A consensus model based on rough bipolar fuzzy approximations

Abstract

The uncertainty in the data is a hurdle in decision-making problems. Rough set theory and fuzzy set theory are built to handle the uncertainty in data. We introduce the rough bipolar fuzzy sets as a hybridization of rough sets and the bipolar fuzzy sets. After that, we discuss a group decision making problem with the data having fuzziness endowed with bipolarity and iron out this problem by applying the rough bipolar fuzzy sets. We also propose an algorithm for this problem, which yields the best decision, as well as, the worst decision between some objects.

Keywords

Rough sets bipolar fuzzy sets approximation space lower and upper approximations group decision making

1 Introduction

While modeling the real world problems in different disciplines, we often have to face uncertain and vague data. The complexity and dimensions of ambiguity in the data is increasing rapidly. Many theories are built to deal with these uncertainties and ambiguities in the data. We mention, here, the rough set theory [17, 18], the fuzzy set theory [30], the hesitant fuzzy set theory [26] and the bipolar fuzzy set theory [33]. Fuzzy sets are studied in numerous directions, see [2 , 20–25]. Some applications of different fuzzy structures in decision-making problems are discussed in [1 , 31]. Some useful group decision-making (GDM) models based on the linguistic term sets and the hesitant fuzzy sets are also discussed in [4–7 , 32]. The linguistic term sets grade the objects as “poor”, “average” and “good” etc., with respect to some property (or phenomena). But, to measure the degree of satisfaction of the objects to these properties would be really fruitful. This is accomplished with the help of fuzzy sets and the bipolar fuzzy sets. If the data is, moreover, endowed by uncertainty or vagueness, the rough sets are proved to be very auspicious to deal such type of data. For example, let us estimate the sweetness in some food items. A linguistic term set can characterize a particular food item as “medium”, “sweet” or “sour” etc. A fuzzy set can better describe the sweetness by associating a degree (between 0 and 1), say 0.6, to a food item. But, the sourness in that item may not be equal to 0.4. It may be less than 0.4 (or even zero, in some cases). So, a much better approach to clarify this phenomena is the use of bipolar fuzzy sets [33]. These sets demonstrate the degree to which the food item is sweet, as well as, the degree to which the food item is sour. The bipolar fuzzy sets are defined with the help of two membership functions: a positive membership function ranging in [0,1], which indicates the degree of fulfilment of the objects for some property and a negative membership function ranging in [-1,0], which measures the degree of fulfilment of those objects to the counter-property. For instance, let a bipolar fuzzy set λ, describing the sweetness of food, associate the value (0.6, -0.1) to a particular food item ‘a’. It means that the degree of sweetness in the food ‘a’ is 0.6 and the degree of sourness in ‘a’ is -0.1. The bipolar fuzzy sets, indeed, have potential to handle the bipolarity of the information, and so, have strong impacts in many fields. The incentive to study such bipolarity is, that, the human decisions are customarily built on the bipolar judgment. For example, hardness and softness of rocks, sourness and sweetness in food, compliance and resistance, sympathy and enmity are the counter aspects of the knowledge in decision problems. Lee [13] defined some basic operations on the bipolar fuzzy sets. Lee [14] also made a comparison of the bipolar fuzzy sets with some other extensions of the fuzzy sets. Dubois and Prade [10] discussed main kinds of the bipolarity.

Rough sets portray a distinctive mathematical approach to the uncertainty. In rough set theory, the membership or the degree of fulfilment is not the primary concept. These sets discriminate the definite and uncertain parts in the membership values or the degrees of satisfaction of some objects to some properties, with the help of the lower and upper approximations. In the above example, the sweetness of food ‘a’ according to the bipolar fuzzy set λ is 0.6, which is an upshot of estimation. If the lower approximation of λ gives the degree 0.4 to the food ‘a’, it means, that, out of 0.6, the food ‘a’ is definitely sweet up to the degree 0.4 and the remaining degree depicts the uncertain or doubtful membership. Knowing about the definite and uncertain parts in the membership degrees, will surely help to refine the decisions. Hence, the decisions made on the basis of the rough approximations are more efficient and reliable.

Rough sets are hybridized with fuzzy sets in [8, 11]. Malik and Shabir [16] investigated roughness in fuzzy bipolar soft sets and applied this concept to solve a decision making problem. In this study, we have adapted the Pawlak’s concept of roughness to hybridize the rough sets with the bipolar fuzzy sets, leading to the notion of rough bipolar fuzzy sets. We also address a GDM problem, where a team of decision makers has to decide between some objects, keeping in view the degree of positive membership, as well as, the degree of negative membership observed in those objects. The bipolar fuzzy sets serve to measure the degrees of positive membership and the negative membership in the objects, related to some attribute of interest. While, the rough approximations demarcate the definite (confirmed) and uncertain parts in these degrees. Since, the definite parts of the membership degrees is more important, it must be given more weightage while making a decision, as compared to the uncertain part. The core of our study is that we work out our GDM problem applying the rough approximations of the bipolar fuzzy sets, giving double weightage to the definite membership as compared to the uncertain membership. We also propose an efficient computational algorithm to solve this problem. This approach makes our algorithm dignified, to iron out such type of GDM problems with uncertain data.

The rest of the paper is arranged in the manner such that Section 2 looks over some basic definitions and concepts. In Section 3, we define the rough bipolar fuzzy sets and investigate some of their characteristics. The application of the rough bipolar fuzzy sets in a GDM problem and an algorithm for its solution are discussed in Section 4. To illustrate the steps of this algorithm, we construct a GDM problem in Section 5 and solve this problem using the algorithm designed in the previous section. The concluding remarks are in the last section.

2 Basic concepts

This section recalls some basic concepts and definitions related to the rough set theory, the fuzzy set theory, and the bipolar fuzzy set theory, which are required to understand this paper.

2.1 Rough sets

The rough set theory [17] provides a systematic procedure for dealing with vagueness in data due to indiscernibility in a situation with incomplete or doubtful information or a lack of knowledge. Let A (≠ φ) be the initial universe of discourse and $R$ be an equivalence relation defined on A . Then, the pair $(A, R)$ is the Pawlak approximation space (shortly written as a P-Ap space). The relation $R$ defines a partition $A / R$ of the universe A into the equivalence classes. These equivalence classes work as the elementary building blocks in the data analysis. In the partition $A / R$ , we denote the equivalence class of the element a ∈ A by $[a]_{R}$ (or by [a], for convenience). On a subset X of A, the relation $R$ defines the following two operators. $\begin{matrix} \underline{X} & = & {a \in A : [a]_{R} \subseteq X}, \\ \bar{X} & = & {a \in A : [a]_{R} \cap X \neq φ} . \end{matrix}$ These two operators assign two subsets $\underline{X}$ and $\bar{X}$ of A, to any subset X of A, known as the lower and upper rough approximations of X according to $R$ , respectively. The set $\underline{X}$ contains the objects which are definitely members of the set X, while, $\bar{X}$ contains the objects which can possibly be in X, subject to the available information. Moreover, the difference $\bar{X} - \underline{X}$ , denoted by ${Bnd}_{R} X$ , only contains the objects whose membership in X is uncertain (doubtful). This is called the boundary of the set X. A rough set always has a nonempty boundary. Now we are ready to present the formal definition of a rough set.

