Covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and applications

Abstract

As an extension of the classical Pawlak rough sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough set models have been researched, in which objects approximated are crisp sets or accurate concepts of the universe of discourse, and there are only two states, which are disjoint and opposite each other for an accurate concept of the universe of discourse for a decision-making problem. However, the objects of many decision-making problems can have more than two states in practice. Moreover, the states of the decision object are not necessarily disjoint and opposite each other, but these decision states are fuzzy descriptions of the states of the object on the universe. In order to deal with decision-making problems in which decision states are fuzzy sets, covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models are proposed in this paper. In addition, the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets are investigated and characterized. Furthermore, the relationships between covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models are discussed. Finally, an example is presented to illustrate the practicality and generality of the proposed models.

Keywords

Decision-theoretic rough sets fuzzy sets multigranulation trapezoidal fuzzy number covering

1 Introduction

Rough set theory, proposed by Pawlak [43] in 1982, has become an important method and tool for insufficient and incomplete information system [17 , 43] in a wide variety of applications related to decision-making [1 , 46], granular computing [10 , 39], clustering analysis [40], feature selection [2 , 28], machine learning [14, 26] and so on. As an extension of the classical Pawlak rough sets, Yao et al. [41] proposed decision-theoretic rough set, which is constructed by positive region, boundary region and negative region respectively. Many scholars have been interested in decision-theoretic rough set and many valuable researches have been done in recent years. For example, Herbert and Yao [15, 16] studied the combination of the decision-theoretic rough set and the game rough set. Li and Zhou [11, 12] presented a multi-perspective explanation of the decision-theoretic rough set and discussed attribute reduction and its application for the decision-theoretic rough. Liu et al. [4 –7] investigated multiple-category classification with decision-theoretic rough set and its applications in the areas of management science. Ma and Sun [29, 30] studied the decision-theoretic rough set theory over two universes. Qian [38] proposed a new decision-theoretic rough set called multigranulation decision-theoretic rough set based on the paper [37] proposed by Qian in 2006, which provides a new perspective for the study of decision-theoretic rough sets.

However, an equivalence relation or a partition of the universe of discourse is restrictive for many real-world applications. To overcome this limitation, many scholars replaced a partition of the universe with a covering [9 , 44]. In the realistic decision process, some influencing factors also result in decision makers not to provide precise values, e.g. tight deadlines, limited budgets and so on. As an extension of precise numerical values, fuzzy set [22] is considered to deal with vague, imprecise and uncertain problems. Therefore, the value of loss function with the measurement of fuzzy set is more realistic. Based on this consideration, Liang [8] proposed triangular fuzzy decision-theoretic rough sets. Due to the better generality and flexibility of trapezoidal fuzzy number dealing with problems than triangular fuzzy number, we assume the loss functions are trapezoidal fuzzy numbers in this paper. In general, the objects approximated are crisp sets or accurate concepts of the universe of discourse in existing decision-theoretic rough sets. For a decision-making problem, there are only two states, which are disjoint and opposite each other for an accurate concept of the universe of discourse. However, the objects of many decision-making problems can have more than two states in practice. Moreover, the states of the decision object are not necessarily disjoint and opposite each other, but these decision states are fuzzy descriptions of the states of the object on the universe. In order to solve this limitation, Sun [1] proposed decision-theoretic rough fuzzy set model and application.

Based on these considerations, in order to make multigranulation decision-theoretic rough set models deal with decision-making problems in which decision states are fuzzy sets, we propose covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models. In addition, the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets are investigated and characterized. Furthermore, the relationships between covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models are discussed. Finally, an illustration is given.

The rest of this paper is organized as follows. To make our analysis possible, Section 2 provides some basic concepts about Pawlak rough sets and trapezoidal fuzzy number. In Section 3, we give the process of the calculation of three probabilistic thresholds and the derivation of decision rules. In Section 4, we shall firstly propose the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets respectively. Furthermore, some characterization theorems are given. Section 5 discusses the relationships between covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models. In Section 6, an example is presented to illustrate the practicality and generality of the proposed models. Finally, we conclude our research in Section 7.

2 Preliminaries and basic concepts

Firstly, we provide a notation-list to explain the meanings of all used symbols in this paper.

U represents a non-empty finite universe, R represents an equivalence relation, C represents a covering, $\underline{R}$ and $\bar{R}$ represent the lower and upper approximations respectively, pos, bn and neg represent the positive, boundary and negative regions respectively, Md (x) represents the minimal description of x, Mag represents the magnitude of trapezoidal fuzzy number, P represents a probability measure, $P (\tilde{A} | \cap {Md}_{C} (x))$ represents the conditional probability of $\tilde{A}$ given the description ∩Md_C (x) , $\tilde{λ}$ represents a loss function, r represents a decision action, Ω represents the set of states, $\tilde{R} (r_{i} | \cap {Md}_{C} (x))$ represents the expected loss, F (U) denotes all the fuzzy subsets of U.

In the following, we briefly review some basic concepts, such as Pawlak rough sets, covering, trapezoidal fuzzy number, etc.

Let U be a non-empty finite universe and R be an equivalence relation of U × U . The equivalence relation R induces a partition of U, denoted by [x] _R, and U/R = {[x] |x ∈ U} represents the equivalence of x . Then (U, R) is the Pawlak approximation space.

For any X ⊆ U, its lower and upper approximations are defined:

$\begin{matrix} \underline{R} (X) & = & {x \in U | [x]_{R} \subseteq X}, \\ \bar{R} (X) & = & {x \in U | [x]_{R} \cap X \neq \emptyset} . \end{matrix}$ (1)

The positive, boundary and negative regions of X are defined as follows:

$\begin{matrix} pos (X) & = & \underline{R} (X), bn (X) = \bar{R} (X) - \underline{R} (X), \\ neg (X) & = & U - \bar{R} (X) . \end{matrix}$ (2)

Let U be a universe of discourse, C = {X|X ⊆ U} be a family of subsets of U . If no element of C is empty, and ∪_X∈CX = U, then C is called a covering of U . The ordered pair (U, C) is called a covering approximation space. It is clear that a partition of U is certainly a covering of U, so the concept of covering is an extension of the concept of a partition.

Let (U, C) be a covering approximation space, x ∈ U, then Md (x) = {K ∈ C|x ∈ K ∧ (∀ S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)} is called the minimal description of x .

The trapezoidal fuzzy number $\tilde{u} = (x_{0}, y_{0}, σ, δ)$ with two defuzzifiers x₀, y₀, left fuzziness σ > 0 and right fuzziness δ > 0, is a special fuzzy number, and its parametric form is $(\underline{u}, \bar{u}),$ where $\underline{u} (r) = x_{0} - σ + σ r, \bar{u} (r) = y_{0} + δ - δ r .$

The addition and scalar multiplication of trapezoidal fuzzy numbers are defined by the extension principle as follows.

