( α,β )-Multi-granulation bipolar fuzzified rough sets and their applications to multi criteria group decision making

Abstract

Pawlak’s rough set theory based on single granulation has been extended to multi-granulation rough set structure in recent years. Multi-granulation rough set theory has become a flouring research direction in rough set theory. In this paper, we propose the notion of (α, β)-multi-granulation bipolar fuzzified rough set ((α, β)-MGBFRSs). For this purpose, a collection of bipolar fuzzy tolerance relations has been used. In the framework of multi-granulation, we proposed two types of (α, β)-multi-granulation bipolar fuzzified rough sets model. One is called the optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^o-MGBFRSs) and the other is called the pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^p-MGBFRSs). Subsequently, a number of important structural properties and results of proposed models are investigated in detail. The relationships among the (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs are also established. In order to illustrate our proposed models, some examples are considered, which are helpful for applying this theory in practical issues. Moreover, several important measures associated with (α, β)-multi-granulation bipolar fuzzified rough set like the measure of accuracy, the measure of precision, and accuracy of approximation are presented. Finally, we construct a new approach to multi-criteria group decision-making method based on (α, β)-MGBFRSs, and the validity of this technique is illustrated by a practical application. Compared with the existing results, we also expound its advantages.

Keywords

Rough set,multi-granulation rough approximations bipolar fuzzy tolerance relation decision making method

1 Introduction

The rough set theory proposed by Pawlak [32, 33], has become a well-established mechanism for dealing with vagueness, imprecision and uncertainty in data analysis. The theory has found its successive applications in several fields such as medical diagnosis, engineering, economics, management sciences, pattern recognition, image processing, feature selection, conflict analysis, artificial intelligence, decision making, data mining, machine learning, neural computing, knowledge discovery, data analysis, algebra and many more.

The fuzzy set theory [62] is another important and successful approach to cope with uncertainties. It depends on the fuzzy membership function, with the help of which we can determine the membership degree of an element with respect to a set. The higher the membership degree, the greater the belongingness of that element to the corresponding set. There are several generalizations of the fuzzy set such as vague set, neutrosophic fuzzy set, intuitionistic fuzzy set, interval valued fuzzy set etc.

Bipolar fuzzy set [64] is another extension of the fuzzy set given by Zhang in 1994. In this theory, the membership degree is enlarged from the interval [0, 1] to the interval [-1, 1]. In a bipolar fuzzy set, the membership degree 0 of an element implies that the element is irrelevant to the corresponding property, the membership degree in (0, 1] of an element shows that the element somewhat satisfies the property and the membership degree in [-1, 0) of an element demonstrates that the element somewhat satisfies the implicit counter-property.

The rough set theory proposed by Pawlak [32, 33] is another important and successful mathematical tool to examine the uncertainty in data analysis. In this theory, uncertainty is represented by a boundary region (area of uncertainty) of a set. Pawlak used the upper and the lower approximations of a collection of objects to investigate how close the objects are to the information attached to them. The theory of rough sets was based upon a single equivalence relation, which is too restrictive for many practical applications, especially in handling real-valued or symbolic attribute values. The theory was further extended by using binary relations (serial, reflexive or tolerance), a set-valued map, a soft equivalence relation, similarity relation, soft binary relation instead of an equivalence relation to getting more generalized models of rough set theory. These theories were based on single granulation, that is, they used single binary relation to find the required approximations. In many real-world problems, where data may be distributed across multiple locations, Hu et al. [15] considered distributed information systems. When we deal with distributed information systems or high dimensional data, single granulation rough set faces some difficulties. In order to overcome these difficulties, Qian et al. [35] generalized Pawlak’s approach by using more than one equivalence relation and name this as multi-granulation rough set theory, which provides a new field of vision from the angle of granular computing. It has become an attractive topic in artificial intelligence and management science and has attracted a broad range of studies from both theoretical and application aspects.

Now multi-granulation rough set approach has proved to provide a new type of information fusion strategy compared to the single granulation rough set. With regard to some special information systems, such as multi-source information systems, distributive information systems, groups of intelligent agents and multiple attribute group decision making, the existing single granularity rough sets can not be used to deal with data from these information systems and uncertainty decision making, but the multi-granulation rough set can. In multi-granulation rough set theory, optimistic multi-granulation and pessimistic multi-granulation are two fundamental ways of research. For the lower approximation of a multi-granulation rough set, the view of optimistic multi-granulation reflects that there exists at least one granular structure such that elements surely belong to a given concept, and the view of pessimistic multi-granulation shows that elements surely belong to a given concept in each granular structure.

Related work and motivation. So for, multi-granulation rough set has become a hot topic in artificial intelligence and management sciences and has attracted a broad range of studies from both theoretical and applied points of view. In literature, many researchers have extended the multi-granulation rough set theory. Xu et al. [53] proposed two new types of multiple granulation rough set. Xu et al. [51] proposed a multigranulation fuzzy rough set model, multigranulation rough sets based on tolerance relations (Xu et al. [52]), a multigranulation rough set model in ordered information systems (Xu et al. [49]) and a multigranulation fuzzy rough set in a fuzzy tolerance approximation space (Xu et al. [54]). Zhan and Xu [63] proposed two types of coverings based multigranulation rough fuzzy sets and discuss their applications to decision making. Yang et al. introduced the hierarchical structure properties of the multigranulation rough sets (Yang et al. [58]) and a test cost sensitive multigranulation rough set model (Yang et al. [56]). Lin et al. [24] presented a neighborhood-based multigranulation rough set. Xu et al. [50] proposed the notion of generalized multigranulation rough sets. Sun et al. [46] proposed multigranulation vague rough set over two universes and discuss its application to group decision making. She et al. [45] explored the topological structures and the properties of multigranulation rough sets. Qian et al. [37] introduced three kinds of multigranulation decision- theoretic rough set models. Feng and Mi [11] studied variable precision multigranulation fuzzy decision-theoretic rough sets in an information system. A novel membership degree based on single granulation rough sets and two operators based on this membership degree were investigated in their study. Li et al. [19] proposed a double-quantitative multigranulation decision-theoretic rough fuzzy set model. Zhang et al. [65] established four kinds of constructive methods of rough approximation operators from existing rough sets and studied the non-dual multigranulation rough sets and hybrid multigranulation rough sets. Tan et al. [48] employed the belief and plausibility functions from evidence theory to characterize the set approximations and attribute reductions in multigranulation rough set theory in incomplete information systems, and an attribute reduction algorithm for multigranulation rough sets was proposed based on evidence theory. Lin et al. [23] proposed a two-grade fusion approach involved in the evidence theory and multigranulation rough set theory based on a well-defined distance function among granulation structures, and presented three types of covering based multigranulation rough sets whose set approximations were defined by different covering approximation operators. Pan et al. [31] proposed multi-granulation preference relation rough set for ordinal system. Zhang et al. [66] proposed multi-granulation hesitant fuzzy rough sets and their corresponding application in decision making. Huang et al. [16] developed a new multigranulation rough set model that was called intuitionistic fuzzy multigranulation rough set (IFMGRS) and three types of IFMGRSs that are generalizations of three existing intuitionistic fuzzy rough set models built. Yang et al. [57] first explored the updating of the multigranulation rough approximations. Liang et al. [21] proposed an efficient rough feature selection algorithm for large-scale data sets, which was stimulated from multi-granulation rough sets. Ali et al. [2] proposed new types of dominance based multi-granulation rough sets and discuss their applications in conflict analysis problems. Rehman et al. [38] put forward the notion of soft dominance based multi granulation rough sets and discussed their applications in conflict analysis problems. You et al. [60] proposed relative reduction of neighborhood-covering pessimistic multigranulation rough set based on evidence theory. W. Yu et al. [61] proposed a consensus model for MAGDM based on multi-granular hesitant fuzzy linguistic term sets.

It is well known that the existing multi-granulation rough set models, including multi-granulation bipolar fuzzy rough sets, can handle some group decision-making problems to quantify the ideas of decision-makers by using a crisp number. However, when facing the problem, where the basic features of decision-making activities are described by several bipolar fuzzy relations to depict the information of decision-makers, we cannot offer a comprehensive, accurate, and flexible solution by using the existing multi-granulation bipolar fuzzy rough sets. Therefore, on the basis of the preceding analysis, it is natural for us to explore a novel multi-granulation rough set model under bipolar fuzzy environment according to the above discussion from a practical aspect. In this context, we introduce the novel idea of (α, β)-multi-granulation bipolar fuzzified rough sets, as well as discuss their applications in multi-criteria group decision making.

Aim and organization of the paper. The main purpose of this article is to extend the notion of (α, β)-bipolar fuzzified rough set theory to a new type of model using a collection of bipolar fuzzy tolerance relations. We call this new model “(α, β)-multi-granulation bipolar fuzzified rough set ((α, β)-MGBFRSs)”. In this approach, approximation operators have been constructed with a viewpoint of clustering objects of the universe with respect to bipolar fuzzy tolerance relations. The parameters α and β have been introduced which allows us to choose positive and negative degrees among the elements of the universe as close as we wish to without being exactly similar. Furthermore, two types of (α, β)-multi-granulation bipolar fuzzified rough sets model, called the (α, β) ^o-MGBFRSs and the (α, β) ^p-MGBFRSs are constructed. Finally, in view of the above-mentioned needs of practical applications of the (α, β)-MGBFRSs, we attempt to build up a general framework of the decision methodology based on the (α, β)-MGBFRSs. The validity of this approach is also verified by a practical example.

The remaining part of this article is organized as follows. Section 2 gives a brief review of some preliminary ideas related to rough set, multi-granulation rough set, fuzzy set, bipolar fuzzy set and (α, β)-bipolar fuzzified rough set. In section 3, the notion of (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β)-MGBFRSs) is introduced, where set approximations are described by a family of bipolar fuzzy tolerance relations. Then, some of their basic structural properties are investigated. In section 4, we discuss some important measures associated with (α, β)-multi-granulation bipolar fuzzified rough sets. Section 5 introduces optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^o-MGBFRSs) and their structural properties. In section 6, we introduce the notion of pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^p-MGBFRSs) and investigate their structural properties. In section 7, we establish the relationships among the (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs. Section 8 establishes a decision-making technique based on (α, β)-MGBFRSs and verifies the main steps of the decision-making technique by using a practical example. In section 9, we describe a sensitivity analysis of our proposed model. Finally, in section 10 we conclude our contribution with a summary and an outlook for further research.

2 Preliminaries

In this section, we review several fundamental definitions and notions related to our research. Throughout this paper, we will use $U$ for an initial universe, E for the set of parameters, A for a finite non-empty subset of the parameters set E and $P (U)$ for thepower set of $U$ , unless stated otherwise.

Definition 2.1. [32] Let $U$ be a non-empty finite universe, and R be an equivalence relation on $U$ . Then the pair $P = (U, R)$ is called a Pawlak approximation space.

If $X \subseteq U$ , then X may or may not be written as a union of some equivalence classes of $U$ . If X is written as a union of some equivalence classes, then X is said to be R-definable; otherwise it is called R-undefinable. If X is R-undefinable then it can be approximated with the help of the following two definable (crisp) subsets:

$R_{*} (X) = {x \in U ∣ [x]_{R} \subseteq X},$ (1) $R^{*} (X) = {x \in U ∣ [x]_{R} \cap X \neq \emptyset} .$ (2)

Equations (1) and (2) are called lower and upper approximations of X with respect to the Pawlak approximation space $P = (U, R)$ respectively, where the equivalence class [x] _R of an element $x \in U$ is the set consists of all objects $y \in U$ such that (x, y) ∈ R, that is, $[x]_{R} = {y \in U ∣ (x, y) \in R} .$ (3)

Moreover, the set $B {nd}_{R} (X) = R^{*} (X) - R_{*} (X)$ is called as the R-boundary region of X.

Thus set X is said to be crisp (definable with respect to R) if and only if R_* (X) = R^* (X); equivalently $B {nd}_{R} (X) = \emptyset$ . And set X is said to be rough (undefinable with respect to R) if and only if R_* (X) ≠ R^* (X); equivalently $B {nd}_{R} (X) \neq \emptyset$ .

Pawlak’s rough set theory is based on a single granulation. Using more than one binary relations, Qian et al. [35] proposed the notion of the multi-granulation rough set which is stated below.

Definition 2.2. [35] Let R₁, R₂, ⋯ , R_n be n independent equivalence relations over a universe $U$ and $X \subseteq U$ . The lower and upper multi-granulation rough approximations of X in U are defined respectively as: $\begin{matrix} (\underline{\sum_{i = 1}^{n} R_{i}}) (X) \\ = {x \in U : [x]_{R_{i}} \subseteq X for some i = 1, 2, \dots, n}, \end{matrix}$ (4) $\begin{matrix} (\bar{\sum_{i = 1}^{n} R_{i}}) (X) \\ = {x \in U : [x]_{R_{i}} \cap X \neq \emptyset for all i = 1, 2, \dots, n} . \end{matrix}$ (5) The boundary region of $X \subseteq U$ under multi-granulation rough set environment is defined as: ${BND}_{\sum_{i = 1}^{n} R_{i}} (X) = (\bar{\sum_{i = 1}^{n} R_{i}}) (X) - (\underline{\sum_{i = 1}^{n} R_{i}}) (X) .$ (6)

The properties satisfied by lower and upper multi-granulation rough approximations can be seen in [35].

Definition 2.3. [62] Let $U$ be a non-empty finite set, called the universe. A fuzzy set (or fuzzy subset) λ on $U$ is a function from $U$ into the unit closed interval [0, 1], that is, $λ : U ⟶ [0, 1] .$

The value λ (u) of λ at $u \in U$ denotes the membership degree of u in λ.

λ (u) =1 means full membership.

λ (u) =0 means non-membership.

0 < λ (u) <1 means partial membership.

For the two extreme cases ∅ (the empty fuzzy set) and

U

(the fuzzy entire set), the membership degree functions are defined by

\forall u \in U

, ∅ (u) =0 and

U (u) = 1

, respectively.

The bipolar fuzzy set is an extension of Zadeh’s fuzzy set. The bipolar fuzzy models deliver more accuracy, flexibility and comparability to the system as compared to the classical and fuzzy models.

