A decision making approach based on multi-fuzzy bipolar soft sets

Abstract

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with vagueness and uncertainties. It is worth noting that the decision making theory has been proven a more effective tool in the real-world problems under imprecise environments. In this paper, we introduce a new fuzzy soft model, called the multi-fuzzy bipolar soft set model. Some operations of this notion are first investigated and some of the related properties are studied. Finally, an algorithm based on multi-fuzzy bipolar soft set is presented and an illustrative example is given to analyze the application of the proposed algorithm.

Keywords

Soft set bipolar soft set fuzzy soft set multi-fuzzy set multi-fuzzy bipolar soft set

1 Introduction

The real world is full of uncertainty, imprecision and vagueness occured in several branches of sciences. Actually most of the concepts we meet in everyday life are vague and imprecise. Dealing with uncertainties is a major problem in many areas such as economics, engineering, environmental science, medical science and social science. So many authors have been engaged in modeling vagueness in recent several years. Classical theories like probability theory, ordinary fuzzy set theory [17], rough set theory [2], and many others are well known theories used in modeling uncertainty and play important roles in modeling vagueness. However, with the rapidly increasing quantity and types of uncertainties, these theories show their inherent difficulties as pointed out by Molodtsov in [1]. In 1999, Molodtsov initiated soft set theory as a completely new mathematical tool for dealing with uncertainties that is free from the difficulties affecting the existing methods [1]. Recently, Sebastain and Ramakarishnan in [13], proposed the concept of a multi-fuzzy set which is a more general form of an ordinary fuzzy set (or Zadeh’s fuzzy set) and Zadeh’s fuzzy set theory is a building block for multi-fuzzy set [17]. The membership function of a multi-fuzzy set is an ordered sequence of ordinary fuzzy membership function. The notion of multi-fuzzy sets provides a new method to solve some problems which are difficult to handle by ordinary fuzzy set or other extension of Zadeh’s fuzzy set. For example, colour of a pixel cannot be represented by a membership function of an ordinary fuzzy set, but it is possible to characterize by a three dimensional membership function (μ_r,μ_g,μ_b); μ_r, μ_g, and μ_b are the membership functions from 1,...,m×1,...,n into [0, 1]. So a colour image can be approximated by a collection of pixels with a multi-membership function (μ_r,μ_g,μ_b) [14]. On the other hand, a soft set has many extensions, for example fuzzy soft set, intuitionistic fuzzy soft set, bipolar soft set, double-framed soft set, bipolar soft, fuzzy bipolar soft set and many others. In [8], Shabir et al. presented an extension of fuzzy soft set, called fuzzy bipolar soft set and applied this concept in a decision making problem. In [8], Shabir et al., followed the method of Roy et al., and presented an algorithm for the identification of an object, which is based on the comparison of different objects in the context of fuzzy bipolar soft set theory. In this paper, we follow Feng et al. [11] method and present a novel approach to fuzzy bipolar soft set based decision making problems by using level bipolar soft sets.

The purpose of this paper is to combine the multi-fuzzy set and soft set, from which we can obtain a new soft set model named multi-fuzzy bipolar soft set model. To facilitate our study, we first review some background of soft set, fuzzy soft set and fuzzy bipolar soft set in Section 2. In Section 3, the concept of a multi-fuzzy set and recall its basic properties in detail. In Section 4, we develop the concept of a multi-fuzzy bipolar soft set and investigate the basic properties of multi-fuzzy bipolar soft set. Finally, we conclude the paper for future work to be addressed.

2 Preliminaries

2.1 Soft sets

Let E be a non-empty finite set of attributes (parameters,characteristics or properties) which the objects in U possess and let P(U) denote the family of all subsets of U. Then a soft set is defined with the help of a set-valued mapping as given below:

2.1.1 Definition

(Molodtsov 1999 [1]) A pair (F,A) is called a soft set over U, where A ⊆ E and F:A⟶P(U) is a set-valued mapping.

In other words, a soft set (F,A) over U is a paramterized family of subsets of U where each parameter eɛA is associated with a subset F(e) of U. The set F(e) contains the objects of U having the property e and is called the set of e-approximate elements in (F,A).

2.1.2 Definition

(Shabir et al. [9]) A triplet (F,G,A) is called a bipolar soft set over U, where F and G are mappings, given by F:A⟶P(U) and G:A⟶P(U) such that F (e)∩ G (¬ e) = ∅ for all e ∈ A

In other words, a bipolar soft set over U gives two parametrized families of subsets of the universe U and the condition F (e)∩ G (¬ e) = ∅ for all e ∈ A, is imposed as a consistency constraint. For each e ∈ A, F(e) and G (¬ e) are regarded as the set of e-approximate elements of the bipolar soft set (F,G,A).

2.2 Fuzzy bipolar soft sets

A fuzzy bipolar soft set (Naz and Shabir 2014) is an extension of fuzzy soft set, infact, a fuzzy bipolar soft set is obtained with the help of two set-valued mappings by considering not only a set of parameters but also an allied set of carefully chosen parameters with opposite measuring termed as “not set of parameters”. The materials included in this section, are taken from Naz and Shabir (2014) [8].

2.2.1 Definition

(Naz and Shabir 2014 [8, 10]) A triplet (F,G,A) is called a fuzzy bipolar soft set over U, where A ⊆ E and F,G are mappings given by F:A⟶FP(U) and G:A⟶FP(U) such that F(e)∈[0, 1],G(e)∈[0, 1] and F (e) (u) + G (¬ e) (u) ≤1 for all e ∈ A and u ∈ U.

2.2.2 Example

Let U = {h₁, h₂, h₃, h₄, h₅} be a set of houses and E=e₂=expensive, e₂=beautiful, e₃ = wooden, e₄ = ingreen surrounding, e₅ = ingoodrepair is a set of parameters for U. Let “the not set of E” be ¬E, where ¬E=¬e₁=cheap, ¬e₂=ugly, ¬e₃=not wooden, ¬e₄=in commercial area, ¬e₅=in bad repair.

Define a fuzzy bipolar soft set (F, G, A) over U, where A = {e₁, e₂, e₃} describing the opinion of Mr. X who wants to purchase a house possessing the attributes of A = {e₁, e₂, e₃} Assuming that Mr. X wants to give the membership values {0.8, 0.7, 0.6, 0.5, 0.4} and {0.1, 0.2, 0.4, 0.5, 0.6} to houses of U for the attributes e_i(i = 1, 2, 3) denoting the degrees of memberships stands for (expensive,cheap), (beautiful,ugly), (wooden, not wooden) respectively. Then, F(e_i) (i = 1,2,3) and G(e_i) (i = 1,2,3) are given as follows: $F (e_{1}) = {h_{1} / 0.7, h_{2} / 0.6, h_{3} / 0.8, h_{4} / 0.5, h_{5} / 0.6}$ $G (\neg e_{1}) = {h_{1} / 0.2, h_{2} / 0.4, h_{3} / 0.2, h_{4} / 0.5, h_{5} / 0.4}$ $F (e_{2}) = {h_{1} / 0.8, h_{2} / 0.7, h_{3} / 0.8, h_{4} / 0.6, h_{5} / 0.3}$ $G (\neg e_{2}) = {h_{1} / 0.2, h_{2} / 0.2, h_{3} / 0.1, h_{4} / 0.4, h_{5} / 0.7$ $F (e_{3}) = {h_{1} / 0.4, h_{2} / 0.6, h_{3} / 0.4, h_{4} / 0.6, h_{5} / 0.5}$ $G (\neg e_{3}) = {h_{1} / 0.5, h_{2} / 0.4, h_{3} / 0.6, h_{4} / 0.4, h_{5} / 0.5}$

2.3 Multi-fuzzy sets

2.3.1 Definition [12, 14]

Let k be a positive integer, a multi-fuzzy set A of dimension k in U is a set of ordered sequences $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . . μ_{k} (u)) : u \in U}$ where μ_i ∈ FP (U) , . . . i = 1, 2, . . . , k.

The function $μ_{\tilde{A}} = (μ_{1}, μ_{2}, . . ., μ_{k} (u))$ is called the multi-membership function of multi-fuzzy set $\tilde{A}$ , k is called dimension of $\tilde{A}$ . The set of all multi-fuzzy sets of dimension k in U is denoted by M^k FP(U).

Clearly, a multi-fuzzy set of dimension 1 is an ordinary fuzzy set, and a multi-fuzzy set of dimension 2 with μ₁ (μ) + μ₂ (μ) ≤1 is an Atanassov intuitionistic fuzzy set.