For any $\tilde{u} = (x_{0}, y_{0}, σ, δ), \tilde{v} = (x_{0}^{'}, y_{0}^{'}, σ^{'}, δ^{'})$ and scalar k, we define addition $\tilde{u} \oplus \tilde{v}$ and scalar multiplication $k \otimes \tilde{u}$ as:

Addition: $\tilde{u} \oplus \tilde{v} = (x_{0} + x_{0}^{'}, y_{0} + y_{0}^{'}, σ + σ^{'}, δ + δ^{'});$

Scalar Multiplication: $k \otimes \tilde{u} = {\begin{matrix} ({kx}_{0}, {ky}_{0}, k σ k δ), k \geq 0; \\ ({ky}_{0}, {kx}_{0}, k δ, k σ), k \leq 0 . \end{matrix}$

For an arbitrary trapezoidal fuzzy number $\tilde{u} = (x_{0}, y_{0}, σ, δ),$ with parametric form $(\underline{u}, \bar{u}),$ the magnitude of $\tilde{u}$ is defined as [27] $Mag (\tilde{u}) = \frac{1}{2} \int_{0}^{1} (\underline{u} (r) + \bar{u} (r) + x_{0} + y_{0}) f (r) dr,$ (3) where the function f (r) is a non-negative and increasing function on [0, 1] with f (0) =0, f (1) =1 and $\int_{0}^{1} f (r) dr = \frac{1}{2} .$ Obviously, f (r) can be considered as a weighting function. In actual applications, f (r) can be chosen according to the actual situations. In this paper, we use f (r) = r . That is, $Mag (\tilde{u}) = \frac{1}{2} \int_{0}^{1} (\underline{u} (r) + \bar{u} (r) + x_{0} + y_{0}) f (r) dr = \frac{1}{2} (x_{0} + y_{0}) - \frac{1}{12} (σ - δ) .$

In the trapezoidal fuzzy number space, we define partial relations by Mag (·) as follows [27].

$\tilde{u} ≻ \tilde{v}$ is to say $Mag (\tilde{u}) > Mag (\tilde{v}),$

$\tilde{u} ≺ \tilde{v}$ is to say $Mag (\tilde{u}) < Mag (\tilde{v}),$

$\tilde{u} \sim \tilde{v}$ is to say $Mag (\tilde{u}) = Mag (\tilde{v}) .$

3 The calculation of three probabilistic thresholds and the derivation of decision rules

Given an object x, let Md (x) = {K ∈ C|x ∈ K ∧ (∀ S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)} be the minimal description of x and ∩Md (x) be a description of the object x . Let Ω = {w₁, w₂, ⋯ , w_s} be a finite set of s states, A = {r₁, r₂, ⋯ , r_m} be a finite set of m possible actions, and P (w_j| ∩ Md (x)) be the conditional probability of x with description ∩Md (x) being in state w_j, and let the loss function λ (r_i|w_j) denote the loss(or cost)for taking the action r_i when the state is w_j . For an object x with description ∩Md (x) , the expected cost associated with taking action r_i is given by $\tilde{R} (r_{i} | \cap Md (x)) = \sum_{j = 1}^{s} λ (r_{i} | w_{j}) P (w_{j} | \cap Md (x)) .$

Let (U, R) be the Pawlak approximation space, P is a probability measure defined on the σ-algebra of the subset family of universe U . Then (U, R, P) is a called a probabilistic approximation space. As is well known, the conditional probability between the target set and the description of the object is the key concept in the Bayesian decision procedure. Hence, we firstly give the definition of conditional probability of any fuzzy event in the probabilistic space.

Definition 3.1. Let U = {x₁, x₂, ⋯ , x_n, ⋯} . Denote P (x_n) (n = 1, 2, ⋯) as the probability of x_n and satisfy $P (x_{n}) > 0, \sum_{n = 1}^{\infty} P (x_{n}) = 1 .$ For any $\tilde{A} \in F (U)$ (where F (U) denotes all the fuzzy subsets of U), the probability of the fuzzy event $\tilde{A}$ is defined as follows: $P (\tilde{A}) = \sum_{n = 1}^{\infty} \tilde{A} (x_{n}) P (x_{n}),$ (4) where $\tilde{A} (x_{n})$ stands for the membership function of fuzzy set $\tilde{A} .$

If the probabilistic space is continuous, then the probability of fuzzy event $\tilde{A}$ is defined as follows: $P (\tilde{A}) = \int_{U} \tilde{A} (x) dP = E (\tilde{A} (x)),$ (5) where dP is the Lebegue-Stieltjes integral.

By this definition and the concept of conditional probability of classical measurement theory, the conditional probability of a fuzzy event given the description of a crisp set is defined as follows.

Definition 3.2. Let U be a non-empty finite universe and C be a covering of U . Denote P as the probabilistic measure. For any $\tilde{A} \in F (U),$ $P (\tilde{A} | \cap {Md}_{C} (x))$ is called the conditional probability of fuzzy event $\tilde{A}$ given the description ∩Md_C (x) . Define $P (\tilde{A} | \cap {Md}_{C} (x)) = \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |},$ (6) where | · | stands for the cardinality of a set and x ∈ U .

The $P (\tilde{A} | \cap {Md}_{C} (x))$ also can be understood as the probability that any objet x ∈ U belongs to the fuzzy concept $\tilde{A}$ given the description ∩Md_C (x) .

In the following, we give the concept of fuzzy partition of the universe of discourse.

Definition 3.3. Let U be a non-empty finite universe. For any ${\tilde{A}}_{t} \in F (U), t \in Γ$ (where Γ is the index set), the family of set ${{\tilde{A}}_{t} \in F (U) | t \in Γ}$ is called a fuzzy partition of the universe U if the following condition holds: $\sum_{t \in Γ} {\tilde{A}}_{t} (x) = 1, x \in U .$ (7)

For the rough approximations of a fuzzy set on the probabilistic approximation space, the decision-making problem is described as follows.

The set of states is given by $Ω = {\tilde{A}, \tilde{B}, \tilde{C}},$ where $\tilde{A}, \tilde{B}, \tilde{C} \in F (U) .$ For any x ∈ U, there is $\tilde{A} (x) + \tilde{B} (x) + \tilde{C} (x) = 1 .$ That is, the given states of decision objects form a fuzzy partition of the universe of discourse.

The set of actions is given by A = {r_P, r_B, r_N} , where r_P, r_B and r_N represent the three actions in classifying an object, namely, deciding $pos (\tilde{A}),$ deciding $bn (\tilde{A})$ and $neg (\tilde{A}),$ respectively.