Definition 2.4. [40] Let $U$ be a finite non-empty universe. A bipolar fuzzy set λ over $U$ is defined as: $λ = {(x, λ^{P} (x), λ^{N} (x)) : x \in U},$ where $λ^{P} : U ⟶ [0, 1]$ and $λ^{N} : U ⟶ [- 1, 0]$ are mappings, which are called positive membership degree and negative membership degree, respectively.

The positive membership degree λ^P (x) denotes the satisfaction degree of an element x to the property and the negative membership degree λ^N (x) denotes the satisfaction degree of x to somewhat implicit counter-property.

In bipolar fuzzy sets λ,

If λ^P (x) ≠0 and λ^N (x) =0, is the situation that x is regarded as having only positive satisfaction for λ.

If λ^P (x) =0 and λ^N (x) ≠0, is the situation that x does not satisfy the property of λ but somewhat satisfies the counter property.

It is possible for an element x to have λ^P (x) ≠0 and λ^N (x) ≠0 when the membership function of the property overlaps that of its counter property over some portion of $U$ .

If λ^P (x) =0 and λ^N (x) =0, then it is an indeterministic situation to investigate the property of x in λ.

The collection of all bipolar fuzzy sets over the universe

U

is represented by

BF (U)

Definition 2.5. [40] Let $λ, ξ \in BF (U)$ . Then, λ is said to be contained in ξ, that is, λ ⊆ ξ if and only if λ^P (x) ≤ ξ^P (x) and λ^N (x) ≥ ξ^N (x) for all $x \in U$ .

Definition 2.6. [40] Let $λ = {(x, λ^{P} (x), λ^{N} (x)) : x \in U}$ be a bipolar fuzzy set over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then we define (α, β) - cut level set of λ to be the crisp set $λ_{β}^{α} = {x \in U : λ^{P} (x) \geq α and λ^{N} (x) \leq β} .$

Definition 2.7. [58] Let $U$ be a non-empty finite universe and $R = {(x, λ^{P} (x), λ^{N} (x)) : x \in U}$ be a bipolar fuzzy set over $U \times U$ , that is, $R = (λ^{P}, λ^{N}) : U \times U ⟶ [0, 1] \times [- 1, 0]$ is said to be a bipolar fuzzy relation (or bipolar fuzzy binary relation) over $U$ , where λ^P (x, y) ∈ [0, 1] and λ^N (x, y) ∈ [-1, 0] for all $x, y \in U$ .

Definition 2.8. [58] A bipolar fuzzy relation R = (λ^P, λ^N) over the universe $U$ is said to be bipolar fuzzy reflexive relation if λ^P (x, x) =1 and λ^N (x, x) = -1 for each $x \in U$ .

Definition 2.9. [58] A bipolar fuzzy relation R = (λ^P, λ^N) over the universe $U$ is said to be bipolar fuzzy symmetric relation if λ^P (x, y) = λ^P (y, x) and λ^N (x, y) = λ^N (y, x) for all $x, y \in U$ .

Definition 2.10. A bipolar fuzzy reflexive and bipolar fuzzy symmetric relation over $U$ is called a bipolar fuzzy tolerance relation (also called bipolar fuzzy proximity relation or bipolar fuzzy compatibility relation).

Definition 2.11. Let $U$ be a non-empty finite universe and R be a bipolar fuzzy relation over $U$ . Then the pair $ℙ = (U, R)$ is called a bipolar fuzzy approximation space $(BFA - space$ ).

Definition 2.12. [13] Let R = (λ^P (x, y) , λ^N (x, y)) be a bipolar fuzzy relation over $U$ , where $U = {x_{1}, x_{2}, \dots, x_{n}}$ . By considering a_ij = λ^P (x_i, y_j) and b_ij = λ^N (x_i, y_j), i = 1, 2, . . . , n; j = 1, 2, . . . , n, the bipolar fuzzy relation R can be represented with the help of a pair of matrices given as: $λ^{P} = [a_{ij}]_{n \times n}^{P} = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{nn} \end{matrix}]$ and $λ^{N} = [b_{ij}]_{n \times n}^{N} = [\begin{matrix} b_{11} & b_{12} & \dots & b_{1 n} \\ b_{21} & b_{22} & \dots & b_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{n 1} & b_{n 2} & \dots & b_{nn} \end{matrix}] .$

These matrices are called the positive membership matrix and the negative membership matrix, respectively.

Remark 2.13.

Let R = (λ^P (x, y) , λ^N (x, y)) be a bipolar fuzzy relation over $U$ , where $U$ is a finite universe. If $λ^{P} = (a_{ij})_{n \times n}^{P}$ and $λ^{N} = (b_{ij})_{n \times n}^{N}$ , then bipolar fuzzy reflexivity implies that a_ii = 1 and b_ii = -1. As a result, we can observe the numbers on the principal diagonal of λ^P and λ^N to judge whether R is bipolar fuzzy reflexive or not.

R is bipolar fuzzy symmetric relation if and only if λ^P and λ^N are symmetric as a matrix.

Recently, Gul and Shabir [13] proposed the notion of (α, β)-bipolar fuzzified rough set based on bipolar fuzzy tolerance relation which is stated below.

Definition 2.14. [13] Let $ℙ = (U, R)$ be a bipolar fuzzy approximation space, where $U$ is a non-empty finite set of objects and R is a bipolar fuzzy tolerance relation characterized by its positive and negative membership functions given as $μ_{R}^{+} : U \times U ⟶ [0, 1]$ and $μ_{R}^{-} : U \times U ⟶ [- 1, 0]$ . For any α ∈ (0, 1] and β ∈ [-1, 0), the fuzzified lower α - positive, upper α - positive, lower β - negative and upper β - negative bipolar rough approximations of $X \subseteq U$ are defined as: $\begin{matrix} {\underline{R}}_{α}^{+} (X) & = {x \in U : μ_{R}^{+} (x, y) < α for all y \in X^{c}}, \\ {\bar{R}}_{α}^{+} (X) & = {x \in U : μ_{R}^{+} (x, y) \geq α for some y \in X}, \\ {\underline{R}}_{β}^{-} (X) & = {x \in U : μ_{R}^{-} (x, y) \leq β for some y \in X}, \\ {\bar{R}}_{β}^{-} (X) & = {x \in U : μ_{R}^{-} (x, y) > β for all y \in X^{c}} . \end{matrix}$ The two pairs given as: $\begin{matrix} {\underline{FB}}_{ℙ} (X) & = ({\underline{R}}_{α}^{+} (X), {\underline{R}}_{β}^{-} (X)), \end{matrix}$ (7) $\begin{matrix} {\bar{FB}}_{ℙ} (X) & = ({\bar{R}}_{α}^{+} (X), {\bar{R}}_{β}^{-} (X)) \end{matrix}$ (8) are called (α, β)-bipolar fuzzified rough approximations of X with respect to $BFA$ -space $ℙ = (U, R)$ .

Moreover, the boundary region (area of uncertainty) of $X \subseteq U$ under (α, β)-bipolar fuzzified rough approximations environment is defined as:

$\begin{matrix} F {BND}_{ℙ} (X) & = {\bar{FB}}_{ℙ} (X) - {\underline{FB}}_{ℙ} (X) \\ = ({\bar{R}}_{α}^{+} (X) \ {\underline{R}}_{α}^{+} (X), {\underline{R}}_{β}^{-} (X) \ {\bar{R}}_{β}^{-} (X)) . \end{matrix}$ (9)

The knowledge regarding an element x of $U$ depicted by the above-defined operators is as follows:

${\underline{R}}_{α}^{+} (X)$ represents a crisp set that contains elements $x \in U$ equivalent to all elements y ∈ X^c with a positive membership degree less than to a certain α ∈ [0, 1].

${\bar{R}}_{α}^{+} (X)$ represents a crisp set that contains elements $x \in U$ equivalent to at least one element y ∈ X with a positive membership degree greater than or equal to a certain α ∈ [0, 1].

${\underline{R}}_{β}^{-} (X)$ represents a crisp set that contains elements $x \in U$ equivalent to at least one element y ∈ X with a negative membership degree less than or equal to a certain β ∈ [-1, 0].

${\bar{R}}_{β}^{-} (X)$ represents a crisp set that contains elements $x \in U$ equivalent to all elements y ∈ X^c with a negative membership degree greater than to a certain β ∈ [-1, 0].

Some properties satisfied by the fuzzified lower α-positive, upper α- positive, lower β-negative, and upper β- negative bipolar rough approximations of $X \subseteq U$ have listed in the following theorem the proofs of which can be seen in [13].

Theorem 2.15. Let $ℙ = (U, R)$ be a bipolar fuzzy approximation space, α ∈ (0, 1] and β ∈ [-1, 0). Then for $X, Y \subseteq U$ we have

${\underline{R}}_{α}^{+} (X) \subseteq X \subseteq {\bar{R}}_{α}^{+} (X)$ ;(1b). ${\bar{R}}_{β}^{-} (X) \subseteq X \subseteq {\underline{R}}_{β}^{-} (X)$ ;

${\underline{R}}_{α}^{+} (\emptyset) = \emptyset = {\bar{R}}_{α}^{+} (\emptyset)$ ;(2b). ${\bar{R}}_{β}^{-} (\emptyset) = \emptyset = {\underline{R}}_{β}^{-} (\emptyset)$ ;

${\underline{R}}_{α}^{+} (U) = U = {\bar{R}}_{α}^{+} (U)$ ;(3b). ${\bar{R}}_{β}^{-} (U) = U = {\underline{R}}_{β}^{-} (U)$ ;

${\underline{R}}_{α}^{+} (X^{c}) = ({\bar{R}}_{α}^{+} (X))^{c}$ ; (4b). ${\underline{R}}_{β}^{-} (X^{c}) = ({\bar{R}}_{β}^{-} (X))^{c}$ ;

${\bar{R}}_{α}^{+} (X^{c}) = ({\underline{R}}_{α}^{+} (X))^{c}$ ; (5b). ${\bar{R}}_{β}^{-} (X^{c}) = ({\underline{R}}_{β}^{-} (X))^{c}$ ;

${\underline{R}}_{α}^{+} (X \cap Y) = {\underline{R}}_{α}^{+} (X) \cap {\underline{R}}_{α}^{+} (Y)$ ; (6b). ${\underline{R}}_{β}^{-} (X \cap Y) \subseteq {\underline{R}}_{β}^{-} (X) \cap {\underline{R}}_{β}^{-} (Y)$ ;

${\underline{R}}_{α}^{+} (X \cup Y) \supseteq {\underline{R}}_{α}^{+} (X) \cup {\underline{R}}_{α}^{+} (Y)$ ; (7b). ${\underline{R}}_{β}^{-} (X \cup Y) = {\underline{R}}_{β}^{-} (X) \cup {\underline{R}}_{β}^{-} (Y)$ ;

${\bar{R}}_{α}^{+} (X \cup Y) = {\bar{R}}_{α}^{+} (X) \cup {\bar{R}}_{α}^{+} (Y)$ ; (8b). ${\bar{R}}_{β}^{-} (X \cup Y) \supseteq {\bar{R}}_{β}^{-} (X) \cup {\bar{R}}_{β}^{-} (Y)$ ;

${\bar{R}}_{α}^{+} (X \cap Y) \subseteq {\bar{R}}_{α}^{+} (X) \cap {\bar{R}}_{α}^{+} (Y)$ (9b). ${\bar{R}}_{β}^{-} (X \cap Y) = {\bar{R}}_{β}^{-} (X) \cap {\bar{R}}_{β}^{-} (Y)$ ;

$X \subseteq Y \Rightarrow {\underline{R}}_{α}^{+} (X) \subseteq {\underline{R}}_{α}^{+} (Y)$ ; (10b). $X \subseteq Y \Rightarrow {\underline{R}}_{β}^{-} (X) \subseteq {\underline{R}}_{β}^{-} (Y)$ ;

$X \subseteq Y \Rightarrow {\bar{R}}_{α}^{+} (X) \subseteq {\bar{R}}_{α}^{+} (Y)$ ; (11b). $X \subseteq Y \Rightarrow {\bar{R}}_{β}^{-} (X) \subseteq {\bar{R}}_{β}^{-} (Y)$ .

3 (α, β)-Multi-granulation bipolar fuzzified rough sets ((α, β)-MGBFRSs)

In this section, we extend the multi-granulation rough set theory to (α, β)-multi-granulation bipolar fuzzified rough set. This extension is based on using a collection of bipolar fuzzy tolerance relations instead of a single bipolar fuzzy tolerance relation in (α, β)-bipolar fuzzified rough approximations. We introduce the notion of (α, β)-multi-granulation indiscernible objects. Also, we discuss some fundamental structural properties of (α, β)-multi-granulation bipolar fuzzified rough set in detail with some constructive examples.

Throughout this section {R_i} _i∈I will be considered as bipolar fuzzy tolerance relations until otherwise specified.

Definition 3.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ characterized by its positive and negative membership functions given as $μ_{R_{i}^{+}} : U \times U ⟶ [0, 1]$ and $μ_{R_{i}^{-}} : U \times U ⟶ [- 1, 0]$ . For any α ∈ (0, 1] and β ∈ [-1, 0), the multi - granulation fuzzified lower α - positive, upper α - positive, lower β - negative and upper β - negative bipolar rough approximations of $X \subseteq U$ are defined as: $\begin{matrix} (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) < α for all y \in X^{c}}, \\ (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) \geq α for some y \in X}, \\ (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) \leq β for some y \in X}, \\ (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) > β for all y \in X^{c}} . \end{matrix}$

The pairs $\begin{matrix} \underline{MGFB} (X) & = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)), \end{matrix}$ (10) $\begin{matrix} \bar{MGFB} (X) & = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X), (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)) \end{matrix}$ (11) are referred to as (α, β)-multi-granulation bipolar fuzzified lower approximation and (α, β)- multi-granulation bipolar fuzzified upper approximation of X, respectively.

Generally, we call $\underline{MGFB} (X)$ and $\bar{MGFB} (X)$ as (α, β)-multi-granulation bipolar fuzzified rough approximations of X. Moreover, if $\underline{MGFB} (X) \neq \bar{MGFB} (X)$ , then X is said to be (α, β)-multi-granulation bipolar fuzzified rough set with respect to {R_i} _i∈I. Otherwise, X is called (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I.

The information regarding an object x of $U$ interpreted by the above-defined operators is as follows:

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ represents a crisp set that contains objects $x \in U$ equivalent to all objects y ∈ X^c with a positive membership degree less than to a specific α ∈ [0, 1] for some i ∈ I.

