic information of the universe of discourse is given by the Euclidean distance. Next, the multigranulation probabilistic rough set model and the multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation are proposed, respectively. The related properties are discussed from two aspects: optimistic multigranulation probabilistic rough set and pessimistic multigranulation probabilistic rough set. Several interesting properties for these two models are investigated in detail. After that, the obtained model is compared with the original multigranulation model, and the relationship among them is obtained by changing the parameters. Then, based on the Multimoora (Multiplicative and Multi-Objective Optimization by Ratio Analysis) method, a consistent ranking result is obtained. Therefore, a multiple attribute group decision making method based on δ-neighborhood relation multigranulation probabilistic fuzzy rough set and Multimoora method is proposed. Also, The specific decision making process and the principles of the established method are given. Finally, the proposed method is illustrated through an numerical example.

This paper is outlined as follows. In Section 2, we briefly provides the rough set and fuzzy set theory, multigranulation fuzzy rough set model, linguistic variables and its properties. In the third section, a multigranulation probabilistic rough set model and a multigranulation probabilistic fuzzy rough set model based on neighborhood relation with linguistic information are proposed. In Section 4, a kind of multiple attribute group decision making problem with linguistic information is considered and then a new approach to multiple attribute group decision making based on multigranulation probabilistic rough set and Multimoora method is proposed. In section 5, according to the established model and method, the example of supplier selection in the context of modern e-commerce is used to validate the effectiveness of the proposed method. Finally, the conclusion and the direction of further development is presented in Section 6.

2 Preliminaries

In this section, the concept of the rough set [15] and fuzzy set [34], multigranulation rough set and its basic operations [22] as well as the linguistic terms and its operations [28] are briefly reviewed.

2.1 Rough set and Fuzzy set

Let U be a finite universe that is not empty and R be an equivalence relation of U × U. The equivalence relation R induces a partition of U, denoted by [x] _R or [x], and U/R = {[x] |x ∈ U} stands for the equivalence classes of x. Then (U, R) is the Pawlak approximation space.

For any X ⊆ U, the lower approximation and upper approximations are defined as follows [15]:

$\underline{R} (X) = {x \in U | [x] \subseteq X} = \cup {[x] | [x] \subseteq X},$ $\bar{R} (X) = {x \in U | [x] \cap X \neq \emptyset} = \cup {[x] | [x] \cap X \neq \emptyset}$ .

The lower approximation $\underline{R} (X)$ is the union of all elementary sets and the upper approximation $\bar{R} (X)$ is the union of all elementary sets which have a non-empty intersection with X.

On this basis, for any A ⊆ U, we introduce a characteristic function A (x) as follows: $A (x) = f_{A} (x) = {\begin{matrix} 1, & x \in A; \\ 0, & x \notin A . \end{matrix}$

The characteristic function A (x) is a mapping from U to {0, 1}. Then any characteristic function on universe U determines a classical subset of U. That is, A = {x ∈ U|A (x) =1}.

In the following, we present the concept of fuzzy set by using the definition of classical crisp set.

Definition 2.1. [34] Let U be a non-empty finite universe. A fuzzy set of U is defined by the mapping A (•) : U ⟶ [0, 1] , where the A (x) denotes the membership of the element x (x ∈ U) with fuzzy set A. Let F (U) stands for all fuzzy subsets of universe U. For any A ∈ F (U) , the r level set and strong r level set of A will be denoted by A_r and A_r+, respectively. That is, A_r = {x ∈ U|A (x) ≥ r} and A_r+ = {x ∈ U|A (x) > r}, where r ∈ [0, 1] , the unit interval, A₀ = U and A₁₊ =∅.

2.2 Multigranulation rough set

Now we introduce the definition of multigranulation rough set model established by Qian et al. [22].

Definition 2.2. [22] Let K = (U, R) be a knowledge base. Let P, Q be two equivalence relations of universe U, which can be regards as two granulations of universe of discourse. For any X ⊆ U, the lower and upper approximations of X included by P and Q are defined as follows, respectively.

${\underline{X}}_{P + Q} = {x \in U | [x]_{P} \subseteq X \lor [x]_{Q} \subseteq X},$ ${\bar{X}}^{P + Q} = ({\underline{X^{c}}}_{P + Q})^{c}$ . Where X^c stands for the complementary of X.

Meanwhile, the boundary of X with respect to the equivalence relation P and Q is defined as:

${Bn}_{P + Q} = {\bar{X}}^{P + Q} - {\underline{X}}_{P + Q}$ .

Generally speaking, we call X definable with respect to equivalence relation P and Q if ${\underline{X}}_{P + Q} = {\bar{X}}^{P + Q};$ Otherwise, X is a rough set with respect to K = (U, R). Moreover, it is clear that any one equivalence relation will form a partition of universe U, i.e., any one equivalence relation of the universe forms a granularity structure of the universe of discourse. So, we call X the multigranulation rough set when ${\underline{X}}_{P + Q} \neq {\bar{X}}^{P + Q}$ because the granularity structure of the universe U is generated by two different equivalence relations P and Q.

2.3 Linguistic terms and its operations

In group decision making with linguistic information, the preference information of decision makers are expressed in the form of linguistic values by means of linguistic variables [30].

Formally, the results of the modeling process for decision making preference information do not match the elements in S. In order to facilitate computation and preserve all the given information, the discrete term set S is extended to a continuous term set S ={ s_α|s_-q ≤ s_α ≤ s_q, α ∈ [- q, q] } (where q (q > t) is a sufficiently large positive integer) whose elements also meet all the characteristics above [28]. If s_α ∈ S, we call s_α ∈ S the original term and α the original term index; otherwise, we call s_α the virtual term and α the virtual term index. Specifically, the decision maker uses the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in operations.

Given a term set S, for any linguistic variables s_α, s_β ∈ S and μ₁, μ₂ ∈ [0, 1], the following operational laws hold [30]:

s_α ⊕ s_β = s_α+β,

s_α ⊕ s_β = s_β ⊕ s_α,

μs_α = s_μα,

(μ₁ + μ₂) s_α = μ₁s_α ⊕ μ₂s_α,

μ (s_α ⊕ s_β) = μs_α ⊕ μs_β.

3 Multigranulation probability fuzzy rough set based on δ- neighborhood relation with linguistic preference information

In this section, the related definitions and some properties of neighborhood rough set are introduced firstly. Then, a multiple fuzzy decision-making linguistic-valued information system is defined. Secondly, the definition of a conversion function that converts a linguistic-valued into a real-valued is given. At the same time, the vector division in the matrix is given. These two steps lay the foundation for the definition of the next distance measure. Then, the definition of the distance measure for any two alternatives with respect to any attribute with linguistic information of the universe of discourse is presented. After that, the multigranulation probability rough set model and the multigranulation probability fuzzy rough set model are proposed.

3.1 Neighborhood rough sets [10]

Let <U, Δ>be a non-empty metric space, where U is a set of non-empty finite objects {x₁, x₂, … , x_n }, called a universe, Δ is a distance function.

For x_i ∈ U, the neighborhood δ (x_i) of x_i is defined as $δ (x_{i}) = {x_{j} | x_{j} \in U, Δ (x_{i}, x_{j}) \leq δ}$

For ∀x₁, x₂, x₃ ∈ U, it satisfies the following relationship:

Δ (x₁, x₂) ≥ 0, Δ (x₁, x₂) = 0 if and only if x₁ = x₂;

Δ (x₁, x₂) = Δ (x₂, x₁);

Δ (x₁, x₃) ≤ Δ (x₁, x₂) + Δ (x₂, x₃).

3.2 Multiple attribute fuzzy decision-making information system and notions

We call six-tuple (U, C, D, F, ω, μ) a multiple attribute fuzzy decision-making linguistic information system, where U = {x₁, x₂, ⋯ , x_n} is the universe of discourse (i.e., the objects set or the alternatives set), C = {c₁, c₂, ⋯ , c_m} is the attribute set and ω = (ω₁, ω₂, ⋯ , ω_m) ^T be the weight vector of attributes where $ω_{h} \geq 0, \sum_{h = 1}^{m} ω_{h} = 1,$ D = {d₁, d₂, ⋯ , d_l} is the set of decision-makers and μ = (μ₁, μ₂, ⋯ , μ_l) ^T be the weight vector of decision-makers, where $\sum_{i = 1}^{l} μ_{k} = 1$ . F = {f_k|k = 1, 2, ⋯ , l} is a family of mapping sets between universe U and attribute set C, where f_k : U × C → V_k and V_k = ∪ _{c_h∈C}V_h (h = 1, 2, ⋯ , m) is the domain of attribute set C under the conditional of decision-maker d_k ∈ D. That is, ∀x_i ∈ U, c_h ∈ C, d_k ∈ D, f_k (x_i, c_h) ∈ V_h (h = 1, 2, ⋯ , m) (i.e., f_k (x_i, c_h) is the evaluation of the alternative x_i with respect to the attribute c_h given by the decision-maker d_k.In order to facilitate computation, the decision maker d_k compares these alternatives with respect to a single criterion using the linguistic terms in the set S ={ S_α|α = - t, …, -1, 0, 1, …, t }. Then the decision maker constructs the linguistic preference relation $B_{ij} = {(b_{ij}^{(k)})}_{n \times n} (k = 1, 2, \dots, l)$ .

To illustrate the above symbols and their definitions, there is a multiple fuzzy decision-making linguistic information system (U, C, D, F, ω, μ) in Table 1.

Table 1
A multiple fuzzy decision-making linguistic information system

U d ₁ d ₂ ⋯ d _l

c ₁ c ₂ ⋯ c _m c ₁ c ₂ ⋯ c _m c ₁ c ₂ ⋯ c _m c ₁ c ₂ ⋯ c _m

x ₁ s _-3 s ₂ ⋯ s ₄ s _-3 s ₂ ⋯ s ₄ .. .. ⋯ .. s _-3 s ₁ ⋯ s _-3

x ₂ s ₁ s ₀ ⋯ s _-2 s ₃ s ₄ ⋯ s _-1 .. .. ⋯ .. s ₁ s _-1 ⋯ s ₃

x ₃ s _-2 s ₀ ⋯ s _-1 s ₂ s ₃ ⋯ s _-4 .. .. ⋯ .. s ₁ s ₂ ⋯ s ₄

x ₄ s ₂ s _-1 ⋯ s ₁ s _-3 s _-2 ⋯ s ₀ .. .. ⋯ .. s ₂ s ₀ ⋯ s _-2

x ₅ s ₂ s _-4 ⋯ s ₀ s _-2 s _-3 ⋯ s ₁ .. .. ⋯ .. s _-2 s ₂ ⋯ s ₄

x ₆ s ₀ s ₂ ⋯ s ₂ s ₃ s ₁ ⋯ s _-2 .. .. ⋯ .. s _-3 s ₃ ⋯ s ₁

U	d ₁	d ₂	⋯	d _l
x ₁	s _-3	s ₂	⋯	s ₄	s _-3	s ₂	⋯	s ₄	..	..	⋯	..	s _-3	s ₁	⋯	s _-3
x ₂	s ₁	s ₀	⋯	s _-2	s ₃	s ₄	⋯	s _-1	..	..	⋯	..	s ₁	s _-1	⋯	s ₃
x ₃	s _-2	s ₀	⋯	s _-1	s ₂	s ₃	⋯	s _-4	..	..	⋯	..	s ₁	s ₂	⋯	s ₄
x ₄	s ₂	s _-1	⋯	s ₁	s _-3	s _-2	⋯	s ₀	..	..	⋯	..	s ₂	s ₀	⋯	s _-2
x ₅	s ₂	s _-4	⋯	s ₀	s _-2	s _-3	⋯	s ₁	..	..	⋯	..	s _-2	s ₂	⋯	s ₄
x ₆	s ₀	s ₂	⋯	s ₂	s ₃	s ₁	⋯	s _-2	..	..	⋯	..	s _-3	s ₃	⋯	s ₁

Based on the description of the multiple fuzzy decision-making linguistic information system, the distance measure over (U, C, D, F, ω, μ) is presented in detail. In order to simplify the calculation, it is convenient to deal with the evaluation given by the decision maker with linguistic information.