Definition 2.1. [18]. Let $(A, R)$ be a P-Ap space. A subset X ⊆ A is said to be a rough set if $\underline{X} \neq \bar{X}$ ; otherwise, X is $R -$ definable set.

2.2 Fuzzy sets and bipolar fuzzy sets

Fuzzy sets [30] measure the inaccuracy of the attributes (or properties) of the objects with the help of a mapping ω : A ⟶ [0, 1], known as the membership function. This function ω assigns a membership degree ω (a) to each object a ∈ A . That is, the degree to which ‘a’ pertains the property of ω. The membership degrees in the bipolar fuzzy sets are expressed by twin membership functions.

Definition 2.2. [33]. A bipolar fuzzy set ω in a nonempty universe A is defined as: $ω = {(a, ω^{P} (a), ω^{N} (a)) : a \in A}$ where ω ^P : A ⟶ [0, 1] and ω ^N : A ⟶ [- 1, 0] are the positive membership function and the negative membership function, respectively.

The positive membership value ω ^P (a) gives the degree of fulfilment of an object ‘a’ to the property of the bipolar fuzzy set ω, while the negative membership value |ω ^N (a) | measures the degree of fulfilment of ‘a’ to the property opposite to ω. If ω ^P (a) =0 = ω ^N (a), the object a is inappropriate to the property of ω. We write Ω for the set of all bipolar fuzzy sets in A and ω (a) = (ω ^P (a) , ω ^N (a)) for (a, ω ^P (a) , ω ^N (a)) ∈ ω. Some basic operations on the bipolar fuzzy sets are defined below.

Definition 2.3. [13]. Let ω ₁, ω ₂ ∈ Ω. Then ω ₁ is contained in ω ₂, that is, ω ₁ ⊆ ω ₂, if $ω_{1}^{P} (a) \leq ω_{2}^{P} (a)$ and $ω_{1}^{N} (a) \geq ω_{2}^{N} (a)$ for all a ∈ A . Clearly, ω ₁ = ω ₂ if and only if ω ₁ ⊆ ω ₂ and ω ₂ ⊆ ω ₁.

Definition 2.4. [13]. The whole bipolar fuzzy set in A is denoted by $\hat{I} = ({\hat{I}}^{P}, {\hat{I}}^{N})$ and is defined as ${\hat{I}}^{P} (a) = 1$ and ${\hat{I}}^{N} (a) = 0$ for all a ∈ A . The null bipolar fuzzy set in A is denoted by $\hat{O} = ({\hat{O}}^{P}, {\hat{O}}^{N})$ and is defined as ${\hat{O}}^{P} (a) = 0$ and ${\hat{O}}^{N} (a) = - 1$ for all a ∈ A . Thus, $\hat{I} (a) = (1, 0)$ and $\hat{O} (a) = (0, - 1)$ for all a ∈ A .

Definition 2.5. [13]. Let ω ₁, ω ₂ ∈ Ω. The union and intersection of ω ₁ and ω ₂ are the bipolar fuzzy sets in A, defined for all a ∈ A as below. $\begin{matrix} (ω_{1} \cup ω_{2}) (a) & = & (ω_{1}^{P} (a) \lor ω_{2}^{P} (a), ω_{1}^{N} (a) \land ω_{2}^{N} (a)), \\ (ω_{1} \cap ω_{2}) (a) & = & (ω_{1}^{P} (a) \land ω_{2}^{P} (a), ω_{1}^{N} (a) \lor ω_{2}^{N} (a)) . \end{matrix}$

Definition 2.6. [13]. The compliment ω′ of ω ∈ Ω is defined for all a ∈ A as below. $ω^{'} (a) = (1 - ω^{P} (a), - 1 - ω^{N} (a)) .$

3 Rough bipolar fuzzy sets

In this section, we crossbreed bipolar fuzzy sets with the Pawlak’s rough sets and introduce the notion of rough bipolar fuzzy (RBF) sets by defining the lower and upper RBF approximations of the bipolar fuzzy sets.

Definition 3.1. $(A, R)$ be a P-Ap space and let ω ∈ Ω. The lower and upper RBF approximations of ω with respect to $(A, R)$ are the bipolar fuzzy sets $\underline{R} (ω)$ and $\bar{R} (ω)$ in A, respectively, defined for all a ∈ A as: $\underline{R} (ω) (a) = (\underset{b \in [a]_{R}}{\land} ω^{P} (b), \underset{b \in [a]_{R}}{\lor} ω^{N} (b));$ (1) $\bar{R} (ω) (a) = (\underset{b \in [a]_{R}}{\lor} ω^{P} (b), \underset{b \in [a]_{R}}{\land} ω^{N} (b)) .$ (2)

If $\underline{R} (ω) \neq \bar{R} (ω)$ , then ω is said to be an RBF (rough bipolar fuzzy) set; otherwise, ω is an $R -$ definable bipolar fuzzy set in A .

Let $\underline{R} (ω) (a)$ and $\bar{R} (ω) (a)$ be denoted by $(\underline{ω^{P} (a)}, \underline{ω^{N} (a)})$ and $(\bar{ω^{P} (a)}, \bar{ω^{N} (a)})$ , respectively. Then, the information about the object “a” interpreted by these RBF approximations is as follows:

The degree of definite fulfilment of “a” to the property of ω is given by $\underline{ω^{P} (a)}$ .

The degree of definite fulfilment of “a” to the counter property of ω is given by $\underline{ω^{N} (a)}$ .

The degree of maximum possible fulfilment of “a” to the property of ω is given by $\bar{ω^{P} (a)}$ .

The degree of maximum possible fulfilment of “a” to the counter property of ω is given by $\bar{ω^{N} (a)}$ .

Theorem 3.2. Let $(A, R)$ be a P-Ap space and let ω, ω ₁, ω ₂ ∈ Ω. Then, the following assertions hold.

$\underline{R} (ω) \subseteq ω \subseteq \bar{R} (ω);$

$\underline{R} (\hat{I}) = \hat{I} = \bar{R} (\hat{I});$

$\underline{R} (\hat{O}) = \hat{O} = \bar{R} (\hat{O});$

$\underline{R} (\underline{R} (ω)) = \underline{R} (ω) = \bar{R} (\underline{R} (ω));$

$\underline{R} (\bar{R} (ω)) = \bar{R} (ω) = \bar{R} (\bar{R} (ω));$

$\bar{R} (ω^{'}) = (\underline{R} (ω))^{'};$

$\underline{R} (ω^{'}) = (\bar{R} (ω))^{'};$

$\underline{R} (ω_{1} \cap ω_{2}) = \underline{R} (ω_{1}) \cap \underline{R} (ω_{2});$

$\underline{R} (ω_{1} \cup ω_{2}) \supseteq \underline{R} (ω_{1}) \cup \underline{R} (ω_{2});$

$\bar{R} (ω_{1} \cup ω_{2}) = \bar{R} (ω_{1}) \cup \bar{R} (ω_{2});$

$\bar{R} (ω_{1} \cap ω_{2}) \subseteq \bar{R} (ω_{1}) \cap \bar{R} (ω_{2});$

ω ₁ ⊆ ω ₂ implies that $\underline{R} (ω_{1}) \subseteq \underline{R} (ω_{2})$ and $\bar{R} (ω_{1}) \subseteq \bar{R} (ω_{2})$ .