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ represents a crisp set that contains objects $x \in U$ equivalent to at least one object y ∈ X with a positive membership degree greater than or equal to a specific α ∈ [0, 1] for all i ∈ I.

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ represents a crisp set that contains objects $x \in U$ equivalent to at least one object y ∈ X with a negative membership degree less than or equal to a specific β ∈ [-1, 0] for all i ∈ I.

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ represents a crisp set that contains objects $x \in U$ equivalent to all objects y ∈ X^c with a negative membership degree greater than to a specific β ∈ [-1, 0] for some i ∈ I.

Definition 3.2. Let $\underline{MGFB} (X)$ and $\bar{MGFB} (X)$ be the (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ . Then the sets,

${POS}_{\sum_{i = 1} R_{i}} (X) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X), (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)),$

${BND}_{\sum_{i = 1} R_{i}} (X) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) - (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) - (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)),$

${NEG}_{\sum_{i = 1} R_{i}} (X) = (U, U) - \bar{MGFB} (X) = (((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c}, ((\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c}),$

are known as the (α, β)-multi-granulation bipolar fuzzified positive region, (α, β)-multi-granulation bipolar fuzzified boundary region and (α, β)-multi-granulation bipolar fuzzified negative region of X, respectively.

Remark 3.3.

From Definition 3.1, we immediately have that $X \subseteq U$ is (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I if and only if BND_{∑_i=1R_i} (X) = (∅ , ∅) .

If R₁ = R₂ = ⋯ = R_i, then the four operators defined in Definition 3.1 degenerates into (α, β)-bipolar fuzzified rough approximations of a set X proposed by Gul and Shabir [13].

Here, we employ an example to understand the concepts (α, β)-multi-granulation bipolar fuzzified lower and upper approximations of $X \subseteq U$ .

Example 3.4. As an illustration, let R₁ = (λ^P (x, y) , λ^N (x, y)) be a bipolar fuzzy tolerance relation over $U$ , where $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ defined by the following pair of matrices:

$λ^{P} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & 1 & 0.9 & 0.6 & 0.2 & 0.8 \\ u_{2} & 0.9 & 1 & 0.7 & 0.3 & 0.5 \\ u_{3} & 0.6 & 0.7 & 1 & 0.4 & 0.1 \\ u_{4} & 0.2 & 0.3 & 0.4 & 1 & 0.8 \\ u_{5} & 0.8 & 0.5 & 0.1 & 0.8 & 1 \end{matrix}$ and $λ^{N} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & - 1 & - 0.7 & - 0.5 & - 0.4 & - 0.9 \\ u_{2} & - 0.7 & - 1 & - 0.3 & - 0.2 & - 0.1 \\ u_{3} & - 0.5 & - 0.3 & - 1 & - 0.4 & - 0.6 \\ u_{4} & - 0.4 & - 0.2 & - 0.4 & - 1 & - 0.9 \\ u_{5} & - 0.9 & - 0.1 & - 0.6 & - 0.9 & - 1 \end{matrix}$ .

Also, assume that R₂ = (ξ^P (x, y) , ξ^N (x, y)) ia a bipolar fuzzy tolerance relation over $U$ , defined by the following pair of matrices:

$ξ^{P} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & 1 & 0.8 & 0.2 & 0.4 & 0.7 \\ u_{2} & 0.8 & 1 & 0.3 & 0.6 & 0.2 \\ u_{3} & 0.2 & 0.3 & 1 & 0.5 & 0.9 \\ u_{4} & 0.4 & 0.9 & 0.5 & 1 & 0.4 \\ u_{5} & 0.7 & 0.2 & 0.9 & 0.4 & 1 \end{matrix}$ and $ξ^{N} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & - 1 & - 0.6 & - 0.8 & - 0.5 & - 0.3 \\ u_{2} & - 0.6 & - 1 & - 0.1 & - 0.2 & - 0.4 \\ u_{3} & - 0.8 & - 0.1 & - 1 & - 0.5 & - 0.7 \\ u_{4} & - 0.5 & - 0.2 & - 0.5 & - 1 & - 0.3 \\ u_{5} & - 0.3 & - 0.4 & - 0.7 & - 0.3 & - 1 \end{matrix}$ .

Multi-granulation fuzzified lower α-positive approximations for X = {u₁, u₄, u₅} with α = 0.8 will be $(\underline{\sum_{i \in I} R_{i}^{+}})_{0.8} (X) = {u_{4}, u_{5}} .$ Also, multi-granulation fuzzified upper α- positive approximations for X = {u₁, u₄, u₅} with α = 0.8 will be $(\bar{\sum_{i \in I} R_{i}^{+}})_{0.8} (X) = {u_{1}, u_{2}, u_{4}, u_{5}} .$ Now, multi-granulation fuzzified lower β-negative approximations for X = {u₁, u₄, u₅} with β = -0.3 will be $(\underline{\sum_{i \in I} R_{i}^{-}})_{- 0.3} (X) = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}} .$

Similarly, multi-granulation fuzzified upper β-negative approximations for X = {u₁, u₄, u₅} with β = -0.3 will be $(\bar{\sum_{i \in I} R_{i}^{-}})_{- 0.3} (X) = {} .$

Thus (α, β)-multi-granulation bipolar fuzzified rough approximations of X with α = 0.8 and β = -0.3 are: $\begin{matrix} \underline{MGFB} (X) & = ({u_{4}, u_{5}}, {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}), \\ \bar{MGFB} (X) & = ({u_{1}, u_{2}, u_{4}, u_{5}}, {}) . \end{matrix}$ Since, $\underline{MGFB} (X) \neq \bar{MGFB} (X)$ , so X is a (α, β)-multi-granulation bipolar fuzzified rough set with respect to R₁ and R₂.

Moreover, (α, β)-multi-granulation bipolar fuzzified positive region, boundary region and negative region of X are respectively given as follows:

POS_{∑_i=1R_i} (X) = ({u₄, u₅} , {} ) ,

BND_{∑_i=1R_i} (X) = ({u₁, u₂} , {u₁, u₂, u₃, u₄, u₅} ) ,

NEG_{∑_i=1R_i} (X) = ({u₃} , ∅ ) .

Definition 3.5. Let α ∈ (0, 1] and β ∈ [-1, 0) and {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ characterized by its positive and negative membership functions given as $μ_{R_{i}^{+}} : U \times U ⟶ [0, 1]$ and $μ_{R_{i}^{-}} : U \times U ⟶ [- 1, 0]$ . Then the objects u and v in $U$ will be called (α, β)- multi-granulation indiscernible if $μ_{R_{i}^{+}} (u, v) \geq α and μ_{R_{i}^{-}} (u, v) \leq β, i \in I .$ □

In order to discover the relationship of containment between the multi-granulation fuzzified lower and upper α-positive bipolar rough approximations of $X \subseteq U$ , whenever α₁, α₂ ∈ (0, 1] is such that α₁ ≤ α₂, the following properties are given.

Proposition 3.6. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and α₁, α₂ ∈ (0, 1] be such that α₁ ≤ α₂. Then,

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X)$ .

Proof.

Let $x \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X)$ . Then by Definition 3.1, we have

$μ_{R_{i}^{+}} (x, y) < α_{1} for all y \in X^{c} and for some i \in I .$

Since α₁ ≤ α₂, we have

$μ_{R_{i}^{+}} (x, y) < α_{1} \leq α_{2} for all y \in X^{c} and for some i \in I .$

This implies that $x \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X)$ . Hence, $(\underline{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X)$ .

Let $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X)$ . Then by Definition 3.1, we have

$μ_{R_{i}^{+}} (x, y) \geq α_{2} for some y \in X and for all i \in I .$

But since α₁ ≤ α₂, we have

$μ_{R_{i}^{+}} (x, y) \geq α_{2} \geq α_{1} for some y \in X and for all i \in I .$

Thus, $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X)$ so that $(\bar{\sum_{i \in I} R_{i}^{+}})_{α_{2}} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{1}} (X)$ .

In order to discover the relationship of containment between the multi-granulation fuzzified lower and upper β-negative bipolar rough approximations of $X \subseteq U$ , whenever β₁, β₂ ∈ [-1, 0) is such that β₁ ≤ β₂, the following properties are given.

Proposition 3.7. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and β₁, β₂ ∈ [-1, 0) be such that β₁ ≤ β₂. Then,

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X)$ .

Proof.

Let $x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X)$ . Then by Definition 3.1, we have

$μ_{R_{i}^{-}} (x, y) \leq β_{1} for some y \in X and for all i \in I .$

But since β₁ ≤ β₂, we have

$μ_{R_{i}^{-}} (x, y) \leq β_{1} \leq β_{2} for some y \in X and for all i \in I .$

This implies that $x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X)$ . Hence, $(\underline{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X)$ .

Let $x \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X)$ . Then by Definition 3.1, we have

$μ_{R_{i}^{-}} (x, y) > β_{2} for all y \in X^{c} and for some i \in I .$

Since β₁ ≤ β₂, we have

$\begin{matrix} μ_{R_{i}^{-}} (x, y) > β_{2} \geq β_{1} for some y \in X^{c} and for some \\ i \in I . \end{matrix}$

Thus, $x \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X)$ so that $(\bar{\sum_{i \in I} R_{i}^{-}})_{β_{2}} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{1}} (X)$ .

□

Example 3.8. To illustrate Proposition 3.6 and Proposition 3.7, let us consider bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) as given in Example 3.4. Let $X = {u_{1}, u_{4}, u_{5}} \subseteq U$ . Then X^c = {u₂, u₃}. For α₁ = 0.8 and α₂ = 0.9, we have $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α_{1}} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α_{1}} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{+} + R_{2}^{+}})_{α_{2}} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α_{2}} (X) & = {u_{1}, u_{4}, u_{5}} . \end{matrix}$ Clearly we can see that $(\underline{R_{1}^{+} + R_{2}^{+}})_{α_{1}} (X) \subseteq (\underline{R_{1}^{+} + R_{2}^{+}})_{α_{2}} (X)$ and $(\bar{R_{1}^{+} + R_{2}^{+}})_{α_{2}} (X) \subseteq (\bar{R_{1}^{+} + R_{2}^{+}})_{α_{1}} (X)$ .

Similarly for β₁ = -0.7 and β₂ = -0.3, we have $\begin{matrix} (\underline{R_{1}^{-} + R_{2}^{-}})_{β_{1}} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β_{1}} (X) & = {u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β_{2}} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β_{2}} (X) & = {} . \end{matrix}$ Clearly we can see that $(\underline{R_{1}^{-} + R_{2}^{-}})_{β_{1}} (X) \subseteq (\underline{R_{1}^{-} + R_{2}^{-}})_{β_{2}} (X)$ and $(\bar{R_{1}^{-} + R_{2}^{-}})_{β_{2}} (X) \subseteq (\bar{R_{1}^{-} + R_{2}^{-}})_{β_{1}} (X)$ .□

In order to discover the relationship between the multi-granulation fuzzified lower and upper α-positive bipolar rough approximations of $X, Y \subseteq U$ , such that X ⊆ Y, the following properties are given.

Proposition 3.9. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and α ∈ (0, 1]. Then for any $X, Y \subseteq U$ , we have

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

Proof.

For any $x \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ , by Definition 3.1, it follows that

$μ_{R_{i}^{+}} (x, y) < α for all y \in X^{c} and for some i \in I .$

Since X ⊆ Y implies Y^c ⊆ X^c. Thus in particular, we have

$μ_{R_{i}^{+}} (x, y) < α for all y \in Y^{c} and for some i \in I .$

This implies that $x \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ . Hence, $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

Let $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ . Then by Definition 3.1, we have

$μ_{R_{i}^{+}} (x, y) \geq α for some y \in X and for all i \in I .$

Since X ⊆ Y and y ∈ X implies y ∈ Y. Thus it follows that,

$μ_{R_{i}^{+}} (x, y) \geq α for some y \in X \subseteq Y and for all i \in I,$

which shows that, $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ . Hence, $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

In order to discover the relationship between the multi-granulation fuzzified lower and upper β-negative bipolar rough approximations of $X, Y \subseteq U$ , such that X ⊆ Y, the following properties are given.

Proposition 3.10. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and β ∈ [-1, 0). Then for any $X, Y \subseteq U$ , we have

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

Proof.

For any $x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ , by Definition 3.1 we have

$μ_{R_{i}^{-}} (x, y) \leq β for some y \in X and for all i \in I .$

As X ⊆ Y, so

$μ_{R_{i}^{-}} (x, y) \leq β for some y \in X \subseteq Y and for all i \in I .$

This implies that $x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ . Hence, $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

Let $x \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ . Then by Definition 3.1,

$μ_{R_{i}^{-}} (x, y) > β for all y \in X^{c} and for some i \in I .$

Since X ⊆ Y, so Y^c ⊆ X^c. Thus in particular,

$μ_{R_{i}^{-}} (x, y) > β for some y \in Y^{c} and for some i \in I,$

which shows that, $x \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ . Hence, $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

□

Example 3.11. To illustrate Proposition 3.9 and Proposition 3.10, let us consider bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) as given in Example 3.4. Let $X, Y \subseteq U$ be such that X = {u₁, u₄, u₅} and Y = {u₁, u₂, u₄, u₅}. Then for α = 0.8, we have $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{+} + R_{2}^{+}})_{α} (Y) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α} (Y) & = {u_{1}, u_{2}, u_{4}, u_{5}} . \end{matrix}$ Clearly we can see that $(\underline{R_{1}^{+} + R_{2}^{+}})_{α} (X) \subseteq (\underline{R_{1}^{+} + R_{2}^{+}})_{α} (Y)$ and $(\bar{R_{1}^{+} + R_{2}^{+}})_{α} (X) \subseteq (\bar{R_{1}^{+} + R_{2}^{+}})_{α} (Y)$ .

Similarly by taking β = -0.7, we have $\begin{matrix} (\underline{R_{1}^{-} + R_{2}^{-}})_{β} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β} (X) & = {u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β} (Y) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β} (Y) & = {u_{1}, u_{2}, u_{4}, u_{5}} . \end{matrix}$ Clearly we can see that $(\underline{R_{1}^{-} + R_{2}^{-}})_{β} (X) \subseteq (\underline{R_{1}^{-} + R_{2}^{-}})_{β} (Y)$ and $(\bar{R_{1}^{-} + R_{2}^{-}})_{β} (X) \subseteq (\bar{R_{1}^{-} + R_{2}^{-}})_{β} (Y)$ .