2.3.2 Remark

If $(F, G, A) \tilde{⊑} (F_{1}, G_{1}, B)$ $\sum_{i = 1}^{k} μ_{i} (u) \leq 1$ fuzzy set of dimension k is called a normalized multi-fuzzy set. If

$\sum_{i = 1}^{k} μ_{i} (u) = l$ >1, for some u ∈ U, we redefine the multi-membership function (μ₁, μ₂, . . . , μ_k (u)) as $(\frac{1}{l})$ (μ₁, μ₂, . . . , μ_k (u)), then the non-normalized multi-fuzzy set can be changed into a normalized multi-fuzzy set.

2.3.3 Definition [14]

Let $\tilde{A} \in M^{k}$ FP(U). If $\tilde{A} = {u / (0, 0, . . ., 0) : u \in U}$ , then $\tilde{A}$ is called the null multi-fuzzy set of dimension k, denoted by. ${\tilde{\emptyset}}_{k}$ If $\tilde{A} = {u / (1, 1, . . ., 1) : u \in U}$ , then $\tilde{A}$ is called the absolute multi-fuzzy set of dimension k, denoted by ${\tilde{1}}_{k}$ .

2.3.4 Example [14]

Suppose that a color image is approximated by an m×n matrix of pixels. Let U be the set of all pixels of the color image. For any pixel x in U, the membership values μ_r (x),μ_g (x),μ_b (x) being the normalized red value, green value and blue value of the pixel x respectively, hence the color image can be approximated by the collection of pixels with the multi-membership function (μ_r,μ_g,μ_b) and it can be represented as a multi-fuzzy set

$\tilde{A} = {x / (μ_{r} (x), μ_{g} (x), μ_{b} (x)) : x \in U}$

In a two dimensional image, color of pixels cannot be characterized by a membership function of an ordinary fuzzy set, but it can be characterized by a three dimensional membership function (μ_r, μ_g, μ_b). Thus, a multi-fuzzy set is a more generalized form of an ordinary fuzzy set and can be used in situations where a fuzzy set of single membership value is not possible.

2.3.5 Definition [13]

Let $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . ., μ_{k} (u) : u \in U$ and $\tilde{B} = {u / (γ_{1} (u), γ_{2} (u), . . ., γ_{k} (u))}$ be two multi-fuzzy set of dimension k in U. Then

$\tilde{A} ⊑ \tilde{B} iff μ_{i} (u) \leq γ_{i} (u), \forall u \in U and 1 \leq i \leq k$

$\tilde{A} ⊑ \tilde{B} iff μ_{i} (u) = γ_{i} (u), \forall u \in U and 1 \leq i \leq k$

$\tilde{A} ⊔ \tilde{B} = {u / (μ_{1} (u) \lor γ_{1} (u), μ_{2} (u) \lor γ_{2} (u), . . ., μ_{k} u) \lor γ_{k} (u)) : u \in U}$

$\tilde{A} ⊓ \tilde{B} = {u / (μ_{1} (u) \land γ_{1} (u), μ_{2} (u) \land γ_{2} (u), . . ., μ_{k} u) \land γ_{k} (u)) : u \in U}$

${\tilde{A}}^{c} = {u / (μ_{1}^{c} (u), μ_{2}^{c} (u)) : u \in U$

2.4 Multi-fuzzy soft set

2.4.1 Definition [7, 14]

A pair ( $F, A$ ) is called a multi-fuzzy soft set of dimension k over U, where $F$ is a mapping given by $F : A \to M^{k}$ FP(U).

A multi-fuzzy soft set is a mapping from parameters to M^kFP(U). It is a parameterized family of multi-fuzzy subsets of U. For e ∈ A, $F (e)$ may be considered as e-approximate elements of a multi-fuzzy soft set ( $F, A$ ).

2.4.2 Example

Suppose that U = {b₁, b₂, b₃, b₄, b₅} is the set of motorcycles under consideration. A = {e₁, e₂, e₃} is the set of parameters, where e₁ stands for the parameter colour, which consists of red, green and blue, e₂ stands for the parameter manufacturers, which consists of Atlas Honda Bikes, DYL Motorcycles and Pak Suzuki Motors, e₃ stands for the parameter price, which can be high, medium and low. We define a multi-fuzzy soft set of dimension 3 as follows:

$\begin{matrix} F (e_{1}) = {b_{1} / (0.4, 0.3, 0.2), b_{2} / (0.2, 0.4, 0.3), \\ b_{3} / (0.1, 0.3, 0.5), b_{4} / (0.3, 0.2, \\ 0.5), b_{5} / (0.4, 0.2, 0.4)} \\ F (e_{2}) = {b_{1} / (0.1, 0.3, 0.5), b_{2} / (0.4, 0.2, 0.4), \\ b_{3} / (0.2, 0.4, 0.3), b_{4} / (0.4, 0.3, 0.2), \\ b_{5} / (0.0, 0.5, 0.4)} \\ F (e_{3}) = {b_{1} / (0.7, 0.2, 0.1), b_{2} / (0.3, 0.5, 0.1), \\ b_{3} / (0.4, 0.3, 0.3), b_{4} / (0.5, 0.2, 0.3), \\ b_{5} / (0.4, 0.3, 0.1)} \end{matrix}$

2.4.3 Definition [14]

Let A, B ⊆ E. Let ( $F, A$ ) and (G, B) be two multi-fuzzy soft sets of dimension k over U. $F, A$ is said to be a multi-fuzzy soft subset of (G, B) if:

A ⊆ B

$F (e) ⊑ G (e) \forall e \in A$

In this case, we write $(F, A) \tilde{⊑} (G, B)$

2.4.4 Definition [14]

(Null multi-fuzzy soft sets) A multi-fuzzy soft set ( $F, A$ ) of dimension k over U is said to be null multi-fuzzy soft set, denoted by $\tilde{Φ_{A}^{k}}$ if $F (e) = \tilde{Φ_{A}^{k}}$ for all e ∈ A.

2.4.5 Definition [14]

(Absolute multi-fuzzy soft sets) A multi-fuzzy soft set ( $F, A$ ) of dimension k over U is said to be absolute multi-fuzzy soft set, denoted by $\tilde{U_{A}^{k}}$ if $F (e) = \tilde{1_{k}}$ for all e ∈ A.

3 Multi-fuzzy bipolar soft sets

In this section, we define multi-fuzzy bipolar soft set and give the basic properties of this notion.

3.1 The concept of a multi-fuzzy bipolar soft set

3.1.1 Definition

A triplet $(F, G, A)$ is called a multi-fuzzy bipolar soft set of dimension k over U, where $F$ and $G$ are mappings given by $F : A \to M^{k}$ :A⟶M^k FP(U) and $G$ :A⟶M^k FP(U).

In other words, a multi-fuzzy bipolar soft set are mappings from parameters to M^k FP(U). These are parameterized families of multi-fuzzy subsets of U. For e ∈ A, $F (e)$ and G (¬ e) may be considered as the set of e-approximate elements of the multi-fuzzy set $(F, G, A)$ over U.

3.1.2 Example

Let U = {h₁, h₂, h₃, h₄, h₅} be a universe containing five houses under consideration. A = {e₁, e₂, e₃} is the set of parameters, where e₁ stands for the parameter “types” which consists of expensive, beautiful, and well furnished, e₂ stands for the parameter “location” which consists of green surrounding, city area, commercial area, and e₃ stands for the parameter “material” which consists of wooden, concrete and bricks, respectively. Let the “not set of A” be ¬A = {¬ e₁, ¬ e₂, ¬ e₃}, where ¬e₁ stands for the parameter, “type” which consists of cheap, ugly, and not well furnished, ¬e₂ stands for the parameter “location” village area, hill area, and town area, and ¬e₃ stands for the parameter “material” which consists of not wooden, not concrete, and not bricks, respectively. We define a multi-fuzzy bipolar soft set of dimension 3 over U as follows: $\begin{matrix} F (e_{1}) = {h_{1} / (0.4, 0.2, 0.3), h_{2} / (0.2, 0.1, 0.6), \\ h_{3} / (0.1, 0.3, 0.4), h_{4} / (0.3, 0.1, 0.5), \\ h_{5} / (0 ., 0.1, 0.2)} \end{matrix}$ $\begin{matrix} G (\neg e_{1}) = {h_{1} / (0.3, 0.1, 0.5), h_{2} / (0.7, 0.1, 0.2), \\ h_{3} / (0.4, 0.5, 0.1), h_{4} / (0.5, 0.3, 0.1), \\ h_{5} / (0.6, 0.3, 0.1)} \end{matrix}$ $\begin{matrix} F (e_{2}) = {h_{1} / (0.5, 0.2, 0.3), h_{2} / (0.4, 0.0, 0.6), \\ h_{3} / (0.3, 0.4, 0.2), h_{4} / (0.4, 0.1, 0.4), \\ h_{5} / (0.6, 0.2, 0.2)} \end{matrix}$ $\begin{matrix} G (\neg e_{2}) = {h_{1} / (0.4, 0.3, 0.4), h_{2} / (0.5, 0.1, 0.4), \\ h_{3} / (0.3, 0.2, 0.4), h_{4} / (0.5, 0.2, 0.3), \\ h_{5} / (0.4, 0.3, 0.2)} \end{matrix}$ $\begin{matrix} F (e_{3}) = {h_{1} / (0.5, 0.3, 0.2), h_{2} / (0.4, 0.5, 0.1), \\ h_{3} / (0.3, 0.4, 0.3), h_{4} / (0.3, 0.3, 0.3), \\ h_{5} / (0.2, 0.3, 0.2)} \end{matrix}$ $\begin{matrix} G (\neg e_{3}) = {h_{1} / (0.5, 0.2, 0.3), h_{2} / (0.5, 0.4, 0.1), \\ h_{3} / (0.4, 0.3, 0.3), h_{4} / (0.4, 0.3, 0.3), \\ h_{5} / (0.1, 0.4, 0.2)} \end{matrix}$