Let $\tilde{λ} (r_{i} | \tilde{A}) = (x_{i \tilde{A}}, y_{i \tilde{A}}, σ_{i \tilde{A}}, δ_{i \tilde{A}})$ (i takes P, B, N respectively) be trapezoidal fuzzy number and denote the loss incurred for taking action r_i when an object belongs to fuzzy set $\tilde{A},$ $\tilde{λ} (r_{i} | \tilde{B}) = (x_{i \tilde{B}}, y_{i \tilde{B}}, σ_{i \tilde{B}}, δ_{i \tilde{B}})$ (i takes P, B, N respectively) be trapezoidal fuzzy number and denote the loss incurred for taking action r_i when an object belongs to fuzzy set $\tilde{B},$ and $\tilde{λ} (r_{i} | \tilde{C}) = (x_{i \tilde{C}}, y_{i \tilde{C}}, σ_{i \tilde{C}}, δ_{i \tilde{C}})$ (i takes P, B, N respectively) be trapezoidal fuzzy number and denote the loss incurred for taking action r_i when an object belongs to fuzzy set $\tilde{C} .$

For any object x with description ∩Md_C (x) , the expected loss $\tilde{R} (r_{i} | \cap {Md}_{C} (x))$ associated with taking action r_i can be expressed as: $\begin{matrix} \tilde{R} (r_{P} | \cap {Md}_{C} (x)) \\ = {\tilde{λ}}_{P \tilde{A}} P (\tilde{A} | \cap {Md}_{C} (x)) \\ + {\tilde{λ}}_{P \tilde{B}} P (\tilde{B} | \cap {Md}_{C} (x)) + {\tilde{λ}}_{P \tilde{C}} P (\tilde{C} | \cap {Md}_{C} (x)), \end{matrix}$ (8) $\begin{matrix} \tilde{R} (r_{B} | \cap {Md}_{C} (x)) \\ = {\tilde{λ}}_{B \tilde{A}} P (\tilde{A} | \cap {Md}_{C} (x)) \\ + {\tilde{λ}}_{B \tilde{B}} P (\tilde{B} | \cap {Md}_{C} (x)) + {\tilde{λ}}_{B \tilde{C}} P (\tilde{C} | \cap {Md}_{C} (x)), \end{matrix}$ (9) $\begin{matrix} \tilde{R} (r_{N} | \cap {Md}_{C} (x)) \\ = {\tilde{λ}}_{N \tilde{A}} P (\tilde{A} | \cap {Md}_{C} (x)) \\ + {\tilde{λ}}_{N \tilde{B}} P (\tilde{B} | \cap {Md}_{C} (x)) + {\tilde{λ}}_{N \tilde{C}} P (\tilde{C} | \cap {Md}_{C} (x)), \end{matrix}$ (10)

where ${\tilde{λ}}_{i \tilde{A}} = \tilde{λ} (r_{i} | \tilde{A}), {\tilde{λ}}_{i \tilde{B}} = \tilde{λ} (r_{i} | \tilde{B}), {\tilde{λ}}_{i \tilde{C}} = \tilde{λ} (r_{i} | \tilde{C}),$ and i takes P, B, N respectively.

Ranking the expected losses is an important process of decision making in covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and because the expected losses are trapezoidal fuzzy numbers, ranking function is a key element during ranking fuzzy numbers which maps fuzzy numbers into real numbers. Hence, we take ranking function of trapezoidal fuzzy number $\tilde{u} = (x_{0}, y_{0}, σ, δ)$ as $Mag (\tilde{u}) = \frac{1}{2} \int_{0}^{1} (\underline{u} (r) + \bar{u} (r) + x_{0} + y_{0}) f (r) dr = \frac{1}{2} (x_{0} + y_{0}) - \frac{1}{12} (σ - δ),$ and we get the following minimum risk decision rules.

(P) $Mag (\tilde{R} (r_{P} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{B} | \cap {Md}_{C} (x)))$ and $Mag (\tilde{R} (r_{P} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{N} | \cap {Md}_{C} (x)))$ decide $pos (\tilde{A}) .$

(B) $Mag (\tilde{R} (r_{B} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{P} | \cap {Md}_{C} (x)))$ and $Mag (\tilde{R} (r_{B} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{N} | \cap {Md}_{C} (x)))$ decide $bn (\tilde{A}) .$

(N) $Mag (\tilde{R} (r_{N} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{P} | \cap {Md}_{C} (x)))$ and $Mag (\tilde{R} (r_{N} | \cap {Md}_{C} (x))) \leq Mag (\tilde{R} (r_{B} | \cap {Md}_{C} (x)))$ decide $neg (\tilde{A}) .$

By using the equation $\tilde{A} (x) + \tilde{B} (x) + \tilde{C} (x) = 1$ for any x ∈ U, we can simplify the above decision rules so that only the conditional probability $P (\tilde{A} | \cap {Md}_{C} (x))$ of the fuzzy $\tilde{A}$ is included and classify any object under the description ∩Md_C (x) based only on the conditional probability $P (\tilde{A} | \cap {Md}_{C} (x))$ of the fuzzy $\tilde{A}$ . Furthermore, we can calculate three probabilistic thresholds.

Suppose that a kind of constraint for ${\tilde{λ}}_{ij}$ (i takes P, B, N respectively and j takes $\tilde{A}, \tilde{B}, \tilde{C}$ )is as follows:

$\begin{matrix} Mag ({\tilde{λ}}_{P \tilde{C}}) > Mag ({\tilde{λ}}_{B \tilde{C}}) \geq Mag ({\tilde{λ}}_{N \tilde{C}}), \\ Mag ({\tilde{λ}}_{N \tilde{A}}) > Mag ({\tilde{λ}}_{B \tilde{A}}) \geq Mag ({\tilde{λ}}_{P \tilde{A}}), \\ Mag ({\tilde{λ}}_{P \tilde{B}}) - Mag ({\tilde{λ}}_{P \tilde{C}}) \\ = Mag ({\tilde{λ}}_{B \tilde{B}}) - Mag ({\tilde{λ}}_{B \tilde{C}}) \\ = Mag ({\tilde{λ}}_{N \tilde{B}}) - Mag ({\tilde{λ}}_{N \tilde{C}}) . \end{matrix}$ (11)

By virtue of the above decision rules, we obtain the decision rules (P) , (B) , (N) are indefinite equations according to the formulations $Mag (\tilde{R} (r_{P} | \cap {Md}_{C} (x))), Mag (\tilde{R} (r_{B} | \cap {Md}_{C} (x))), Mag (\tilde{R} (r_{N} | \cap {Md}_{C} (x)))$ because there are three conditional probabilities $P (\tilde{A} | \cap {Md}_{C} (x)), P (\tilde{B} | \cap {Md}_{C} (x)), P (\tilde{C} | \cap {Md}_{C} (x))$ for three fuzzy concepts $\tilde{A}, \tilde{B}$ and $\tilde{C}$ respectively and only two degrees of freedom. Therefore, for any object x under the description ∩Md_C (x) , we can take one of the coefficients as zero for the conditional probabilities of three fuzzy concepts when solving the indefinite equations (P) , (B) and (N) . Then, we get the following results, respectively.