Proof.

These assertions follow from Definitions 2.3, 2.4 and 3.1.

We have $\underline{R} (ω) (b) = (\underset{c \in [b]}{\land} ω^{P} (c), \underset{c \in [b]}{\lor} ω^{N} (c))$ for all b ∈ A. Equation (1) gives $\begin{matrix} \underline{R} (\underline{R} (ω)) (a) \\ = (\underset{b \in [a]}{\land} (\underline{R} (ω))^{P} (b), \underset{b \in [a]}{\lor} (\underline{R} (ω))^{N} (b)) \\ = (\underset{b \in [a]}{\land} (\underset{c \in [b]}{\land} ω^{P} (c)), \underset{b \in [a]}{\lor} (\underset{c \in [b]}{\lor} ω^{N} (c))) \end{matrix}$ for all a ∈ A . As $R$ is equivalence relation and b ∈ [a], so [a] = [b]. This yields $\begin{matrix} \underline{R} (\underline{R} (ω)) (a) & = & (\underset{c \in [a]}{\land} ω^{P} (c), \underset{c \in [a]}{\lor} ω^{N} (c)) \\ = & \underline{R} (ω) (a) \end{matrix}$ for all a ∈ A . Thus, $\underline{R} (\underline{R} (ω)) = \underline{R} (ω) .$ Similarly $\bar{R} (\underline{R} (ω)) = \underline{R} (ω)$ .

It can be proved similar to (4).

We prove it by using Definition 2.6, as below. $\begin{matrix} \bar{R} (ω^{'}) (a) \\ = (\underset{b \in [a]}{\lor} (ω^{'})^{P} (b), \underset{b \in [a]}{\land} (ω^{'})^{N} (b)) \\ = (\underset{b \in [a]}{\lor} (1 - ω^{P} (b)), \underset{b \in [a]}{\land} (- 1 - ω^{N} (b))) \end{matrix}$

$\begin{matrix} = (1 - \underset{b \in [a]}{\land} ω^{P} (b), - 1 - \underset{b \in [a]}{\lor} ω^{N} (b)) \\ = (\underline{R} (ω))^{'} (a) \end{matrix}$ for all a ∈ A . Thus, $\bar{R} (ω^{'}) = (\underline{R} (ω))^{'} .$

It can be proved similar to (6).

We prove it by using Definition 2.5, as below. $\begin{matrix} \underline{R} (ω_{1} \cap ω_{2})^{P} (a) \\ = \underset{b \in [a]}{\land} (ω_{1}^{P} (b) \land ω_{2}^{P} (b)) \\ = (\underset{b \in [a]}{\land} ω_{1}^{P} (b)) \land (\underset{b \in [a]}{\land} ω_{2}^{P} (b)) \\ = (\underline{R} (ω_{1}) \cap \underline{R} (ω_{2}))^{P} (a) \end{matrix}$ for all a ∈ A . Similarly, $\underline{R} (ω_{1} \cap ω_{2})^{N} (a) = (\underline{R} (ω_{1}) \cap \underline{R} (ω_{2}))^{N} (a)$ for all a ∈ A . Thus, $\underline{R} (ω_{1} \cap ω_{2}) = \underline{R} (ω_{1}) \cap \underline{R} (ω_{2}) .$

Again by using Definition 2.5, we get $\begin{matrix} \underline{R} (ω_{1} \cup ω_{2})^{P} (a) \\ = \underset{b \in [a]}{\land} (ω_{1}^{P} (b) \lor ω_{2}^{P} (b)) \\ \geq (\underset{b \in [a]}{\land} ω_{1}^{P} (b)) \lor (\underset{b \in [a]}{\land} ω_{2}^{P} (b)) \\ = (\underline{R} (ω_{1}) \cup \underline{R} (ω_{2}))^{P} (a) \end{matrix}$ for all a ∈ A . Similarly $\underline{R} (ω_{1} \cup ω_{2})^{N} (a) \leq (\underline{R} (ω_{1}) \cup \underline{R} (ω_{2}))^{N} (a)$ for all a ∈ A . Thus, proved that $\underline{R} (ω_{1} \cup ω_{2}) \supseteq \underline{R} (ω_{1}) \cup \underline{R} (ω_{2}) .$

It can be proved similar to (8).

It can be proved similar to (9).

This assertion follows directly from Definition 2.3, Definition 3.1 and the assumption ω ₁⊆ ω ₂ □.

Following is a simple example to illustrate the above theorem.

Example 3.3. Let A = {a ₁, a ₂, a ₃, a ₄, a ₅, a ₆, a ₇} be a collection of some food items, such that, a ₁, a ₂ are cakes, a ₃, a ₄ are some chilli products and a ₅, a ₆, a ₇ are pies. Take a binary relation $R$ on A such that $(a_{i}, a_{j}) \in R$ if and only if a _i and a _j are of same type. Then, $R$ defines classes {a ₁, a ₂} , {a ₃, a ₄} and {a ₅, a ₆, a ₇}. In this example, we calculate the lower and upper RBF approximations of some bipolar fuzzy sets, and verify the assertions (1) and (11) of Theorem 3.2. We first take two bipolar fuzzy sets ω ₁ and ω ₂ describing the sweetness of food items in A, as below. $\begin{matrix} ω_{1} = {(a_{1}, 1, 0), (a_{2}, 0.8, - 0.18), \\ (a_{3}, 0.1, - 0.8), (a_{4}, 0, - 0.9), (a_{5}, 0.4, - 0.4), \\ (a_{6}, 0.5, - 0.2), (a_{7}, 0.4, - 0.35)}, \\ ω_{2} = {(a_{1}, 0.7, - 0.2), (a_{2}, 0.7, - 0.2), \\ (a_{3}, 0.1, - 0.9), (a_{4}, 0.1, - 0.9), (a_{5}, 0.5, - 0.4), \\ (a_{6}, 0.5, - 0.4), (a_{7}, 0.5, - 0.4)} . \end{matrix}$