In matrix form, the above multi-fuzzy bipolar soft set can be expressed as follows:

$= [\begin{matrix} \begin{matrix} (0 . 4, 0 . 2, 0 . 3) (0 . 2, 0 . 1, 0 . 6) (0 . 1, 0 . 3, 0 . 4) \begin{matrix} (0 . 3, 0 . 1, 0 . 5) (0 . 7, 0 . 1, 0 . 2) \end{matrix} \end{matrix} \\ \begin{matrix} (0 . 5, 0 . 2, 0 . 3) (0 . 4, 0 . 0, 0 . 6) (0 . 3, 0 . 4, 0 . 2) \begin{matrix} (0 . 4, 0 . 1, 0 . 4) (0 . 6, 0 . 2, 0 . 2) \end{matrix} \end{matrix} \\ \begin{matrix} (0 . 5, 0 . 3, 0 . 2) (0 . 4, 0 . 5, 0 . 1) (0 . 3, 0 . 4, 0 . 3) \begin{matrix} (0 . 3, 0 . 3, 0 . 3) (0 . 2, 0 . 3, 0 . 2) \end{matrix} \end{matrix} \end{matrix}]$ $G = [\begin{matrix} \begin{matrix} (0 . 3, 0 . 1, 0 . 5) (0 . 7, 0 . 1, 0 . 2) (0 . 4, 0 . 5, 0 . 1) \begin{matrix} (0 . 5, 0 . 3, 0 . 1) (0 . 6, 0 . 3, 0 . 1) \end{matrix} \end{matrix} \\ \begin{matrix} (0 . 4, 0 . 3, 0 . 4) (0 . 5, 0 . 1, 0 . 4) (0 . 3, 0 . 2, 0 . 4) \begin{matrix} (0 . 5, 0 . 2, 0 . 3) (0 . 4, 0 . 3, 0 . 2) \end{matrix} \end{matrix} \\ \begin{matrix} (0 . 5, 0 . 2, 0 . 3) (0 . 5, 0 . 4, 0 . 1) (0 . 4, 0 . 3, 0 . 3) \begin{matrix} (0 . 4, 0 . 3, 0 . 3) (0 . 1, 0 . 4, 0 . 2) \end{matrix} \end{matrix} \end{matrix}]$ where the ith row vector represents $F (e_{i})$ and G (e_i), the ith column vector represents h_i and is called the membership matrices of $F (e_{i})$ and G (e_i), respectively.

3.1.3 Definition

Let A, B ⊆ E. Let $(F, G, A)$ and $(F_{1}, G_{1}, B)$ be two multi-fuzzy bipolar soft set of dimension k over U. Then $(F, G, A)$ is called a multi-fuzzy bipolar soft subset of $(F_{1}, G_{1}, B)$ if:

A ⊆ B, and

$F (e) ⊑ F_{1} (e)$ and G₁ (e) ⊑ G (e), for all e ∈ A.

Here, we write $(F, G, A) \tilde{⊑} (F_{1}, G_{1}, B)$ .

3.1.4 Example

Let U = {h₁, h₂, h₃, h₄}, A = {e₁, e₂} and B = {e₁, e₂, e₃}. Clearly, A ⊆ B. Suppose that $(F, G, A)$ and $(F_{1}, G_{1}, B)$ are two multi-fuzzy bipolar soft set of dimension 3 over U, defined as follows. $\begin{matrix} F (e_{1}) = {h_{1} / (0.3, 0.3, 0.1), h_{2} / (0.4, 0.3, 0.2), \\ h_{3} / (0.5, 0.2, 0.1), h_{4} / (0.4, 0.2, 0.2), \\ h_{5} / (0.4, 0.2, 0.3)} \end{matrix}$ $\begin{matrix} G (\neg e_{1}) = {h_{1} / (0.5, 0.4, 0.1), h_{2} / (0.4, 0.4, 0.2), \\ h_{3} / (0.5, 0.2, 0.3), h_{4} / (0.4, 0.5, 0.1), \\ h_{5} / (0.4, 0.3, 0.3)} \end{matrix}$ $\begin{matrix} F_{1} (e_{1}) = {h_{1} / (0.4, 0.3, 0.2), h_{2} / (0.5, 0.3, 0.2), \\ h_{3} / (0.5, 0.3, 0.2), h_{4} / (0.4, 0.3, 0.3), \\ h_{5} / (0.4, 0.3, 0.3)} \end{matrix}$

$\begin{matrix} G_{1} (\neg e_{1}) = {h_{1} / (0.4, 0.3, 0.1), h_{2} / (0.3, 0.3, 0.2), \\ h_{3} / (0.4, 0.1, 0.3), h_{4} / (0.3, 0.4, 0.1), \\ h_{5} / (0.3, 0.3, 0.3)} \end{matrix}$ $\begin{matrix} F (e_{2}) = {h_{1} / (0.4, 0.4, 0.1), h_{2} / (0.5, 0.2, 0.2), \\ h_{3} / (0.4, 0.1, 0.1), h_{4} / (0.3, 0.1, 0.2), \\ h_{5} / (0.4, 0.3, 0.3)} \end{matrix}$ $\begin{matrix} G (- e_{2}) = {h_{1} / (0.5, 0.3, 0.2), h_{2} / (0.4, 0.5, 0.1), \\ h_{3} / (0.4, 0.3, 0.3), h_{4} / (0.3, 0.4, 0.2), \\ h_{5} / (0.4, 0.2, 0.3)} \end{matrix}$ $\begin{matrix} F_{1} (e_{2}) = {h_{1} / (0.5, 0.4, 0.1), h_{2} / (0.5, 0.3, 0.2), \\ h_{3} / (0.4, 0.2, 0.2), h_{4} / (0.3, 0.3, 0.3), \\ h_{5} / (0.4, 0.4, 0.3)} \end{matrix}$ $\begin{matrix} G_{1} (\neg e_{2}) = {h_{1} / (0.4, 0.3, 0.2), h_{2} / (0.4, 0.4, 0.1), \\ h_{3} / (0.3, 0.3, 0.3), h_{4} / (0.2, 0.3, 0.2), \\ h_{5} / (0.3, 0.2, 0.2)} \end{matrix}$

From above multi-fuzzy sets, we observe that $F (e_{i}) ⊑ F_{1} (e_{i}) and G_{1} (e_{i}) ⊑ G (e_{i})$ . Therefore, $(F, G, A) \tilde{⊑} (F_{1}, G_{1}, B)$ .

3.1.5 Definition

Two multi-fuzzy bipolar soft sets $(F, G, A)$ and $(F_{1}, G_{1}, B)$ of dimension k over a common universe U are said to be multi-fuzzy bipolar soft equal if $(F, G, A)$ is a multi-fuzzy bipolar soft subset of $(F_{1}, G_{1}, B)$ and $(F_{1}, G_{1}, B)$ is a multi-fuzzy bipolar soft subset of $(F, G, A)$ .

3.1.6 Definition

(Null multi-fuzzy bipolar soft set) A multi-fuzzy bipolar soft set (F,G,A) of dimension k over U is said to be null multi-fuzzy bipolar soft set, denoted by ${\tilde{Φ}}_{A}^{k}$ if $F (e) = {\tilde{Φ}}_{k}$ and $G (\neg e) = {\tilde{1}}_{k}$ for all e ∈ A.

3.1.7 Definition

(Absolute multi-fuzzy bipolar soft set) A multi-fuzzy bipolar soft set $(F, G, A)$ of dimension k over U is said to be absolute multi-fuzzy soft set, denoted by ${\tilde{U}}_{A}^{k}$ if $F (e) = {\tilde{1}}_{k}$ and $G (\neg e) = {\tilde{Φ}}_{k}$ for all e ∈ A.