For decision rule (P) , we obtain $\begin{matrix} P (\tilde{A} | \cap {Md}_{C} (x)) \\ \geq \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}})}{(Mag ({\tilde{λ}}_{B \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}}))}, \\ P (\tilde{A} | \cap {Md}_{C} (x)) \\ \geq \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))} . \end{matrix}$

For decision rule (B) , we obtain $\begin{matrix} P (\tilde{A} | \cap {Md}_{C} (x)) \\ \geq \frac{Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{B \tilde{A}})) + (Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))}, \\ P (\tilde{A} | \cap {Md}_{C} (x)) \\ \leq \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}})}{(Mag ({\tilde{λ}}_{B \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}}))} . \end{matrix}$

For decision rule (N) , we obtain $\begin{matrix} P (\tilde{A} | \cap {Md}_{C} (x)) \\ \leq \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))}, \\ P (\tilde{A} | \cap {Md}_{C} (x)) \\ \leq \frac{Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{B \tilde{A}})) + (Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))} . \end{matrix}$

Hence, we can get three probabilistic thresholds $\begin{matrix} α = \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}})}{(Mag ({\tilde{λ}}_{B \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{B \tilde{C}}))}, \end{matrix}$ (12) $\begin{matrix} β = \frac{Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{B \tilde{A}})) + (Mag ({\tilde{λ}}_{B \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))}, \end{matrix}$ (13) $\begin{matrix} γ = \frac{Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}})}{(Mag ({\tilde{λ}}_{N \tilde{A}}) - Mag ({\tilde{λ}}_{P \tilde{A}})) + (Mag ({\tilde{λ}}_{P \tilde{C}}) - Mag ({\tilde{λ}}_{N \tilde{C}}))}, \end{matrix}$ (14)

where $\begin{matrix} Mag ({\tilde{λ}}_{P \tilde{A}}) = \frac{1}{2} (x_{P \tilde{A}} + y_{P \tilde{A}}) - \frac{1}{12} (σ_{P \tilde{A}} - δ_{P \tilde{A}}), \\ Mag ({\tilde{λ}}_{B \tilde{A}}) = \frac{1}{2} (x_{B \tilde{A}} + y_{B \tilde{A}}) - \frac{1}{12} (σ_{B \tilde{A}} - δ_{B \tilde{A}}), \\ Mag ({\tilde{λ}}_{N \tilde{A}}) = \frac{1}{2} (x_{N \tilde{A}} + y_{N \tilde{A}}) - \frac{1}{12} (σ_{N \tilde{A}} - δ_{N \tilde{A}}), \\ Mag ({\tilde{λ}}_{P \tilde{C}}) = \frac{1}{2} (x_{P \tilde{C}} + y_{P \tilde{C}}) - \frac{1}{12} (σ_{P \tilde{C}} - δ_{P \tilde{C}}), \\ Mag ({\tilde{λ}}_{B \tilde{C}}) = \frac{1}{2} (x_{B \tilde{C}} + y_{B \tilde{C}}) - \frac{1}{12} (σ_{B \tilde{C}} - δ_{B \tilde{C}}), \\ Mag ({\tilde{λ}}_{N \tilde{C}}) = \frac{1}{2} (x_{N \tilde{C}} + y_{N \tilde{C}}) - \frac{1}{12} (σ_{N \tilde{C}} - δ_{N \tilde{C}}) . \end{matrix}$

According to the above derivation, when α > β, we have α > γ > β . Then, we can obtain the decision rules as follows.

(P) If $P (\tilde{A} | \cap {Md}_{C} (x)) \geq α,$ decides $pos (\tilde{A});$

(B) If $β < P (\tilde{A} | \cap {Md}_{C} (x)) < α,$ decides $bn (\tilde{A});$

(N) If $P (\tilde{A} | \cap {Md}_{C} (x)) \leq β,$ decides $neg (\tilde{A}) .$

Therefore, we can present the positive region, boundary region and negative region of fuzzy set $\tilde{A} .$ $\begin{matrix} pos (\tilde{A}) & = & \underline{apr} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) \geq α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \geq α}, \\ bn (\tilde{A}) & = & \bar{apr} (\tilde{A}) - \underline{apr} (\tilde{A}) \\ = & {x \in U | β < P (\tilde{A} | \cap {Md}_{C} (x)) < α} \\ = & {x \in U | β < \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} < α}, \\ neg (\tilde{A}) & = & U - \bar{apr} (\tilde{A}) \\ = & {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) \leq β} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \leq β} . \end{matrix}$

Based on the relationships between the lower and upper approximations and the three regions, we can obtain the probabilistic rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ . $\begin{matrix} \underline{apr} (\tilde{A}) & = & {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) \geq α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \geq α}, \\ \bar{apr} (\tilde{A}) & = & U - {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) \leq β} \\ = & U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \leq β} . \end{matrix}$

When α = β, we have α = γ = β . Then, we have the following decision rules.

(P) If $P (\tilde{A} | \cap {Md}_{C} (x)) > α,$ decides $pos (\tilde{A});$

(B) If $P (\tilde{A} | \cap {Md}_{C} (x)) = α,$ decides $bn (\tilde{A});$

(N) If $P (\tilde{A} | \cap {Md}_{C} (x)) < α,$ decides $neg (\tilde{A}) .$

Similarly, we give the positive region, boundary region and negative region of fuzzy set $\tilde{A} .$ $\begin{matrix} pos (\tilde{A}) & = & \underline{apr} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) > α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} > α}; \\ bn (\tilde{A}) & = & \bar{apr} (\tilde{A}) - \underline{apr} (\tilde{A}) \\ = & {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) = α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} = α}; \\ neg (\tilde{A}) & = & U - \bar{apr} (\tilde{A}) \\ = & {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) < α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} < α} . \end{matrix}$

And, the probabilistic rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ are gained in the following. $\begin{matrix} \underline{apr} (\tilde{A}) & = & {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) > α} \\ = & {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} > α}, \\ \bar{apr} (\tilde{A}) & = & U - {x \in U | P (\tilde{A} | \cap {Md}_{C} (x)) < α} \\ = & U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} < α} . \end{matrix}$

4 Covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models

In this section, we propose the mean, optimistic and pessimistic covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets respectively and give some characterization theorems.