Firstly, divide the minimum and maximum values of the row vectors of any linguistic preference row vector given by the decision maker d_k, respectively [26]. The distance measure is calculated by the relationship between two variables.

Definition 3.1. Let $β_{k}^{(i)}$ stand for the ith row vector of linguistic preference relations $B_{ij} = {(b_{ij}^{(k)})}_{n \times n)}$ , i.e., $β_{k}^{(i))} = (b_{i 1}^{k}, b_{i 2}^{k}, \dots, b_{in}^{k})$ . For any linguistic preference row vector $β_{k}^{(i)}$ of B_k, we use ${\underline{β}}_{k}^{(i)} = min_{j} {b_{ij}^{k} | b_{ij}^{k} \in B_{k}},$

${\bar{β}}_{k}^{(i)} = max_{j} {b_{ij}^{k} | b_{ij}^{k} \in B_{k}}$ stand for the minimum and maximum values given by the decision maker d_k ∈ D for the alternative x_i ∈ U, respectively.

At the same time, due to the nature of linguistic variables, the data obtained by the above formula cannot be added or subtracted directly. To facilitate the calculation, a preprocessing on the linguistic terms is defined. That is, converting the linguistic terms into the corresponding term indices. In the following content, the definition of conversion function [30] presented.

Definition 3.2. Let S = {s_α|s_-q ≤ s_α ≤ s_q, α ∈ [- q, q]} be a set of extended continuous linguistic terms. For any $s_{α} \in \bar{S}$ , then the corresponding term index α can be got by the function I as follows: $I : \bar{S} \to [- q, q], and I (S_{α}) = α, for any s_{α} \in \bar{S} .$ (1)

By using the conversion function I between linguistic terms and term indices, the operations between the linguistic terms are transformed into the real-valued operations. Then it will help us to deal with the decision-making problems with linguistic preference relations under uncertainty.

To define the distance measure between any two alternatives, the concept of Minkowsky distance is introduced to measure the distance between two objects of the universe of discourse [9].

Definition 3.3. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. For any x₁, x₂ ∈ U (i, j = 1, 2, ⋯ , n), c_h ∈ C (h = 1, 2, ⋯ , m) and d_k ∈ D (k = 1, 2, ⋯ , l), $φ_{C} (x_{1}, x_{2}) = {(\sum_{h = 1}^{m} {| f_{k} (c_{h} (x_{1})) - f_{k} (c_{h} (x_{2})) |}^{p})}^{\frac{1}{p}}$ is called the Minkowsky distance of objects x₁ and x₂ with respect to attribute set C. In particular,

If p = 1, φ_C (x₁, x₂) is called Manhattan distance;

If p = 2, φ_C (x₁, x₂) is called Euclidean distance; and

If p = ∞ , φ_C (x₁, x₂) is called Chebychev distance.

Here, the Euclidean distance is used to define the distance measure. Based on the Definition 3.1 and the formula (1), the definition of the distance measure between alternatives x₁ and x₂ of universe U can be given over the multiple attribute fuzzy decision-making linguistic information system.

Definition 3.4. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. For any x_i, x_j ∈ U (i, j = 1, 2, ⋯ , n), c_h ∈ C (h = 1, 2, ⋯ , m) and d_k ∈ D (k = 1, 2, ⋯ , l) , $ξ_{ij}^{k} = \sqrt{[I ({\bar{β}}_{k}^{(i)}) - I ({\bar{β}}_{k}^{(j)})]^{2} + [I ({\underline{β}}_{k}^{(i)}) - I ({\underline{β}}_{k}^{(j)})]^{2}}$ (2) is called the distance measure between alternatives x_i and x_j with linguistic information of universe U with respect to the attribute c_h ∈ C based on the evaluation of decision-maker d_k ∈ D.

For any s_α, μ ∈ [0, 1], it can be easily seen that the operational laws μs_α = s_μα is hold [30].

Here, by considering weighting with respect to the attribute set C, the following formula is obtained.

$\begin{matrix} b_{ij}^{kh} = ω_{i} b_{ij}^{k}, {\bar{β}}_{kh}^{(i)} = max_{j} (b_{i 1}^{kh}, b_{i 2}^{kh}, \dots, b_{in}^{kh}) and \\ {\underline{β}}_{kh}^{(i)} = min_{j} (b_{i 1}^{kh}, b_{i 2}^{kh}, \dots, b_{in}^{kh}) \end{matrix}$ (3)

Furthermore, $ξ_{ij}^{kC} = \sqrt{[I ({\bar{β}}_{kh}^{(i)}) - I ({\bar{β}}_{kh}^{(j)})]^{2} + [I ({\underline{β}}_{kh}^{(i)}) - I ({\underline{β}}_{kh}^{(j)})]^{2}}$ (4) is called the weighted distance measure between alternatives x_i and x_j with linguistic information of universe U with respect to the attribute c_h ∈ C based on the evaluation of decision-maker d_k ∈ D.

According to the definition of the distance measure given in formula (2) and the definition of the weighted distance measure given in formula (4), it is easy to know that there exists x_i, x_j ∈ U (i, j = 1, 2, …, n), $ξ_{ij}^{kC} \notin [0, 1]$ . From the definition of neighborhood relationship, the date must be normalized in order to establish its neighborhood relation of any alternatives given the linguistic-valued object of universe of discourse. Then the definition of the standard distance measure is as following.

Definition 3.5. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic-valued information system. $ξ_{ij}^{kC}$ is the weighted distance measure between alternatives x_i and x_j with linguistic information of universe U with respect to the attribute C based on the evaluation of decision-maker d_k ∈ D. $(ξ_{ij}^{kC})_{min}$ is the minimum values given in formula (3), $(ξ_{ij}^{kC})_{max}$ is maximum values given in formula (3). For any x_i, x_j ∈ U (i, j = 1, 2, ⋯ , n) , c_h ∈ C (h = 1, 2, ⋯ , m) , and d_k ∈ D (k = 1, 2, ⋯ , l), we call $(ξ_{ij}^{kC})^{*} = \frac{ξ_{ij}^{kC} - (ξ_{ij}^{kC})_{min}}{(ξ_{ij}^{kC})_{max} - (ξ_{ij}^{kC})_{min}}$ (5) the standard distance measure between alternatives x_i and x_j with linguistic information of universe U with respect to the attribute C, the value of the original distance measure is mapped to [0, 1] by normalizing.

Then, according to the principles of neighborhood rough set [10], the definition of δ-neighborhood relation of any linguistic preference alternatives over universe U based on the standard distance measure is proposed.

Definition 3.6. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic-valued information system. $(ξ_{ij}^{kC})^{*}$ is the standard distance measure between alternatives x_i and x_j with linguistic information of universe U with respect to the attribute C. Then,

$\begin{matrix} δ_{C} (x_{i}^{k}) = {y | y \in U, (ξ_{ij}^{kc})^{*} \leq δ, δ \in [0, 1], \\ k = 1, 2, \dots, l} x_{i} \in U \end{matrix}$ (6) is defined as δ-neighborhood relation classes of alternative x_i ∈ U over (U, C, D, F, ω, μ).

From this definition of δ-neighborhood relation classes, the following results are clear.

Proposition 3.1. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. $δ_{C} (x_{i}^{k}) (k = 1, 2, \dots, l)$ is the δ-neighborhood relation classes of universe U. Then the following properties are satisfied:

For any x_i ∈ U, there is $x_{i} \in δ_{C} (x_{i}^{k}),$ i.e., $δ_{C} (x_{i}^{k})$ is reflexivity;

For any x_i, x_j ∈ U, there is $x_{j} \in δ_{C} (x_{i}^{k}) \Leftrightarrow x_{i} \in δ_{C} (x_{j}^{k}) (k = 1, 2, \dots, l),$ i.e., $δ_{C} (x_{i}^{k})$ is symmetry.

Proof. It can be directly derived from the above definitions.

From the definition of the δ-neighborhood relation classes, for any x_i, x_j ∈ U and satisfy $x_{i} \in δ_{C} (x_{i}^{k}),$ there exists x₀ ∈ U satisfies $x_{0} \in δ_{C} (x_{i}^{k})$ and $x_{0} \notin δ_{C} (x_{j}^{k})$ . That is, the δ-neighborhood relation class does not satisfy the transitivity. So, the δ-neighborhood relation classes is a similarity relation of the universe of discourse,which satisfy the properties of reflexivity and symmetry.

At the same time, according to the definition of the distance function, the following results are also satisfied for the neighborhood relation.□

Proposition 3.2. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. $δ_{C} (x_{i}^{k}) (k = 1, 2, \dots, l)$ is the δ-neighborhood relation classes of universe U. Then, there are

For any x_i ∈ U, there is $x_{i} \in δ_{C} (x_{i}^{k});$

For any x_i ∈ U, there is $δ_{C} (x_{i}^{k}) \neq \emptyset;$

For any x_i ∈ U, $⋃_{i = 1}^{n} δ_{C} (x_{i}^{k}) = U;$

For any x_i ∈ U, δ₁, δ₂ ∈ [0, 1] and satisfies δ₁ ≤ δ₂, there is $δ_{1} (x_{i}^{k}) \subseteq δ_{2} (x_{i}^{k})$ .

Proof. The results given in Proposition 3.1 and it can be directly derived from the Definition 3.6.

In this section, the multiple attribute fuzzy decision-making linguistic information system is presented and then the δ-neighborhood relation of the universe of discourse is constructed. Based on the above definition, the multigranulation probabilistic rough set model can be defined over the multiple attribute fuzzy decision-making linguistic information system.□

3.3 Multigranulation probabilistic rough set based on δ-neighborhood relation

In this section, under the framework of Qian’s multigranulation rough set [21], the rough approximation of decision objects with linguistic preference information in multiple attribute fuzzy decision information system (U, C, D, F, ω, μ) will be discussed in detail. Thus, a multigranulation probabilistic rough set model based on neighborhood relation is constructed. Further, optimistic and pessimistic multigranulation probabilistic rough set based on neighborhood relation are defined, respectively.

Because of the similarities in the definitions, in order to facilitate the research, the definitions and propositions given in the following content are only proved for the optimistic multigranulation probabilistic rough set, because the pessimistic multigranulation probabilistic rough set is also the same, and will not be described here.

Firstly, for generalized probabilistic approximation, the definition of the probabilistic rough set based on neighborhood relation of the universe of discourse is given.

Definition 3.7. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (k = 1, 2, ⋯ , l) is the δ-neighborhood relation class of universe U. Then, for any A ∈ P (U) , 0 ≤ β ≤ α ≤ 1, the α lower and β upper approximation of the probabilistic rough set with respect to (U, C, D, F, ω, μ) based on neighborhood relation are defined as follows, respectively. ${\underline{Apr}}_{α} (A) = {x \in U | P (A | δ_{C} (x^{k})) \geq α},$ ${\bar{Apr}}_{β} (A) = {x \in U | P (A | δ_{C} (x^{k})) > β} .$

Here $P (A | δ_{C} (x^{k})) = \frac{| δ_{C} (x^{k}) \cap A |}{| δ_{C} (x^{k}) |},$ it is easy to obtain that there are P (A|δ_C (x^k)) =0 hold for δ_C (x^k) = ∅ , k = 1, 2, …, 1.