3.2 Operations on multi-fuzzy bipolar soft sets

3.2.1 Definition

The complement of a multi-fuzzy bipolar soft set (F, G, A) of dimension k over U is denoted

by (F, G^c, A) ^c and is defined by (F, G, A) ^c = (F^c , G, A) ^c = (F^c , G^c, A) where F^c and G^c are mappings given by F^c (e) = F (e)) ^c and G^c (¬ e) = . G (¬ e)) ^c for all e ∈ AClearly, ((F, G, A) ^c) ^c is the same as (F, G, A),i.e., complement of a multi-fuzzy soft bipolar soft set is involute in nature. Suppose $φ_{A}^{\tilde{k}} ▪$ and $U_{A}^{\tilde{k}}$ are multi-fuzzy bipolar soft sets of dimension k over U, then $(φ_{A}^{\tilde{k}})^{c} = U_{A}^{\tilde{k}}$ and $(U_{A}^{\tilde{k}})^{c} = φ_{A}^{\tilde{k}}$ .

3.2.2 Example

Reconsider Example 3.1.2, we have (F, G, A) ^c = (F^c, G^c, A)as follows:

$\begin{matrix} F^{c} (e_{1}) = {h_{1} / (0.7, 0.7, 0.9), h_{2} / (0.6, 0.7, 0.8), \\ h_{3} / (0.5, 0.8, 0.9), h_{4} / (0.6, 0.8, 0.8), \\ h_{5} / (0.6, 0.8, 0.7)} \end{matrix}$ $\begin{matrix} G^{c} (\neg e_{1}) = {h_{1} / (0.5, 0.6, 0.9), h_{2} / (0.6, 0.6, 0.8), \\ h_{3} / (0.5, 0.8, 0.7), h_{4} / (0.6, 0.5, 0.9), \\ h_{5} / (0.6, 0.7, 0.7)} \end{matrix}$ $\begin{matrix} F^{c} (e_{2}) = {h_{1} / (0.6, 0.6, 0.9), h_{2} / (0.5, 0.8, 0.8), \\ h_{3} / (0.6, 0.9, 0.9), h_{4} / (0.7, 0.9, 0.8), \\ h_{5} / (0.6, 0.7, 0.7)} \end{matrix}$ $\begin{matrix} G^{c} (\neg e_{2}) = {h_{1} / (0.5, 0.7, 0.8), h_{2} / (0.6, 0.5, 0.9), \\ h_{3} / (0.6, 0.7, 0.7), h_{4} / (0.7, 0.6, 0.8), \\ h_{5} / (0.6, 0.8, 0.7)} \end{matrix}$

In matrix form this can be expressed as follows: $F^{'} = [\begin{matrix} (0.7, 0.7, 0.9) (0.6, 0.7, 0.8) (0.5, 0.8, 0.9) (0.6, 0.8, 0.8) (0.6, 0.8, 0.7) \\ (0.6, 0.6, 0.9) (0.5, 0.8, 0.8) (0.7, 0.9, 0.8) (0.7, 0.9, 0.8) (0.6, 0.7, 0.7) \end{matrix}]$ $G^{'} = [\begin{matrix} (0.5, 0.6, 0.9) (0.6, 0.6, 0.8) (0.5, 0.8, 0.7) (0.6, 0.5, 0.9) (0.6, 0.7, 0.7) \\ (0.5, 0.7, 0.8) (0.6, 0.5, 0.9) (0.6, 0.7, 0.7) (0.7, 0.6, 0.8) (0.6, 0.8, 0.7) \end{matrix}]$

Now, we present the notion of AND and OR operations on two multi-fuzzy bipolar soft sets of dimension k over U, as follows:

3.2.3 Definition

If (F, G, A) and (F₁, G₁, B) are two multi-fuzzy bipolar soft sets of dimension k over U, then “ (F, G, A) AND (F₁, G₁, B)” denoted by (F, G, A) ∧ (F₁, G₁, B) is defined by(F, G, A) ∧ (F₁, G₁, B) = (H, I, A × B), where H (α, β) = F (α) ⊓ F₁ (β) and I (¬ α, ¬ β) = G (¬ α) ⊔ G₁ (¬ β) for all (α, β) ∈ A × B.

3.2.4 Definition

If (F, G, A) and (F₁, G₁, B) are two multi-fuzzy bipolar soft sets of dimension k over U, then “ (F, G, A) OR (F₁, G₁, B)” denoted by (F, G, A) ∨ (F₁, G₁, B) is defined by(F, G, A) ∨ (F₁, G₁, B) = (O, J, A × B), where O (α, β) = F (α) ⊔ F₁ (β) and J (¬ α, ¬ β) = G (¬ α) ⊓ G₁ (¬ β) for all (α, β) ∈ A × B

3.2.5 Example

Let U = {u₁, u₂, u₃}, A = {e₁, e₂} and B = {e₂, e₄, e₅} Suppose that (F, G, A) and (F₁, G₁, B) are two multi-fuzzy bipolar soft sets of dimension 2 over U, defined as follows:

$F (e_{1}) = {u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), u_{3} / (0.7, 0.3)}$ $G (\neg e_{1}) = {u_{1} / (0.7, 0.2), u_{2} / (0.8, 0.1), u_{3} / (0.6, 0.3)}$ $F (e_{2}) = {u_{1} / (0.6, 0.4), u_{2} / (0.5, 0.4), u_{3} / (0.8, 0.2)}$ $G (\neg e_{2}) = {u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.2), u_{3} / (0.8, 0.2)}$ $F_{1} (e_{2}) = {u_{1} / (0.8, 0.2), u_{2} / (0.6, 0.4), u_{3} / (0.5, 0.3)}$ $G_{1} (\neg e_{2}) = {u_{1} / (0.5, 0.4), u_{2} / (0.3, 0.7), u_{3} / (0.5, 0.5)}$ $F_{1} (e_{4}) = {u_{1} / (0.4, 0.5), u_{2} / (0.2, 0.7), u_{3} / (0.3, 0.5)}$ $G_{1} (\neg e_{4}) = {u_{1} / (0.5, 0.3), u_{2} / (0.3, 0.6), u_{3} / (0.4, 0.6)}$ $F_{1} (e_{5}) = {u_{1} / (0.1, 0.8), u_{2} / (0.3, 0.7), u_{3} / (0.4, 0.5)}$ $G_{1} (\neg e_{5}) = {u_{1} / (0.2, 0.6), u_{2} / (0.4, 0.5), u_{3} / (0.5, 0.3)}$

Then we have (F, G, A) ∧ (F₁, G₁, B) = (H, I, A × B) and

(F,G,A) ∨ (F₁, G₁, B) = (O, J, A × B)as follows:

$H (e_{1}, e_{2}) = {u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), u_{3} / (0.7, 0.2)}$ $I (\neg e_{1}, \neg e_{2}) = {u_{1} / (0.7, 0.4), u_{2} / (0.8, 0.2), u_{3} / (0.8, 0.3)}$ $H (e_{1}, e_{4}) = {u_{1} / (0.4, 0.4), u_{2} / (0.2, 0.4), u_{3} / (0.3, 0.3)}$ $I (\neg e_{1}, \neg e_{4}) = {u_{1} / (0.5, 0.6), u_{2} / (0.7, 0.7), u_{3} / (0.8, 0.5)}$ $H (e_{1}, e_{5}) = {u_{1} / (0.1, 0.4), u_{2} / (0.3, 0.4), u_{3} / (0.4, 0.3)}$ $\begin{matrix} I (\neg e_{1}, \neg e_{5}) = {u_{1} / (0.5, 0.6), . . . . . . . . . . . . . u_{2} / (0.7, 0.5), \\ u_{3} / (0.8, 0.3)} \end{matrix}$ $H (e_{2}, e_{2}) = {u_{1} / (0.6, 0.2), u_{2} / (0.5, 0.4), u_{3} / (0.5, 0.2)}$ $I (\neg e_{2}, \neg e_{2}) = {u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.7), u_{3} / (0.8, 0.5)}$ $H (e_{2}, e_{4}) = {u_{1} / (0.4, 0.4), u_{2} / (0.2, 0.4), u_{3} / (0.3, 0.2)}$ $I (\neg e_{2}, \neg e_{4}) = {u_{1} / (0.5, 0.4), u_{2} / (0.3, 0.7), u_{3} / (0.5, 0.6)}$ $H (e_{2}, e_{5}) = {u_{1} / (0.1, 0.4), u_{2} / (0.3, 0.4), u_{3} / (0.4, 0.2)}$ $I (\neg e_{2}, \neg e_{5}) = {u_{1} / (0.8, 0.8), u_{2} / (0.6, 0.7), u_{3} / (0.5, 0.5)}$

Similarly, we can find $(F, G, A) \lor (F_{1}, G_{1}, B) = (O, J, A \times B)$

3.2.6 Theorem

Let (F, G, A) and (F₁, G₁, B) be two multi-fuzzy bipolar soft sets of dimension k over U. Then