Let C₁, C₂, ⋯ , C_m be m coverings of the universe of discourse U, then C_i is a basic granular of U and can deduce a granular structure. In multigranulation decision-theoretic rough sets, the conditional probability of an object x within a target concept in m granular structures can be computed by its mathematic expectation. That is,

$\begin{matrix} E (P (X | x)) & = & (P (X | \cap {Md}_{C_{1}} (x)) + P (X | \cap {Md}_{C_{2}} (x)) \\ + \dots + P (X | \cap {Md}_{C_{m}} (x))) / m . \end{matrix}$ (15)

Based on this idea, we propose a covering mean multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set model.

Definition 4.1. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any $\tilde{A} \in F (U),$ the covering mean multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ are defined respectively as follows. $\begin{matrix} {\sum_{_}}_{i = 1}^{m} C_{i}^{M, α} (\tilde{A}) \\ = {x \in U | (P (\tilde{A} | \cap {Md}_{C_{1}} (x)) + P (\tilde{A} | \cap {Md}_{C_{2}} (x)) \\ + \dots + P (\tilde{A} | \cap {Md}_{C_{m}} (x))) / m \geq α} \\ = {x \in U | (\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \\ + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots \\ + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m \geq α}; \end{matrix}$ $\begin{matrix} {\sum^{¯}}_{i = 1}^{m} C_{i}^{M, β} (\tilde{A}) \\ = U - {x \in U | (P (\tilde{A} | \cap {Md}_{C_{1}} (x)) \\ + P (\tilde{A} | \cap {Md}_{C_{2}} (x)) + \dots \\ + P (\tilde{A} | \cap {Md}_{C_{m}} (x))) / m \leq β} \\ = U - {x \in U | (\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \\ + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} \\ + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m \leq β} . \end{matrix}$

Similar to the decision rules in Section 3, when α > β, we can obtain the following decision rules.

(MP1) If $(\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m \geq α,$ decides $pos (\tilde{A});$

(MB1) If $β < (\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m < α,$ decides $bn (\tilde{A});$

(MN1) If $(\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m \leq β,$ decides $neg (\tilde{A}) .$

When α = β, the covering mean multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set has the following decision rules.

(MP2) If $(\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m > α,$ decides $pos (\tilde{A});$

(MB2) If $(\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m = α,$ decides $bn (\tilde{A});$

(MN2) If $(\frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} + \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} + \dots + \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |}) / m < α,$ decides $neg (\tilde{A}) .$

Example 4.1. Let U = {x₁, x₂, x₃, x₄, x₅, x₆} be the universe of discourse. C₁ = {{x₁, x₂, x₃} , {x₂, x₃, x₄, x₅} , {x₃, x₄, x₆} , {x₄, x₅}} , C₂ = {{x₁, x₂, x₅} , {x₂, x₃, x₄, x₅} , {x₃, x₄, x₆}} are two coverings of U. Let $\tilde{A} = {\frac{0.3}{x_{1}} + \frac{0.5}{x_{2}} + \frac{0.4}{x_{3}} + \frac{0.6}{x_{4}} + \frac{0.2}{x_{5}} + \frac{0.9}{x_{6}}}$ be a fuzzy set on U.

From the formula(6), we have

$P (\tilde{A} | \cap {Md}_{C_{1}} (x_{1})) = 0.4,$ $P (\tilde{A} | \cap {Md}_{C_{1}} (x_{2})) = 0.45,$ $P (\tilde{A} | \cap {Md}_{C_{1}} (x_{3})) = 0.4,$ $P (\tilde{A} | \cap {Md}_{C_{1}} (x_{4})) = 0.6,$

$P (\tilde{A} | \cap {Md}_{C_{1}} (x_{5})) = 0.4,$ $P (\tilde{A} | \cap {Md}_{C_{1}} (x_{6})) = 0.63,$ $P (\tilde{A} | \cap {Md}_{C_{2}} (x_{1})) = 0.33,$ $P (\tilde{A} | \cap {Md}_{C_{2}} (x_{2})) = 0.35,$

$P (\tilde{A} | \cap {Md}_{C_{2}} (x_{3})) = 0.5,$ $P (\tilde{A} | \cap {Md}_{C_{2}} (x_{4})) = 0.5,$ $P (\tilde{A} | \cap {Md}_{C_{2}} (x_{5})) = 0.35,$ $P (\tilde{A} | \cap {Md}_{C_{2}} (x_{6})) = 0.63 .$

Suppose that α = 0.49, β = 0.41 .

From the above definition, we obtain

${\sum_{_}}_{i = 1}^{2} C_{i}^{M, α} (\tilde{A}) = {x_{4}, x_{6}}, {\sum^{¯}}_{i = 1}^{2} C_{i}^{M, β} (\tilde{A}) = {x_{3}, x_{4}, x_{6}} .$

Furthermore, according to the above decision rules, we can present the positive, boundary and negative regions of fuzzy set $\tilde{A}$ as follows:

$pos (\tilde{A}) = {x_{4}, x_{6}}, bn (\tilde{A}) = {x_{3}}, neg (\tilde{A}) = {x_{1}, x_{2}, x_{5}} .$

Naturally, the covering optimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations and the covering pessimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ can be given by the Definition 4.2 and Definition 4.3 respectively.

Definition 4.2. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any $\tilde{A} \in F (U),$ the covering optimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ are defined respectively as follows.

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C_{1}} (x)) \geq α \lor P (\tilde{A} | \cap {Md}_{C_{2}} (x)) \geq α \lor \dots \lor P (\tilde{A} | \cap {Md}_{C_{m}} (x)) \geq α} = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α};$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | P (\tilde{A} | \cap {Md}_{C_{1}} (x)) \leq β \land P (\tilde{A} | \cap {Md}_{C_{2}} (x)) \leq β \land \dots \land P (\tilde{A} | \cap {Md}_{C_{m}} (x)) \leq β} = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} \leq β \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} .$

Similarly, when α > β, we can obtain the following decision rules.

(OP1) If there exits i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \geq α,$ decides $pos (\tilde{A});$

(ON1) If for any i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \leq β,$ decides $neg (\tilde{A});$

(OB1) Otherwise, decides $bn (\tilde{A}) .$

When α = β, the covering optimistic multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set has the following decision rules.

(OP2) If there exits i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} > α,$ decides $pos (\tilde{A});$

(ON2) If for any i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} < α,$ decides $neg (\tilde{A});$

(OB2) Otherwise, decides $bn (\tilde{A}) .$

Example 4.2. Let us consider Example 4.1 again. According to the Definition 4.2, we have

${\sum_{_}}_{i = 1}^{2} C_{i}^{O, α} (\tilde{A}) = {x_{3}, x_{4}, x_{6}}, {\sum^{¯}}_{i = 1}^{2} C_{i}^{O, β} (\tilde{A}) = {x_{2}, x_{3}, x_{4}, x_{6}} .$

Similarly, we obtain

$pos (\tilde{A}) = {x_{3}, x_{4}, x_{6}}, bn (\tilde{A}) = {x_{2}}, neg (\tilde{A}) = {x_{1}, x_{5}} .$

Definition 4.3. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any $\tilde{A} \in F (U),$ the covering pessimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A}$ are defined respectively as follows.