The lower and upper RBF approximations of ω ₁ with respect to $R$ are calculated by using Equation (1), as below. $\begin{matrix} \underline{R} (ω_{1}) = {(a_{1}, 0.8, 0), (a_{2}, 0.8, 0), \\ (a_{3}, 0, - 0.8), (a_{4}, 0, - 0.8), (a_{5}, 0.4, - 0.2), \\ (a_{6}, 0.4, - 0.2), (a_{7}, 0.4, - 0.2)}, \\ \bar{R} (ω_{1}) = {(a_{1}, 1, - 0.18), (a_{2}, 1, - 0.18), \\ (a_{3}, 0.1, - 0.9), (a_{4}, 0.1, - 0.9), (a_{5}, 0.5, - 0.4), \\ (a_{6}, 0.5, - 0.4), (a_{7}, 0.5, - 0.4)} . \end{matrix}$

Comparing the values of $\underline{R} (ω_{1})$ and $\bar{R} (ω_{1})$ , we note that $\underline{R} (ω_{1}) \neq \bar{R} (ω_{1})$ . So, ω ₁ is a rough bipolar fuzzy set in A . Note that $\underline{R} (ω_{1}) \subseteq ω_{1} \subseteq \bar{R} (ω_{1})$ . Also notice that $\underline{R} (ω_{2}) = ω_{2} = \bar{R} (ω_{2})$ , by Definition 2.3. Which indicates that ω ₂ is $R -$ definable. For the verification of the assertion (11) of Theorem 3.2, we take another bipolar fuzzy set ω ₃ as below. $\begin{matrix} ω_{3} = {(a_{1}, 0.3, - 0.4), (a_{2}, 0.4, - 0.5), \\ (a_{3}, 0.6, - 0.3), (a_{4}, 0.2, - 0.5), (a_{5}, 0.5, - 0.5), \\ (a_{6}, 0.6, - 0.2), (a_{7}, 0.8, - 0.1)} . \end{matrix}$

Now we calculate the following quantities. $\begin{matrix} \bar{R} (ω_{3}) = {(a_{1}, 0.4, - 0.5), (a_{2}, 0.4, - 0.5), \\ (a_{3}, 0.6, - 0.5), (a_{4}, 0.6, - 0.5), (a_{5}, 0.8, - 0.5), \\ (a_{6}, 0.8, - 0.5), (a_{7}, 0.8, - 0.5)}, \\ ω_{1} \cap ω_{3} = {(a_{1}, 0.3, 0), (a_{2}, 0.4, - 0.18), \\ (a_{3}, 0.1, - 0.3), (a_{4}, 0, - 0.5), (a_{5}, 0.4, - 0.4), \\ (a_{6}, 0.5, - 0.2), (a_{7}, 0.4, - 0.1)}, \\ \bar{R} (ω_{1} \cap ω_{3}) = {(a_{1}, 0.4, - 0.18), (a_{2}, 0.4, - 0.18), \end{matrix}$

$\begin{matrix} (a_{3}, 0.1, - 0.5), (a_{4}, 0.1, - 0.5), (a_{5}, 0.5, - 0.4), \\ (a_{6}, 0.5, - 0.4), (a_{7}, 0.5, - 0.4)}, \\ \bar{R} (ω_{1}) \cap \bar{R} (ω_{3}) = {(a_{1}, 0.4, - 0.5), (a_{2}, 0.4, - 0.5), \\ (a_{3}, 0.1, - 0.5), (a_{4}, 0.1, - 0.5), (a_{5}, 0.5, - 0.4), \\ (a_{6}, 0.5, - 0.4), (a_{7}, 0.5, - 0.4)} . \end{matrix}$

Comparing the values of $\bar{R} (ω_{1} \cap ω_{3})$ and $\bar{R} (ω_{1}) \cap \bar{R} (ω_{3})$ , we observe that $\bar{R} (ω_{1} \cap ω_{3}) \subseteq \bar{R} (ω_{1}) \cap \bar{R} (ω_{3}) .$ Similarly, the other assertions can also be observed.

The assertions (8 -11) of Theorem 3.2 hold for any number of bipolar fuzzy sets in U . Thus, we have the following assertions for any indexing set I.

Theorem 3.4. Let $(A, R)$ be a P-Ap space and let {ω _i : i ∈ I} ⊆ Ω. Then, the following statements hold.

$\underline{R} (\underset{i \in I}{\cap} ω_{i}) = \underset{i \in I}{\cap} \underline{R} (ω_{i});$

$\underline{R} (\underset{i \in I}{\cup} ω_{i}) \supseteq \underset{i \in I}{\cup} \underline{R} (ω_{i});$

$\bar{R} (\underset{i \in I}{\cup} ω_{i}) = \underset{i \in I}{\cup} \bar{R} (ω_{i});$

$\bar{R} (\underset{i \in I}{\cap} ω_{i}) \subseteq \underset{i \in I}{\cap} \bar{R} (ω_{i}) .$

Proof. (1) Take ω _i ∈ Ω, where i ∈ I . Then, $\underset{i \in I}{\cap} ω_{i} = {(a, \underset{i \in I}{\land} ω_{i}^{P} (a), \underset{i \in I}{\lor} ω_{i}^{N} (a)) : a \in A} .$ By using Equation (1), we get $\begin{matrix} \underline{R} (\underset{i \in I}{\cap} ω_{i}) (a) \\ = (\underset{b \in [a]}{\land} (\underset{i \in I}{\land} ω_{i}^{P} (b)), \underset{b \in [a]}{\lor} (\underset{i \in I}{\lor} ω_{i}^{N} (b))) \\ = (\underset{i \in I}{\land} (\underset{b \in [a]}{\land} ω_{i}^{P} (b)), \underset{i \in I}{\lor} (\underset{b \in [a]}{\lor} ω_{i}^{N} (b))) \\ = \underset{i \in I}{\cap} \underline{R} (ω_{i}) (a) \end{matrix}$ for all a ∈ A . Hence, proved that $\underline{R} (\underset{i \in I}{\cap} ω_{i}) = \underset{i \in I}{\cap} \underline{R} (ω_{i}) .$

(2) For ω _i ∈ Ω, where i ∈ I, we have $\underset{i \in I}{\cup} ω_{i} = {(a, \underset{i \in I}{\lor} ω_{i}^{P} (a), \underset{i \in I}{\land} ω_{i}^{N} (a)) : a \in A} .$

By using Equation (1), we get $\begin{matrix} \underline{R} {(\underset{i \in I}{\cup} ω_{i})}^{P} (a) & = & \underset{b \in [a]}{\land} (\underset{i \in I}{\lor} ω_{i}^{P} (b)) \\ \geq & \underset{i \in I}{\lor} (\underset{b \in [a]}{\land} ω_{i}^{P} (b)) \\ = & \underset{i \in I}{\cup} \underline{R} (ω_{i})^{P} (a) \end{matrix}$ for all a ∈ A . Similarly $\underline{R} (\underset{i \in I}{\cup} ω_{i})^{N} (a) \leq \underset{i \in I}{\cup} \underline{R} (ω_{i})^{N} (a)$ for all a ∈ A . Hence, by Definition 2.3, it is proved that $\underline{R} (\underset{i \in I}{\cup} ω_{i}) \supseteq \underset{i \in I}{\cup} \underline{R} (ω_{i}) .$

(3) The proof is similar to the proof of (1).