Remark 3.12. The purpose of introducing degree of relationship α and β is to give the margin to the data sets in which no two objects has exactly the same attribute values, instead of taking them to be exactly similar.

One can obtain the following properties of the multi-granulation fuzzified lower and upper α-positive bipolar rough approximations of $X \subseteq U$ .

Theorem 3.13. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and α ∈ (0, 1]. Then for any $X, Y \subseteq U$ , we have

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq X \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (U) = U = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (U)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cap (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cap (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cup (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cup (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

Proof.

By definition, $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq X$ is trivial. For the other inclusion, suppose $x \in X \subseteq U$ . Then

$μ_{R_{i}^{+}} (x, x) = 1 \geq α for some x \in X and for all i \in I .$

This shows that $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ . Thus, $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq X \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ .

Direct consequence of Definition 3.1.

The proof is analogous to the proof of part (1).

For any $x \in U$ , $\begin{matrix} x \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) & \Leftrightarrow μ_{R_{i}^{+}} (x, y) < α for all y \in (X^{c})^{c} = X and for some i \in I \\ \Leftrightarrow μ_{R_{i}^{+}} (x, y) ≱ α for any y \in X and for some i \in I \\ \Leftrightarrow x \notin (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \\ \Leftrightarrow x \in ((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c} . \end{matrix}$ Hence, $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c}$ .

For any $x \in U$ , $\begin{matrix} x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) & \Leftrightarrow μ_{R_{i}^{+}} (x, y) \geq α for some y \in X^{c} and for all i \in I \\ \Leftrightarrow μ_{R_{i}^{+}} (x, y) / < α for any y \in X^{c} and for all i \in I \\ \Leftrightarrow x \notin (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \\ \Leftrightarrow x \in ((\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c} . \end{matrix}$

Hence, $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X))^{c}$ .

Since we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (i) of Proposition 3.9 that $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ and $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ . Thus, it follows that $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cap (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

Since we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (ii) of Proposition 3.9 that $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ and $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ . Hence, we have

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cap (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y)$ .

As we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so it follows from part (i) of Proposition 3.9 that $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ and $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ . Thus, we have $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cup (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ .

Since we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so it follows from part (ii) of Proposition 3.9 that $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ and $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ . Thus, we have $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \cup (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y)$ .

□

One can obtain the following properties of the multi-granulation fuzzified lower and upper β-negative bipolar rough approximations of $X \subseteq U$ .

Theorem 3.14. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and β ∈ [-1, 0). Then for any $X, Y \subseteq U$ , we have

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq X \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (U) = U = (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (U)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cap (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cap (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cup (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cup (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

Proof.

By definition, $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq X$ is obvious. For the other inclusion, suppose $x \in X \subseteq U$ . Then

$μ_{R_{i}^{-}} (x, x) = - 1 \leq β for some x \in X and for all i \in I .$

This shows that $x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ . Hence, $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq X \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ .

Direct consequence of Definition 3.1.

The proof is analogous to the proof of part (1).

For any $x \in U$ ,

$\begin{matrix} x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) & \Leftrightarrow μ_{R_{i}^{-}} (x, y) \leq β for some y \in X^{c} and for all i \in I \\ \Leftrightarrow μ_{R_{i}^{-}} (x, y) / > β for any y \in X^{c} and for all i \in I \\ \Leftrightarrow x \notin (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \\ \Leftrightarrow x \in ((\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c} . \end{matrix}$ Hence, $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c}$ .

For any $x \in U$ , $\begin{matrix} x \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) & \Leftrightarrow μ_{R_{i}^{-}} (x, y) > β for all y \in (X^{c})^{c} = X and for some i \in I \\ \Leftrightarrow μ_{R_{i}^{-}} (x, y) ≰ β for any y \in X and for some i \in I \\ \Leftrightarrow x \notin (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \\ \Leftrightarrow x \in ((\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X))^{c} . \end{matrix}$ Hence, $(\bar{\sum_{i \in I} R_{i}^{-}})_{α} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{-}})_{α} (X))^{c}$ .

As we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (i) of Proposition 3.10 that $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ and $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ . Hence, we have $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cap (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

As we know that X ∩ Y ⊆ X and X ∩ Y ⊆ Y, so it follows from part (ii) of Proposition 3.10 that $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ and $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq ({\bar{\sum_{i \in I} R_{i}^{-}}}^{-})_{β} (Y)$ . Thus, we have $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cap (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y)$ .

Since we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so from part (i) of Proposition 3.10 it implies that $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ and $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ . Hence we get $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cup (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ .

As we know that X ⊆ X ∪ Y and Y ⊆ X ∪ Y, so from part (ii) of Proposition 3.10 it implies that

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ and $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ . Hence we get $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \cup (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y)$ .

□

Corollary 3.1. If α = 1 and β = -1, then

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) = X = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) = X = (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ .

Proof.

From part (1) of Theorem 3.13 for any $X \subseteq U$ , we have $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq X \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ .

Also for any x ∈ X, we know that $μ_{R_{i}^{+}} (x, z) < α = 1 for all z \in X^{c} for some i \in I$ ,

which implies that $X \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ and thus it follows that

$X = (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) .$ (12) Similarly, for any $x \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ , we have $μ_{R_{i}^{+}} (x, y) \geq α = 1 for some y \in X for all i \in I$ .

But since $μ_{R_{i}^{+}} (x, y) \in [0, 1]$ , so we obtain $μ_{R_{i}^{+}} (x, y) = 1$ . This implies x = y. Thus x ∈ X as x ∈ Y, which further implies that $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq X$ . Therefore we have $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) = X .$ (13) Hence from Equations (12) and (13), it follows that $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) = X = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ , as required.

From part (1) of Theorem 3.14 for any $X \subseteq U$ , we have $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq X \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ .

Also we know that for any x ∈ X, $μ_{R_{i}^{-}} (x, z) > β = - 1 for all z \in X^{c} for some i \in I$ ,

which implies that $X \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ and therefore it follows that

X = (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) .

(14) Similarly, for any

x \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)

, we have

μ_{R_{i}^{-}} (x, z) \leq β = - 1 for some y \in X for all i \in I

But since $μ_{R_{i}^{-}} (x, y) \in [- 1, 0]$ , so we get $μ_{R_{i}^{-}} (x, y) = - 1$ . This implies x = y. Therefore x ∈ X as x ∈ Y, which further implies that $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) \subseteq X$ . Therefore we have $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) = X .$ (15) Thus from Equation (14) and (15), it follows that $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) = X = (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ , as required.□

The following example shows that the inclusion in parts (6), (7), (8) and (9) of Theorem 3.13 and Theorem 3.14 might be strict.

Example 3.15. Consider the bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ as given in Example 3.4. Let $X, Y \subseteq U$ be such that X = {u₁, u₄, u₅} and Y = {u₂, u₄}. Then X ∩ Y = {u₄} and X ∪ Y = {u₁, u₂, u₄, u₅}. For α = 0.7, we have $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) & = {u_{4}, u_{5}}, \\ (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) & = {u_{4}}, \\ (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) & = {}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) & = {u_{1}, u_{2}, u_{4}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) & = {u_{4}}, \\ (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}} . \end{matrix}$ Clearly we can observe that $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) = {}$ and $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cap (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) = {u_{4}}$ . So $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) \subset (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cap (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y)$ , which shows that the inclusion in part (6) of Theorem 3.13 may hold strictly.

Similarly, we have $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) = {u_{4}}$ and $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cap (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) = {u_{1}, u_{2}, u_{4}}$ . So we have $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cap Y) \subset (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cap (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y)$ , which shows that the inclusion in part (7) of Theorem 3.13 might be strict.

Also, $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cup (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) = {u_{4}, u_{5}}$ and $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) = {u_{1}, u_{2}, u_{4}, u_{5}}$ . So $(\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) \supset (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cup (\underline{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y)$ , which shows that the inclusion in part (8) of Theorem 3.13 may hold strictly.

Moreover, $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cup (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y) = {u_{1}, u_{2}, u_{4}, u_{5}}$ and $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ . So $(\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X \cup Y) \supset (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (X) \cup (\bar{R_{1}^{+} + R_{2}^{+}})_{0.7} (Y)$ , which shows that the inclusion in part (9) of Theorem 3.13 may hold strictly.

Again consider $X, Y \subseteq U$ be such that X = {u₁, u₄, u₅} and Y = {u₂, u₄}. Then X ∩ Y = {u₄}.

Taking β = -0.3, we have $\begin{matrix} (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (Y) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X \cap Y) & = {u_{1}, u_{3}, u_{4}, u_{5}} . \end{matrix}$ Clearly we can observe that $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X \cap Y) = {u_{1}, u_{3}, u_{4}, u_{5}}$ and $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X) \cap (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (Y) = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ . So $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X \cap Y) \subset (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (X) \cap (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.3} (Y)$ , which shows that the inclusion in part (6) of Theorem 3.14 might be strict.

Similarly, for the same X and Y, taking β = -0.2, we have $\begin{matrix} (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X) & = {}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (Y) & = {}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X \cup Y) & = {u_{2}} . \end{matrix}$

Clearly we can see that $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X \cup Y) = {u_{2}}$ and $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X) \cup (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (Y) = {}$ . So $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X \cup Y) \supset (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X) \cup (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (Y)$ , which shows that the inclusion in part (9) of Theorem 3.14 might be strict.

Again consider $X, Y \subseteq U$ be such that X = {u₁, u₂, u₅} and Y = {u₁, u₃, u₄}. Then X ∩ Y = {u₁}.

Taking β = -0.7, we have $\begin{matrix} (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (X) & = {u_{1}, u_{2}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (Y) & = {u_{1}, u_{3}, u_{4}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (X \cap Y) & = {} . \end{matrix}$ Clearly we can see that $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (X \cap Y) = {}$ and $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X) \cap (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (Y) = {u_{1}}$ . So $(\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (X \cap Y) \subset (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.2} (X) \cap (\bar{R_{1}^{-} + R_{2}^{-}})_{- 0.7} (Y)$ , which shows that the inclusion in part (7) of Theorem 3.14 may hold strictly.

Again consider $X, Y \subseteq U$ be such that X = {u₁} and Y = {u₂}. Then X ∪ Y = {u₁, u₂}. By taking β = -0.5, we have $\begin{matrix} (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X) & = {u_{1}, u_{2}, u_{3}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (Y) & = {u_{1}, u_{2}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X \cup Y) & = {u_{1}, u_{2}, u_{3}, u_{4}} . \end{matrix}$

Clearly we can see that $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X \cup Y) = {u_{1}, u_{2}, u_{3}, u_{4}}$ and $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X) \cup (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (Y) = {u_{1}, u_{2}, u_{3}}$ . So $(\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X \cup Y) \supset (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (X) \cup (\underline{R_{1}^{-} + R_{2}^{-}})_{- 0.5} (Y)$ , which shows that the inclusion in part (8) of Theorem 3.14 might be strict.

Proposition 3.16. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then for any $X \subseteq U$ , we have

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) = ⋃_{i \in I} {\underline{(R_{i}^{+})}}_{α} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) = ⋂_{i \in I} {\bar{(R_{i}^{+})}}_{α} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) = ⋂_{i \in I} {\underline{(R_{i}^{-})}}_{β} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) = ⋃_{i \in I} {\bar{(R_{i}^{-})}}_{β} (X)$ .

Proof. Straightforward.□

Proposition 3.17. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then for any $X, Y \subseteq U$ , we have

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) = ⋃_{i \in I} [{\underline{(R_{i}^{+})}}_{α} (X) \cap {\underline{(R_{i}^{+})}}_{α} (Y)]$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y) = ⋂_{i \in I} [{\bar{(R_{i}^{+})}}_{α} (X) \cup {\bar{(R_{i}^{+})}}_{α} (Y)]$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y) = ⋂_{i \in I} [{\underline{(R_{i}^{-})}}_{β} (X) \cup {\underline{(R_{i}^{-})}}_{β} (Y)]$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) = ⋃_{i \in I} [{\bar{(R_{i}^{-})}}_{β} (X) \cap {\bar{(R_{i}^{-})}}_{β} (Y)]$ .

Proof.

From part (1) of Proposition 3.16, we have $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X \cap Y) = ⋃_{i \in I} {\underline{(R_{i}^{+})}}_{α} (X \cap Y) = ⋃_{i \in I} [{\underline{(R_{i}^{+})}}_{α} (X) \cap {\underline{(R_{i}^{+})}}_{α} (Y)]$ .

From part (2) Proposition 3.16, we get $(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X \cup Y) = ⋂_{i \in I} {\bar{(R_{i}^{+})}}_{α} (X \cup Y) = ⋂_{i \in I} [{\bar{(R_{i}^{+})}}_{α} (X) \cup {\bar{(R_{i}^{+})}}_{α} (Y)]$ .

From part (3) of Proposition 3.16, we have $(\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X \cup Y) = ⋂_{i \in I} {\underline{(R_{i}^{-})}}_{β} (X \cup Y) = ⋂_{i \in I} [{\underline{(R_{i}^{-})}}_{β} (X) \cup {\underline{(R_{i}^{-})}}_{β} (Y)]$ .

From part (4) of Proposition 3.16, it implies that $(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X \cap Y) = ⋃_{i \in I} {\bar{(R_{i}^{-})}}_{β} (X \cap Y) = ⋃_{i \in I} [{\bar{(R_{i}^{-})}}_{β} (X) \cap {\bar{(R_{i}^{-})}}_{β} (Y)]$ .

□

4 Measures associated with (α, β)-multi-granulation bipolar fuzzified rough sets

Generally speaking, the uncertainty of a set is due to the existence of the boundary region. The wider the boundary region of a set is, the lower the accuracy of the set is. To express the idea precisely, in this section, we present some important measures associated with (α, β)-multi-granulation bipolar fuzzified rough sets, like the measure of accuracy, the measure of precision, and the accuracy of an approximation. The main concern is to measure completeness of knowledge provided by a given bipolar fuzzy tolerance relation.