That is, $({\underline{Apr}}_{α} (A), {\bar{Apr}}_{β} (A))$ is the probabilistic rough set based on neighborhood relation of the universe of discourse. Meanwhile, we have ${pos}_{δ}^{α} (A) = {\underline{Apr}}_{α} (A) = {x \in U | P (A | δ_{C} (x^{k})) \geq α} |;$ $\begin{matrix} {neg}_{δ}^{β} (A) & = U - {\bar{Apr}}_{β} (A) \\ = {x \in U | P (A | δ_{C} (x^{k})) \leq β}; \end{matrix}$ $\begin{matrix} {bn}_{δ}^{α, β} (A) & = {\bar{Apr}}_{β} (A) - {\underline{Apr}}_{α} (A) \\ = {x \in U | β < P (A | δ_{C} (x^{k})) < α} . \end{matrix}$

In the following, we extend the above concept of definition to the case of multiple granularity, then we present the definition of the optimistic multigranulation probabilistic rough set based on neighborhood relation and pessimistic multigranulation probabilistic rough set based on neighborhood relation of the universe of discourse.

Definition 3.8. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , c_h ∈ C (h = 1, 2, ⋯ , m), d_k ∈ D (k = 1, 2, ⋯ , l) , and 0 ≤ β ≤ α ≤ 1, the α lower and β upper approximations with respect to (U, C, D, F, ω, μ) of optimistic multigranulation probabilistic rough set based on neighborhood relation are defined as follows, respectively.

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq α \lor P (A | δ_{C} (x^{2})) \geq α \lor \dots \lor P (A | δ_{C} (x^{l})) \geq α};$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) = {x \in U | P (A | δ_{C} (x^{1})) > β \land P (A | δ_{C} (x^{2})) > β \land \dots \land P (A | δ_{C} (x^{l})) > β};$

If there is ${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) \neq {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A)$ for any object A ∈ P (U). So the set-pair $({\underline{Apr}}_{\sum_{k = 1}^{1} δ_{C} (x^{k})}^{O, α} (A), {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A))$ is called optimistic multigranulation probabilistic rough set based on neighborhood relation.

Furthermore, the positive region, negation region and the boundary region of the optimistic multigranulation probabilistic rough set model base on δ-neighborhood relation are given as follows, respectively. ${Pos}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) = {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A);$ $\begin{matrix} {Neg}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) & = {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) \\ - {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A); \end{matrix}$ $\begin{matrix} {Bn}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α, β} (A) = ({Pos}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) \\ ⋃ {Neg}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A))^{c} . \end{matrix}$ Similarly, the definition of the pessimistic multigranulation probabilistic rough set based on neighborhood relation of the universe of discourse is presented.

Definition 3.9. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , c_h ∈ C (h = 1, 2, ⋯ , m), d_k ∈ D (k = 1, 2, ⋯ , l) , and 0 ≤ β ≤ α ≤ 1, the α lower and β upper approximation of the pessimistic multigranulation probabilistic rough set with respect to (U, C, D, F, ω, μ) based on neighborhood relation are defined as follows, respectively.

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq α \land P (A | δ_{C} (x^{2})) \geq α \land \dots \land P (A | δ_{C} (x^{l})) \geq α};$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A) = {x \in U | P (A | δ_{C} (x^{1})) > β \lor P (A | δ_{C} (x^{2})) > β \lor \dots \lor P (A | δ_{C} (x^{l})) > β};$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A) = {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (\sim A)$ .

So the set-pair $({\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A)$ , ${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A))$ is called pessimistic multigranulation probabilistic rough set based on neighborhood relation.

Furthermore, the positive region, the negation region and the boundary region of the pessimistic multigranulation probabilistic rough set model based on δ-neighborhood relation are given as follows, respectively. ${Pos}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A) = {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A);$ ${Neg}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A) = U - {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A);$ $\begin{matrix} {Bn}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, α, β} (A) & = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{P, β} (A) \\ - {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A) . \end{matrix}$

As discussed for the relationship between the multigranulation probabilistic rough set model and the other original multigranulation rough set models, the results are also hold for the multigranulation probabilistic rough set model based on δ-neighborhood relation established in former. In the following, the original multigranulation rough set models can be deduced by using the optimistic and pessimistic multigranulation rough set model based on δ-neighborhood relation.

Remark 3.1. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (k = 1, 2, …, l) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , and 0 ≤ β ≤ α ≤ 1. Then

(1) If α = 1, β = 0, δ = 0, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1 \lor P (A | δ_{C} (x^{2})) \geq 1 \lor \dots \lor P (A | δ_{C} (x^{l})) \geq 1}$

= {x ∈ U|δ_C (x¹) ⊆ A ∨ δ_C (x²) ⊆ A ∨ … ∨ δ_C (x^l) ⊆ A}

= {x ∈ U| [x¹] ⊆ A ∨ [x²] ⊆ A ∨ … ∨ [x^l] ⊆ A} $= {\underline{Apr}}_{Σ_{k = 1}^{l} [x^{k}]}^{O} (A);$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 0} (A) = {x \in U | P (A | δ_{c} (x^{1})) > 0 \land P (A | δ_{C} (x^{2})) > 0 \land \dots \land P (A | δ_{C} (x^{l})) > 0}$

= {x ∈ U|δ_C (x¹) ∩ A ≠ ∅ ∧ δ_C (x²) ∩ A ≠ ∅ ∧ … ∧ δ_C (x^l) ∩ A ≠ ∅}

= {x ∈ U| [x¹] ∩ A ≠ ∅ ∧ [x²] ∩ A ≠ ∅ ∧ … ∧ [x^l] ∩ A ≠ ∅} $= {\bar{Apr}}_{Σ_{k = 1}^{l} [x^{k}]}^{O} (A)$ .

This is the optimistic multigranulation rough set model based on similarity relation.

At this time, the optimistic multigranulation rough set model based on δ-neighborhood relation degenerates into the optimistic multigranulation rough set model proposed by Qian [21].

(2) If α = 1, β = 0, δ = 0, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1 \land P (A | δ_{C} (x^{2})) \geq 1 \land \dots \land P (A | δ_{C} (x^{l})) \geq 1}$

= {x ∈ U|δ_C (x¹) ⊆ A ∧ δ_C (x²) ⊆ A ∧ … ∧ δ_C (x^l) ⊆ A}

= {x ∈ U| [x¹] ⊆ A ∧ [x²] ⊆ A ∧ … ∧ [x^l] ⊆ A} $= {\underline{Apr}}_{Σ_{k = 1}^{l} [x^{k}]}^{P} (A);$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 0} (A) = {x \in U | P (A | δ_{C} (x^{1})) > 0 \lor P (A | δ_{C} (x^{2})) > 0 \lor \dots \lor P (A | δ_{C} (x^{l})) > 0}$

= {x ∈ U|δ_C (x¹) ∩ A ≠ ∅ ∨ δ_C (x²) ∩ A ≠ ∅ ∨ … ∨ δ_C (x^l) ∩ A ≠ ∅}

= {x ∈ U| [x¹] ∩ A ≠ ∅ ∨ [x²] ∩ A ≠ ∅ ∨ … ∨ [x^l] ∩ A ≠ ∅} $= {\bar{Apr}}_{Σ_{k = 1}^{l} [x^{k}]}^{P} (A)$ . This is the pessimistic multigranulation rough set model based on similarity relation.

At this time, the pessimistic multigranulation rough set model based on δ-neighborhood relation degenerates into the pessimistic multigranulation rough set model proposed by Qian [21].

Therefore, it is easy obtain that both the optimistic and pessimistic multigranulation rough set are special case of multigranulation probabilistic rough set based on δ-neighborhood relation.

Remark 3.2. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (k = 1, 2, …, l) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , and 0 ≤ β ≤ α ≤ 1. Then

(1) If α = 1, β = 0, l = 1, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1}$ = {x ∈ U|δ_C (x¹) ⊆ A}

$= {x \in U | δ_{C} (x^{1}) \leq δ} = {\underline{Apr}}_{δ}^{O, 1} (A);$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 0} (A) = {x \in U | P (A | δ_{C} (x^{1})) > 0}$

$= {x \in U | δ_{C} (x^{1}) \cap A \neq \emptyset} = {\bar{Apr}}_{δ}^{O, 0} (A)$ .

This is the rough set model based on neighborhood relation. At this time, the optimistic multigranulation rough set model based on δ-neighborhood relation degenerates into the rough set model based on neighborhood relation.

(2) If α = 1, β = 0, l = 1, δ = 0, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1} = {x \in U | δ_{C} (x^{1}) \subseteq A}$

$= {x \in U | [x^{1}] \subseteq A} = {\underline{Apr}}_{λ}^{O} (A)$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, 0} (A) = {x \in U | P (A | δ_{C} (x^{1})) > 0} = {x \in U | δ_{C} (x^{1}) \cap A \neq \emptyset}$

$= {x \in U | [x^{1}] \cap A \neq \emptyset} = {\bar{Apr}}_{λ}^{O} (A)$ .

This is the original rough set model based on similarity relation, i.e, the classic rough set. At this time, the optimistic multigranulation rough set model based on δ-neighborhood relation degenerates into the original rough set model proposed by Pawlak [115].

(3) If α = 1, β = 0, l = 1, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1} = {x \in U | δ_{C} (x^{1}) \subseteq A}$

$= {x \in U | δ_{C} (x^{1}) \leq δ} = {\underline{Apr}}_{δ}^{P} (A)$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 0} (A) = {x \in U | P (A | δ_{C} (x^{1})) > 0} = {x \in U | δ_{C} (x^{1}) \cap A \neq \emptyset}$

$= {\bar{Apr}}_{δ}^{P} (A)$

(4) If α = 1, β = 0, l = 1, δ = 0, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq 1} = {x \in U | δ_{C} (x^{1}) \subseteq A}$

$= {x \in U | [x^{1}] \subseteq A} = {\underline{Apr}}_{λ}^{P} (A)$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 0} (A) = {x \in U | P (A | δ_{C} (x^{1})) > 0} = {x \in U | δ_{C} (x^{1}) \cap A \neq \emptyset}$

$= {x \in U | [x^{1}] \cap A \neq \emptyset} = {\bar{Apr}}_{λ}^{P} (A)$ .

This is the original rough set model based on similarity relation.

At this time, the pessimistic multigranulation rough set model based on δ-neighborhood relation degenerates into the original rough set model proposed by Pawlak [15].

By analyzing several important properties of the neighborhood-based multigranulation probability rough set, it can be found that when the size of the neighborhood is equal to zero, the new rough set degenerates into the original MGRS. Obviously, the multigranulation probabilistic rough set model based on δ-neighborhood relation is the natural generalization of Yao’s probabilistic rough set model [33] and original Pawlak rough set [15], respectively. A more adaptable model can be obtained by setting different parameters. Therefore, the optimistic and pessimistic multigranulation probabilistic rough set based on δ-neighborhood relation defined in this section is a reasonable generalization of the existing model.

Next, taking the optimistic multigranulation probabilistic rough set based on δ-neighborhood relation as an example, some properties are discussed in detail.

Theorem 3.2. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U, k = 1, 2, …, l) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , c_h ∈ C (h = 1, 2, ⋯ , m) and d_k ∈ D (k = 1, 2, ⋯ , l) , and 0 ≤ β ≤ α ≤ 1. Then

${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) \subseteq U,$

${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (φ) = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (\emptyset) = \emptyset,$

${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A^{c}) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A^{c}) \subseteq U,$

${({\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A))}^{c} = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A^{c})$ .

Proof. According to the definition of optimistic multigranulation probabilistic rough set based on δ-neighborhood relation, for any A ∈ P (U) , 0 ≤ β ≤ α ≤ 1, we have ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) \subseteq U,$ and ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) \subseteq U$ . Due to the definition of the upper and lower approximation, we prove that ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) \subseteq U$ . Similarly, ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (φ) = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (\emptyset) = \emptyset$ .