((F, G, A) ∧ (F₁, G₁, B) ^c = (F, G, A) ^c ∨ (F₁, G₁, B) ^c

((F, G, A) ∨ (F₁, G₁, B) ^c = (F, G, A) ^c ∧ (F₁, G₁, B) ^c

Proof. (1) Suppose that $(F, G, A) \land (F_{1}, G_{1}, B) = (H, I, A \times B)$

Then $\begin{matrix} (F, G, A) \land (F_{1}, G_{1}, B))^{c} = \\ (F^{c}, G^{c}, A)^{c} \lor (F_{1}^{c}, G_{1}^{c}, B)^{c} = (O, J, A, B) \end{matrix}$

Now, $\begin{matrix} (F, G, A) \lor (F_{1}, G_{1}, B))^{c} = \\ (F^{c}, G^{c}, A)^{c} \lor (F_{1}^{c}, G_{1}^{c}, B)^{c} = (O, J, A, B) \end{matrix}$

Let (α, β) ∈ A × B, then $\begin{matrix} H^{c} (α, β) = (H (α, β))^{c} = F (α) ⊓ F_{1} (β)^{c} \\ = F^{c} (α) ⊔ F_{1}^{c} (β) . . . . = O (α, β) \end{matrix}$ and $\begin{matrix} I^{c} (\neg α, \neg β) = (I (\neg α, \neg β))^{c} = (F^{c} (\neg α) ⊔ F_{1} (\neg β))^{c} \\ = F^{c} (\neg α) ⊓ F_{1}^{c} (\neg β) = J (\neg α, \neg β) \end{matrix}$

Hence, H^c (α, β) = O (α, β) and I^c I^c (¬ α, ¬ β) = J (¬ α, ¬ β)

(2) The result can be proved in a similar way as part (1).

3.2.7 Definition

Union of two multi-fuzzy bipolar soft sets (F, G, A) and (F₁, G₁, B) sof dimension k over U is the multi-fuzzy bipolar soft set (H, I, C), where C = A ∪ B and for all e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B \\ F_{1} (e), if e \in B - A \\ F (e) ⊔ F_{1} (e), if e \in A \cap B \end{matrix}$ $I (\neg e) = {\begin{matrix} G (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G (\neg e) ⊓ G_{2} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

We write $(F, G, A) \tilde{⊔} (F_{1}, G_{1}, B) = (H, I, C)$

3.2.8 Example

Reconsider Example 3.1.4, we have $(F, G, A) \tilde{⊔} (F_{1}, G_{1}, B) = (H, I, C)$ where C = A ∪ B = {e₁, e₂, e₄, e₅} $\begin{matrix} H (e_{1}) = F (e_{1}) = {u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), \\ u_{3} / (0.7, 0.3)} \\ H (e_{4}) = F_{1} (e_{4}) = {u_{1} / (0.4, 0.5), u_{2} / (0.2, 0.7), \\ u_{3} / (0.3, 0.5)} \\ H (e_{5}) = F_{1} (e_{5}) = {u_{1} / (0.1, 0.8), u_{2} / (0.3, 0.7), \\ u_{3} / (0.4, 0.5)} \\ H (e_{2}) = F (e_{2}) ⊔ F_{1} (e_{2}) = {u_{1} / (0.8, 0.4), \\ u_{2} / (0.6, 0.4), u_{3} / (0.8, 0.3)} \\ I (\neg e_{1}) = G (\neg e_{1}) = {u_{1} / (0.7, 0.2), u_{2} / (0.8, 0.1), \\ u_{3} / (0.6, 0.3)} \\ I (\neg e_{4}) = G (\neg e_{4}) = {u_{1} / (0.5, 0.3), u_{2} / (0.3, 0.6), \\ u_{3} / (0.4, 0.6)} \end{matrix}$

$\begin{matrix} I (\neg e_{5}) = G (\neg e_{5}) = {u_{1} / (0.2, 0.6), u_{2} / (0.4, 0.5), \\ u_{3} / (0.5, 0.3)} \\ I (\neg e_{2}) = G (\neg e_{2}) ⊓ G_{1} (\neg e_{2}) = {u_{1} / (0.7, 0.2), \\ u_{2} / (0.8, 0.1), u_{3} / (0.6, 0.3)} \end{matrix}$

3.2.9 Definition

Intersection of two multi-fuzzy bipolar soft sets (F, G, A) and (F₁, G₁, B) of dimension k with A ∩ B ≠ φ over U is the multi-fuzzy bipolar soft set (H, I, C), where C = A ∩ B and for all e ∈ C, $H (e) = F (e) ⊓ 𝔽_{1} (e)$ and I (¬ e) = G (¬ e) ⊔ G₁ (¬ e).

We write $(F, G, A) \tilde{⊓} (F_{1}, G_{1}, B) = (H, I, C) .$

3.2.10 Example

Reconsider Example 3.1.4, we have A ∩ B = {e₂} and $(F, G, A) \tilde{⊓} (F_{1}, G_{1}, B) = (H, I, C) .$ implies $\begin{matrix} H (e_{2}) = F (e_{2}) ⊓ F_{1} (e_{2}) = \\ {u_{1} / (0.6, 0.2), u_{2} / (0.5, 0.4), u_{3} / (0.8, 0.2) \\ I (\neg e_{2}) = G (\neg e_{2}) ⊔ G_{1} (\neg e_{2}) = \\ {u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.7), u_{3} / (0.8, 0.5) \end{matrix}$

The following results are obvious.

3.2.11 Theorem

Let (F, G, A) and (F₁, G₁, B) be two multi-fuzzy bipolar soft sets of dimension k over U. Then

$(F, G, A) \tilde{⊔} (F, G, A) = (F, G, A)$ .

$(F, G, A) \tilde{⊓} (F, G, A)) = (F, G, A)$

$(F, G, A) \tilde{⊔} \tilde{φ_{A}^{K}} = (F, G, A)$

$(F, G, A) \tilde{⊓} \tilde{φ_{A}^{K}} = \tilde{φ_{A}^{K}}$

$(F, G, A) \tilde{⊔} \tilde{U_{A}^{K}} = \tilde{U_{A}^{K}}$

$(F, G, A) \tilde{⊓} \tilde{U_{A}^{K}} = (F, G, A)$

$(F, G, A) \tilde{⊔} (F_{1}, G_{1}, B) = (F_{1}, G_{1}, B) \tilde{⊔} (F, G, A)$

$(F, G, A) \tilde{⊓} (F_{1}, G_{1}, B) = (F_{1}, G_{1}, B) \tilde{⊓} (F, G, A)$

3.2.12 Theorem

Let (F, G, A) and (F₁, G₁, B) be two multi-fuzzy bipolar soft sets of dimension k over U. Then

${((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c} \tilde{⊑}$

${(F, G, A)}^{c} \tilde{⊔} {(F_{1}, G_{1}, B)}^{c}$

${(F, G, A)}^{c} \tilde{⊓} {(F_{1}, G_{1}, A)}^{c} \tilde{⊑}$

${((F, G, A) \tilde{⊓} (F_{1}, G_{1}, B))}^{c}$

Proof (1) Suppose that $(F, G, A) \tilde{⊔} (F_{1}, G_{1}, B) = (H, I, C)$ where C = A ∪ B, and for all e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B \\ F_{1} (e), if e \in B - A \\ F (e) ⊔ F_{1} (e), if e \in A \cap B \end{matrix}$ $I (\neg e) = {\begin{matrix} G (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G (\neg e) ⊓ G_{1} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

Thus ${((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c} = {(H, I, C)}^{c} = (H^{c}, I^{c}, C)$ ,and $H^{c} (e) = {\begin{matrix} F^{'} (e), if e \in A - B \\ F_{1}^{'} (e), if e \in B - A \\ F^{'} (e) ⊓ F_{1}^{'} (e), if e \in A \cap B \end{matrix}$ $I (\neg e) = {\begin{matrix} G^{'} (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1}^{'} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G^{'} (\neg e) ⊓ G_{1}^{'} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

Again suppose that $\begin{matrix} {(F, G, A)}^{c} \tilde{⊔} {(F_{1}, G_{1}, B)}^{c} = (F^{c}, G^{c}, A) \tilde{⊔} \\ (F_{1}^{c}, G_{1}^{c}, B) = (K, L, J), \end{matrix}$ whereJ = A ∪ B, and for all e ∈ J, $K (e) = {\begin{matrix} F^{'} (e), if e \in A - B \\ F_{1}^{'} (e), if e \in B - A \\ F^{'} (e) ⊔ F_{1}^{'} (e), if e \in A \cap B \end{matrix}$ $L (\neg e) = {\begin{matrix} G^{'} (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1}^{'} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G^{'} (\neg e) ⊓ G_{1}^{'} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

We see that C = J and for all e ∈ C (orJ),

H^c (e) ⊑ K (e)and L (¬ e) ⊑ I^c (¬ e)Thus

${((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c} \tilde{⊑} {(F, G, A)}^{c} \tilde{⊔} {(F_{1}, G_{1}, B)}^{c}$

(2) Proof is similar to part (1).