Pawlak [32] presented the accuracy and roughness measures associated with rough set approximations. The accuracy measure is the ratio of the cardinality of lower approximation to the cardinality of upper approximation while the roughness measure is the complement of the accuracy measure. The purpose of introducing this measure is to capture the degree of completeness of knowledge about the set X or to express the quality of an approximation. The roughness measure which is a complement of accuracy measure is interpreted as the degree of incompleteness of knowledge about the set X. As a generalization of these measures, we introduce accuracy measure and roughness measure using (α, β)-Multi-granulation bipolar fuzzified rough sets as follows:

Definition 4.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then measure of accuracy for (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined by an ordered pair: $A_{α, β} (X, R_{i}) = (X^{α^{+}}, X^{β^{-}}),$ (16) where $X^{α^{+}} = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) |}{| (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) |}$ and $X^{β^{-}} = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) |}{| (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) |} .$ Here | • | denote the cardinality of the set.

Similarly, measure of roughness for (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined as: $\begin{matrix} R_{α, β} (X, R_{i}) & = (1, 1) - A_{α, β} (X, R_{i}) \\ = (1, 1) - (X^{α^{+}}, X^{β^{-}}) \\ = (1 - X^{α^{+}}, 1 - X^{β^{-}}) \\ = (1 - \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) |}{| (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) |}, \\ 1 - \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) |}{| (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) |}) . \end{matrix}$ (17)

From the above definition, we conclude that;

0 ≤ X^α
⁺ ≤ 1 and 0 ≤ X^{β
^-} ≤ 1 for any non-empty subset X of $U$ .

Also it can be seen that $A_{α, β} (X, R_{i}) = (1, 1) if and only if X = U$ .

$A_{α, β} (X, R_{i}) = (0, 0)$ if and only if $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}$ .

When α = 1 and β = -1 then for any $\emptyset \neq X \subseteq U$ , $A_{α, β} (X, R_{i}) = (1, 1)$ and $R_{α, β} (X, R_{i}) = (0, 0)$ .

If $X, Y \subseteq U$ such that X ⊆ Y, then $A_{α, β} (X, R_{i}) \leq A_{α, β} (Y, R_{i})$ .

If α₁ ≤ α₂ and β₁ ≤ β₂ then $X^{α_{1}^{+}} \leq X^{α_{2}^{+}}$ and $X^{β_{2}^{-}} \leq X^{β_{1}^{-}}$ .

Remark 4.2. $X \subseteq U$ is (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I if and only if $R_{α, β} (X, R_{i}) = (0, 0) .$

The following example shows that measure of accuracy of a set defined by using (α, β)-multi-granulation bipolar fuzzified rough set is always much better than that defined by using a single granulation.

Example 4.3. Consider the bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ as given in Example 3.4. Let $X = {u_{1}, u_{2}, u_{5}} \subseteq U$ . Then for α = 0.7 and β = -0.8, measure of accuracy for (α, β)-bipolar fuzzified rough set with respect to R₁, R₂ and R₁ + R₂ are respectively given as: $\begin{matrix} A_{α, β} (X, R_{1}) & = (0.2, 0.5), \\ A_{α, β} (X, R_{2}) & = (0.4, 0.5), \\ A_{α, β} (X, R_{1} + R_{2}) & = (0.4, 1) . \end{matrix}$ Clearly, it follows from the above calculations that $A_{α, β} (X, R_{1} + R_{2}) \geq A_{α, β} (X, R_{1}) and A_{α, β} (X, R_{1} + R_{2}) \geq A_{α, β} (X, R_{2}) .$

Proposition 4.4. Let $X, Y \subseteq U$ . Then

$A_{α, β} (X, R_{i}) \leq A_{α, β} (X \cup Y, R_{i})$ and $A_{α, β} (Y, R_{i}) \leq A_{α, β} (X \cup Y, R_{i})$ .

$A_{α, β} (X, R_{i}) \geq A_{α, β} (X \cap Y, R_{i})$ and $A_{α, β} (Y, R_{i}) \geq A_{α, β} (X \cap Y, R_{i})$ .

Proof. Straightforward.□

Gediga and Düntsch [12] defined the notion of measure of precision of approximation of $\emptyset \neq X \subseteq U$ , which is the ratio of the cardinality of lower approximation of X to the cardinality of X. In the sense of (α, β)-multi-granulation bipolar fuzzified rough approximations it can be extended as follows:

Definition 4.5. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then measure of precision for (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined by an order pair:

$Π_{α, β} (X, R_{i}) = (X_{π}^{α^{+}}, X_{π}^{β^{-}}),$ (18) where $X_{π}^{α^{+}} = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) |}{| X |}$ and $X_{π}^{β^{-}} = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) |}{| X |} .$

From the above definition, we conclude that;

$0 \leq X_{π}^{α^{+}} \leq 1$ and $0 \leq X_{π}^{β^{-}} \leq 1$ for any non-empty subset X of $U$ .

$Π_{α, β} (X, R_{i}) \geq A_{α, β} (X, R_{i})$ , that is, $X_{π}^{α^{+}} \geq X^{α^{+}}$ and $X_{π}^{β^{-}} \geq X^{β^{-}}$ .

Also it can be seen that $Π_{α, β} (X, R_{i}) = (1, 1) if and only if X = U$ .

Π_α,β (X, R_i) = (0, 0) if and only if $(\underline{\sum_{i \in I} R_{i}^{+}})_{α} = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}$ .

When α = 1 and β = -1 then for any $\emptyset \neq X \subseteq U$ , Π_α,β (X, R_i) = (1, 1).

If α₁ ≤ α₂ and β₁ ≤ β₂ then $X_{π}^{α_{1}^{+}} \leq X_{π}^{α_{2}^{+}}$ and $X_{π}^{β_{2}^{-}} \leq X_{π}^{β_{1}^{-}}$ .

If $X, Y \subseteq U$ such that X ⊆ Y, then Π_α,β (X, R_i) ≤ Π_α,β (Y, R_i).

It may be noted that measure of precision requires complete information about the set X, whereas measure of accuracy does not.

Proposition 4.6. Let $X, Y \subseteq U$ . Then

Π_α,β (X, R_i) ≤ Π_α,β (X ∪ Y, R_i) and Π_α,β (Y, R_i) ≤ Π_α,β (X ∪ Y, R_i).

Π_α,β (X, R_i) ≥ Π_α,β (X ∩ Y, R_i) and Π_α,β (Y, R_i) ≥ Π_α,β (X ∩ Y, R_i).

Proof. Straightforward.□

Yao [56] pointed out that the term “accuracy” should be defined and interpreted accurately as it might be misleading in some cases. He revised some of the properties of the accuracy measure given by Pawlak [32, 33] and proposed another measure is known as the completeness of knowledge, or the accuracy of approximations, which is given as follows: $α_{R} (X) = \frac{| R_{*} (X) | + | R_{*} (X^{c}) |}{| U |} .$ (19) In (α, β)-multi-granulation bipolar fuzzified rough approximations environment it can be extended as follows:

Definition 4.7. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then accuracy of approximation for (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined by an order pair: $ϒ_{α, β} (X, R_{i}) = (X_{γ}^{α^{+}}, X_{γ}^{β^{-}}),$ (20) where $X_{γ}^{α^{+}} = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) | + | (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X^{c}) |}{| U |}$ and $X_{γ}^{β^{-}} = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) | + | (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X^{c}) |}{| U |} .$ The corresponding measure of roughness is defined as:

$Γ_{α, β} (X, R_{i}) = (1, 1) - ϒ_{α, β} (X, R_{i}) = (1 - X_{γ}^{α^{+}}, 1 - X_{γ}^{β^{-}}) .$ (21)

From the above definition, we conclude that;

$0 < X_{γ}^{α^{+}} \leq 1$ and $0 < X_{γ}^{β^{-}} \leq 1$ for any non-empty subset X of $U$ .

When α = 1 and β = -1 then for any $\emptyset \neq X \subseteq U$ , ϒ_α,β (X, R_i) = (1, 1) and Γ_α,β (X, R_i) = (0, 0).

ϒ_α,β (X, R_i) cannot be zero for any $X \subseteq U$ .

If α₁ ≤ α₂ and β₁ ≤ β₂ then $X_{γ}^{α_{1}^{+}} \leq X_{γ}^{α_{2}^{+}}$ and $X_{γ}^{β_{2}^{-}} \leq X_{γ}^{β_{1}^{-}}$ .

If $X, Y \subseteq U$ such that X ⊆ Y, then ϒ_α,β (X, R_i) ≤ ϒ_α,β (Y, R_i).

Proposition 4.8. ϒ_α,β (X, R_i) = (1, 1) if and only if X =∅ or $X = U$ .

Proof. We prove the required result in two cases.□

When X =∅, then $\begin{matrix} X_{γ}^{α^{+}} & = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset) | + | (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset^{c}) |}{| U |} \\ = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset) | + | (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (U) |}{| U |} \\ = \frac{| \emptyset | + | U |}{| U |} \\ = \frac{0 + | U |}{| U |} \\ = 1 \end{matrix}$ Similarly, we have $\begin{matrix} X_{γ}^{β^{-}} & = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset) | + | (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset^{c}) |}{| U |} \\ = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset) | + | (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (U) |}{| U |} \\ = \frac{| \emptyset | + | U |}{| U |} \\ = \frac{0 + | U |}{| U |} \\ = 1 \end{matrix}$ Thus, it follows that ϒ_α,β (X, R_i) = (1, 1).

When $X = U$ , then $\begin{matrix} X_{γ}^{α^{+}} & = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (U) | + | (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (U^{c}) |}{| U |} \\ = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (U) | + | (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (\emptyset) |}{| U |} \\ = \frac{| U | + | \emptyset |}{| U |} \\ = \frac{| U | + 0}{| U |} \\ = 1 \end{matrix}$ Similarly, $\begin{matrix} X_{γ}^{β^{-}} & = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (U) | + | (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (U^{c}) |}{| U |} \\ = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (U) | + | (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (\emptyset) |}{| U |} \\ = \frac{| U | + | \emptyset |}{| U |} \\ = \frac{| U | + 0}{| U |} \\ = 1 \end{matrix}$ This implies that ϒ_α,β (X, R_i) = (1, 1).

Hence, in both cases we get ϒ_α,β (X, R_i) = (1, 1).

□

Proposition 4.9. Let $X, Y \subseteq U$ . Then

ϒ_α,β (X, R_i) ≤ ϒ_α,β (X ∪ Y, R_i) and ϒ_α,β (Y, R_i) ≤ ϒ_α,β (X ∪ Y, R_i).

ϒ_α,β (X, R_i) ≥ ϒ_α,β (X ∩ Y, R_i) and ϒ_α,β (Y, R_i) ≥ ϒ_α,β (X ∩ Y, R_i).

Proof. Straightforward.□

Here, we employ an example to illustrate the concepts of the measure of accuracy, the measure of precision and the accuracy of approximation of (α, β)-multi-granulation bipolar fuzzified rough set.

Example 4.10. Let us consider R₁ = (λ^P (x, y) , λ^N (x, y)), R₂ = (ξ^P (x, y) , ξ^N (x, y)) and R₃ = (η^P (x, y) , η^N (x, y)) be three bipolar fuzzy tolerance relations over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ defined by the following pair of matrices, respectively:

$η^{P} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & 1 & 0.7 & 0.4 & 0.1 & 0.9 \\ u_{2} & 0.7 & 1 & 0.5 & 0.2 & 0.3 \\ u_{3} & 0.4 & 0.5 & 1 & 0.6 & 0.8 \\ u_{4} & 0.1 & 0.2 & 0.6 & 1 & 0.9 \\ u_{5} & 0.9 & 0.3 & 0.8 & 0.9 & 1 \end{matrix}$ and $η^{N} = \begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ u_{1} & - 1 & - 0.8 & - 0.4 & - 0.5 & - 0.2 \\ u_{2} & - 0.8 & - 1 & - 0.1 & - 0.2 & - 0.3 \\ u_{3} & - 0.4 & - 0.1 & - 1 & - 0.5 & - 0.7 \\ u_{4} & - 0.5 & - 0.2 & - 0.5 & - 1 & - 0.2 \\ u_{5} & - 0.2 & - 0.3 & - 0.7 & - 0.2 & - 1 \end{matrix}$ .

Take $X = {u_{1}, u_{4}, u_{5}} \subseteq U$ , then for α = 0.8 and β = -0.7, we have $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+} + R_{3}^{+}})_{α} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+} + R_{3}^{+}})_{α} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{+} + R_{2}^{+} + R_{3}^{+}})_{α} (X^{c}) & = {u_{2}, u_{3}}, \\ (\underline{R_{1}^{-} + R_{2}^{-} + R_{3}^{-}})_{β} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-} + R_{3}^{-}})_{β} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-} + R_{3}^{-}})_{β} (X^{c}) & = {u_{2}, u_{3}} . \end{matrix}$ So the values of the measure of accuracy, the measure of precision, and accuracy of approximation for (α, β)-multi-granulation bipolar fuzzified rough set with respect to X are respectively given as:

$\begin{matrix} A_{α, β} (X, R_{1} + R_{2} + R_{3}) & = (\frac{3}{3}, \frac{2}{3}) = (1, 0.66), \\ Π_{α, β} (X, R_{1} + R_{2} + R_{3}) & = (\frac{3}{3}, \frac{2}{3}) = (1, 0.66), \\ ϒ_{α, β} (X, R_{1} + R_{2} + R_{3}) & = (\frac{3 + 2}{5}, \frac{2 + 2}{5}) = (1, 0.8) . \end{matrix}$

Hence, X^{α
⁺} and X^{β
^-} describes the objects of the universe $U$ accurately up to degrees 1 and 0.66, respectively. Similarly $X_{π}^{α^{+}}$ and $X_{π}^{β^{-}}$ describes the objects of the universe $U$ accurately up to degrees 1 and 0.66, respectively. Moreover, $X_{γ}^{α^{+}}$ and $X_{γ}^{β^{-}}$ describes the objects of the universe $U$ accurately up to degrees 1 and 0.8, respectively.

5 Optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^o-MGBFRSs)

According to two different approximations, Qian et al. [34, 35] developed two different multi-granulation rough sets including optimistic and pessimistic ones. In this section, based on the idea of optimistic multi-granulation rough set, we propose the notion of optimistic (α, β)-multi-granulation bipolar fuzzified rough sets and investigate some of their important structural properties.