${\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C_{1}} (x)) \geq α \land P (\tilde{A} | \cap {Md}_{C_{2}} (x)) \geq α \land \dots \land P (\tilde{A} | \cap {Md}_{C_{m}} (x)) \geq α} = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α};$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} (\tilde{A}) = U - {x \in U | P (\tilde{A} | \cap {Md}_{C_{1}} (x)) \leq β \lor P (\tilde{A} | \cap {Md}_{C_{2}} (x)) \leq β \lor \dots \lor P (\tilde{A} | \cap {Md}_{C_{m}} (x)) \leq β} = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \lor \frac{\sum_{y \in \cap {Md}_{C_{2}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{2}} (x) |} \leq β \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} .$

Similarly, when α > β, we can obtain the following decision rules.

(PP1) If for any i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \geq α,$ decides $pos (\tilde{A});$

(PN1) If there exists i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \leq β,$ decides $neg (\tilde{A});$

(PB1) Otherwise, decides $bn (\tilde{A}) .$

When α = β, the covering pessimistic multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set has the following decision rules.

(PP2) If for any i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} > α,$ decides $pos (\tilde{A});$

(PN2) If there exists i ∈ {1, 2, ⋯ , m} such that $\frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} < α,$ decides $neg (\tilde{A});$

(PB2) Otherwise, decides $bn (\tilde{A}) .$

Example 4.3. Let us continue Example 4.1. From the Definition 4.3, we have $\begin{matrix} {\sum_{_}}_{i = 1}^{2} C_{i}^{P, α} (\tilde{A}) = {x_{4}, x_{6}}, \\ {\sum^{¯}}_{i = 1}^{2} C_{i}^{P, β} (\tilde{A}) = {x_{4}, x_{6}} . \end{matrix}$

Analogously, we obtain $\begin{matrix} pos (\tilde{A}) = {x_{4}, x_{6}}, bn (\tilde{A}) = \emptyset, \\ neg (\tilde{A}) = {x_{1}, x_{2}, x_{3}, x_{5}} . \end{matrix}$

For the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets, it is easy to prove the following characterization theorems.

Theorem 4.1. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any $\tilde{A} \in F (U),$ we have

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) \supseteq {\underline{C_{i}}}_{α} (\tilde{A}), i \leq m;$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) \subseteq {\bar{C_{i}}}_{β} (\tilde{A}), i \leq m;$

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = \cup_{i = 1}^{m} {\underline{C_{i}}}_{α} (\tilde{A}), i \leq m;$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = \cap_{i = 1}^{m} {\bar{C_{i}}}_{β} (\tilde{A}), i \leq m;$

${\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} (\tilde{A}) \subseteq {\underline{C_{i}}}_{α} (\tilde{A}), i \leq m;$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} (\tilde{A}) \supseteq {\bar{C_{i}}}_{β} (\tilde{A}), i \leq m;$

${\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} (\tilde{A}) = \cap_{i = 1}^{m} {\underline{C_{i}}}_{α} (\tilde{A}), i \leq m;$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} (\tilde{A}) = \cup_{i = 1}^{m} {\bar{C_{i}}}_{β} (\tilde{A}), i \leq m;$

where ${\underline{C_{i}}}_{α} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C_{i}} (x)) \geq α} = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \geq α},$ ${\bar{C_{i}}}_{β} (\tilde{A}) = {x \in U | P (\tilde{A} | \cap {Md}_{C_{i}} (x)) \geq β} = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{i}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{i}} (x) |} \geq β} .$

Theorem 4.2. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any ${\tilde{A}}_{1}, {\tilde{A}}_{2} \in F (U),$ and ${\tilde{A}}_{1} \subseteq {\tilde{A}}_{2},$ we have

${\sum_{_}}_{i = 1}^{m} C_{i}^{M, α} ({\tilde{A}}_{1}) \subseteq {\sum_{_}}_{i = 1}^{m} C_{i}^{M, α} ({\tilde{A}}_{2});$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{M, β} ({\tilde{A}}_{1}) \subseteq {\sum^{¯}}_{i = 1}^{m} C_{i}^{M, β} ({\tilde{A}}_{2});$

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} ({\tilde{A}}_{1}) \subseteq {\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} ({\tilde{A}}_{2});$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} ({\tilde{A}}_{1}) \subseteq {\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} ({\tilde{A}}_{2});$

${\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} ({\tilde{A}}_{1}) \subseteq {\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} ({\tilde{A}}_{2});$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} ({\tilde{A}}_{1}) \subseteq {\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} ({\tilde{A}}_{2}) .$

Theorem 4.3. Let C₁, C₂, ⋯ , C_m be m coverings of U, for any $\tilde{A} \in F (U),$ we have $\begin{matrix} {\sum_{_}}_{i = 1}^{m} C_{i}^{P, α} (\tilde{A}) \subseteq {\sum_{_}}_{i = 1}^{m} C_{i}^{M, α} (\tilde{A}) \\ \subseteq {\sum_{_}}_{i = 1}^{m} C_{i}^{M, α} (\tilde{A}); \\ {\sum^{¯}}_{i = 1}^{m} C_{i}^{P, β} (\tilde{A}) \subseteq {\sum^{¯}}_{i = 1}^{m} C_{i}^{M, β} (\tilde{A}) \\ \subseteq {\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) . \end{matrix}$

5 The relationships between the covering multi-granulation trapezoidal fuzzy decision theoretic rough fuzzy set models and the existing models

In the following, we only illustrate the covering optimistic multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set model, other models can be elucidated similarly.

Case 1. The case of degenerating into the covering multigranulation trapezoidal fuzzy decision-theoretic rough sets.