(4) The proof is similar to the proof of (2).□

Theorem 3.5. Let $(A, R)$ be a P-Ap space and let ω ∈ Ω. Then, the following statements are equivalent.

$\bar{R} (ω) \subseteq ω;$

$ω \subseteq \underline{R} (ω);$

ω is $R -$ definable.

Proof. This follows from Theorem 3.2.□

The following theorem is note worthy as it highlights an interesting relation between lower and upper RBF approximations of a bipolar fuzzy set ω, when the equivalence relation $R$ in the P-Ap space is replaced by another equivalence relation σ on A, which contains $R$ .

Theorem 3.6. Let A be a nonempty set with two equivalence relations $R$ and σ on A, such that, $R \subseteq σ$ . Then, the assertions $\underline{σ} (ω) \subseteq \underline{R} (ω)$ and $\bar{R} (ω) \subseteq \bar{σ} (ω)$ hold for all ω ∈ Ω.

Proof. Take any ω ∈ Ω. Since $R \subseteq σ$ , we get $[a]_{R} \subseteq [a]_{σ}$ for all a ∈ A . Thus, $\underset{b \in [a]_{σ}}{\land} ω^{P} (b) \leq \underset{b \in [a]_{R}}{\land} ω^{P} (b)$ and $\underset{b \in [a]_{σ}}{\lor} ω^{N} (b) \geq \underset{b \in [a]_{R}}{\lor} ω^{N} (b)$ hold for all a ∈ A . So, Definition 2.3 gives $\underline{σ} (ω) \subseteq \underline{R} (ω)$ . Similarly, one can verify that $\bar{R} (ω) \subseteq \bar{σ} (ω)$ .□

Theorem 3.7. Let A be a nonempty set with two equivalence relations $R$ and σ. Then, the following assertions hold for all ω ∈ Ω.

$\bar{(R \cap σ)} (ω) \subseteq \bar{R} (ω) \cap \bar{σ} (ω);$

$\underline{(R \cap σ)} (ω) \supseteq \underline{R} (ω) \cup$ $\underline{σ} (ω) .$

Proof. (1) Let $R$ and σ be two equivalence relations on A. Then, $R \cap σ$ is also an equivalence relation on A. It is also clear that $R \cap σ \subseteq R$ and $R \cap σ \subseteq σ$ . By Theorem 3.6, we have $\bar{(R \cap σ)} (ω) \subseteq \bar{R} (ω)$ and $\bar{(R \cap σ)} (ω) \subseteq \bar{σ} (ω)$ for any ω ∈ Ω. Which yields $\bar{(R \cap σ)} (ω) \subseteq \bar{R} (ω) \cap \bar{σ} (ω)$ for all ω ∈ Ω.

(2) The proof is similar to the proof of (1).□

Corresponding to each ordered pair (x, y), where x ∈ [0, 1] and y ∈ [-1, 0], we can define a bipolar fuzzy set $ω_{xy} = (ω_{xy}^{P}, ω_{xy}^{N})$ in A, such that, $ω_{xy}^{P}$ and $ω_{xy}^{N}$ are constant maps defined by $ω_{xy}^{P} (a) = x$ and $ω_{xy}^{N} (a) = y$ for all a ∈ A . Such a bipolar fuzzy set is termed as the constant bipolar fuzzy set.

Theorem 3.8. Let $(A, R)$ be a P-Ap space.

The constant bipolar fuzzy set ω _xy in A is $R -$ definable for each x ∈ [0, 1] and y ∈ [-1, 0].

Let $R$ be the universal binary relation on A . A bipolar fuzzy set ω in A is $R -$ definable if and only if ω is a constant bipolar fuzzy set in A.

Each bipolar fuzzy set ω in A is $R -$ definable, whenever $R$ is the diagonal binary relation on A .

Proof. (1) Take a constant bipolar fuzzy set ω _xy in A, where $ω_{xy}^{P} (b) = x$ and $ω_{xy}^{N} (b) = y$ for all b ∈ A . Then, $\underset{b \in [a]}{\land} ω_{xy}^{P} (a) = x = \underset{b \in [a]}{\lor} ω_{xy}^{P} (a)$ and $\underset{b \in [a]}{\land} ω_{xy}^{N} (b) = y = \underset{b \in [a]}{\lor} ω_{xy}^{N} (b)$ for any a ∈ A . Thus, we have $\begin{matrix} \underline{R} (ω_{xy}) & = & {(a, \underset{b \in [a]}{\land} ω_{xy}^{P} (b), \underset{b \in [a]}{\lor} ω_{xy}^{N} (b)) : a \in A} \\ = & {(a, \underset{b \in [a]}{\lor} ω_{xy}^{P} (b), \underset{b \in [a]}{\land} ω_{xy}^{N} (b)) : a \in A} \\ = & \bar{R} (ω_{xy}) . \end{matrix}$ Which shows that ω _xy is $R -$ definable.

(2) Let ω be $R -$ definable. Then $\underline{R} (ω) = \bar{R} (ω)$ . Which gives $\underline{R} (ω) (a) = \bar{R} (ω) (a)$ for all a ∈ A. So, for all a ∈ A, we have $\underset{b \in [a]}{\land} ω^{P} (b) = \underset{b \in [a]}{\lor} ω^{P} (b)$ and $\underset{b \in [a]}{\lor} ω^{N} (b) = \underset{b \in [a]}{\land} ω^{N} (b)$ by Definition 3.1. Also, [a] = A for all a ∈ A, as $R = A \times A .$ So, we get $\underset{b \in A}{\land} ω^{P} (b) = \underset{b \in A}{\lor} ω^{P} (b)$ and $\underset{b \in A}{\land} ω^{N} (b) = \underset{b \in A}{\lor} ω^{N} (b) .$ This clearly indicates that ω is a constant bipolar fuzzy set ω _xy, where $x = \underset{b \in A}{\lor} ω^{P} (b)$ and $y = \underset{b \in A}{\lor} ω^{N} (b)$ .

Converse holds by (1).

(3) Straightforward.□

The converse of (1) of the above theorem does not hold, in general. This can be seen in Example 3.3, where the bipolar fuzzy set ω ₂ is $R -$ definable, but not constant.

4 An algorithm for a GDM problem

Data analysis in many areas necessitates different problems related to the decision making. Many algorithms are designed by the researchers, in this regard, to find a best decision. All those algorithms provide the decision to choose the best object. But, in some circumstances, the best decision becomes difficult to be taken and we have to look for another better option. So, it will always be advantageous if the poor decision becomes apparent, in order to avoid the poor decision, as well. We present an efficient computational algorithm which provides the best, as well as, the poor decision. The collection of k objects to be considered is denoted by A = {a _j : 1 ≤ j ≤ k} and the collection of the bipolar fuzzy sets describing the opinions of m decision makers is denoted by ð = {ω _i : 1 ≤ i ≤ m}. The information about the objects a _j, provided by each ω _i, is represented by a table (called the table of ð) with (i, j) th entry as ω _i (a _j) = (x _ij, y _ij). First, we assign the indiscernibility grades to each object under consideration relative to each bipolar fuzzy set ω _i and then, we define the indiscernibility relations between the objects.