According to the definition of optimistic multigranulation probabilistic rough set based on δ-neighborhood relation and Proposition 3.1, for any A ∈ P (U) , 0 ≤ β ≤ α ≤ 1, we have ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A^{c}) \subseteq U,$ and ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A^{c}) \subseteq U$ . Similarly, ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A^{c}) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A^{c}) \subseteq U$ and ${({\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A))}^{c} = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A^{c})$ are satisfied.□

Theorem 3.3. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U, k = 1, 2, …, l) is the δ-neighborhood relation classes of universe U. For any A ∈ P (U) , c_h ∈ C (h = 1, 2, ⋯ , m) , d_k ∈ D (k = 1, 2, ⋯ , l) , and 0 ≤ β ≤ α ≤ 1. Then

For any α₁ ≤ α₂, then ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{2}} (A) \subseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{1}} (A),$

For any β₁ ≤ β₂, then ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{2}} (A) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{1}} (A),$

For any A₁, A₂ ∈ P (U) and A₁ ⊆ A₂, then ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{1}) \subseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{2}),$ ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{1}) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{2})$ .

Proof. According to the definition of the α lower approximation of optimistic multigranulation probabilistic rough set based on δ-neighborhood relation, for any A ∈ P (U) , 0 ≤ α₁ ≤ α₂ ≤ 1, we have ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{1}} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq α_{1} \lor P (A | δ_{C} (x^{2})) \geq α_{1} \lor \dots \lor P (A | δ_{C} (x^{l})) \geq α_{1}}$ and ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{2}} (A) = {x \in U | P (A | δ_{C} (x^{1})) \geq α_{2} \lor P (A | δ_{C} (x^{2})) \geq α_{2} \lor \dots \lor P (A | δ_{C} (x^{l})) \geq α_{2}}$ . It is easy to know that {x ∈ U|P (A|δ_C (x^k)) ≥ α₂} ⊆ {x ∈ U|P (A|δ_C (x^k)) ≥ α₁} , then ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{2}} (A) \subseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{1}} (A)$ .□

According to the definition of the β upper approximation of optimistic multigranulation probabilistic rough set based on δ-neighborhood relation, for any A ∈ P (U) , 0 ≤ β₁ ≤ β₂ ≤ 1, we have ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{1}} (A) = {x \in U | P (A | δ_{C} (x^{1})) > β_{1} \land P (A | δ_{C} (x^{2})) > β_{1} \land \dots \land P (A | δ_{C} (x^{l})) > β_{1}}$ and ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{2}} (A) = {x \in U | P (A | δ_{C} (x^{1})) > β_{2} \land P (A | δ_{C} (x^{2})) > β_{2} \land \dots \land P (A | δ_{C} (x^{l})) > β_{2}}$ . It is easy to know that {x ∈ U|P (A|δ_C (x^k)) > β₂} ⊆ {x ∈ U|P (A|δ_C (x^k)) > β₁} , then ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{2}} (A) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{1}} (A)$ .

Similarly,for any A₁, A₂ ∈ P (U) and A₁ ⊆ A₂, we have ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{1}) \subseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{2}),$ ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{1}) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{2})$ .

3.4 Multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation

In this section, the rough approximation of a fuzzy decision-making object with respect to multiple attribute fuzzy decision-making linguistic information system is considered, i.e., multigranulation fuzzy rough set model based on δ-neighborhood relation. Then, the multigranulation probability fuzzy rough set model is constructed. Conveniently, the properties of the optimistic versions are discussed mainly. The pessimistic versions can be done similarly. After that, the relationship between multigranulation rough set model and multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation will be discussed in detail.

As is well known, the key concept in probabilistic rough set is the conditional probability between the target set and the equivalence classes. According to the concept of the probabilistic rough set based on neighborhood relation of the universe of discourse given in Section 3.2, we firstly give the definition of conditional probability of any fuzzy event A given the description of a crisp set based on δ-neighborhood relation.

Definition 3.10. Let U be a non-empty finite universe and δ_C (x) (x ∈ U) be the δ- neighborhood relation classes of universe U. P is the probabilistic measure. For any A ∈ F (U) , the conditional probability of fuzzy event A is defined as follows: $P (A | δ_{C} (x)) = \frac{\sum_{y \in δ_{C} (x)} A (y)}{| δ_{C} (x) |}, x \in U$

The P (A|δ_C (x)) also can be understood as the probability that a randomly selected object x ∈ U belongs to the fuzzy concept given the description δ_C (x).

By this definition, the following properties are clear.

Proposition 3.3. Let U be a non-empty finite universe and δ_C (x) (x ∈ U) be the δ- neighborhood relation classes of universe U. P is the probabilistic measure. Then the following conclusions hold.

0 ≤ P (A|δ_C (x)) ≤1.

If A, B ∈ F (U) and A ⊆ B, then P (A|δ_C (x)) ⊆ P (B|δ_C (x)).

P (A^c|δ_C (x)) = 1 - P (A|δ_C (x)).

Based on the above definition of conditional probability of a fuzzy event given the δ- neighborhood relation class of any object, we present the multigranulation probabilistic fuzzy rough set based on δ- neighborhood relation classes with linguistic preference information.

Definition 3.11. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U) is the δ-neighborhood relation classes of universe U. For any A ∈ F (U) , and d_k ∈ D (k = 1, 2, ⋯ , l) , 0 ≤ β ≤ α ≤ 1, the α lower and β upper approximations of optimistic multigranulation probabilistic fuzzy rough set based on neighborhood relation with respect to(U, C, D, F, ω, μ) are defined as follows, respectively.

${\underline{Apr}}_{\sum_{k = 1}^{1} δ_{C} (x^{k})}^{O, α} (A) (x) = min {A (x) | P (A | δ_{C} (x^{1})) \geq α \lor P (A | δ_{C} (x^{2})) \geq α \lor \dots \lor P (A | δ_{C} (x^{l})) \geq α};$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) (x) = max {A (x) | P (A | δ_{C} (x^{1})) > β \land P (A | δ_{C} (x^{2})) > β \land \dots \land P (A | δ_{C} (x^{l})) > β}$ .

Here the minimum and maximum will become inf and sup when universe U is infinite set.

Let ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) (x) = 0,$ ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) (x)$ = 1, ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) (x)$ and ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) (x)$ are two fuzzy sets of universe U. When ${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) (x) = {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) (x),$ a definable fuzzy set with respect to multiple attribute fuzzy decision-making linguistic information system (U, C, D, F, ω, μ). Then, a set pair $({\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A) (x), {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A) (x))$ , is called optimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation.

Similarly, we can present the definition of the pessimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation of the universe of discourse.

Definition 3.12. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic-valued information system. δ_C (x^k) (x ∈ U) is the δ-neighborhood relation class of universe U. For any A ∈ F (U) , and 0 ≤ β ≤ α ≤ 1, the α lower and β upper approximations of the pessimistic multigranulation probabilistic fuzzy rough set with respect to(U, C, D, F, ω, μ) based on neighborhood relation are defined as follows, respectively.

${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{P, α} (A) (x) = min {A (x) | P (A | δ_{C} (x^{1})) \geq α \land P (A | δ_{C} (x^{2})) \geq α \land \dots \land P (A | δ_{C} (x^{l})) \geq α};$

${\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{k})}^{P, β} (A) (x) = max {A (x) | P (A | δ_{C} (x^{1})) > β \lor P (A | δ_{C} (x^{2})) > β \lor \dots \lor P (A | δ_{C} (x^{l})) > β}$ .

Next, taking the optimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation as an example, some properties are discussed in detail.

Theorem 3.4. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U, k = 1, 2, …, l) is the δ-neighborhood relation classes of universe U. For any A ∈ F (U) , d_k ∈ D (k = 1, 2, ⋯ , l) , and 0 ≤ β ≤ α ≤ 1. Then

For any α₁ ≤ α₂, then ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{1}} (A) (x) \supseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α_{2}} (A) (x),$

For any β₁ ≤ β₂, then ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{1}} (A) (x) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β_{2}} (A) (x),$

For any A₁, A₂ ∈ F (U) and A₁ ⊆ A₂, then ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{1}) \subseteq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, α} (A_{2}),$ ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{1}) \subseteq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{k})}^{O, β} (A_{2})$ .

Proof. It can be derived directly by the definition of optimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation.

Similar to the multigranulation probabilistic rough set model based on δ-neighborhood relation given in Section 3.2, it is easy to obtain that the relationship between optimistic and pessimistic multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation and the original multigranulation fuzzy rough set model by deducing the basic model.□

Remark 3.3. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U) is the δ-neighborhood relation class of universe U. For any A ∈ P (U) , and 0 ≤ β ≤ α ≤ 1,.

(1) If $α = \frac{1}{l}, β = \frac{1}{l}, δ = 0,$ there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, \frac{1}{l}} (A) (x) = min {A (x) | P (X | δ_{C} (x^{1})) \geq \frac{1}{l} \lor P (X | δ_{C} (x^{2})) \geq \frac{1}{l} \lor \dots \lor P (X | δ_{C} (x^{l})) \geq \frac{1}{l}}$

$= min {A (x) | \frac{Σ_{y \in δ_{C} (x^{k})} A (y)}{| δ_{C} (x^{k}) |} \geq \frac{1}{l}, x \in U}$

$= min {A (x) | \frac{Σ_{y \in δ_{C} (x^{k})} A (y)}{l} \geq \frac{1}{l}, x \in U}$

= min {A (x) |Σ_{y∈δ_C(x^k)}A (y) ≥1, x ∈ U}

= min {A (x) |A (y) =1, ∃ k = 1, 2, …, l, x ∈ U}

= min {A (x) |A^k (x) ≥ λ, ∃ k = 1, 2, …, l, x ∈ U}

= min {A (x) |A¹ (x) ≥ λ ∨ A² (x) ≥ λ ∨ … ∨ A^l (x) ≥ λ, ∀ k = 1, 2, …, l, x ∈ U}

$= {\underline{Apr}}_{\sum_{k = 1}^{l} [x^{k}]}^{O} (A)$

Similarly, we can obtain the upper approximation as following form.

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, \frac{1}{l}} (A) (x) = max {A (x) | P (X | δ_{C} (x^{1})) > \frac{1}{l} \land P (X | δ_{C} (x^{2})) > \frac{1}{l} \land \dots \land P (X | δ_{C} (x^{l})) > \frac{1}{l}}$

$= {\bar{Apr}}_{\sum_{k = 1}^{l} [x^{k}]}^{O} (A)$

This is the optimistic multigranulation fuzzy rough set based on λ-similarity relation.At this time, the optimistic multigranulation fuzzy rough set model based on δ-neighborhood relation degenerates into the optimistic multigranulation rough set model proposed by Qian [21].

(2) If α = 1, β = 1, δ = 0, there are

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) (x) = min {A (x) | P (X | δ_{C} (x^{1})) \geq 1 \land P (X | δ_{C} (x^{2})) \geq 1 \land \dots \land P (X | δ_{C} (x^{l})) \geq 1}$

$= min {A (x) | \frac{Σ_{y \in δ_{C} (x^{k})} A (y)}{| δ_{C} (x^{k}) |} \geq 1, x \in U}$

$= min {A (x) | \frac{Σ_{y \in δ_{C} (x^{k})} A (y)}{l} \geq 1, x \in U}$

= min {A (x) |Σ_{y∈δ_C(x^k)}A (y) ≥ l, x ∈ U}

= min {A (x) |A (y) =1, ∀ k = 1, 2, …, l, x ∈ U}

= min {A (x) |A^k (x) ≥ λ, ∀ k = 1, 2, …, l, x ∈ U}

= min {A (x) |A¹ (x) ≥ λ ∧ A² (x) ≥ λ ∧ … ∧ A^l (x) ≥ λ, ∀ k = 1, 2, …, l, x ∈ U}

$= {\underline{Apr}}_{\sum_{k = 1}^{l} [x^{k}]}^{P} (A)$

Similarly, we can obtain the upper approximation as following form.

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) (x) = max {A (x) | P (X | δ_{C} (x^{1})) > 1 \lor P (X | δ_{C} (x^{2})) > 1 \lor \dots \lor P (X | δ_{C} (x^{l})) > 1}$

$= {\bar{Apr}}_{\sum_{k = 1}^{l} [x^{k}]}^{P} (A)$

This is the pessimistic multigranulation fuzzy rough set based on λ-similarity relation.At this time, the pessimistic multigranulation fuzzy rough set model based on δ-neighborhood relation degenerates into the pessimistic multigranulation rough set model proposed by Qian [21].