When the approximation concept $\tilde{A}$ degenerates into the crisp set A of U and the fuzzy partition $Ω = {\tilde{A}, \tilde{B}, \tilde{C}}$ of U degenerates into the crisp partition of U, the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets degenerate into the covering multigranulation trapezoidal fuzzy decision-theoretic rough sets, namely,

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α} \Rightarrow {\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (A) = {x \in U | \frac{| A \cap (\cap {Md}_{C_{1}} (x)) |}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{| A \cap (\cap {Md}_{C_{m}} (x)) |}{| \cap {Md}_{C_{m}} (x) |} \geq α},$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} \Rightarrow {\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (A) = U - {x \in U | \frac{| A \cap (\cap {Md}_{C_{1}} (x)) |}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \dots \land \frac{| A \cap (\cap {Md}_{C_{m}} (x)) |}{| \cap {Md}_{C_{m}} (x) |} \leq β} .$

Case 2. The case of degenerating into the multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets

When C_i (i = 1, 2, ⋯ , m) are the partitions of U, the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets degenerate into the multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets, that is,

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α} \Rightarrow {\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in [x]_{C_{1}}} \tilde{A} (y)}{| [x]_{C_{1}} |} \geq α \lor \dots \lor \frac{\sum_{y \in [x]_{C_{m}}} \tilde{A} (y)}{| [x]_{C_{m}} |} \geq α},$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} \Rightarrow {\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in [x]_{C_{1}}} \tilde{A} (y)}{| [x]_{C_{1}} |} \leq β \land \dots \land \frac{\sum_{y \in [x]_{C_{m}}} \tilde{A} (y)}{| [x]_{C_{m}} |} \leq β} .$

Case 3. The case of degenerating into the covering multigranulation decision-theoretic rough fuzzy sets.

When the loss functions ${\tilde{λ}}_{ij}$ (i takes P, B, N respectively and j takes $\tilde{A}, \tilde{B}, \tilde{C}$ ) degenerate into the precise values, the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets degenerate into the covering multigranulation decision-theoretic rough fuzzy sets, namely,

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α} \Rightarrow {\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α^{'} \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α^{'}},$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} \Rightarrow {\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β^{'} \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β^{'}},$

where $α^{'} = \frac{{\tilde{λ}}_{P \tilde{C}} - {\tilde{λ}}_{B \tilde{C}}}{({\tilde{λ}}_{B \tilde{A}}) - {\tilde{λ}}_{P \tilde{A}}) + ({\tilde{λ}}_{P \tilde{C}} - {\tilde{λ}}_{B \tilde{C}})},$ (16) $β^{'} = \frac{{\tilde{λ}}_{B \tilde{C}} - {\tilde{λ}}_{N \tilde{C}}}{({\tilde{λ}}_{N \tilde{A}} - {\tilde{λ}}_{B \tilde{A}}) + ({\tilde{λ}}_{B \tilde{C}} - {\tilde{λ}}_{N \tilde{C}})},$ (17) $γ^{'} = \frac{{\tilde{λ}}_{P \tilde{C}} - {\tilde{λ}}_{N \tilde{C}}}{({\tilde{λ}}_{N \tilde{A}} - {\tilde{λ}}_{P \tilde{A}}) + ({\tilde{λ}}_{P \tilde{C}} - {\tilde{λ}}_{N \tilde{C}})} .$ (18)

Case 4. The case of degenerating into the covering trapezoidal fuzzy decision-theoretic rough fuzzy sets.

When the multiple granular structures degenerate into the simple granular structure, the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets degenerate into the covering trapezoidal fuzzy decision-theoretic rough fuzzy sets, that is,

${\sum_{_}}_{i = 1}^{m} C_{i}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \geq α \lor \dots \lor \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \geq α} \Rightarrow {\underline{C}}^{O, α} (\tilde{A}) = {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \geq α},$

${\sum^{¯}}_{i = 1}^{m} C_{i}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C_{1}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{1}} (x) |} \leq β \land \dots \land \frac{\sum_{y \in \cap {Md}_{C_{m}} (x)} \tilde{A} (y)}{| \cap {Md}_{C_{m}} (x) |} \leq β} \Rightarrow {\bar{C}}^{O, β} (\tilde{A}) = U - {x \in U | \frac{\sum_{y \in \cap {Md}_{C} (x)} \tilde{A} (y)}{| \cap {Md}_{C} (x) |} \leq β} .$

6 An illustration

In this section, we present a simplified example to illustrate the generality of the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models established in Section 4. Below, Table 1 gives a decision information system for responding to a credit card applicant.

In the following discussion, we show the process and results of decision-making for the credit card applicant by using the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models.

The universe U = {x₁, x₂, ⋯ , x₆} is six applicants for a credit card. The conditional attribute set is {Salary(a), Degrees(b), Bad records(c)}. From the Table 1, we can obtain C₁ and C₂ are two coverings of U and C₁ = {{x₁, x₅} , {x₃, x₄} , {x₂, x₆} , {x₁, x₄, x₅} , {x₂, x₃} , {x₆}} , C₂ = {{x₁, x₄, x₅} , {x₂, x₃} , {x₆} , {x₁, x₃, x₆} , {x₄, x₅} , {x₂}} . The set of states is given by $Ω = {\tilde{A}, \tilde{B}, \tilde{C}},$ where $\tilde{A} = CRS : Excellent$ , $\tilde{B} = CRS : Good$ and $\tilde{C} = CRS : Medium$ are three fuzzy sets on universe U, and CRS stands for the Credit rating states for every applicant. Here, we discuss the decision-procedure of fuzzy set $\tilde{A}$ and present the decision-making results. The decision-making procedures of fuzzy set $\tilde{B}$ and $\tilde{C}$ are similar to $\tilde{A} .$

Denote $pos (\tilde{A}), bn (\tilde{A})$ and $neg (\tilde{A})$ as the positive region(accept), boundary region(further evaluation) and the negative region(reject) of the fuzzy set $\tilde{A} .$ With respect to these three regions, the set of actions is given by A = {r_P, r_B, r_N} , where r_P, r_B, r_N represent the three actions in classify an applicant, that is, deciding $pos (\tilde{A}),$ deciding $bn (\tilde{A})$ and deciding $neg (\tilde{A})$ respectively. The loss functions $\tilde{λ} (r_{i} | \tilde{A}), \tilde{λ} (r_{i} | \tilde{B})$ and $\tilde{λ} (r_{i} | \tilde{C})$ are consistent with the definitions in Section 3 and satisfy with the formula (11).

Here, we consider the following loss functions: $\begin{matrix} {\tilde{λ}}_{P \tilde{A}} & = & {0.3, 0.5, 0.2, 0.3}, \\ {\tilde{λ}}_{B \tilde{A}} & = & {0.3, 0.6, 0.2, 0.2}, \\ {\tilde{λ}}_{N \tilde{A}} & = & {0.4, 0.7, 0.3, 0.1}, \\ {\tilde{λ}}_{P \tilde{C}} & = & {0.4, 0.7, 0.2, 0.1}, \\ {\tilde{λ}}_{B \tilde{C}} & = & {0.4, 0.6, 0.2, 0.2}, \\ {\tilde{λ}}_{N \tilde{C}} & = & {0.3, 0.6, 0.2, 0.3}, \\ {\tilde{λ}}_{P \tilde{B}} & = & {0.8, 0.9, 0.2, 0.3}, \\ {\tilde{λ}}_{B \tilde{B}} & = & {0.8, 1, 0.3, 0.3}, \\ {\tilde{λ}}_{N \tilde{B}} & = & {0.7, 0.8, 0.2, 0.1} . \end{matrix}$