3.2.13 Theorem

Let $(F, G, A)$ and $(F_{1}, G_{1}, B)$ be two multi-fuzzy bipolar soft sets of dimension k over U. Then

$(F, G, A) \tilde{⊓} {(F_{1}, G_{1}, B)}^{c} \tilde{⊑} {((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c}$

${((F, G, A) \tilde{⊓} (F_{1}, G_{1}, B))}^{c} \tilde{⊑} (F, G, A) \tilde{⊔} {(F_{1}, G_{1}, B)}^{c}$

Proof

(1) Suppose that(F, G, A) ⊔ (F₁, G₁, B) = (H, I, C), where C = A ∪ B, and for all e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B \\ F_{1} (e), if e \in B - A \\ F (e) ⊔ F_{1} (e), if e \in A \cap B \end{matrix}$ $I (\neg e) = {\begin{matrix} G (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G (\neg e) ⊓ G_{1} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

Thus ${((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c} = {(H, I, C)}^{c} = (H^{c}, I^{c}, C)$ , and $H^{c} (e) = {\begin{matrix} F^{'} (e), if e \in A - B \\ F_{1}^{'} (e), if e \in B - A \\ F^{'} (e) ⊓ F_{1}^{'} (e), if e \in A \cap B \end{matrix}$ $I (\neg e) = {\begin{matrix} G^{'} (\neg e), if \neg e \in (\neg A) - (\neg B) \\ G_{1}^{'} (\neg e), if \neg e \in (\neg B) - (\neg A) \\ G^{'} (\neg e) ⊓ G_{1}^{'} (\neg e), if \neg e \in (\neg A) \cap (\neg B) \end{matrix}$

Again suppose that ${(F, G, A)}^{c} \tilde{⊓} {(F_{1}, G_{1}, B)}^{c} = (F^{c}, G^{c}, A) \tilde{⊓} (F_{1}^{c}, G_{1}^{c}, B) = (K, L, J)$ , where J = A ∪ B, and for all e ∈ J,

$K (e) = F^{c} (e) ⊓ F_{1}^{c} (e) = H^{c} (e)$ and $L (\neg e) = G^{c} (\neg e) ⊔ G_{1}^{c} (\neg e) = I^{c} (\neg e)$ . Thus J ⊆ C and for alle ∈ J, K (e) = H^c (e) and L (¬ e) = I^c (¬ e) Proved.

(2) Proof follows from part (1).

From the above discussion, we know that De Morgan laws are invalid for multi-fuzzy bipolar soft sets with the different parameter sets, but they are true for multi-fuzzy bipolar soft sets with the same parameter set, as given in the following theorem.

3.2.14 Theorem

Let(F, G, A) and (F₁, G₁, A) be two multi-fuzzy bipolar soft sets of dimension k over U. Then

${((F, G, A) \tilde{⊔} (F_{1}, G_{1}, B))}^{c} = {(F, G, A)}^{c} \tilde{⊔} {(F_{1}, G_{1}, B)}^{c}$

${((F, G, A) \tilde{⊓} (F_{1}, G_{1}, B))}^{c} = (F, G, A) \tilde{⊓} {(F_{1}, G_{1}, B)}^{c}$

Proof

Proof is straightforward.

4 Application of multi-fuzzy soft set in decision making

In [8], Shabir et al., followed the method of Roy et al., given in [25, 26] and presented an algorithm for the identification of an object, which is based on the comparison of different objects in the context of fuzzy bipolar soft set theory. In this section, we follow Feng et al. algorithm [11] and Yang’s method [7, 14] and present a novel approach to fuzzy bipolar soft set based decision making problems by using level bipolar soft sets.

4.1 Level soft sets of fuzzy bipolar soft set

4.1.1 Definition

Let (F, G, A)be a fuzzy bipolar soft set over U, where A ⊆ E. For t, s ∈ [0, 1], the t-level and the s-level subsets of the fuzzy soft sets (F, A) and (G, A) are crisp soft sets defined as follows:

F_t (e) = LF (e) ; t) = {x ∈ U : F (e) (x) ≥ t} and G_s = L (G (¬ e) ; s) = {x ∈ U : G (¬ e) (x) ≤ s} for all e ∈ A

To illustrate the above definition, we recall fuzzy bipolar soft set of Example 4.

4.1.2 Example

(The 0.5-level soft sets of the fuzzy bipolar soft set (F, G, A) As discussed in Example 4, the fuzzy bipolar soft set (F,G,A) can be viewed as follows: $\begin{matrix} F (e_{1}) = expensive houses = \\ {h_{1} / 0.7, h_{2} / 0.6, h_{3} / 0.8, h_{4} / 0.5, h_{5} / 0.6} \end{matrix}$ $\begin{matrix} G (\neg e_{1}) = cheap houses = \\ {h_{1} / 0.2, h_{2} / 0.4, h_{3} / 0.2, h_{4} / 0.5, h_{5} / 0.4} \end{matrix}$ $\begin{matrix} F (e_{2}) = beautiful houses = \\ {h_{1} / 0.8, h_{2} / 0.7, h_{3} / 0.8, h_{4} / 0.6, h_{5} / 0.3} \end{matrix}$ $\begin{matrix} G (\neg e_{2}) = ugly houses = \\ {h_{1} / 0.2, h_{2} / 0.2, h_{3} / 0.1, h_{4} / 0.4, h_{5} / 0.7} \end{matrix}$ $\begin{matrix} F (e_{3}) = wooden houses = \\ {h_{1} / 0.4, h_{2} / 0.6, h_{3} / 0.4, h_{4} / 0.6, h_{5} / 0.5} \end{matrix}$ $\begin{matrix} G (\neg e_{3}) = not wooden houses = \\ {h_{1} / 0.5, h_{2} / 0.4, h_{3} / 0.6, h_{4} / 0.4, h_{5} / 0.5} \end{matrix}$

By taking t = 0.5 and s = 0.5, we obtain the 0.5-level soft sets of the fuzzy sets F (e₁), F (e₂), F (e₃), G (¬ e₁), G (¬ e₂) and G (¬ e₃) as follows: $\begin{matrix} L (F (e_{1}); 0.5) = {h_{1}, h_{2}, h_{3}, h_{4}, h_{5}}, \\ L (F (e_{2}); 0.5) = {h_{1}, h_{2}, h_{3}, h_{4}} \\ L (F (e_{3}); 0.5) = {h_{2}, h_{4}, h_{5}}, \\ L (G (\neg e_{1}); 0.5) = {h_{1}, h_{2}, h_{3}, h_{4}, h_{5}}, \\ L (G (\neg e_{2}); 0.5) = {h_{1}, h_{2}, h_{3}, h_{4}}, \\ L (G (\neg e_{3}); 0.5) = {h_{1}, h_{2}, h_{3}, h_{4}, h_{5}} . \end{matrix}$

The 0.5-level bipolar soft set of (F, G, A) is a bipolar soft set L (F, G ; 0.5) = (F_0.5,G_0.5,A), where the set-valued mappings F_0.5 : A → P (U) and G_0.5 : A → P (U) defined by (F_0.5 (e₁) = L (F (e₁) _0.5 and G_0.5 (e_i)=L(G(e_i),0.5) for fori = 1, 2, 3 .. Tables 1 and 2 give the tabular representation of level soft sets L (F, G ; 0.5) .

Table 1
Tabular representation of the level soft sets L(F;0.5) and L(G;0.5)

U e ₁ e ₂ e ₃

h ₁ 1 1 0

h ₂ 1 1 1

h ₃ 1 1 0

h ₄ 1 1 1

h ₅ 1 0 1

U	e ₁	e ₂	e ₃
h ₁	1	1	0
h ₂	1	1	1
h ₃	1	1	0
h ₄	1	1	1
h ₅	1	0	1

Table 2

Tabular representation of the level soft sets L(G;0.5)

U	e ₁	e ₂	e ₃
h ₁	1	1	0
h ₂	1	1	1
h ₃	1	1	1
h ₄	1	1	1
h ₅	1	0	1

4.1.3 Definition

In definition of a t-level soft set, the level (or threshold) assigned to each parameter is a constant value t ∈ [0, 1]. But in some decision making problems, it may happen that decision makers would like to take different thresholds on different decision parameters. To overcome on this issue, we use a function instead of a constant value as the threshold on membership values.