Remark 3.4. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. δ_C (x^k) (x ∈ U) is the δ-neighborhood relation class of universe U. For any A ∈ P (U) , and 0 ≤ β ≤ α ≤ 1,

(1) If $α = \frac{1}{l}, β = \frac{1}{l}, δ = 0, l = 1$ there are $\begin{matrix} {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, \frac{1}{l}} (A) (x) \\ = min {A (x) | P (X | δ_{C} (x^{1}) \geq \frac{1}{l}} \\ = min {A (x) | Σ_{y \in δ_{C} (x^{k})} A (y) \geq 1, x \in U} \end{matrix}$ $\begin{matrix} = min {A (x) | A^{1} (x) \geq λ, x \in U} \\ = {\underline{Apr}}_{{[x^{k}]}^{λ}}^{O} (A) \end{matrix}$

Similarly, we can obtain the upper approximation as following form. $\begin{matrix} {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{O, \frac{1}{l}} (A) (x) \\ = max {A (x) | P (X | δ_{C} (x^{1})) > \frac{1}{l}} \\ = {\bar{Apr}}_{{[x^{k}]}^{λ}}^{O} (A) \end{matrix}$

This is the fuzzy rough set model based on similarity relation. Obviously, this λ-similarity relation fuzzy rough set also is generalization of the classical fuzzy rough set model [5].

(2) If α = 1, β = 1, δ = 0, l = 1 there are $\begin{matrix} {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) (x) \\ = min {A (x) | P (X | δ_{C} (x^{1})) \geq 1} \\ = min {A (x) | Σ_{y \in δ_{C} (x^{k})} A (y) \geq 1, x \in U} \\ = min {A (x) | A^{1} (x) \geq λ, x \in U} \\ = {\underline{Apr}}_{{[x^{k}]}^{λ}}^{P} (A) \end{matrix}$

Similarly, we can obtain the upper approximation as following form. $\begin{matrix} {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x^{k})}^{P, 1} (A) (x) \\ = max {A (x) | P (X | δ_{C} (x^{1})) > 1} \\ = {\bar{Apr}}_{[x^{k}]^{λ}}^{P} (A) \end{matrix}$

This is the fuzzy rough set model based on similarity relation. Obviously, this λ-similarity relation fuzzy rough set also is generalization of the classical fuzzy rough set model [4]. At the same time, we will prove that the optimistic and pessimistic multigranulation fuzzy rough set model based on δ-neighborhood relation degenerates into the multigranulation rough set model based on λ-similarity relation according to Definition 3.8 and Definition 3.9. That is, the multigranulation fuzzy rough set model based on δ-neighborhood relation is more extensive than the multigranulation rough set model [21] if the fuzzy decision-making object degenerates into crisp decision-making object over the universe of discourse.

Based on Remarks 3.1 to 3.4, the relationship between the optimistic multigranulation probabilistic fuzzy rough set model based on neighborhood relation as well as the pessimistic multigranulation probabilistic fuzzy rough set model based on neighborhood relation and the original multigranulation rough set model as well as the classical rough set model are obtained. In practice, the values of the parameters α, β reflect the degree of preference of the decision maker for risk in the decision-making process. For different requirements of decision makers, the corresponding parameter settings are not same. Therefore, the value of the precision parameter α, β (0 ≤ β ≤ α ≤ 1) is determined by the decision maker’s preference or by investigating in advance. At the same time, according to the different purposes of decision-making, choose the appropriate model for research.

So far, a multigranulation probabilistic fuzzy rough set based on neighborhood relation has been established on multiple attribute fuzzy decision information systems with language preference information. This model provides an effective approach for problem solving in the context of multi granulations. Also, it can be seen that the multigranulation probabilistic fuzzy rough set based on neighborhood relation is a more generalized multigranulation rough set form, which is also a useful extension to the existing multigranulation rough set model. Further, it provides a new tool and idea for subsequent processing of uncertain, inaccurate, incomplete data or decision systems.

4 δ-neighborhood relation-based multigranulation probabilistic fuzzy rough set approach to MAGDM with linguistic information

In this section, the principle and approach to multiple attribute group decision-making with linguistic information based on multigranulation probabilistic fuzzy rough set with respect to δ-neighborhood relation established in Section 3 will be introduced. Also, the decision making algorithm and the general steps for the established method will be presented in detail.

4.1 Description of the multiple attribute group decision-making problem with linguistic information

Let U = {x₁, x₂, ⋯ , x_n} be a discrete set of alternatives, C = {c₁, c₂, ⋯ , c_m} be the set of attributes, ω = (ω₁, ω₂, ⋯ , ω_m) ^T be the weight vector of attributes where $ω_{j} \geq 0, \sum_{j = 1}^{m} ω_{j} = 1$ . Let D = {d₁, d₂, ⋯ , d_l} be the set of decision makers, and μ = (μ₁, μ₂, ⋯ , μ_l) ^T be the weight vector of decision-makers, where μ_k ≥ 0, k = 1, 2, ⋯ , l, and satisfies $\sum_{i = 1}^{l} μ_{k} = 1$ . The decision-maker d_k ∈ D present the evaluation for all alternatives with respect to all the attributes C by a family of mapping F = {f_k|k = 1, 2, ⋯ , l} from universe U and attribute set C, where f_k : U × C → V_k and V_k = ∪ _{c_h∈C}V_h (h = 1, 2, ⋯ , m) is the domain of attribute set C. The decision maker d_k ∈ D compares these alternatives with respect to a single criterion using the linguistic terms in the set S ={ S_α|α = - t, …, -1, 0, 1, …, t } is the linguistic terms set. Then, different decision-makers present different evaluations for the same alternative with respect to the attribute set. Then, the decision maker constructs the linguistic preference relation $B_{ij} = {(b_{ij}^{(k)})}_{n \times n} (k = 1, 2, \dots, l)$ . So, the multiple attribute fuzzy group decision-making linguistic information system (U, C, D, F, ω, μ) is constructed. Then, the decision-making problem is how to select the optimal alternative or ranking all alternatives according to the linguistic information evaluations given by different decision-makers with different weights.

4.2 The model and decision-making methodology

According to the previous description of the multiple attribute group decision-making problem with linguistic information given in Section 4.1, this subsection will present the decision model and steps by using multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation for the considered multiple attribute group decision-making problem with linguistic information. On the basis of the multigranulation probabilistic fuzzy rough set model based on neighborhood relation and the basic idea and structure of multiple attribute group decision making problem with language preference information, the proposed decision-making model mainly includes three steps.

The first process is to calculate the conditional probability under the δ-neighborhood relation of the universe of universe. In order to determine the domain of A, the Multimoora method is used to sort the scheme. The second process is computing the lower and upper approximations of the fuzzy decision-making object of universe with the precision parameter α, β (0 ≤ β ≤ α ≤ 1). Finally, making the result of ranking for all alternatives by using the decision-making principle based on the former two steps and then give the optimal decision making (i.e., the optimal alternative).

In the following, according to the three processes given in above, the model and method of the multiple attribute group decision-making with respect to multigranulation probabilistic fuzzy set based on δ-neighborhood relation is proposed one by one.

Firstly, calculating the conditional probability under the δ-neighborhood relation of the universe of universe. As is described in Section 4.1, it is necessary to sort all alternatives based on the evaluations given by all decision-makers with respect to attribute set. But, the values of evaluation given by decision-makers are linguistic-valued (i.e. using the linguistic terms in the set S ={ S_α|α = - t, …, -1, 0, 1, …, t }. In order to establish the ranking function over the multiple attribute fuzzy decision-making linguistic-valued information system, we make a transformation between the linguistic-valued and the real-valued over [0,1]. Then, the definition is presented as follows.

Definition 4.1. [14, 24] Let S = {s_α|s_-q≤ s_α ≤ s_q, α ∈ [- q, q]} be a set of extended continuous linguistic terms. For any $S_{α} \in \bar{S}$ , a continuous mapping T from continuous linguistic terms $\bar{S}$ to real-valued interval [0, 1] is defined as follows: $T : \bar{S} \to [0, 1] and T (s_{α}) = \frac{| I (s_{α}) - I (s_{- α}) |}{2 q} .$ (7)

In fact, the conversion between the linguistic value and the real value [0, 1] can also be rephrased as follows: $T (s_{α}) = \frac{| I (s_{α}) - I (s_{0}) |}{q} = 0.5 + \frac{I (s_{α})}{2 q} .$ (7′)

Then, the continuous mapping T (•) is called the transformation between $\bar{S}$ and [0, 1]. At the same time, according to the definition of T (•), the following properties are clear.

Proposition 4.1. Let S = {s_α|s_-q≤ s_α ≤ s_q, α ∈ [- q, q]} be a set of extended continuous linguistic terms. T (•) is a transformation between the linguistic-valued and the real-valued over [0, 1]. Then

T (s_α) ∈ [0, 1] for any $s_{α} \in \bar{S};$

T (s_-q) = 0, T (s₀) = 0.5, T (s_q) = 1 ;

T is increased over $\bar{S}$ with the increasing of s_α for any α ∈ [- q, q].

Proof. It can be verified directly by the definition.□

According to Definition 4.1, T (•) is a transformation between $\bar{S}$ and [0, 1]. For any x_i ∈ U (i = 1, 2, ⋯ , n), c_h ∈ C (h = 1, 2, ⋯ , m), and d_k ∈ D (k = 1, 2, ⋯ , l) , the definition of the conversion function that gets the corresponding evaluation value is presented. $f_{k} (c_{h} (x_{i})) = T (b_{ih}^{k}) .$ (8)

Next, it is feasible to use the above formula to convert the value of the variable to get the value, which determines the fuzzy decision-making object of the universe of discourse. That is, the evaluation of all alternatives with respect to any attribute is a fuzzy set over the attribute set.

Then, the idea of Multimoora [2] (Multiplicative and Multi-Objective Optimization by Ratio Analysis) principle is used to obtain the best fuzzy decision-making object and the worst fuzzy decision-making object in order to avoid the subjective preference of individual decision-maker as much as possible.

The steps to construct the optimal index function of fuzzy decision-making objects of the universe of discourse by using Multimoora are as follows. It has three parts, namely Ratio System, Reference Point and Full Multiplicative Form. Then, the rank of these three parts is determined by using the formula of the Multimoora.

According to the formula (7)and the formula (8), the corresponding evaluation value is obtained by the definition of the conversion function, and then standardized by the following formula. Ratio system of MOORA defines data normalization by comparing alternative of an objective to all values of the objective: $x_{ij}^{*} = \frac{T (b_{ih}^{k})}{\sqrt{\sum_{i = 1}^{m} T {(b_{ih}^{k})}^{2}}}$ (9) where $x_{ij}^{*}$ represents the dimensionless number representing the normalized response of alternative j (j = 1, 2, ⋯ , n) on objective i (i = 1, 2, ⋯ , m) , $T (b_{ih}^{k})$ represents the conversion function. It is easy to know that the normalized results of the alternatives with respect to the attributes belong to the interval [0, 1].

These indicators are added (if desirable value of indicator is maxima) or subtracted (if desirable value is minima). Thus, summary index of each alternative is derived in this way: $y_{i} = \sum_{j = 1}^{g} x_{ij}^{*} - \sum_{j = g + 1}^{n} x_{ij}^{*}$ (10) where j = 1, 2, …, g denotes the number of objectives to be maximized; j = g + 1, g + 2, …, n denotes the number of objectives to be minimized; y_i is represents the normalized assessment of alternative i (i = 1, 2, ⋯ , m) with respect to all objectives.

So, the optimal solution under the Ratio System Approach is $x_{RS}^{*} = {x_{i} | max_{i} y_{i} |}$ .

The Reference Point of MOORA is based on the ratio system. The Maximal Objective Reference Point is established by the ratio relationship obtained by formula (9). According to the deviation from the reference point and the Min-Max metric of Tchebycheff, the final ranks of the alternatives are given.