Definition 5.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ characterized by its positive and negative membership functions given as $μ_{R_{i}^{+}} : U \times U ⟶ [0, 1]$ and $μ_{R_{i}^{-}} : U \times U ⟶ [- 1, 0]$ . For any α ∈ (0, 1] and β ∈ [-1, 0), the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ are defined in the following manners, respectively. $\begin{matrix} \underline{{MGFB}^{o}} (X) \\ = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)), \end{matrix}$ (22) $\begin{matrix} \bar{{MGFB}^{o}} (X) \\ = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X), (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)), \end{matrix}$ (23) where $\begin{matrix} (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) < α for all y \in X^{c}}, \end{matrix}$ (24) $\begin{matrix} (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) \geq α for some y \in X}, \end{matrix}$ (25) $\begin{matrix} (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) \leq β for some y \in X}, \end{matrix}$ (26) $\begin{matrix} (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) > β for all y \in X^{c}} . \end{matrix}$ (27) Moreover, if $\underline{{MGFB}^{o}} (X) \neq \bar{{MGFB}^{o}} (X)$ , then X is said to be optimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to {R_i} _i∈I. Otherwise, X is called optimistic (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I.

The boundary region (area of uncertainty) of $X \subseteq U$ under optimistic (α, β)-multi-granulation bipolar fuzzified rough sets environment is defined as: $\begin{matrix} {BND}_{\sum_{i = 1} R_{i}}^{o} (X) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) - (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} \\ (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) - (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)) . \end{matrix}$ (28)

Remark 5.2. From Definition 5.1, we immediately have that $X \subseteq U$ is optimistic (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I if and only if ${BND}_{\sum_{i = 1} R_{i}}^{o} (X) = (\emptyset, \emptyset) .$

Here, we employ an example in order to understand the concepts of the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ .

Example 5.3. As an illustration, consider the bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ as given in Example 3.4. Let $X = {u_{1}, u_{4}, u_{5}} \subseteq U$ . Then for α = 0.8 and β = -0.7, the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ are given as follows: $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α}^{o} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α}^{o} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β}^{o} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β}^{o} (X) & = {u_{4}} . \end{matrix}$ Thus optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ with α = 0.8 and β = -0.7 are: $\begin{matrix} \underline{{MGFB}^{o}} (X) & = ({u_{4}, u_{5}}, {u_{1}, u_{4}, u_{5}}), \\ \bar{{MGFB}^{o}} (X) & = ({u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, {u_{4}}) . \end{matrix}$ Since, $\underline{{MGFB}^{o}} (X) \neq \bar{{MGFB}^{o}} (X)$ , so X is an optimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to R₁ and R₂. Moreover, the boundary region of $X \subseteq U$ under optimistic (α, β)-multi-granulation bipolar fuzzified rough approximations is: ${BND}_{\sum_{i = 1} R_{i}}^{o} (X) = ({u_{1}, u_{2}, u_{3}}, {u_{1}, u_{5}}) .$

In order to discover the relationship between the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of a single set and the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified roug approximations of two sets described on the universe $U$ , the following properties are given.

Theorem 5.4. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and α ∈ (0, 1]. Then for any $X, Y \subseteq U$ , the optimistic (α, β)-multi-granulation bipolar fuzzified rough approximation satisfy the following properties.

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \subseteq X \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \subseteq X \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (\emptyset)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (\emptyset)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (U) = U = (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (U)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (U) = U = (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (U)$ ;

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \cap (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \cap (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \cap (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \cap (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \cup (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \cup (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) \cup (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) \cup (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (Y)$ .

Proof. It can be directly derived from Definition 5.1.□

In order to discover the relationship of containment between the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ , whenever α₁, α₂ ∈ (0, 1] and β₁, β₂ ∈ [-1, 0) are such that α₁ ≤ α₂ and β₁ ≤ β₂, the following properties are given.

Proposition 5.5. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and $X \subseteq U$ . If α₁, α₂ ∈ (0, 1] and β₁, β₂ ∈ [-1, 0) are such that α₁ ≤ α₂ and β₁ ≤ β₂, then,

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α_{1}}^{o} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{2}}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α_{1}}^{o} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{2}}^{o} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β_{1}}^{o} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{2}}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β_{1}}^{o} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{2}}^{o} (X)$ .

Proof. Straightforward.□

Definition 5.6. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then the measure of accuracy for optimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined by an ordered pair: $A_{α, β} (X, R_{i})^{o} = (X_{o}^{α^{+}}, X_{o}^{β^{-}}),$ (29) where $X_{o}^{α^{+}} = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) |}{| (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) |}$ and $X_{o}^{β^{-}} = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) |}{| (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) |} .$ Here | • | denote the cardinality of the set.

Similarly, the measure of roughness for optimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined as: $\begin{matrix} R_{α, β} (X, R_{i})^{o} & = (1, 1) - A_{α, β} (X, R_{i})^{o} \\ = (1, 1) - (X_{o}^{α^{+}}, X_{o}^{β^{-}}) \\ = (1 - X_{o}^{α^{+}}, 1 - X_{o}^{β^{-}}) . \end{matrix}$ (30)

Example 5.7. (Continued from Example 5.3) We can calculate the measure of accuracy and the measure of roughness for optimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X as follows: $\begin{matrix} A_{α, β} (X, R_{1} + R_{2})^{o} & = (\frac{2}{5}, \frac{1}{3}) = (0.4, 0.33), \\ R_{α, β} (X, R_{1} + R_{2})^{o} & = (1, 1) - (0.4, 0.33) \\ = (0.6, 0.67) . \end{matrix}$

Proposition 5.8. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then, the measure of accuracy $A_{α, β} (X, R_{i})^{o} = (X_{o}^{α^{+}}, X_{o}^{β^{-}})$ satisfies the following properties:

$(0, 0) \leq A_{α, β} (X, R_{i})^{o} \leq (1, 1)$ for any non-empty subset X of $U$ .

$A_{α, β} (X, R_{i})^{o} = (1, 1) if and only if X = U$ .

$A_{α, β} (X, R_{i})^{o} = (0, 0)$ if and only if $(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o}$ .

When α = 1 and β = -1 then for any $\emptyset \neq X \subseteq U$ , $A_{α, β} (X, R_{i})^{o} = (1, 1)$ .

If $X, Y \subseteq U$ such that X ⊆ Y, then $A_{α, β} (X, R_{i})^{o} \leq A_{α, β} (Y, R_{i})^{o}$ .

If α₁ ≤ α₂ and β₁ ≤ β₂ then, $X_{o}^{α_{1}^{+}} \leq X_{o}^{α_{2}^{+}}$ and $X_{o}^{β_{2}^{-}} \leq X_{o}^{β_{1}^{-}}$ .

Proof. Straightforward.□

6 Pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^p-MGBFRSs)

In this section, based on the idea of the pessimistic multi-granulation rough set, we propose the notion of pessimistic (α, β)-multi-granulation bipolar fuzzified rough sets and investigate some of their important structural properties.

Definition 6.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ characterized by its positive and negative membership functions given as $μ_{R_{i}^{+}} : U \times U ⟶ [0, 1]$ and $μ_{R_{i}^{-}} : U \times U ⟶ [- 1, 0]$ . For any α ∈ (0, 1] and β ∈ [-1, 0), the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ are defined in the following manners, respectively. $\begin{matrix} \underline{{MGFB}^{p}} (X) & = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X)), \end{matrix}$ (31) $\begin{matrix} \bar{{MGFB}^{p}} (X) & = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X), (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X)), \end{matrix}$ (32) where $\begin{matrix} (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) < α for all y \in X^{c}}, \end{matrix}$ (33) $\begin{matrix} (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \\ = ⋂_{i \in I} {x \in U : μ_{R_{i}^{+}} (x, y) \geq α for some y \in X}, \end{matrix}$ (34) $\begin{matrix} (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) \leq β for some y \in X}, \end{matrix}$ (35) $\begin{matrix} (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \\ = ⋃_{i \in I} {x \in U : μ_{R_{i}^{-}} (x, y) > β for all y \in X^{c}} . \end{matrix}$ (36) Moreover, if $\underline{{MGFB}^{p}} (X) \neq \bar{{MGFB}^{p}} (X)$ , then X is said to be pessimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to {R_i} _i∈I. Otherwise, X is called pessimistic (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I.

The boundary region (area of uncertainty) of $X \subseteq U$ under optimistic (α, β)-multi-granulation bipolar fuzzified rough sets environment is defined as: $\begin{matrix} {BND}_{\sum_{i = 1} R_{i}}^{p} (X) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) - (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} \\ (X), (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) - (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X)) . \end{matrix}$ (37)

Remark 6.2. From Definition 6.1, we immediately have that $X \subseteq U$ is pessimistic (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I if and only if ${BND}_{\sum_{i = 1} R_{i}}^{p} (X) = (\emptyset, \emptyset) .$

Here, we employ an example in order to understand the concepts of the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ .

Example 6.3. As an illustration, consider the bipolar fuzzy tolerance relations R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ as given in Example 3.4. Let $X = {u_{1}, u_{4}, u_{5}} \subseteq U$ . Then for α = 0.8 and β = -0.7, the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ are given as follows: $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α}^{p} (X) & = {}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α}^{p} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β}^{p} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β}^{p} (X) & = {u_{4}, u_{5}} . \end{matrix}$ Thus pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ with α = 0.8 and β = -0.7 are: $\begin{matrix} \underline{{MGFB}^{p}} (X) & = ({}, {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}), \\ \bar{{MGFB}^{p}} (X) & = ({u_{1}, u_{2}, u_{4}, u_{5}}, {u_{4}, u_{5}}) . \end{matrix}$ Since, $\underline{{MGFB}^{p}} (X) \neq \bar{{MGFB}^{p}} (X)$ , so X is a pessimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to R₁ and R₂. Moreover, the boundary region of $X \subseteq U$ under pessimistic (α, β)-multi-granulation bipolar fuzzified rough approximations is: ${BND}_{\sum_{i = 1} R_{i}}^{p} (X) = ({u_{1}, u_{2}, u_{4}, u_{5}}, {u_{1}, u_{2}, u_{3}}) .$

In order to discover the relationship between the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of a single set and the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified roug approximations of two sets described on the universe $U$ , the following properties are given.

Theorem 6.4. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and α ∈ (0, 1]. Then for any $X, Y \subseteq U$ , the pessimistic (α, β)-multi-granulation bipolar fuzzified rough approximation satisfy the following properties.

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq X \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \subseteq X \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (\emptyset)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (\emptyset) = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (\emptyset)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (U) = U = (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (U)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (U) = U = (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (U)$ ;

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$X \subseteq Y \Rightarrow (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$X \subseteq Y \Rightarrow (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X^{c}) = ((\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X))^{c}$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X^{c}) = ((\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X))^{c}$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \cap (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X \cap Y) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \cap (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \cap (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X \cap Y) \subseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \cap (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \cup (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X \cup Y) \supseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \cup (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \cup (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (Y)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X \cup Y) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \cup (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (Y)$ .

Proof. It can be directly derived from Definition 6.1.□

In order to discover the relationship of containment between the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ , whenever α₁, α₂ ∈ (0, 1] and β₁, β₂ ∈ [-1, 0) are such that α₁ ≤ α₂ and β₁ ≤ β₂, the following properties are given.

Proposition 6.5. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ and $X \subseteq U$ . If α₁, α₂ ∈ (0, 1] and β₁, β₂ ∈ [-1, 0) are such that α₁ ≤ α₂ and β₁ ≤ β₂, then,

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α_{1}}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α_{2}}^{p} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α_{1}}^{p} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α_{2}}^{p} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β_{1}}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β_{2}}^{p} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β_{1}}^{p} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β_{2}}^{p} (X)$ .

Proof. Straightforward.□

Definition 6.6. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then the measure of accuracy for pessimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined by an ordered pair: $A_{α, β} (X, R_{i})^{p} = (X_{p}^{α^{+}}, X_{p}^{β^{-}}),$ (38) where $X_{p}^{α^{+}} = \frac{| (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) |}{| (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) |}$ and $X_{p}^{β^{-}} = \frac{| (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) |}{| (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) |} .$ Here | • | denote the cardinality of the set.

Similarly, the measure of roughness for pessimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X is defined as: $\begin{matrix} R_{α, β} (X, R_{i})^{p} & = (1, 1) - A_{α, β} (X, R_{i})^{p} \\ = (1, 1) - (X_{p}^{α^{+}}, X_{p}^{β^{-}}) \\ = (1 - X_{p}^{α^{+}}, 1 - X_{p}^{β^{-}}) . \end{matrix}$ (39)

Example 6.7. (Continued from Example 6.3) We can calculate the measure of accuracy and the measure of roughness for pessimistic (α, β)-multi-granulation bipolar fuzzified rough set with respect to X as follows: $\begin{matrix} A_{α, β} (X, R_{1} + R_{2})^{p} & = (\frac{0}{4}, \frac{2}{5}) = (0, 0.4), \\ R_{α, β} (X, R_{1} + R_{2})^{p} & = (1, 1) - (0, 0.4) = (1, 0.6) . \end{matrix}$

Proposition 6.8. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1], β ∈ [-1, 0) and $\emptyset \neq X \subseteq U$ . Then, the measure of accuracy $A_{α, β} (X, R_{i})^{p} = (X_{p}^{α^{+}}, X_{p}^{β^{-}})$ satisfies the following properties:

$(0, 0) \leq A_{α, β} (X, R_{i})^{p} \leq (1, 1)$ for any non-empty subset X of $U$ .

$A_{α, β} (X, R_{i})^{p} = (1, 1) if and only if X = U$ .

$A_{α, β} (X, R_{i})^{p} = (0, 0)$ if and only if $(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} = \emptyset = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p}$ .

When α = 1 and β = -1 then for any $\emptyset \neq X \subseteq U$ , $A_{α, β} (X, R_{i})^{p} = (1, 1)$ .

If $X, Y \subseteq U$ such that X ⊆ Y, then $A_{α, β} (X, R_{i})^{p} \leq A_{α, β} (Y, R_{i})^{p}$ .

If α₁ ≤ α₂ and β₁ ≤ β₂ then, $X_{p}^{α_{1}^{+}} \leq X_{p}^{α_{2}^{+}}$ and $X_{p}^{β_{2}^{-}} \leq X_{p}^{β_{1}^{-}}$ .

Proof. Straightforward.□

7 Relationships among the (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs

In sections 5 and 6, we mainly discuss the properties of the (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs. Now in this section, the relationships among the (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs will be further established.