Definition 4.1. The indiscernibility grades G _ij corresponding to each object a _j ∈ A and each bipolar fuzzy set ω _i ∈ ð , are assigned using the formula given below. $G_{i j} = {\begin{matrix} P & i f x_{i j} \underset{\neq}{>} | y_{i j} | \\ N & i f x_{i j} \underset{\neq}{<} | y_{i j} | \\ O & i f x_{i j} = | y_{i j} | \end{matrix}$ (3) where (x _ij, y _ij) = ω _i (a _j).∥The indiscernibility grades depict the following information about the objects.

If G _ij = P, the object a _j has positive membership value x _ij higher than the negative membership value |y _ij|, with respect to ω _i.

If G _ij = N, the object a _j has negative membership value |y _ij| higher than the positive membership value x _ij, with respect to ω _i.

If G _ij = O, the object a _j has positive membership x _ij equal to the negative membership |y _ij|, with respect to ω _i.

∥Now we give the concept of indiscernibility relations on A associated with the bipolar fuzzy sets in ð. We say that two objects a _j and a _k are indiscernible, written as a _j ∼ a _k, if and only if they have same grades for each ω _i. Thus, when we say that the objects a _j and a _k are indiscernible, it means that, either both the objects have positivity higher than the negativity, or both the objects have negativity higher than the positivity, or both the objects have equal measures of positivity and negativity. The indiscernibility relation

R

between the objects of A is constructed using the formula:

R = {(a_{j}, a_{k}) \in A \times A : a_{j} \sim a_{k}} .

(4) Surely,

R

is an equivalence relation on A .∥Now, we proceed to the decision values d _j for the objects. For the equivalence relation

R

, we write

\underline{R} (ω_{i}) (a_{j}) = (\underline{x_{ij}}, \underline{y_{ij}})

and

\bar{R} (ω_{i}) (a_{j}) = (\bar{x_{ij}}, \bar{y_{ij}})

. Denote

\underline{z_{j}} = \overset{m}{\underset{i = 1}{Σ}} (\underline{x_{ij}} - | \underline{y_{ij}} |),

(5)

\bar{z_{j}} = \overset{m}{\underset{i = 1}{Σ}} (\bar{x_{ij}} - | \bar{y_{ij}} |) .

(6) ∥Then,

\underline{z_{j}}

represents the definite fulfilment of the object a _j, while,

\bar{z_{j}}

represents the maximum possible fulfilment of the object a _j, towards all decision makers {ω _i : 1 ≤ i ≤ m}. Thus, the uncertain (or doubtful) fulfilment of a _j is given by the difference

\bar{z_{j}} - \underline{z_{j}}

. The decision will be taken on the basis of the decision parameter, as defined below.∥Definition 4.2. The decision parameter D has the values d _j corresponding to each object a _j ∈ A, given by

d_{j} = \underline{z_{j}} + \bar{z_{j}} .

(7) This value gives the definite fulfilment

\underline{z_{j}}

of the object a _j, a double weightage than to the uncertain (doubtful) fulfilment

\bar{z_{j}} - \underline{z_{j}}

, because we have

d_{j} = \underline{z_{j}} + \bar{z_{j}} = 2 \underline{z_{j}} - \underline{z_{j}} + \bar{z_{j}},

d_{j} = 2 \underline{z_{j}} + (\bar{z_{j}} - \underline{z_{j}}) .

(8) We can rewrite Equation (8) as:

d_{j} = \overset{m}{\underset{i = 1}{Σ}} (2 (\underline{x_{ij}} - | \underline{y_{ij}} |) + (\bar{x_{ij}} - | \bar{y_{ij}} |) - (\underline{x_{ij}} - | \underline{y_{ij}} |))

(9) From Equation (9), it is clear that the higher the definite positive fulfilment

\underline{x_{ij}}

of a _j the larger the value d _j. Also, the higher the definite negative fulfilment

| \underline{y_{ij}} |

of a _j the smaller the value d _j. In this way, we identify the poor objects having lowest value of d _j. These are the objects with high definite negative fulfilment to ω _i. Hence, our algorithm has the following main advantages.

It manipulates technically the fuzziness of the data enriched with the bipolarity of information.

It accommodates the opinions of any (finite) number of decision makers about any (finite) number of objects.

It gives double weightage to the definite fulfilment of the objects, than to the uncertain fulfilment.

It yields a wise decision, containing the best, as well as, the poor decision, so that, one can sidestep the poor decision.

∥Main steps of the algorithm are as follows.∥ The algorithm to decide for the best and poor objects in A, is given below.∥Step I: Input the bipolar fuzzy sets ð.∥Step II: Find out the equivalence relation

R

on A, using Formula 4.∥Step III: Evaluate

\underline{R} (ω_{i})

and the values

\underline{z_{j}}

using Equation (1) and Equation (5), respectively.∥Step IV: Evaluate

\bar{R} (ω_{i})

and the values

\bar{z_{j}}

using Equation (2) and Equation (6), respectively.∥Step V: Find the decision values d _j for each object a _j ∈ A, using Formula 7.∥Step VI: Construct the decision table with rows of A and the decision parameter D in the descending order with respect to the values of D. Choose p and q, so that

d_{p} = max_{j}

d _j and

d_{q} = min_{j}

d _j. Then, a _p is the best optimal object, while, a _q is the poor object to be decided. If p has more than one values, any one of a _p’s can be selected.

5 An illustrative GDM problem

In this section, we consider a GDM problem and apply the algorithm designed in the previous section to solve it. First, we discuss our problem, then we apply and illustrate all the steps of the Algorithm 4 to our GDM problem, as follows.

A GDM problem:

Let A = {a ₁, a ₂, a ₃, a ₄, a ₅, a ₆} be a collection of some similar products and a company X wishes to decide for one product to manufacture. Let ð = {ω ₁, ω ₂, ω ₃, ω ₄} be a collection of the bipolar fuzzy sets describing the opinions of four independent experts, who are assigned by the company to decide in the favour of a single product. Here, we have 1 ≤ i ≤ 4 and 1 ≤ j ≤ 6.

Step I:

The information about the objects a _j, provided by each ω _i, is represented by the Table 1, where the (i, j) th entry ω _i (a _j) = (x _ij, y _ij) describes the opinion of ω _i about the product a _j. The value x _ij represents the degree to which the product a _j is suitable for being manufactured and the value y _ij represents the degree to which a _j is not favorable for production, according to the opinion of the expert ω _i.