Then, the optimal solution under the Reference Point Approach is $x_{RP}^{*} = {x_{i} | min_{i} (max$ $| r_{j} - x_{ij}^{*} |)},$ in that

$r_{j} = {\begin{matrix} max_{i} x_{ij}^{*}, j \in S \\ min_{i} x_{ij}^{*}, j \in C \end{matrix}$ (11) where r_j represents the j (j = 1, 2, ⋯ , n)th coordinate of the reference point, $x_{ij}^{*}$ represents the normalized attribute i (i = 1, 2, ⋯ , m) of alternative j (j = 1, 2, ⋯ , n) , S is benefit attribute, C is cost attribute.

Brauers and Zavadskas proposed updating MOORA by a method of fully multiplication, which embodies maximization and minimization of the purely multiplication utility function. This became the full multiplicative form of Multimoora. The overall utility of the ith alternative is represented as dimensionless numbers: $U_{i} = \frac{\prod_{j = 1}^{g} x_{ij}^{*}}{\prod_{j = g + 1}^{n} x_{ij}^{*}}$ (12) where $x_{ij}^{*}$ represents the dimensionless number representing the normalized response of alternative j (j = 1, 2, ⋯ , n) on objective i (i = 1, 2, ⋯ , m), and U_i denotes the overall utility of alternative i (i = 1, 2, ⋯ , m).

Then, the optimal solution under the Full Multiplicative Form is $x_{MF}^{*} = {x_{i} | max_{i} U_{i} |}$ .

In this way, the Multimoora method summarizes the ratio system and reference point of MOORA in the full multiplicative form. Then the three different sorts corresponding to the three parts of the Multimoora method are combined into a single sort by the dominating theory [2]. After that, determine the comprehensive ranking of the alternatives.

Based on the three different ranking results, a final ranking result is obtained according to Equations (9)–(12). According to the need of actual decision-making, select the appropriate number of alternatives as the domain of A choose the right amount of alternatives. So, the domain of A given by Multimoora method is constructed.

In order to obtain the membership function of a fuzzy set A, we use the idea of TOPSIS principle to obtain the best optimal fuzzy decision-making object and the worst optimal fuzzy decision-making object. This method avoids the subjective preference of individual decision-maker as much as possible. So, the best and worst fuzzy decision-making objects of the universe of discourse given by individual decision-maker with respect to attribute set is constructed as follows, respectively.

$\begin{matrix} A_{k}^{+} (x_{i}) = max {f_{k} (c_{h} (x_{i})) | x_{i} \in U, k = 1, 2, \dots, l; \\ h = 1, 2, \dots, m; i = 1, 2, \dots, n} \end{matrix}$ (13)

$\begin{matrix} A_{k}^{-} (x_{i}) = min {f_{k} (c_{h} (x_{i})) | x_{i} \in U, k = 1, 2, \dots, l; \\ h = 1, 2, \dots, m; i = 1, 2, \dots, n} \end{matrix}$ (14)

It can be easily seen that the decision-making objects $A_{k}^{+}$ and $A_{k}^{-}$ are fuzzy set of universe U.

Since both optimistic multigranulation probabilistic fuzzy rough set and pessimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation are similar when defining, next calculation takes optimistic multigranulation probabilistic fuzzy rough set based on δ-neighborhood relation as an example.

Secondly, we compute the lower and upper approximations of the best and worst fuzzy decision-making objects, given by group decision-makers with respect to attribute set based on the results given by individual decision-maker in formula (6) - (11), with respect to multiple fuzzy decision-making linguistic-valued information system under the precision parameter α, β (0 ≤ β ≤ α ≤ 1) , respectively.

$\begin{matrix} {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{+}) (x_{i}) \\ = min {A^{+} (x_{i}) | max P (A | δ_{C} (x_{i}^{k})) \geq α} \end{matrix}$ (15)

$\begin{matrix} {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{+}) (x_{i}) \\ = max {A^{+} (x_{i}) | min P (A | δ_{C} (x_{i}^{k})) > β} \end{matrix}$ (16)

$\begin{matrix} {\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{-}) (x_{i}) \\ = min {A^{-} (x_{i}) | max P (A | δ_{C} (x_{i}^{k})) \geq α} \end{matrix}$ (17)

$\begin{matrix} {\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{-}) (x_{i}) \\ = max {A^{-} (x_{i}) | min P (A | δ_{C} (x_{i}^{k})) > β} \end{matrix}$ (18) Proposition 4.2. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy group decision-making linguistic information system. For any x_i ∈ U (i = 1, 2, ⋯ , n) , c_h ∈ C (h = 1, 2, ⋯ , m), d_k ∈ D (k = 1, 2, ⋯ , l), and α, β (0 ≤ β ≤ α ≤ 1), then

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{+}) (x_{i}) \leq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{+}) (x_{i}),$

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{-}) (x_{i}) \leq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{-}) (x_{i});$

${\underline{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{-}) (x_{i}) \leq {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{+}) (x_{i}),$

${\bar{Apr}}_{\sum_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{-}) (x_{i}) \leq {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{+}) (x_{i})$ .

Proof. It can be verified directly by the formulas (15) to (18).

Finally, putting the rules of ranking for all alternatives of the universe of discourse according to the optimistic multigranulation probabilistic lower and upper approximations of the decision-making objects based on δ-neighborhood relation.

Then, the definition of ranking function based on optimistic multigranulation probabilistic fuzzy rough set with respect to δ-neighborhood relation is presented in the follows. Here, the Euclidean distance [9] is used to define the ranking function for any alternative of the universe of discourse.

In the following, according to Definition 4.2 and Proposition 4.2, the concept of ranking function for any alternative of the universe of discourse over the multiple attribute fuzzy group decision-making linguistic-valued information system based on δ-neighborhood relation is given.□

Definition 4.3. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. $δ_{C} (x_{i}^{k})$ is the δ-neighborhood relation class of universe U. For any x_i ∈ U (i = 1, 2, ⋯ , n) , c_h ∈ C (h = 1, 2, ⋯ , m) , d_k ∈ D (k = 1, 2, ⋯ , l) , $A_{k}^{+}$ and $A_{k}^{-}$ are the worst and best fuzzy decision-making objects of the universe, and α, β (0 ≤ β ≤ α ≤ 1) ,

$\begin{matrix} φ_{k} (x_{i}) = [({\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{+}) (x_{i}) \\ - {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{-}) (x_{i}))^{2} \\ + ({\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{+}) (x_{i}) \\ - {\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{-}) (x_{i}))^{2}]^{\frac{1}{2}} \end{matrix}$ (19)

is called the ranking function of alternative x_i ∈ U, over (U, C, D, F, ω, μ).

Here, the Euclidean distance is used to define the ranking function for any alternative of the universe of discourse. Meanwhile, it is easy to prove that φ_k (x_i) ≥ 0 (x_i ∈ U) is hold according to Proposition 4.2.

Next, the concept of optimal index function of any alternative with respect to the fuzzy decision-making object over the multiple fuzzy decision-making linguistic information system is presented.

Definition 4.4. Let (U, C, D, F, ω, μ) be a multiple attribute fuzzy decision-making linguistic information system. $δ_{C} (x_{i}^{k})$ is the neighborhood relation class of universe U. For any x_i ∈ U (i = 1, 2, ⋯ , n), c_h ∈ C (h = 1, 2, ⋯ , m), d_k ∈ D (k = 1, 2, ⋯ , l) , μ = (μ₁, μ₂, ⋯ , μ_l) ^T, and α, β (0 ≤ β ≤ α ≤ 1), $φ (x_{i}) = \sum_{k = 1}^{l} μ_{k} φ_{k} (x_{i}), x_{i} \in U,$ (20) is called the optimal index function of alternative x_i ∈ U over (U, C, D, F, ω, μ).

According to Proposition 4.2 and Definition 4.2, it is easy to verify that φ (x_i) ≥ 0 (x_i ∈ U) is hold. Therefore, there is the final ranking for all alternatives based on the values of the optimal index function φ (x_i).

4.3 The steps of the decision model

In this section, the steps of the proposed multiple attribute group decision-making model with multigranulation probabilistic fuzzy rough set with linguistic information based on δ-neighborhood relation and Multimoora method are given as follows:

Input. Multiple attribute fuzzy group decision-making linguistic-valued information system (U, C, D, F, ω, μ) ;

Output. The sort ordering for all alternatives;

Step 1. Computing the distance measure $(ξ_{ij}^{kC})^{*}$ according to formula (2)–(5);

Step 2. Determining threshold δ (0 ≤ δ ≤ 1) and computing the δ-neighborhood relation classes $δ_{C} (A_{i}^{k})$ using formula (6);

Step 3. Construct the domain of A given by Multimoora method using formula (9)–(12);

Step 4. Construct the membership function of $A_{k}^{+}$ and $A_{k}^{-}$ given by TOPSIS method using formula (13)-(14);

Step 5. Calculate the conditional probability under the δ-neighborhood relation of the universe of discourse;

Step 6. Setting the precision parameter α, β (0≤ β ≤ α ≤ 1) ;

Step 7. Computing the α upper and β lower approximations ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{+}) (x_{i})$ , ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{+}) (x_{i})$ , ${\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, α} (A_{k}^{-}) (x_{i})$ and ${\bar{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x_{i}^{k})}^{O, β} (A_{k}^{-}) (x_{i})$ according to formula (15)–(18);

Step 8. Computing the ranking function φ_k (x_i) according to formula (19);

Step 9. Computing the optimal index function φ (x_i) according to formula (20);

Step 10. Present the ranking according to the values of the optimal index function. In this paper, a multi-attribute group decision model and method based on a multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation with linguistic preference information is proposed. Comparing to the traditional approach to MAGDM problems with linguistic information, the method proposed in this paper has several following advantages. The literature [7] points out that there are three steps to solve the multi-attribute decision making problem under linguistic information, and we present these differences from the previous literature in terms of information expression, information fusion and the ordering of the optimal alternatives.

Firstly, it is a meaningful attempt to apply the multigranulation probabilistic fuzzy rough set based on neighborhood relation to the multiple attribute group decision problem with uncertain linguistic information. Based on the δ-neighborhood relationship, this approach can be well applied to the problem of continuous variables. At the same time, optimistic multigranulation probabilistic rough set and pessimistic multigranulation probabilistic rough set can better reflect the different risk preferences of decision makers in actual decision-making. Different from the rough set method proposed in the existing literature [30], the method proposed in this paper is more adaptable in practical application. Secondly, it provides a new approach to acquire collective preferences based on the individual preference by using given in group decision problems. The upper and lower approximations of the neighborhood relation-based multigranulation rough fuzzy set used in this paper are different from the definitions of the aggregation operators proposed in previous literature [29]. Aggregation operators have the advantage of simple calculation when information fusion. However, when different aggregation operators are used for the same group decision problem, different final optimal results could be obtained. Such as OWA operators [8], OWG operators, the ULHA operators, ULOWA operator. In the proposed model, the individual preference given by different decision maker are considered as a granularity of multiple fuzzy decision information system with linguistic information, and then the collective preference is regarded as the total granularity. The principle of the multigranulation lower and upper approximation is used for the fusion all single granularity (i.e.,corresponds to individual preference).It is well known that a pair of lower approximation and upper approximation of any multigranulation rough set model are deterministic and unique. Undoubtedly, the total granularity (i.e.,collective preference) obtained by any multigranulation rough set model is determined. Therefore, the optimal ordering of the model established in this paper is more stable. Finally, in order to reduce the influence of subjective preferences of individual decision makers as much as possible, the Multimoora method is used to obtain the value of reasonable conditional probability for decision makers. The Multimoora method [2] is more stable than the TOPSIS method, which is convenient for sorting and selecting the scheme. In addition, the accuracy of the final ranking for all alternatives will increase due to improving the effectiveness of the decision-making process and the reliability of sorting results.