Proposition 7.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then, the following properties hold:

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) = (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) = (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) = (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ .

Proof. Direct consequence of Definition 3.1, Definition 5.1 and Definition 6.1.□

Proposition 7.2. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then, for $X \subset U$ the following properties hold:

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)$ .

Proof. Straightforward.□

Proposition 7.2 shows the relationship of containment between the (α, β)-MGBFRSs and (α, β) ^o-MGBFRSs. In other words, the multi-granulation fuzzified upper α-positive bipolar rough approximation of $X \subseteq U$ is contained in the optimistic upper α-positive multi-granulation bipolar fuzzified rough approximation of $X \subseteq U$ . But for the multi-granulation fuzzified upper β-negative bipolar rough approximation of $X \subseteq U$ , the order of the inclusion is just reversed. The properties also reveal that the optimistic upper α-positive multi-granulation bipolar fuzzified rough approximation of $X \subseteq U$ is finer than the multi-granulation fuzzified upper α-positive bipolar rough approximation of $X \subseteq U$ . Similarly, the multi-granulation fuzzified upper β-negative bipolar rough approximation of $X \subseteq U$ is finer than the optimistic upper β-negative multi-granulation bipolar fuzzified rough approximation of $X \subseteq U$ .

Proposition 7.3. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then, for $X \subset U$ the following properties hold:

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \supseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X)$ .

Proof. Straightforward.□

The above proposition shows that, the multi-granulation fuzzified lower α-positive bipolar rough approximation of $X \subseteq U$ is finer than the optimistic lower α-positive multi-granulation bipolar fuzzified rough approximation of $X \subseteq U$ . Similarly, the multi-granulation fuzzified lower β-negative bipolar rough approximation of $X \subseteq U$ is coarser than the optimistic lower β-negative multi-granulation bipolar fuzzified rough approximation of $X \subseteq U$ .

The following theorem shows the relationship of containment among the optimistic and the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations of $X \subseteq U$ .

Theorem 7.4. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). Then, for $X \subseteq U$ the following properties hold:

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) \subseteq (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X)$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \supseteq (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) \supseteq (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X)$ .

Proof. Straightforward.□

The following theorem shows the relationship among the measures accuracy of (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs.

Theorem 7.5. Let $A_{α, β} (X, R_{i}) = (X^{α^{+}}, X^{β^{-}})$ , $A_{α, β} (X, R_{i})^{o} = (X_{o}^{α^{+}}, X_{o}^{β^{-}})$ and $A_{α, β} (X, R_{i})^{p} = (X_{p}^{α^{+}}, X_{p}^{β^{-}})$ be the measures of accuracy of (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs, respectively. Then,

$X_{o}^{α^{+}} \leq X^{α^{+}}$ and $X_{o}^{β^{-}} \leq X^{β^{-}}$ ;

$X_{p}^{α^{+}} \leq X^{α^{+}}$ and $X_{p}^{β^{-}} \leq X^{β^{-}}$ ;

$X_{o}^{α^{+}} \geq X_{p}^{α^{+}}$ and $X_{o}^{β^{-}} \leq X_{p}^{β^{-}}$ .

Proof. Straightforward.□

Example 7.6. As an illustration, we revisit Example 3.4, where R₁ = (λ^P (x, y) , λ^N (x, y)) and R₂ = (ξ^P (x, y) , ξ^N (x, y)) are two bipolar fuzzy tolerance relations over $U = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}$ and $X = {u_{1}, u_{4}, u_{5}} \subseteq U$ . Then for α = 0.8 and β = -0.7, we have $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β} (X) & = {u_{4}, u_{5}} . \end{matrix}$

Similarly,

$\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α}^{o} (X) & = {u_{4}, u_{5}}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α}^{o} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β}^{o} (X) & = {u_{1}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β}^{o} (X) & = {u_{4}} . \end{matrix}$

And, $\begin{matrix} (\underline{R_{1}^{+} + R_{2}^{+}})_{α}^{p} (X) & = {}, \\ (\bar{R_{1}^{+} + R_{2}^{+}})_{α}^{p} (X) & = {u_{1}, u_{2}, u_{4}, u_{5}}, \\ (\underline{R_{1}^{-} + R_{2}^{-}})_{β}^{p} (X) & = {u_{1}, u_{2}, u_{3}, u_{4}, u_{5}}, \\ (\bar{R_{1}^{-} + R_{2}^{-}})_{β}^{p} (X) & = {u_{4}, u_{5}} . \end{matrix}$

Now the measure of accuracy for the (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs with respect to R₁ and R₂ are respectively given as; $\begin{matrix} A_{α, β} (X, R_{1} + R_{2}) & = (X^{α^{+}}, X^{β^{-}}) = (\frac{2}{4}, \frac{2}{3}) \\ = (0.5, 0.66), \\ A_{α, β} (X, R_{1} + R_{2})^{o} & = (X_{o}^{α^{+}}, X_{o}^{β^{-}}) = (\frac{2}{5}, \frac{1}{3}) \\ = (0.4, 0.33), \\ A_{α, β} (X, R_{1} + R_{2})^{p} & = (X_{p}^{α^{+}}, X_{p}^{β^{-}}) = (\frac{0}{4}, \frac{2}{5}) \\ = (0, 0.4) . \end{matrix}$

Clearly, we can see that, $X_{o}^{α^{+}} \leq X^{α^{+}}$ and $X_{o}^{β^{-}} \leq X^{β^{-}}$ . Also, $X_{p}^{α^{+}} \leq X^{α^{+}}$ and $X_{p}^{β^{-}} \leq X^{β^{-}}$ . Moreover, $X_{o}^{α^{+}} \geq X_{p}^{α^{+}}$ and $X_{o}^{β^{-}} \leq X_{p}^{β^{-}}$ .

Proposition 7.7. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ , α ∈ (0, 1] and β ∈ [-1, 0). If α = 1 and β = -1. Then, for any $X \subseteq U$ we have

$(\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) = (\underline{\sum_{i \in I} R_{i}^{+}})_{α} (X) = (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) = X$ ;

$(\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X) = (\bar{\sum_{i \in I} R_{i}^{+}})_{α} (X) = (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{p} (X) = X$ ;

$(\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) = (\underline{\sum_{i \in I} R_{i}^{-}})_{β} (X) = (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) = X$ ;

$(\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X) = (\bar{\sum_{i \in I} R_{i}^{-}})_{β} (X) = (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{p} (X) = X$ .

Proof. Straightforward.□

The above proposition tells us that, the optimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations, the pessimistic lower and upper (α, β)-multi-granulation bipolar fuzzified rough approximations and (α, β)-multi-granulation bipolar fuzzified lower and upper approximations of $X \subseteq U$ coincide. The properties also reveal that the set X becomes optimistic, pessimistic (α, β)-multi-granulation bipolar fuzzified definable set with respect to {R_i} _i∈I if α = 1 and β = -1.

8 Multi criteria group decision making based on (α, β)-MGBFRSs

The increasing complexity of the socio-economic environment, operational research, and industrial engineering force humans to tackle problems crossing many disciplines. Group decision-making (GDM), as one of the effective approaches to handle complex decision-making problems, is defined as a decision problem in which several experts provide their judgment over a set of alternatives. The aim is to reconcile (or compromise) differences of opinion expressed by individual experts to find an alternative (or set of alternatives) that is most acceptable by the group of experts as a whole. In a complex society, group decision-making (GDM) processes must inevitably take many criteria (or factors) into account. Thus, research on group decision-making (GDM) that explicitly incorporates multiple criteria has been a major direction and has made significant progress with the rapid development of operations research, management science, systems engineering, and other disciplines. Hwang and Lin [17] first study to explore systematically how multiple criteria could be used in group decision making (GDM).

In general, multi-criteria group decision-making problems (MCGDM) involve selecting or ranking from all of the feasible alternatives among multiple, conflicting, and interactive criteria. For example, in a decision recruitment problem for engaging a new young employee, the alternatives are the candidates and the criteria are some characteristics useful to give a comprehensive evaluation of the candidates such as educational degree, professional experience, age, and job interview. At the same time, several experts are invited to give a comprehensive evaluation for all candidates according to the given criteria in advance. Then, aggregating all comprehensive evaluations given by the experts based on a determined method and then obtain the ranking for all candidates. Finally, the expected candidate with the highest ranking will be selected, i.e., the optimal decision making is given for this multiple criteria group decision-making problem.

In this section, we try to establish a new approach to multi-criteria group decision-making problem on (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β)-MGBFRSs). We present the basic description of a multi-criteria group decision-making problem under the framework of the (α, β)-MGBFRSs, and then give a general decision making methodology for multi-criteria group decision-making problem by using the (α, β)-MGBFRSs.

8.1 Problem statement

We firstly give the basic description of the considered multi-criteria group decision-making problem in this paper. We present the description by using multi-criteria group decision-making problem in the case of car selection.

Let $U = {u_{1}, u_{2}, \dots, u_{n}}$ be the finite universe of n objects (criteria set). Suppose that $V = {P_{1}, P_{2}, \dots, P_{k}}$ is a set of k invited experts in the group. Like classical group decision-making, every expert provides his evaluation for all criteria $u_{i} \in U$ ; (i = 1, 2, …, n). Generally speaking, the evaluations of experts are in the form of R₁, R₂, …, R_k bipolar fuzzy tolerance relations over the universe $U$ . That is, $μ_{R_{k}^{+}} (x_{i}, y_{i})$ and $μ_{R_{k}^{-}} (x_{i}, y_{i})$ ; (i = 1, 2, …, n) is the evaluation of criteria x_i with evaluation element y_i was given by expert k according to their experience and professional knowledge of himself. Let $X_{1}, X_{2}, \dots, X_{r} \subseteq U$ , represents the interest of r customers in certain criteria. Then the decision-making for this multi-criteria group decision-making problem is how to obtain the evaluation of these particular customers so that the selected criteria (alternative) is optimal for all.

Actually, all three models of (α, β)-MGBFRSs, (α, β) ^o-MGBFRSs and (α, β) ^p-MGBFRSs established in Sections 3, 5 and 6 can be used to discuss this decision problem. In the following, we give an approach to decision making for this kind of multiple criteria group-decision problem with the above-described characteristics by using the theory of (α, β)-multi-granulation bipolar fuzzified rough sets. In this paper, we use the model of optimistic (α, β)-multi-granulation bipolar fuzzified rough sets to present the decision method for multiple criteria group decision-making.

8.2 Decision making methodology

In this subsection, we present the procedure of decision making approach based on optimistic (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β) ^o-MGBFRSs).

Definition 8.1. Let {R_i} _i∈I be a collection of bipolar fuzzy tolerance relations over the universe $U$ . Suppose that $\underline{{MGFB}^{o}} (X_{j}) = ((\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}), (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}))$ and $\bar{{MGFB}^{o}} (X_{j}) = ((\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}), (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}))$ be the optimistic (α, β)-multi-granulation bipolar fuzzified lower and upper approximations of X_j ; (j = 1, 2, …, r) with respect to {R_i} _i∈I bipolar fuzzy tolerance relations, where α ∈ (0, 1] and β ∈ [-1, 0) can be regarded as the given threshold, which are usually given by decision-makers according to their requirement in practical decision-making applications. Then,

$\begin{matrix} {\underline{M}}_{(α, β)^{o}} & = {[\begin{matrix} {{\underline{X_{k}}}_{α}^{r^{+}}, {\underline{X_{k}}}_{β}^{r^{-}}} \end{matrix}]}_{r \times 1}, \end{matrix}$ (40)

and $\begin{matrix} {\bar{M}}_{(α, β)^{o}} & = {[\begin{matrix} {{\bar{X_{k}}}_{α}^{r^{+}}, {\bar{X_{k}}}_{β}^{r^{-}}} \end{matrix}]}_{r \times 1}, \end{matrix}$ (41)

are said to be optimistic (α, β)-multi-granulation bipolar fuzzified lower approximation matrix and (α, β)- multi-granulation bipolar fuzzified upper approximation matrix, respectively. Where,

$\begin{matrix} {\underline{X_{k}}}_{α}^{r^{+}} & = {\begin{matrix} 1 & if {\underline{X_{k}}}_{α}^{r^{+}} \in (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}), \\ 0 & if {\underline{X_{k}}}_{α}^{r^{+}} \notin (\underline{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}) . \end{matrix} \end{matrix}$ (42) $\begin{matrix} {\underline{X_{k}}}_{β}^{r^{-}} & = {\begin{matrix} - \frac{1}{2} & if {\underline{X_{k}}}_{β}^{r^{-}} \in (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}), \\ 0 & if {\underline{X_{k}}}_{β}^{r^{-}} \notin (\underline{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}) . \end{matrix} \end{matrix}$ (43) $\begin{matrix} {\bar{X_{k}}}_{α}^{r^{+}} & = {\begin{matrix} \frac{1}{2} & if {\bar{X_{k}}}_{α}^{r^{+}} \in (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}), \\ 0 & if {\bar{X_{k}}}_{α}^{r^{+}} \notin (\bar{\sum_{i \in I} R_{i}^{+}})_{α}^{o} (X_{j}) . \end{matrix} \end{matrix}$ (44) $\begin{matrix} {\bar{X_{k}}}_{β}^{r^{-}} & = {\begin{matrix} - 1 & if {\bar{X_{k}}}_{β}^{r^{-}} \in (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}), \\ 0 & if {\bar{X_{k}}}_{β}^{r^{-}} \notin (\bar{\sum_{i \in I} R_{i}^{-}})_{β}^{o} (X_{j}) . \end{matrix} \end{matrix}$ (45)

Definition 8.2. Let ${\underline{M}}_{(α, β)^{o}}$ and ${\bar{M}}_{(α, β)^{o}}$ be optimistic (α, β)-multi-granulation bipolar fuzzified lower and upper approximation matrices. Then the optimistic (α, β)-multi-granulation bipolar fuzzified lower approximation vector (represented by ${\underline{V}}_{(α, β)^{o}}$ ) and the optimistic (α, β)-multi-granulation bipolar fuzzified upper approximation vector (represented by ${\bar{V}}_{(α, β)^{o}}$ ) are respectively defined as follows: ${\underline{V}}_{(α, β)^{o}} = \sum_{r} ({\underline{X_{k}}}_{α}^{r^{+}} \oplus {\underline{X_{k}}}_{β}^{r^{-}}),$ (46) ${\bar{V}}_{(α, β)^{o}} = \sum_{r} ({\bar{X_{k}}}_{α}^{r^{+}} \oplus {\bar{X_{k}}}_{β}^{r^{-}}) .$ (47) Here the operation ∑ and ⊕ represent the vector addition.