4.1.4 Definition

Let (F,G,A) be a fuzzy bipolar soft set over U, where A ⊆ E. λ μ:A⟶[0, 1] be fuzzy sets in A which are called threshold fuzzy sets. The level soft sets of the fuzzy soft set (F,G) with respect to the fuzzy sets λ μ are crisp sets

L ((F, G, A) ; λ, μ) = (F_λ, G_μ, A) defined as follows:

F_λ (e) = L (F (e) ; λ (e)) = {X ∈ U : F (e) (X) ≥ λ} and G_μ (¬ e) = L (G (¬ e) ; λ (¬ e)) = {X ∈ U : G (¬ e) (X) ≤ μ}

for all e ∈ A.

4.1.5 Definition

(Mid-level bipolar soft set of a fuzzy bipolar soft set)

Let (F,G,A) be a fuzzy bipolar soft set over U, where A ⊆ E. Based on the fuzzy bipolar soft set (F,G,A) we can define two fuzzy sets (F,G,A), we can define a fuzzy sets mid_F: A → [0, 1] and mid_G: A → [0, 1] by

(e) = (raise0.7ex1 / - lower0.7ex (|U | )) ∑_x∈UF(e)(x)and

(¬ e) = (raise0.7ex1 / - lower0.7ex (|U |)) ∑_x∈UG (¬ e) (X) for all e ∈ A. The fuzzy sets mid_F(e) and mid_G (e) are called the mid-thresholds of the fuzzy bipolar soft set (F, G, A)The level soft set with respect to the mid-thresholds mid_F (e) and mid_G (¬ e)) namely L (F, G, A) mid_F (e) mid_G (¬ e)) is called mid-level bipolar soft set.

4.1.6 Definition

Let U ={ u₁, u₂, . . . u_n }. Let ( $F$ , $G$ ,A) be a multi-fuzzy bipolar soft set of dimension k over U. Then the positive multi-fuzzy soft set and the negative multi-fuzzy soft set induced from ( $F$ , $G$ ,A) denoted by ( $F$ ,A) and ( $G$ ,A), respectively, are defined as follows:

( $F$ ,A)= $F$ (e): e ∈ A where $\begin{matrix} F (e) = {u_{1} / (u_{1} (u_{1}), u_{2} / (u_{1}), . . ., μ_{k} (u_{1})), \\ u_{2} / (u_{1} (u_{2}), μ_{2} (u_{2}), . . ., μ_{k} (u_{2})), . . ., u_{n} / (μ_{1} (u_{n}), \\ μ_{2} (u_{n}), . . ., u_{k} (u_{n}))} \end{matrix}$ and

(G, A) ={ G (¬ e) : ¬ e ∈ ¬ A } where

$\begin{matrix} G (\neg e) = {u_{1} / γ_{1} (u_{1}), γ_{2} (u_{1}), . . ., γ_{k} (u_{1}), u_{2} / γ_{1} (u_{2}), \\ γ_{2} (u_{2}), . . ., γ_{k} (u_{2})), . . ., u_{n} / γ_{1} (u_{n}), γ_{2} (u_{n}), . . ., γ_{k} (u_{n}))} \end{matrix}$

Let U ={ u₁, u₂, . . . u_n } and (F, A) and (G, A) be multi-fuzzy soft sets induced from (F, G, A) of dimension k over U.

For every e ∈ A, F (e) and G (¬ e) are two multi-fuzzy sets which can be expressed in matrix form as follows:

$F (e) = [\begin{matrix} μ_{1} (u_{1}) & μ_{2} (u_{1}) & . . . . . & μ_{k} (u_{1}) \\ μ_{1} (u_{2}) & μ_{2} (u_{2}) & . . . . . & μ_{k} (u_{2}) \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ μ_{1} (u_{n}) & μ_{2} (u_{n}) & . . . . . & μ_{k} (u_{n}) \end{matrix}]$

$G (\neg e) = [\begin{matrix} γ_{1} (u_{1}) & γ_{2} (u_{1}) & . . . . . & γ_{k} (u_{1}) \\ γ_{1} (u_{2}) & γ_{2} (u_{2}) & . . . . . & γ_{k} (u_{2}) \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ γ_{1} (u_{n}) & γ_{2} (u_{n}) & . . . . . & γ_{k} (u_{n}) \end{matrix}]$

Using the Yang’s methods [7], suppose that $ω (e) = (ω_{1}, ω_{2}, . . ., ω_{n})^{T}, (\sum_{i = 1}^{k} ω_{i} = 1)$ , ( $\sum_{i = 1}^{k} ω_{i}$ =1), is the relative weight of parameter e, and assume F (e) and G (¬ e) are normalized, then we can change $F (e)$ and G (¬ e) into two Zadeh’s fuzzy sets μ_F (e) and γ_G (e) respectively, where μ_F (e) and γ_G (e) are given as follows:

$\begin{matrix} μ_{F (e)} = \\ [\begin{matrix} μ_{1} (u_{1}) & μ_{2} (u_{1}) & . . . . . & μ_{k} (u_{1}) \\ μ_{1} (u_{2}) & μ_{2} (u_{2}) & . . . . . & μ_{k} (u_{2}) \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ μ_{1} (u_{n}) & μ_{2} (u_{n}) & . . . . . & μ_{k} (u_{n}) \end{matrix}] [\begin{matrix} ω_{1} \\ ω_{2} \\ . \\ . \\ . \\ ω_{n} \end{matrix}] [\begin{matrix} \sum_{i = 1}^{k} ω_{i} μ_{i} (u_{1}) \\ \sum_{i = 1}^{k} ω_{i} μ_{i} (u_{2}) \\ . \\ . \\ \sum_{i = 1}^{k} ω_{i} μ_{i} (u_{n}) \end{matrix}] \end{matrix}$

and

$\begin{matrix} γ_{G (\neg e)} = \\ [\begin{matrix} γ_{1} (u_{1}) & γ_{2} (u_{1}) & . . . . . & γ_{k} (u_{1}) \\ γ_{1} (u_{2}) & γ_{2} (u_{2}) & . . . . . & γ_{k} (u_{2}) \\ . & . & . & . \\ . & . & . & . \\ . & . & . & . \\ γ_{1} (u_{n}) & γ_{2} (u_{n}) & . . . . . & γ_{k} (u_{n}) \end{matrix}] [\begin{matrix} ω_{1} \\ ω_{2} \\ . \\ . \\ . \\ ω_{n} \end{matrix}] [\begin{matrix} \sum_{i = 1}^{k} ω_{i} γ_{i} (u_{1}) \\ \sum_{i = 1}^{k} ω_{i} γ_{i} (u_{2}) \\ . \\ . \\ \sum_{i = 1}^{k} ω_{i} γ_{i} (u_{n}) \end{matrix}] \end{matrix}$

Thus, if ω (e) is given, we can change the multi-fuzzy sets F (e) and G (e) into induced fuzzy sets F (e) and G (¬ e). Hence, by using this method, we change a multi-fuzzy bipolar soft set into an induced fuzzy bipolar soft set. Once the induced fuzzy bipolar soft set of a multi-fuzzy bipolar soft set has been arrived at, it may be necessary to choose the best alternative from the alternatives based on Feng’s algorithm [11]. Therefore, we can make a decision by the following algorithm.

Algorithm

Step 1. Input a multi-fuzzy bipolar soft set (F, G, A).

Step 2. Find the multi-fuzzy soft set (F, A) and (G, A).

Step 3. Change (F, A) and (G, A) into normalized multi-fuzzy soft set, respectively.

Step 4. Input the relative weight ω (e_i) of parameter e_i.

Compute the fuzzy soft sets Δ_F and Δ_G.

Step 5. Input two threshold fuzzy sets s,t:A → [0.1] (or give two threshold values λ, μ ∈ [0, 1] or choose the mid_F-level mid_G-level decision rules) for decision making.

Step 6. Compute the level soft sets L (Δ_F ; s) and LΔ_G ; t) with respect to the threshold fuzzy sets s, t ∈ [0, 1] (or λ;-level soft set L (Δ_F ; λ) and μ;-level soft set L (Δ_F ; λ) or

mid_ΔF-level soft set L (Δ_F ; λ) and mid_ΔG-level soft set L (Δ_F ; mid_ΔF})

Step 7. Present the level soft sets L (Δ_F ; mid_ΔF) and L(_G;t) (or L (Δ_F ; λ) and L (Δ_G ; mid_ΔG) or

L (Δ_F ; mid_ΔF}) and L (Δ_G ; mid_ΔG) in tabular forms and compute the choice values $c_{i}^{+}$ and $c_{i}^{-}$ of X_i,∀i respectively.

Step 8. Compute the bipolar choice value $ξ_{i} = c_{i}^{+} - c_{i}^{-}$ of X_i ∀ _i. The optimal decision is to select X_j if. $ξ_{i} = max_{i} ξ_{i}$

Step 9. If j has more than one value then any one of X_j may be chosen.