Even though there are many advantages compared with the past literature, the method proposed in this paper has some limitations in practical applications. In practical applications, there may be parallel results that are not conducive to the final comparison when using the Multimoora method to sort the alternatives given by the real data. Because the method is based on the superiority theory for final sorting, decision makers need to make subjective judgments when there are parallel results. Also, information loss may occur during the conversion of linguistic variables using formula(7) proposed in section 4.2, and this situation should be avoided as much as possible in order to make accurate decisions in practical applications. In short, we construct a new way to information representation, the information fusion, and the computing of final optimal results for the decision making problem by using the approach to uncertainty multiple attribute group decision problem with linguistic preference relation based on multigranulation probabilistic rough set and the Multimoora method.

4.4 An illustrative example

In this section, we will apply the model and method for multiple attribute group decision making with linguistic information based on multigranulation probabilistic rough set and Multimoora method to solve problems in real-life situations. And it has been improved in the multi-attribute group decision-making environment. On this basis, we will introduce the basic principles and steps of the method established in this paper by discussing the illustrative example.

With the rapid development of e-commerce, online shopping is becoming more and more common. Among them, the choice of logistics distribution providers is a important issue that e-commerce companies need to consider. Appropriate logistics providers can effectively improve the company’s operating level, in order to meet the income costs and service levels of e-commerce companies. Therefore, an important issue in the management of e-commerce enterprize suppliers is how to evaluate different suppliers, and choose the best supplier to match the existing product distribution. As previously considered, the choice of logistics distribution providers is essentially a multi-attribute group decision problem [19]. However, the current evaluation system and selection method have certain limitations. This paper proposes an approach to multiple attribute group decision making based on multigranulation probabilistic rough set and Multimoora method for e-commerce supplier decision-making.

This section is a multiple attribute group decision-making method based on linguistic information, and proposes another method for evaluating logistics suppliers. The statement of the problem is as follows.

Let U = {x₁, x₂, x₃, x₄, x₅} be five alternative logistics suppliers established in advance of e-commerce and C={logistics service quality (c₁), logistics service cost (c₂), logistics enterprise capability (c₃), informationization degree (c₄), enterprise development prospect (c₅)} be the attribute set of the basic description of logistics suppliers. The weights of every attribute are presented as follows, respectively. $ω_{1} = 0.3, ω_{2} = 0.25, ω_{3} = 0.2, ω_{4} = 0.15, ω_{5} = 0.1 .$

Suppose there are four experts d₁, d₂, d₃ and d₄ which are invited to evaluate the five logistics suppliers based on their respective expertise and preferences, and provide all logistics suppliers for each of the attributes in Table 2. The actual value is described by the linguistic variable form S = {S_α|α = -4, - 3, - 2, - 1, 0, 1, 2, 3, 4}, which finally constitutes the language preference matrix of different indicators of each logistics distribution provider. Meanwhile, the weights of every expert are given as follows, respectively. $μ_{1} = 0.2, μ_{2} = 0.1, μ_{3} = 0.4 μ_{4} = 0.3 .$

Table 2
The evaluation of logistics suppliers fuzzy group decision-making information system

U d ₁ d ₂ d ₃ d ₄

c ₁ c ₂ c ₃ c ₄ c ₅ c ₁ c ₂ c ₃ c ₄ c ₅ c ₁ c ₂ c ₃ c ₄ c ₅ c ₁ c ₂ c ₃ c ₄ c ₅

x ₁ s ₃ s ₁ s _-1 s _-2 s ₂ s ₂ s _-1 s _-4 s ₄ s ₃ s _-2 s ₀ s ₃ s ₂ s _-1 s _-1 s ₄ s ₃ s _-2 s ₂

x ₂ s ₁ s ₄ s ₂ s _-1 s _-3 s ₃ s ₂ s ₁ s _-4 s ₀ s ₃ s ₁ s ₀ s _-3 s _-4 s ₁ s _-3 s ₄ s ₃ s _-4

x ₃ s ₂ s ₁ s _-2 s ₀ s ₁ s ₂ s ₄ s ₀ s _-2 s _-3 s ₀ s _-1 s ₄ s ₁ s ₃ s ₄ s ₀ s _-1 s _-3 s ₁

x ₄ s ₀ s ₄ s _-1 s _-3 s ₂ s _-3 s ₃ s ₀ s _-2 s ₂ s ₀ s _-1 s ₁ s ₄ s ₂ s ₀ s ₄ s _-2 s ₁ s _-1

x ₅ s ₁ s ₃ s _-2 s ₂ s _-1 s ₂ s ₄ s _-2 s ₀ s _-1 s ₀ s ₃ s ₁ s _-1 s _-2 s _-2 s ₂ s ₃ s ₀ s _-3

U	d ₁	d ₂	d ₃	d ₄
x ₁	s ₃	s ₁	s _-1	s _-2	s ₂	s ₂	s _-1	s _-4	s ₄	s ₃	s _-2	s ₀	s ₃	s ₂	s _-1	s _-1	s ₄	s ₃	s _-2	s ₂
x ₂	s ₁	s ₄	s ₂	s _-1	s _-3	s ₃	s ₂	s ₁	s _-4	s ₀	s ₃	s ₁	s ₀	s _-3	s _-4	s ₁	s _-3	s ₄	s ₃	s _-4
x ₃	s ₂	s ₁	s _-2	s ₀	s ₁	s ₂	s ₄	s ₀	s _-2	s _-3	s ₀	s _-1	s ₄	s ₁	s ₃	s ₄	s ₀	s _-1	s _-3	s ₁
x ₄	s ₀	s ₄	s _-1	s _-3	s ₂	s _-3	s ₃	s ₀	s _-2	s ₂	s ₀	s _-1	s ₁	s ₄	s ₂	s ₀	s ₄	s _-2	s ₁	s _-1
x ₅	s ₁	s ₃	s _-2	s ₂	s _-1	s ₂	s ₄	s _-2	s ₀	s _-1	s ₀	s ₃	s ₁	s _-1	s _-2	s _-2	s ₂	s ₃	s ₀	s _-3

Based on the definition of distance measure given in section 3.1, the standard distance measure between any two logistics suppliers with respect to attribute set are calculated as follows: $ξ_{ij}^{1 C} = (\begin{matrix} U & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ x_{1} & 0 & 0.05 & 0.42 & 0.19 & 0.19 \\ x_{2} & - & 0 & 0.59 & 0.14 & 0.34 \\ x_{3} & - & - & 0 & 0.57 & 0.14 \\ x_{4} & - & - & - & 0 & 0.32 \\ x_{5} & - & - & - & - & 0 \end{matrix})$ $ξ_{ij}^{2 C} = (\begin{matrix} U & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ x_{1} & 0 & 0.5 & 0.98 & 0.19 & 0.85 \\ x_{2} & - & 0 & 0.42 & 0.46 & 0.26 \\ x_{3} & - & - & 0 & 1 & 0.05 \\ x_{4} & - & - & - & 0 & 0.84 \\ x_{5} & - & - & - & - & 0 \end{matrix})$ $ξ_{ij}^{3 C} = (\begin{matrix} U & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ x_{1} & 0 & 0.50 & 0.57 & 0.48 & 0.62 \\ x_{2} & - & 0 & 0.19 & 0.46 & 0.31 \\ x_{3} & - & - & 0 & 0.22 & 0 \\ x_{4} & - & - & - & 0 & 0.15 \\ x_{5} & - & - & - & - & 0 \end{matrix})$ $ξ_{ij}^{4 C} = (\begin{matrix} U & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ x_{1} & 0 & 0.26 & 0.31 & 0.05 & 0.57 \\ x_{2} & - & 0 & 0.57 & 0.22 & 0.26 \\ x_{3} & - & - & 0 & 0.23 & 0.95 \\ x_{4} & - & - & - & 0 & 0.59 \\ x_{5} & - & - & - & - & 0 \end{matrix})$

Subsequently, we set the threshold value δ = 0.4. Then, we calculate the δ-simliarity classes for all logistics suppliers are as follows: $\begin{matrix} δ_{C} (x_{1}^{1}) = {x_{2}, x_{3}, x_{4}, x_{5}}, δ_{C} (x_{2}^{1}) = {x_{1}, x_{4}, x_{5}}, \\ δ_{C} (x_{3}^{1}) = {x_{1}, x_{5}}, \end{matrix}$ $δ_{C} (x_{4}^{1}) = {x_{1}, x_{2}, x_{5}}, δ_{C} (x_{5}^{1}) = {x_{1}, x_{2}, x_{3}, x_{4},}$ $\begin{matrix} δ_{C} (x_{1}^{2}) = {x_{2}, x_{4},}, δ_{C} (x_{2}^{2}) = {x_{1}, x_{3}, x_{4}, x_{5}}, \\ δ_{C} (x_{3}^{2}) = {x_{2}, x_{5}}, \end{matrix}$ $δ_{C} (x_{4}^{2}) = {x_{1}, x_{2}, x_{5}}, δ_{C} (x_{5}^{2}) = {x_{2}, x_{3}}$ $\begin{matrix} δ_{C} (x_{1}^{3}) = {x_{4}}, δ_{C} (x_{2}^{3}) = {x_{3}, x_{5}}, \\ δ_{C} (x_{3}^{3}) = {x_{2}, x_{4}, x_{5}}, \end{matrix}$ $δ_{C} (x_{4}^{3}) = {x_{1}, x_{3}, x_{5}}, δ_{C} (x_{5}^{3}) = {x_{2}, x_{3}, x_{5}},$ $\begin{matrix} δ_{C} (x_{1}^{4}) = {x_{4}, x_{5}}, δ_{C} (x_{2}^{4}) = {x_{2}, x_{3}, x_{4}}, \\ δ_{C} (x_{3}^{4}) = {x_{1}, x_{4}, x_{5}}, \end{matrix}$ $δ_{C} (x_{4}^{4}) = {x_{1}, x_{4}}, δ_{C} (x_{5}^{4}) = {x_{2}, x_{3}, x_{5}} .$

At the same time, the best and worst fuzzy decision-making objects of the universe U according to formula (13) and (14) given by four different decision-makers with respect to attribute set are as follows, respectively. $\begin{matrix} A_{1}^{+} & = \frac{0.875}{x_{1}} + \frac{1}{x_{2}} + \frac{0.75}{x_{3}} + \frac{0.75}{x_{4}} + \frac{0.75}{x_{5}}, \\ A_{1}^{-} & = \frac{0.5}{x_{1}} + \frac{0.625}{x_{2}} + \frac{0.25}{x_{3}} + \frac{0.125}{x_{4}} + \frac{0.125}{x_{5}}, \end{matrix}$ $\begin{matrix} A_{2}^{+} & = \frac{0.875}{x_{1}} + \frac{1}{x_{2}} + \frac{0.625}{x_{3}} + \frac{1}{x_{4}} + \frac{0.875}{x_{5}}, \\ A_{2}^{-} & = \frac{0.125}{x_{1}} + \frac{0.375}{x_{2}} + \frac{0}{x_{3}} + \frac{0}{x_{4}} + \frac{0.125}{x_{5}}, \end{matrix}$ $\begin{matrix} A_{3}^{+} & = \frac{0.875}{x_{1}} + \frac{0.875}{x_{2}} + \frac{1}{x_{3}} + \frac{1}{x_{4}} + \frac{0.875}{x_{5}}, \\ A_{3}^{-} & = \frac{0.25}{x_{1}} + \frac{0.375}{x_{2}} + \frac{0.5}{x_{3}} + \frac{0.125}{x_{4}} + \frac{0}{x_{5}}, \end{matrix}$ $\begin{matrix} A_{4}^{+} & = \frac{1}{x_{1}} + \frac{1}{x_{2}} + \frac{1}{x_{3}} + \frac{0.875}{x_{4}} + \frac{0.75}{x_{5}}, \\ A_{4}^{-} & = \frac{0.25}{x_{1}} + \frac{0.125}{x_{2}} + \frac{0.25}{x_{3}} + \frac{0.125}{x_{4}} + \frac{0}{x_{5}}, \end{matrix}$