Table 1
Table of ð

ð a ₁ a ₂ a ₃ a ₄ a ₅ a ₆

ω ₁ (0.6, - 0.3) (0.7, - 0.4) (0.5, - 0.5) (0.6, - 0.6) (0.7, - 0.3) (0.4, - 0.4)

ω ₂ (0.6, - 0.4) (0.5, - 0.5) (0.6, - 0.3) (0.4, - 0.5) (0.5, - 0.5) (0.3, - 0.4)

ω ₃ (0.7, - 0.2) (0.6, - 0.3) (0.5, - 0.2) (0.5, - 0.5) (0.6, - 0.4) (0.4, - 0.4)

ω ₄ (0.5, - 0.4) (0.3, - 0.3) (0.5, - 0.5) (0.5, - 0.3) (0.4, - 0.4) (0.5, - 0.3)

ð	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
ω ₁	(0.6, - 0.3)	(0.7, - 0.4)	(0.5, - 0.5)	(0.6, - 0.6)	(0.7, - 0.3)	(0.4, - 0.4)
ω ₂	(0.6, - 0.4)	(0.5, - 0.5)	(0.6, - 0.3)	(0.4, - 0.5)	(0.5, - 0.5)	(0.3, - 0.4)
ω ₃	(0.7, - 0.2)	(0.6, - 0.3)	(0.5, - 0.2)	(0.5, - 0.5)	(0.6, - 0.4)	(0.4, - 0.4)
ω ₄	(0.5, - 0.4)	(0.3, - 0.3)	(0.5, - 0.5)	(0.5, - 0.3)	(0.4, - 0.4)	(0.5, - 0.3)

Step II:

For the relation $R,$ first we assign the indiscernibility grades to each object, corresponding to each bipolar fuzzy set ω _i∈ ð, using Formula 3, as in Table 2.

Table 2

Assignment of indiscernibility grades G _ij

A	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
G _1j	P	P	O	O	P	O
G _2j	P	O	P	N	O	N
G _3j	P	P	P	O	P	O
G _4j	P	O	O	P	O	P

From Table 2, it can be clearly seen that, a ₂ and a ₅ got the same grades, while, a ₄ and a ₆ received the same grades. So, the Formula 4 leads to the following equivalence relation on A. $\begin{matrix} R & = & {(a_{1}, a_{1}), (a_{2}, a_{2}), (a_{3}, a_{3}), (a_{4}, a_{4}), (a_{5}, a_{5}), \\ (a_{6}, a_{6}), (a_{2}, a_{5}), (a_{5}, a_{2}), (a_{4}, a_{6}), (a_{6}, a_{4})} . \end{matrix}$ This relation serves as our key tool to evaluate the RBF approximations.

Step III:

The lower RBF approximations $\underline{R} (ω_{i})$ of each ω _i and the values $\underline{z_{j}}$ are evaluated in Table 3, by using the Equation (1) and Equation (5), respectively. Step IV:

Table 3

Calculations of $\underline{R} (ω_{i})$ and $\underline{z_{j}}$

$\underline{R} (ω_{i})$	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
$\underline{R} (ω_{1})$	(0.6, - 0.3)	(0.7, - 0.3)	(0.5, - 0.5)	(0.4, - 0.4)	(0.7, - 0.3)	(0.4, - 0.4)
$\underline{R} (ω_{2})$	(0.6, - 0.4)	(0.5, - 0.5)	(0.6, - 0.3)	(0.3, - 0.4)	(0.5, - 0.5)	(0.3, - 0.4)
$\underline{R} (ω_{3})$	(0.7, - 0.2)	(0.6, - 0.3)	(0.5, - 0.2)	(0.4, - 0.4)	(0.6, - 0.3)	(0.4, - 0.4)
$\underline{R} (ω_{4})$	(0.5, - 0.4)	(0.3, - 0.3)	(0.5, - 0.5)	(0.5, - 0.3)	(0.3, - 0.3)	(0.5, - 0.3)
$\underline{z_{j}}$	1.1	0.7	0.6	0.1	0.7	0.1

The upper RBF approximations $\bar{R} (ω_{i})$ of each ω _i and the values $\bar{z_{j}}$ are evaluated in Table 4, by using the Equation (2) and Equation (6), respectively. Step V:

Table 4

Calculations of $\bar{R} (ω_{i})$ and $\bar{z_{j}}$

$\bar{R} (ω_{i})$	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
$\bar{R} (ω_{1})$	(0.6, - 0.3)	(0.7, - 0.4)	(0.5, - 0.5)	(0.6, - 0.6)	(0.7, - 0.4)	(0.6, - 0.6)
$\bar{R} (ω_{2})$	(0.6, - 0.4)	(0.5, - 0.5)	(0.6, - 0.3)	(0.4, - 0.5)	(0.5, - 0.5)	(0.4, - 0.5)
$\bar{R} (ω_{3})$	(0.7, - 0.2)	(0.6, - 0.4)	(0.5, - 0.2)	(0.5, - 0.5)	(0.6, - 0.4)	(0.5, - 0.5)
$\bar{R} (ω_{4})$	(0.5, - 0.4)	(0.4, - 0.4)	(0.5, - 0.5)	(0.5, - 0.3)	(0.4, - 0.4)	(0.5, - 0.3)
$\bar{z_{j}}$	1.1	0.5	0.6	0.1	0.5	0.1

The decision values d _j for each a _j ∈ A are determined in Table 5, using Formula 7.

Table 5

Calculations of decision values d _j

A	a ₁	a ₂	a ₃	a ₄	a ₅	a ₆
$\underline{z_{j}}$	1.1	0.7	0.6	0.1	0.7	0.1
$\bar{z_{j}}$	1.1	0.5	0.6	0.1	0.5	0.1
d _j	2.2	1.2	1.2	0.2	1.2	0.2

Step VI:

Now we construct our decision table by placing the set A in first row and the decision parameter D in the second row. The table is rearranged in the descending order with respect to the values of D. Decision table is given by the Table 6.

Table 6

Decision table

A	a ₁	a ₂	a ₃	a ₅	a ₄	a ₆
D	2.2	1.2	1.2	1.2	0.2	0.2

We get $max_{j}$ d _j = d ₁ = 2.2 and $min_{j}$ d _j = d ₄ = d ₆ = 0.2. Hence p = 1 and q = 4, 6. Thus, the product a ₁ is the best decision, while a ₄ and a ₆ are the poor selections. So, the company can manufacture the product a ₁ for maximum revenue. The second best option may be a ₂, a ₃ or a ₅ . But, in any case, it should not go for a ₄ or a ₆.

6 Conclusion

In this article, we have introduced a general approach for roughness in the bipolar fuzzy sets and presented the RBF sets. Some algebraic properties of the RBF approximations of the bipolar fuzzy sets have also been studied. We have addressed a GDM problem with the data having fuzziness, as well as, bipolarity and applied the RBF approximations to iron out this problem. An algorithm is also proposed for this problem. This algorithm has four main advantages over the existing algorithms. Firstly, it manipulates the fuzziness and bipolarity of the data, endowed with uncertainty. Secondly, the definite fulfilment of the objects is given double weightage than to the uncertain fulfilment. Thirdly, this algorithm accommodates the opinions of any (finite) number of decision makers about any (finite) number of objects. Fourthly, it also yields the poor decision, along with the best decision. Hence, the decisions figured out by this algorithm are more accurate and reliable. Further study can be done to investigate the roughness in different bipolar fuzzy substructures and to establish fruitful algorithms for different kinds of decision making problems.