4.1.7 Remark

In the last step of the above algorithm, one may go back to the fifth step and change the threshold (or decision rule) that one once used so as to adjust the final optimal decision, especially when there are too many “optimal choices” to be chosen.

Illustrative example

Suppose that U ={ p₁, p₂, p₃, p₄, p₅ } is the universe consisting of five kinds of colored drawing paper for engineering and the parameter set A = {e₁, e₂, e₃} where e₁ stands for ‘thickness’ which includes three levels: thick, average and thin, e₂ stands for ‘color’ which consists of red, green and blue, and e₃ stands for ‘ingredient’ which is made from cellulose, hemicellulose and lignin. Let ¬A = {¬ e₁, ¬ e₂, ¬ e₃} be the “not set of A” where ¬e₁ consists on three levels: attenuated, outstanding, and fat, ¬e₂ consists of green, red and orange and ¬e₃ stands for plastic, rubbery, and raw materials.

Suppose that the multi-fuzzy bipolar soft set ( $F$ , $G$ ,A) is given as follows:

$\begin{matrix} F (e_{1}) = {P_{1} / (0.22, 0.42, 0.36), P_{2} / (0.52, 0.31, 0.16), \\ P_{3} / (0.13, 0.31, 0.53), P_{4} / (0.54, 0.22, 0.13), \\ P_{5} / (0.27, 0.62, 0.03)} \end{matrix}$ $\begin{matrix} . . G (\neg e_{1}) = {p_{1} / (0.33, 0.37, 0.24), p_{2} / (0.13, 0.35, 0.48), \\ p_{3} / (0.46, 0.27, 0.18), p_{4} / (0.39, 0.26, 0.32), \\ p_{5} / (0.34, 0.46, 0.16)} \end{matrix}$ $\begin{matrix} F (e_{2}) = {P_{1} / (0.31, 0.24, 0.45), P_{2} / (0.30, 0.42, 0.24), \\ P_{3} / (0.15, 0.38, 0.41), P_{4} / (0.36, 0.28, 0.30), \\ P_{5} / (0.47, 0.20, 0.31)} \end{matrix}$ $\begin{matrix} G (\neg e_{2}) = {p_{1} / (0.47, 0.14, 0.37), p_{2} / (0.38, 0.25, 0.33), \\ p_{3} / (0.44, 0.16, 0.39), p_{4} / (0.50, 0.07, 0.39), \\ p_{5} / (0.31, 0.56, 0.11)} \end{matrix}$ $\begin{matrix} F (e_{3}) = {P_{1} / (0.30, 0.45, 0.25), P_{2} / (0.14, 0.61, 0.24), \\ P_{3} / (0.35, 0.20, 0.07), P_{4} / (0.47, 0.34, 0.18), \\ P_{5} / (0.17, 0.29, 053)} \end{matrix}$ $\begin{matrix} G (\neg e_{3}) = {p_{1} / (0.27, 0.35, 0.38), p_{2} / (0.57, 0.16, 0.24), \\ p_{3} / (0.35, 0.20, 0.07), p_{4} / (0.47, 0.34, 0.18), \\ p_{5} / (0.17, 0.29, 0.53)} \end{matrix}$

Obviously, ( $F$ ,A) and ( $G$ ,A) are normalized. Suppose that a company would like to select one of them depending on its performance, and after a serious discussion, the company has imposed the following weights for the parameters in A:

For the parameter ‘thickness’, ω (e₁) = (0.4, 0.1, 0.5), for the parameter ‘color’, ω (e₂) = (0.6, 0.3, 0.1), and for the parameter ‘ingredient’, ω (e₃) = (0.2, 0.3, 0.5). Thus we can compute the fuzzy soft sets Δ_F and the Δ_G with their tabular representations as in Tables 3 and 4.

Table 3

Tabular representation of the fuzzy soft set Δ

U	ω (e₁) =	ω (e₂) =	ω (e₃) =
U	(0.4, 0.1, 0.5)	(0.6, 0.3, 0.1)	(0.2, 0.3, 0.5)
P ₁	0.310	0.303	0.320
P ₂	0.319	0.330	0.331
P ₃	0.348	0.245	0.315
P ₄	0.315	0.330	0.286
P ₅	0.185	0.373	0.386

Table 4

Tabular representation of the fuzzy soft set Δ

U	ω (e₁) =	ω (e₂) =	ω (e₃) =
U	(0.4, 0.1, 0.5)	(0.6, 0.3, 0.1)	(0.2, 0.3, 0.5)
P ₁	0.289	0.361	0.349
P ₂	0.327	0.336	0.282
P ₃	0.301	0.351	0.264
P ₄	0.342	0.360	0.317
P ₅	0.262	0.365	0.262

As an appropriate approach, different authors use different rules (or the thresholds) in decision making problems. For example, if we deal with this problem by positive and negative mid-level decision rules, it is clear that the mid-thresholds of Δ_F and Δ_G are fuzzy sets as given below: ${mid}_{Δ F} = {(e_{1}, 0.2954), (e_{1}, 0.3162), (e_{1}, 0.3276)}$ and ${mid}_{Δ G} = {\neg e_{1}, 0.3042), (\neg e_{1}, 0.3546), (\neg e_{1}, 0.2948)$

In tabular representations as in Tables 2 and 3, we can get mid-level soft sets L (Δ_F ; mid) of Δ_F and L (Δ_G ; mid) of Δ_G with choice values. From Tables 2 and 3, we have the bipolar choice values of alternatives as given below:

From Tables 5 and 6, we have the bipolar choice values of alternatives as below

\begin{matrix} ξ_{1} = c_{1}^{+} - c_{1}^{-} = 0 \to ξ_{2} = c_{2}^{+} - c_{2}^{-} = \\ 1 \to ξ_{3} = c_{3}^{+} - c_{3}^{-} = - 2 \to ξ_{4} = c_{4}^{+} - c_{4}^{-} = \\ 2 \to ξ_{5} = c_{5}^{+} - c_{5}^{-} = 0 \end{matrix}

Table 5

Tabular representation of the mid-level soft set L(Δ_F;mid) with choice values

U	e ₁	e ₂	e ₃	Choice value c_i⁺
P ₁	1	0	0	1
P ₂	1	1	1	3
P ₃	1	0	0	1
P ₄	1	1	0	2
P ₅	0	1	1	2

Table 6

U	¬e₁	¬e₂	¬e₃	Choice value c_i^-
P ₁	1	0	0	1
P ₂	0	1	1	2
P ₃	1	1	1	3
P ₄	0	0	0	0
P ₅	1	0	1	2

From above bipolar choice values we see that the maximum bipolar value is 2 and clearly the optimal solution is to select p₄ ;. Therefore, the company should select p₄ as the best colored drawing paper for engineering after specifying weights for different parameters.

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A decision making approach based on multi-fuzzy bipolar soft sets

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Soft sets

2.1.1 Definition

2.1.2 Definition

2.2 Fuzzy bipolar soft sets

2.2.1 Definition

2.2.2 Example

2.3 Multi-fuzzy sets

2.3.1 Definition [12, 14]

2.3.2 Remark

2.3.3 Definition [14]

2.3.4 Example [14]

2.3.5 Definition [13]

2.4 Multi-fuzzy soft set

2.4.1 Definition [7, 14]

2.4.2 Example

2.4.3 Definition [14]

2.4.4 Definition [14]

2.4.5 Definition [14]

3 Multi-fuzzy bipolar soft sets

3.1 The concept of a multi-fuzzy bipolar soft set

3.1.1 Definition

3.1.2 Example

3.1.3 Definition

3.1.4 Example

3.1.5 Definition

3.1.6 Definition

3.1.7 Definition

3.2 Operations on multi-fuzzy bipolar soft sets

3.2.1 Definition

3.2.2 Example

3.2.3 Definition

3.2.4 Definition

3.2.5 Example

3.2.6 Theorem

3.2.7 Definition

3.2.8 Example

3.2.9 Definition

3.2.10 Example

3.2.11 Theorem

3.2.12 Theorem

3.2.13 Theorem

3.2.14 Theorem

4 Application of multi-fuzzy soft set in decision making

4.1 Level soft sets of fuzzy bipolar soft set

4.1.1 Definition

4.1.2 Example

Table 1 Tabular representation of the level soft sets L(F;0.5) and L(G;0.5) U e 1 e 2 e 3 h 1 1 1 0 h 2 1 1 1 h 3 1 1 0 h 4 1 1 1 h 5 1 0 1

4.1.4 Definition

4.1.5 Definition

4.1.6 Definition

4.1.7 Remark

References

Table 1
Tabular representation of the level soft sets L(F;0.5) and L(G;0.5)

U e ₁ e ₂ e ₃

h ₁ 1 1 0

h ₂ 1 1 1

h ₃ 1 1 0

h ₄ 1 1 1

h ₅ 1 0 1