Here, select precision parameter α = 0.3, β = 0.1.Thus, by formula (15) - (18), the lower and upper approximations of A with respect to multiple attribute fuzzy group decision-making linguistic-valued information system (U, C, D, F, ω, μ) ; are obtained as follows, respectively. $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{1})}^{0, 0.3} (A_{1}^{+}) (x_{i}) \\ = \frac{0}{x_{1}} + \frac{0}{x_{2}} + \frac{0.38}{x_{3}} + \frac{0}{x_{4}} + \frac{0}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{1})}^{0, 0.1} (A_{1}^{+}) (x_{i}) \\ = \frac{0}{x_{1}} + \frac{0.15}{x_{2}} + \frac{0.25}{x_{3}} + \frac{0.15}{x_{4}} + \frac{0}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{1})}^{0, 0.3} (A_{1}^{-}) (x_{i}) \\ = \frac{0}{x_{1}} + \frac{0}{x_{2}} + \frac{0.44}{x_{3}} + \frac{0}{x_{4}} + \frac{0}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{1})}^{0, 0.1} (A_{1}^{-}) (x_{i}) \\ = \frac{0.16}{x_{1}} + \frac{0.21}{x_{2}} + \frac{0.13}{x_{3}} + \frac{0.21}{x_{4}} + \frac{0.16}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{2})}^{0, 0.3} (A_{2}^{+}) (x_{i}) \\ = \frac{0.38}{x_{1}} + \frac{0}{x_{2}} + \frac{0.38}{x_{3}} + \frac{0}{x_{4}} + \frac{0.38}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{2})}^{0, 0.1} (A_{2}^{+}) (x_{i}) \\ = \frac{0.25}{x_{1}} + \frac{0}{x_{2}} + \frac{0.25}{x_{3}} + \frac{0.15}{x_{4}} + \frac{0.25}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{2})}^{0, 0.3} (A_{2}^{-}) (x_{i}) \\ = \frac{0.44}{x_{1}} + \frac{0}{x_{2}} + \frac{0.44}{x_{3}} + \frac{0}{x_{4}} + \frac{0.44}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{2})}^{0, 0.1} (A_{2}^{-}) (x_{i}) \\ = \frac{0.13}{x_{1}} + \frac{0.16}{x_{2}} + \frac{0.13}{x_{3}} + \frac{0.21}{x_{4}} + \frac{0.13}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{3})}^{0, 0.3} (A_{3}^{+}) (x_{i}) \\ = \frac{1}{x_{1}} + \frac{0.38}{x_{2}} + \frac{0}{x_{3}} + \frac{0}{x_{4}} + \frac{0}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{3})}^{0, 0.1} (A_{3}^{+}) (x_{i}) \\ = \frac{0.74}{x_{1}} + \frac{0.25}{x_{2}} + \frac{0.15}{x_{3}} + \frac{0.15}{x_{4}} + \frac{0.15}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{3})}^{0, 0.3} (A_{3}^{-}) (x_{i}) \\ = \frac{0.88}{x_{1}} + \frac{0.44}{x_{2}} + \frac{0}{x_{3}} + \frac{0}{x_{4}} + \frac{0}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{3})}^{0, 0.1} (A_{3}^{-}) (x_{i}) \\ = \frac{0.25}{x_{1}} + \frac{0.13}{x_{2}} + \frac{0.21}{x_{3}} + \frac{0.21}{x_{4}} + \frac{0.21}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{4})}^{0, 0.3} (A_{4}^{+}) (x_{i}) \\ = \frac{0}{x_{1}} + \frac{0}{x_{2}} + \frac{0.38}{x_{3}} + \frac{0}{x_{4}} + \frac{1}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{4})}^{0, 0.1} (A_{4}^{+}) (x_{i}) \\ = \frac{0.15}{x_{1}} + \frac{0.15}{x_{2}} + \frac{0.25}{x_{3}} + \frac{0.5}{x_{4}} + \frac{0.74}{x_{5}} \end{matrix}$ $\begin{matrix} {\underline{Apr}}_{Σ_{k = 1}^{l} δ_{C} (x^{4})}^{0, 0.3} (A_{4}^{-}) (x_{i}) \\ = \frac{0}{x_{1}} + \frac{0}{x_{2}} + \frac{0.44}{x_{3}} + \frac{0}{x_{4}} + \frac{0.88}{x_{5}}, \\ {\bar{Apr}}_{Σ_{k = 1}^{1} δ_{C} (x^{4})}^{0, 0.1} (A_{4}^{-}) (x_{i}) \\ = \frac{0.21}{x_{1}} + \frac{0.21}{x_{2}} + \frac{0.13}{x_{3}} + \frac{0.21}{x_{4}} + \frac{0.25}{x_{5}} \end{matrix}$

Then, the ranking function φ_k (x_i) (k = 1, 2, 3, 4; i = 1, 2, 3, 4, 5) according to formula (19) are computed as follows, respectively. $\begin{matrix} φ_{1} (x_{1}) = 0.16, φ_{1} (x_{2}) = 0.06, φ_{1} (x_{3}) = 0.13, \\ φ_{1} (x_{4}) = 0.06, φ_{1} (x_{5}) = 0.16, \end{matrix}$ $\begin{matrix} φ_{2} (x_{1}) = 0.40, φ_{2} (x_{2}) = 0.16, φ_{2} (x_{3}) = 0.40, \\ φ_{2} (x_{4}) = 0.06, φ_{2} (x_{5}) = 0.40, \end{matrix}$ $\begin{matrix} φ_{3} (x_{1}) = 0.50, φ_{3} (x_{2}) = 0.13, φ_{3} (x_{3}) = 0.06, \\ φ_{3} (x_{4}) = 0.06, φ_{3} (x_{5}) = 0.06, \end{matrix}$ $\begin{matrix} φ_{4} (x_{1}) = 0.06, φ_{4} (x_{2}) = 0.06, φ_{4} (x_{3}) = 0.13, \\ φ_{4} (x_{4}) = 0.06, φ_{4} (x_{5}) = 0.50 . \end{matrix}$

Finally, the optimal index function φ (x_i) (i = 1, 2, 3, 4, 5) according to formula (20) are obtained as follows: $\begin{matrix} φ (x_{1}) = μ_{1} φ_{1} (x_{1}) & + μ_{2} φ_{2} (x_{1}) + μ_{3} φ_{3} (x_{1}) \\ + μ_{4} φ_{4} (x_{1}) = 0.29, \end{matrix}$ $\begin{matrix} φ (x_{2}) = μ_{1} φ_{1} (x_{2}) & + μ_{2} φ_{2} (x_{2}) + μ_{3} φ_{3} (x_{2}) \\ + μ_{4} φ_{4} (x_{2}) = 0.1, \end{matrix}$ $\begin{matrix} φ (x_{3}) = μ_{1} φ_{1} (x_{3}) & + μ_{2} φ_{2} (x_{3}) + μ_{3} φ_{3} (x_{3}) \\ + μ_{4} φ_{4} (x_{3}) = 0.13, \end{matrix}$ $\begin{matrix} φ (x_{4}) = μ_{1} φ_{1} (x_{4}) & + μ_{2} φ_{2} (x_{4}) + μ_{3} φ_{3} (x_{4}) \\ + μ_{4} φ_{4} (x_{4}) = 0.06, \end{matrix}$ $\begin{matrix} φ (x_{5}) = μ_{1} φ_{1} (x_{5}) & + μ_{2} φ_{2} (x_{5}) + μ_{3} φ_{3} (x_{5}) \\ + μ_{4} φ_{4} (x_{5}) = 0.25 . \end{matrix}$

Therefore, the order of comprehension for all supplier decision-makings according to the optimal index function φ (x_i) (i = 1, 2, 3, 4, 5) obtained as: $x_{1} ≽ x_{5} ≽ x_{3} ≽ x_{2} ≽ x_{4} .$

Therefore, the logistics supplier x₁ is the optimal decision making. At the same time, we complete the decision-making process for the problem of e-commerce supplier decision-making using multigranulation probabilistic rough set theory and Multimoora method.

5 Conclusions

Group decision making with linguistic information is widespread in fields of management science, this paper proposes a new method to deal with the multiple attributes group decision making by using the generalized rough set theory and Multimoora method. Comparing the existing literatures, the new contributions can be concluded as follows: (1) For the multiple attribute group decision-making problem with linguistic information, the preference information in this paper is described by linguistic variables. Based on the original research, the model is extended to the discrete language evaluation set, which makes preference information of the decision makers more comprehensive to describe the problem in real life. This kind of preference information is more reasonable for the evaluation of decision problems than the preference information of numerical variables, thus improving the accuracy and reliability of decision results. (2) A new approach to multiple attribute fuzzy group decision-making problem with linguistic information using the multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation is established. Based on the neighborhood relation, the rough approximation under the framework of multigranulation rough set is systematically discussed. Also, it enriches the binary relationship used in the existing literature, making the rough set theory more widely applicable. On this basis, we have proposed optimistic multigranulation probabilistic rough set and pessimistic multigranulation probabilistic rough set based on neighborhood relation. Through the discussion in Section3.3, we can know that the multigranulation probabilistic fuzzy rough set based on neighborhood relation is a more generalized multigranulation model, which is an extension of the existing models. Further, the multiple attribute fuzzy group decision making method based on the multigranulation probabilistic fuzzy rough set has a meaningful attempt in the multi-attribute group decision making problem. (3) Combining the idea of multigranulation rough set with the Multimoora method, we get a more stable decision making model. In order to avoid the influence of subjective preferences of individual decision makers as much as possible, this paper use the method of Multimoora method to obtain the value of reasonable conditional probability for decision makers. Multimoora method is the sorting result obtained by merging the three methods, and its result is more accurate and precise than TOPSIS method. (4) This paper presents a description of the problem of the uncertain management decision problem of actual management, and gives an example of the multi-granularity probability fuzzy rough set model based on the δ-neighborhood relationship model. Taking the e-commerce enterprise logistics provider selection decision problem as an example, the feasibility and effectiveness of the method are further verified. Decision makers make optimal group decisions according to the needs of enterprises, which have good practical significance and application value. Therefore, this model enrich and develop the theory and method of multiple attribute group decision making with linguistic information, which provides a new idea and method for solving the problem of e-commerce enterprises choosing suppliers.

This paper makes a new perspective to apply multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation to multiple attribute fuzzy group decision-making problems with linguistic information. Also, this paper is focused on group decision making based on the multigranulation probabilistic rough set theory and Multimoora method. Furthermore, we also explore the basic properties and establish some interesting results as well as the interrelationship with the existing generalized rough set theories for the new proposed model. We can illustrate the feasibility of the new model by studying the basic properties and relationships between the newly established model and the original rough set model. The proposed method not only supplements and expands the original rough set theory, but also provides a new method and tool for dealing with group decision problems with linguistic information. On the theoretical level, the extension model of classical rough set theory [15] and multi-granularity rough theory [21] is extended and studied. At the method level, a new fusion method of aggregated individual preferences information is considered to solve the group decision problem. Although, this paper focuses on the basic theory of multiple attribute group decision making multigranulation probabilistic fuzzy rough set model based on δ-neighborhood relation with linguistic information. For further study, it is suggested that the proposed method be further improved so that it can be used to solve more complex decision-making problems in the field of management. At the same time, real data or public database should be used to verify the validity of the proposed method.

Footnotes

Acknowledgments

The authors are very grateful to the Editor-in-Chief Professor Reza Langari, and the anonymous referees for their thoughtful comments and valuable suggestions. Some remarks directly benefit from the referees’ comments. The work was partly supported by the National Natural Science Foundation of China (71571090,61772019), the Fundamental Research Funds for the Central Universities (JB190602), the Youth Innovation Team of Shaanxi Universities, the Interdisciplinary Foundation of Humanities and Information (RW180167).

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