Then, according to the formula (12)–(14), we have $α = \frac{5}{12}, β = \frac{1}{3} .$

According to the formula (6), we also have the following results: $\begin{matrix} P (\tilde{A} | \cap {Md}_{C_{1}} (x_{1})) = P (\tilde{A} | \cap {Md}_{C_{1}} (x_{5})) = 0.55, \\ P (\tilde{A} | \cap {Md}_{C_{1}} (x_{2})) = P (\tilde{A} | \cap {Md}_{C_{2}} (x_{2})) = 0.8, \\ P (\tilde{A} | \cap {Md}_{C_{1}} (x_{3})) = P (\tilde{A} | \cap {Md}_{C_{2}} (x_{3})) = 0.2, \\ P (\tilde{A} | \cap {Md}_{C_{1}} (x_{6})) = P (\tilde{A} | \cap {Md}_{C_{2}} (x_{6})) = 0.7, \\ P (\tilde{A} | \cap {Md}_{C_{2}} (x_{4})) = P (\tilde{A} | \cap {Md}_{C_{2}} (x_{5})) = 0.4, \\ P (\tilde{A} | \cap {Md}_{C_{1}} (x_{4})) = 0.2, P (\tilde{A} | \cap {Md}_{C_{2}} (x_{1})) = 0.5 . \end{matrix}$

Therefore, according to Definition 4.1, we can obtain the covering mean multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A} :$ $\begin{matrix} {\sum_{_}}_{i = 1}^{2} C_{i}^{M, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ {\sum^{¯}}_{i = 1}^{2} C_{i}^{M, β} (\tilde{A}) = U - {x_{3}, x_{4}} = {x_{1}, x_{2}, x_{5}, x_{6}} . \end{matrix}$

From the definitions of the positive region, boundary region and negative region, we have the decision-making results by using the covering mean multigranualtion trapezoidal fuzzy decision-theoretic rough fuzzy set model in the following. $\begin{matrix} pos (\tilde{A}) & = & {\sum_{_}}_{i = 1}^{2} C_{i}^{M, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ bn (\tilde{A}) & = & {\sum^{¯}}_{i = 1}^{2} C_{i}^{M, β} (\tilde{A}) - {\sum_{_}}_{i = 1}^{2} C_{i}^{M, α} (\tilde{A}) = \emptyset, \\ neg (\tilde{A}) & = & U - {\sum^{¯}}_{i = 1}^{2} C_{i}^{M, β} (\tilde{A}) = {x_{3}, x_{4}} . \end{matrix}$

The results show that applicant 1, applicant 2, applicant 5 and applicant 6 should be accepted, and applicant 3 and applicant 4 should be rejected.

From Definition 4.2, we can gain the covering optimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A} :$ $\begin{matrix} {\sum_{_}}_{i = 1}^{2} C_{i}^{O, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ {\sum^{¯}}_{i = 1}^{2} C_{i}^{O, β} (\tilde{A}) = U - {x_{3}} = {x_{1}, x_{2}, x_{4}, x_{5}, x_{6}} . \end{matrix}$

Similarly, we can obtain the decision-making results by using the covering optimistic multigranualtion trapezoidal fuzzy decision-theoretic rough fuzzy set model in the following. $\begin{matrix} pos (\tilde{A}) = {\sum_{_}}_{i = 1}^{2} C_{i}^{O, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ bn (\tilde{A}) = {\sum^{¯}}_{i = 1}^{2} C_{i}^{O, β} (\tilde{A}) - {\sum_{_}}_{i = 1}^{2} C_{i}^{O, α} (\tilde{A}) = {x_{4}}, \\ neg (\tilde{A}) = U - {\sum^{¯}}_{i = 1}^{2} C_{i}^{O, β} (\tilde{A}) = {x_{3}} . \end{matrix}$

The results show that applicant 1, applicant 2, applicant 5 and applicant 6 should be accepted, applicant 4 should be further evaluated and applicant 3 should be rejected.

According to Definition 4.3, we can get the covering pessimistic multigranulation trapezoidal fuzzy rough fuzzy lower and upper approximations of fuzzy set $\tilde{A} :$ $\begin{matrix} {\sum_{_}}_{i = 1}^{2} C_{i}^{P, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ {\sum^{¯}}_{i = 1}^{2} C_{i}^{P, β} (\tilde{A}) = U - {x_{3}, x_{4}} = {x_{1}, x_{2}, x_{5}, x_{6}} . \end{matrix}$

And, we have the decision-making results by using the covering pessimistic multigranualtion trapezoidal fuzzy decision-theoretic rough fuzzy set model in the following. $\begin{matrix} pos (\tilde{A}) = {\sum_{_}}_{i = 1}^{2} C_{i}^{P, α} (\tilde{A}) = {x_{1}, x_{2}, x_{5}, x_{6}}, \\ bn (\tilde{A}) = {\sum^{¯}}_{i = 1}^{2} C_{i}^{P, β} (\tilde{A}) - {\sum_{_}}_{i = 1}^{2} C_{i}^{P, α} (\tilde{A}) = \emptyset, \\ neg (\tilde{A}) = U - {\sum^{¯}}_{i = 1}^{2} C_{i}^{P, β} (\tilde{A}) = {x_{3}, x_{4}} . \end{matrix}$

The results show that applicant 1, applicant 2, applicant 5 and applicant 6 should be accepted, and applicant 3 and applicant 4 should be rejected.

Furthermore, we can obtain the following results: $\begin{matrix} {\sum_{_}}_{i = 1}^{2} C_{i}^{P, α} (\tilde{A}) \subseteq {\sum_{_}}_{i = 1}^{2} C_{i}^{M, α} (\tilde{A}) \subseteq {\sum_{_}}_{i = 1}^{2} C_{i}^{O, α} (\tilde{A}); \\ {\sum^{¯}}_{i = 1}^{2} C_{i}^{P, β} (\tilde{A}) \subseteq {\sum^{¯}}_{i = 1}^{2} C_{i}^{M, β} (\tilde{A}) \subseteq {\sum^{¯}}_{i = 1}^{2} C_{i}^{O, β} (\tilde{A}) . \end{matrix}$

The above results illustrate the Theorem 4.3.

7 Conclusions

In this paper, we propose the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models by calculating the three probabilistic thresholds based on the fuzziness and vagueness of the approximation concept, discuss and characterize the three models, namely, the covering mean, optimistic and pessimistic multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy sets, obtain the relationships between the covering multigranulation trapezoidal fuzzy decision-theoretic rough fuzzy set models and the existing models, give an example and the obtained results illustrate the practicality and generality of the proposed models.

Acknowledgments

The authors would like to thank the referees and Associate Editor, for providing very helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (61262022, 11461062).

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