Definition 8.3. Let ${\underline{V}}_{(α, β)^{o}}$ and ${\bar{V}}_{(α, β)^{o}}$ be the optimistic (α, β)-multi-granulation bipolar fuzzified lower and upper approximation vectors, respectively. Then, $V_{(α, β)^{o}} = {\underline{V}}_{(α, β)^{o}} \oplus {\bar{V}}_{(α, β)^{o}}$ (48) is called the optimistic (α, β)-multi-granulation bipolar fuzzified vector.

Definition 8.4. Let $V_{(α, β)^{o}} = {\underline{V}}_{(α, β)^{o}} \oplus {\bar{V}}_{(α, β)^{o}} = (v_{1}, v_{2}, \dots, v_{n})$ be the optimistic (α, β)-multi-granulation bipolar fuzzified vector. Then each v_i is said to be a weighted number of $u_{i} \in U$ . An element $u_{i} \in U$ is called an optimal element of the universe $U$ if its weighted number is a maximum of v_i for all i = 1, 2, …, n. An element $u_{i} \in U$ is called the worst element of the universe $U$ if its weighted number is a minimum of v_i for all i = 1, 2, …, n. If there are more than one optimal elements of the universe $U$ , then select any one of them.

8.3 Proposed algorithm

In this subsection, we present the algorithm for the established method of considered multi-criteria group decision-making problem in section 8.1.

Take judgment R₁, R₂, …, R_k of invited experts P₁, P₂, …, P_k.

Take personal interest X₁, X₂, …, X_r of different customers.

Compute $\underline{{MGFB}^{o}} (X_{j})$ and $\bar{{MGFB}^{o}} (X_{j})$ for each j = 1, 2, …, r.

Construct optimistic (α, β)-multi-granulation bipolar fuzzified lower approximation matrix ${\underline{M}}_{(α, β)^{o}}$ and optimistic (α, β)- multi-granulation bipolar fuzzified upper approximation matrix ${\bar{M}}_{(α, β)^{o}}$ .

Compute optimistic (α, β)-multi-granulation bipolar fuzzified lower approximation vector ${\underline{V}}_{(α, β)^{o}}$ and optimistic (α, β)- multi-granulation bipolar fuzzified upper approximation vector ${\bar{V}}_{(α, β)^{o}}$ .

Compute optimistic (α, β)-multi-granulation bipolar fuzzified vector $V_{(α, β)^{o}} = {\underline{V}}_{(α, β)^{o}} \oplus {\bar{V}}_{(α, β)^{o}}$ .

Find $max_{i} v_{i}$ ; i = 1, 2, …, n.

Flow chart depiction of the above algorithm is shown in Fig. 1.

Fig. 1

(Flow chart of proposed Algorithm).

8.4 A design example

Here, we consider a multi-criteria group-decision making problem in the case of the best car selection to illustrate the decision method proposed in subsection 8.2. Let $U = {c_{1}, c_{2}, c_{3}, c_{4}, c_{5}}$ be the universe (showroom) consisting of five different brands of cars, where each c_i (i = 1, 2, ⋯ , 5) stands for “Civic”, “Saturn”, “Mazda”, “Toyota” and “Honda” respectively.

Suppose that P₁, P₂ and P₃ are three invited experts. They present their judgment about each brand of car in the form of bipolar fuzzy tolerance relations R₁, R₂, and R₃.

R₁ = (λ^P (x, y) , λ^N (x, y)), R₂ = (ξ^P (x, y) , ξ^N (x, y)) and R₃ = (η^P (x, y) , η^N (x, y)) are given as follows:

$λ^{P} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & 1 & 0.9 & 0.6 & 0.2 & 0.8 \\ c_{2} & 0.9 & 1 & 0.7 & 0.3 & 0.5 \\ c_{3} & 0.6 & 0.7 & 1 & 0.4 & 0.1 \\ c_{4} & 0.2 & 0.3 & 0.4 & 1 & 0.8 \\ c_{5} & 0.8 & 0.5 & 0.1 & 0.8 & 1 \end{matrix}$ and $λ^{N} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & - 1 & - 0.7 & - 0.5 & - 0.4 & - 0.9 \\ c_{2} & - 0.7 & - 1 & - 0.3 & - 0.2 & - 0.1 \\ c_{3} & - 0.5 & - 0.3 & - 1 & - 0.4 & - 0.6 \\ c_{4} & - 0.4 & - 0.2 & - 0.4 & - 1 & - 0.9 \\ c_{5} & - 0.9 & - 0.1 & - 0.6 & - 0.9 & - 1 \end{matrix}$ .

$ξ^{P} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & 1 & 0.8 & 0.2 & 0.4 & 0.7 \\ c_{2} & 0.8 & 1 & 0.3 & 0.6 & 0.2 \\ c_{3} & 0.2 & 0.3 & 1 & 0.5 & 0.9 \\ c_{4} & 0.4 & 0.9 & 0.5 & 1 & 0.4 \\ c_{5} & 0.7 & 0.2 & 0.9 & 0.4 & 1 \end{matrix}$ and $ξ^{N} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & - 1 & - 0.6 & - 0.8 & - 0.5 & - 0.3 \\ c_{2} & - 0.6 & - 1 & - 0.1 & - 0.2 & - 0.4 \\ c_{3} & - 0.8 & - 0.1 & - 1 & - 0.5 & - 0.7 \\ c_{4} & - 0.5 & - 0.2 & - 0.5 & - 1 & - 0.3 \\ c_{5} & - 0.3 & - 0.4 & - 0.7 & - 0.3 & - 1 \end{matrix}$ .

$η^{P} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & 1 & 0.7 & 0.4 & 0.1 & 0.9 \\ c_{2} & 0.7 & 1 & 0.5 & 0.2 & 0.3 \\ c_{3} & 0.4 & 0.5 & 1 & 0.6 & 0.8 \\ c_{4} & 0.1 & 0.2 & 0.6 & 1 & 0.9 \\ c_{5} & 0.9 & 0.3 & 0.8 & 0.9 & 1 \end{matrix}$ and $η^{N} = \begin{matrix} c_{1} & c_{2} & c_{3} & c_{4} & c_{5} \\ c_{1} & - 1 & - 0.8 & - 0.4 & - 0.5 & - 0.2 \\ c_{2} & - 0.8 & - 1 & - 0.1 & - 0.2 & - 0.3 \\ c_{3} & - 0.4 & - 0.1 & - 1 & - 0.5 & - 0.7 \\ c_{4} & - 0.5 & - 0.2 & - 0.5 & - 1 & - 0.2 \\ c_{5} & - 0.2 & - 0.3 & - 0.7 & - 0.2 & - 1 \end{matrix}$ .

Interest of two different customers car’s brand are: $X_{1} = {c_{1}, c_{2}, c_{4}} and X_{2} = {c_{1}, c_{3}, c_{5}} .$

Taking α = 0.8 and β = -0.5. Then,

$\begin{matrix} \underline{{MGFB}^{o}} (X_{1}) & = ({c_{1}, c_{2}, c_{4}}, {c_{1}, c_{2}, c_{3}, c_{4}}), & \bar{{MGFB}^{o}} (X_{1}) & = ({c_{1}, c_{2}, c_{4}, c_{5}}, {c_{2}}) . \\ \underline{{MGFB}^{o}} (X_{2}) & = ({c_{1}, c_{3}, c_{5}}, {c_{1}, c_{2}, c_{3}, c_{4}, c_{5}}), & \bar{{MGFB}^{o}} (X_{2}) & = ({c_{1}, c_{2}, c_{3}, c_{4}, c_{5}}, {}) . \end{matrix}$

Calculate optimistic (α, β)-multi-granulation bipolar fuzzified lower approximation matrix ${\underline{M}}_{(α, β)^{o}}$ and optimistic (α, β)- multi-granulation bipolar fuzzified upper approximation matrix ${\bar{M}}_{(α, β)^{o}}$ by using Equations (39) and (40), we have $\begin{matrix} {\underline{M}}_{(α, β)^{o}} & = (\begin{matrix} {(1, 1, 0, 1, 0), (- \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, 0)} \\ {(1, 0, 1, 0, 1), (- \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2}, - \frac{1}{2})} \end{matrix}), \end{matrix}$ and $\begin{matrix} {\bar{M}}_{(α, β)^{o}} & = (\begin{matrix} {(\frac{1}{2}, \frac{1}{2}, 0, \frac{1}{2}, \frac{1}{2}), (0, - 1, 0, 0, 0)} \\ {(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2}), (0, 0, 0, 0, 0)} \end{matrix}) . \end{matrix}$

Using Equations (45) and (46), the optimistic (α, β)-multi-granulation bipolar fuzzified lower and upper approximation vector can be calculated as follows: $\begin{matrix} {\underline{V}}_{(α, β)^{o}} & = (1, 0, 0, 0, 0.5), \\ {\bar{V}}_{(α, β)^{o}} & = (1, 0, 0.5, 1, 1) . \end{matrix}$

By using Equation (47), the optimistic (α, β) decision vector is obtained as follows:

$V = {\underline{V}}_{(α, β)^{o}} \oplus {\bar{V}}_{(α, β)^{o}} = (2, 0, 0.5, 1, 1.5) .$

As $max_{i \in I_{n}} c_{i} = c_{1} = 2$ and $min_{i \in I_{n}} c_{i} = c_{2} = 0$ . So c₁ (Civic) is the optimal element and c₂ (Saturn) is the worst element.

The pictorial representation for the ranking of the cars is given in the following Fig. 2.

Fig. 2

(Ranking of cars).

9 Sensitivity analysis

9.1 Advantages of the proposed model

Generally, the real-world MCDM and MCGDM problems occur in complex environment under imprecise and uncertain data, which is difficult to handle. The proposed model is highly suitable for the situation when the information is complex, imprecise and uncertain. Specially, when the existing data is based upon the bipolar fuzzy relations by decision-makers. Some advantages of the proposed approach are listed below:

The proposed decision-making technique considers the positive and negative aspects of each individual object in the form of bipolar fuzzy tolerance relation. This hybrid model is more generalized and suitable to handle aggressive decision-making problems.

This method is also preferable because in this method the decision-makers are free from any external conditions and requirements.

As is well known, the opinions or preferences of all decision-makers (or experts) aggregation is the critical step for the traditional group decision-making methods. In our proposed decision-making method, every decision-maker is regarded as a bipolar fuzzy tolerance relation, and then all opinions given by decision-makers are aggregated by using the (α, β)-multi-granulation bipolar fuzzified lower and upper approximations, and then a compromise optimal proposal is obtained. So, the (α, β)-multi-granulation bipolar fuzzified rough set approach to multi-criteria group decision-making provides another way to aggregate the preferences of decision-makers. Therefore, the proposed decision-making method also presents a new tool and perspective to explore group decision-making problems in reality.

9.2 Disadvantages of the proposed model

A little bit flaws are there in the proposed model, including its complex structure, the huge data in the form of bipolar information. Such a huge data is difficult to handle, because of the massive calculations, which are not so easy to perform. However, one can generate a computer programming code to make these complex calculations easier.

9.3 Comparison with existing models

There are several methods in the literature convenient for solving multi-criteria group decision-making (MCGDM) problems. All these MCGDM approaches have their own pros and cons. The capability of each method is depends upon problem under consideration. In this subsection, we make a set based comparison of the proposed MCGDM method with some existing MCGDM methods in fuzzy and bipolar fuzzy environment, and see the importance of the proposed MCGDM method.

We discuss comparison analysis of proposed technique with multi-granulation hesitant fuzzy rough sets [66], double-quantitative multigranulation decision-theoretic rough fuzzy set model [19], fuzzy multi-granulation decision-theoretic rough sets based on fuzzy preference relation [27], and three-way decisions based on multi-granulation support intuitionistic fuzzy probabilistic rough sets [55]. All these models have their own worth in the literature. If we compare all these methods with our proposed method, we explore the following points.

The above-mentioned methods are unable to capture bipolarity in decision-making which is an essential part of human thinking and behavior.

Secondly, these methods do not guarantee harmony in the decision-maker’s opinions.

The values of α and β have a great impact on the decision result. If the values of α and β are changed, the decision result will be affected.

MCGDM approach connected with the above structure makes perfect sense to handle complex cases of decision-making.

10 Conclusion

In this article, we have proposed a new model of (α, β)-multi-granulation bipolar fuzzified rough sets ((α, β)-MGBFRSs) using a collection of bipolar fuzzy tolerance relations to define the approximation operators. Some important structural properties of (α, β)-multi-granulation bipolar fuzzified rough approximation operators have also been investigated. Moreover, some important measures associated with (α, β)-MGBFRSs like the measure of accuracy, the measure of precision and the accuracy of approximation are also provided. At the same time, we present two types of (α, β)-MGBFRSs models: one is the (α, β) ^o-MGBFRSs and the other is the (α, β) ^p-MGBFRSs. Meanwhile, we build up a general framework of multi-criteria group decision-making based on the (α, β)-MGBFRSs. And the validity of this approach is illustrated by a practical application. Lastly, a sensitivity analysis of the proposed model is performed.

The notions of (α, β)-multi-granulation bipolar fuzzified rough sets are preliminary and may result in more sufficient mathematical tools to approximate reasoning in soft computing. The research results of this paper enrich decision analysis theory and methodology and provide new ideas foe solving complex MCGDM problems. In the future, we will establish the links of (α, β)-multi-granulation bipolar fuzzified rough sets to various algebraic structures and try to develop effective decision making methods. Modeling of supported physical phenomenon is our next goal. Another perspective direction is to study the topological properties and similarity measures of (α, β)-multi-granulation bipolar fuzzified rough sets in order to explore for a solid foundation of the research work and development of working methodologies. Further we will focus on the practical applications of the proposed method in solving a wider range of selection problems, such as TOPSIS, VIKOR, ELECTRE, AHP, and PROMETHEE. We will also be looking at the possible hybridization of the proposed method to get more preciseness in results and applying these techniques to real-world problems with big data sets. In this way, we can obtain and show off the usefulness of our proposed framework.

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