Declarations of interest

None.

Funding source

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Footnotes

Acknowledgement

The authors are thankful to the Editor, Associate Editor Prof. Yucheng Dong and the anonymous referees for their useful comments and suggestions, which have helped to improve this paper.

References

Ahsan

, Mordeson

J.N.

, Shabir

, Fuzzy semirings withapplications to automata theory, Springer-Verlag Heidelberg, (2012) ISBN 978-3-642-27641-5.

Akram

, Dudek

W.A.

, Regular bipolar fuzzy graphs, Neural Comput.Appl. 21(1) (2012), 197–205.

Cagman

, Citak

, Enginoglu

, Fuzzy parameterized fuzzy softset theory and its applications, Turk J. Fuzzy Syst. 1 (2010), 21–35.

Chen

, Zhang

, Dong

Y.C.

, The fusion process with heterogeneouspreference structures in group decision making: A survey, Inf. Fusion 24(2015), 72–83.

Dong

Y.C.

, Li

C.C.

, Francisco

, Connecting the linguistichierarchy and the numerical scale for the 2-tuple linguistic model and itsuse to deal with hesitant unbalanced linguistic information, Inf. Sciences 367–368 (2016), 259–278.

Dong

Y.C.

, Zha

, Zhang

, Kou

, Fujita

, Chiclana

, Herrera-Viedma

, Consensus reaching in social network group decisionmaking: Research paradigms and challenges, Knowledge-based syst. (in press) doi: 10.1016/j.knosys.2018.06.036 .

Dong

Y.C.

, Zhang

, Herrera-Viedma

, Consensus reaching modelin the complex and dynamic MAGDM problem, Knowledge-based syst. 106 (2016), 206–219.

Dubois

, Prade

, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst. 17(2–3) (1990), 191–209.

Dubois

, Prade

, Fuzzy sets: a survey of engineeringapplications, Comput. Chem. Engng. 17(1) (1993), 373–380.

10.

Dubois

, Prade

, An introduction to bipolar representationsof information and preferences, Int. J. Intell. Syst. 23(8) (2008), 866–877.

11.

Feng

, Changuing

Li.

, Davvaz

, Ali

M.I.

, Soft sets combined withfuzzy sets and rough sets: a tentative approach, Soft Comput. 14 (2010), 899–911.

12.

Feng

, Jun

Y.B.

, Liu

, Li

, An adjustable approach to fuzzysoft set based decision-making, J. Comput. Appl. Math. 234(1) (2010), 10–20.

13.

Lee

K.M.

, Bipolar-valued fuzzy sets and their basic operations, Proc. Int. Conf. on Intell. Technologies, Bangkok Thailand (2000), 307–317.

14.

Lee

K.M.

, Comparison of interval-valued fuzzy sets,intuitionistic fuzzy sets and bipolar-valued fuzzy sets, J. Fuzzy LogicIntell. Syst. 14 (2004), 125–129.

15.

C.C.

, Rodríguez

R.M.

, Martínez

, Dong

Y.C.

, Francisco

, Consistency of hesitant fuzzy linguistic preference relations: Aninterval consistency index, Inf. Sciences 432 (2018), 347–361.

16.

Malik

, Shabir

, Rough fuzzy bipolar soft sets andapplications in decision-making problems, Soft Comput. (2017) doi: 10.1007/s00500-017-2883-1 .

17.

Pawlak

, Rough sets, Int. J. Inf. Comput. Sci. 11 (1982), 341–356.

18.

Pawlak

, Skowron

, Rudiments of rough sets, Inf. Sci. 177(2007), 3–27.

19.

Perez

I.J.

, Cabrerizo

F.J.

, Alonso

, Dong

Y.C.

, Chiclana

, Herrera-Viedma

, On dynamic consensus processes in group decision makingproblems, Inf. Sciences 459 (2018), 20–35.

20.

Shabir

, Ali

M.I.

, Soft ideals and generalized fuzzy ideals in semigroups, New Mathematics and Neural Computation 5(3) (2009), 599–615.

21.

Shabir

, Jun

Y.B.

, Nawaz

, Characterizations of regular semgroups by (α, β)-fuzzy ideals, Comput. Math. with Appl. 59(1) (2010), 161–175.

22.

Shabir

, Jun

Y.B.

, Nawaz

, Semigroups characterized by (∈, ∈ ∨ qk)-fuzzy ideals, Comput. Math. with Appl. 60(5) (2010), 1473–1493.

23.

Shabir

, Malik

, Mahmood

, Characterizations of hemirings bythe properties of their interval-valued fuzzy ideals, Annals of Fuzzy Math. Inf. 3(2) (2012), 229–242.

24.

Shabir

, Malik

, Mahmood

, On interval-valued fuzzy idealsin hemirings, Annals of Fuzzy Math. Inf. 4(1) (2012), 49–62.

25.

Shabir

, Malik

, Mahmood

, On interval-valued fuzzy h-idealsin hemirings, East Asian Math. J. 28(1) (2012), 49–62.

26.

Torra

, Hesitant fuzzy sets, Int. J. Intell. Syst. doi: 10.1002/int.20418 .

27.

, Li

C.C.

, Chen

, Dong

Y.C.

, Group decision making based onlinguistic distributions and hesitant assessments: Maximizing the supportdegree with an accuracy constraint, Inf. Fusion 41 (2018), 151–160.

28.

Yejun

, Lei

, Rosa

M.R.

, Francisco

, Huimin

, Deriving thepriority weights from incomplete hesitant fuzzy preference relations ingroup decision making, Knowledge-based syst. 99 (2016), 71–78.

29.

Yejun

, Francisco

J.C.

, Herrera

V.E.

, A consensus model forhesitant fuzzy preference relations and its application in water allocationmanagement, Appl. Soft Comput. 58 (2017), 265–284.

30.

Zadeh

L.A.

, Fuzzy sets, Inf. Control 8 (1965), 338–353.

31.

Zhan

, Ali

M.I.

, Mehmood

, Z-soft fuzzy rough set model andcorresponding decision-making methods, Appl. Soft Comput. 56 (2017), 446–457.

32.

Zhang

, Iván

, Dong

Y.C.

, Wang

, Managing non-cooperative behaviors in consensus-based multiple attribute group decision making: Anapproach based on social network analysis, Knowledge-based syst. (in press) doi: 10.1016/j.knosys.2018.06.008 .

33.

Zhang

W.R.

, Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis, Proc. of the Industrial Fuzzy Control and Intell. Syst. Conf. and NASA Joint Technology Workshop on Neural Networks & Fuzzy Logic and Fuzzy Inf. Processing Society Biannual Conf. San Antonio, Teu USA (1994), 305–309.