Generalised multi-fuzzy bipolar soft sets and its application in decision making

Abstract

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainty. In this work, new soft model, called the generalised multi-fuzzy bipolar soft model is proposed, and an algorithm based on this notion is presented. Some basic properties of this concept are studied and the related results are investigated. Finally, an application based on generalised multi-fuzzy soft sets in decision making is analyzed.

Keywords

Soft set bipolar soft set fuzzy soft set multi-fuzzy set generalised multi-fuzzy bipolar soft set 03E72 18B30 54A40

1 Introduction

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]. The real world is full of uncertainty, imprecision and vagueness occurred 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. 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. 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, color 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 color image can be approximated by a collection of pixels with a multi-membership function (μ_r, μ_g, μ_b) [14]. The purpose of this paper is to combine the multi-fuzzy set and the bipolar soft set, from which we can obtain a new soft set model named generalised 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, multi-fuzzy bipolar soft set and recall their basic properties in detail. In Section 4, we develop the concept of a generalised multi-fuzzy bipolar soft set and investigate the basic properties of generalised multi-fuzzy bipolar soft set. Finally, an experimental example is discussed to show the validity of the proposed concept.

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:

Definition 1. (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 parameterized family of subsets of U where each parameter e ∈ A 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).

Definition 2. (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, infect, 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].

Definition 3. (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.

Example 4. Let U = {h₁, h₂, h₃, h₄, h₅} be a set of houses and E = {e₁ =expensive, e₂ =beautiful, e₃ =wooden, e₄ =in green surrounding, e₅ =in good repair} 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: $\begin{matrix} F (e_{1}) = (expensive) = {h_{1} / 0.7, h_{2} / 0.6, h_{3} / 0.8, \\ h_{4} / 0.5, h_{5} / 0.6} \\ G (\neg e_{1}) = (cheap) = {h_{1} / 0.2, h_{2} / 0.4, h_{3} / 0.2, \\ h_{4} / 0.5, h_{5} / 0.4} \\ F (e_{2}) = (beautiful) = {h_{1} / 0.8, h_{2} / 0.7, h_{3} / 0.8, \\ h_{4} / 0.6, h_{5} / 0.3} \end{matrix}$ $\begin{matrix} G (\neg e_{2}) = (ugly) = {h_{1} / 0.2, h_{2} / 0.2, h_{3} / 0.1, \\ h_{4} / 0.4, h_{5} / 0.7} \\ F (e_{3}) = (wooden) = {h_{1} / 0.4, h_{2} / 0.6, h_{3} / 0.4, \\ h_{4} / 0.6, h_{5} / 0.5} \\ G (\neg e_{3}) = (not wooden) = {h_{1} / 0.5, h_{2} / 0.4, h_{3} / 0.6, \\ h_{4} / 0.4, h_{5} / 0.5} . \end{matrix}$

2.3 Multi-fuzzy sets

Definition 5. [12,14, 12,14] Let k be a positive integer, a multi-fuzzy set $\tilde{A}$ of dimension k in U is a set of ordered sequences $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . ., μ_{i} (u), . . ., μ_{k} (u)) : u \in U},$ 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^kFP (U).

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

Remark 6. If $\sum_{i = 1}^{k} μ_{i} (u) \leq 1, \forall u \in U,$ then the multi-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 $\frac{1}{l} (μ_{1}, μ_{2}, . . ., μ_{k} (u))$ , then the non-normalized multi-fuzzy set can be changed into a normalized multi-fuzzy set.

Definition 7. [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 ${\tilde{1}}_{k} .$

Example 8. [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.

Definition 9. [13] Let $\tilde{A} = {u / (μ_{1} (u), μ_{2} (u), . . .,$ μ_k (u)) : u ∈ 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) ≤ γ_i (u) , ∀ u ∈ U and 1 ≤ i ≤ k .

$\tilde{A} = \tilde{B}$ iff μ_i (u) = γ_i (u) , ∀ u ∈ U and 1 ≤ i ≤ k .

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

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

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

3 Multi-fuzzy bipolar soft set

Definition 10. 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 ⟶ M^kFP (U) and G : ¬ A ⟶ M^kFP (U) , where M^kFP (U) is the set of all multi-fuzzy soft sets of dimetion k over U .

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

Example 11. 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: $\begin{matrix} F (e_{1}) \\ = {\frac{h_{1}}{(0.4, 0.2, 0.3)}, \frac{h_{2}}{(0.2, 0.1, 0.6)}, \frac{h_{3}}{(0.1, 0.3, 0.4)}, \\ \frac{h_{4}}{(0.3, 0.1, 0.5)}, \frac{h_{5}}{(0.7, 0.1, 0.2)}} \end{matrix}$ $\begin{matrix} G (\neg e_{1}) \\ = {\frac{h_{1}}{(0.3, 0.1, 0.5)}, \frac{h_{2}}{(0.7, 0.1, 0.2)}, \frac{h_{3}}{(0.4, 0.5, 0.1)}, \\ \frac{h_{4}}{(0.5, 0.3, 0.1)}, \frac{h_{5}}{(0.6, 0.3, 0.1)}} \end{matrix}$ $\begin{matrix} F (e_{2}) \\ = {\frac{h_{1}}{(0.5, 0.2, 0.3)}, \frac{h_{2}}{(0.4, 0.0, 0.6)}, \frac{h_{3}}{(0.3, 0.4, 0.2)}, \\ \frac{h_{4}}{(0.4, 0.1, 0.4)}, \frac{h_{5}}{(0.6, 0.2, 0.2)}} \end{matrix}$ $\begin{matrix} G (\neg e_{2}) \\ = {\frac{h_{1}}{(0.4, 0.3, 0.4)}, \frac{h_{2}}{(0.5, 0.1, 0.4)}, \frac{h_{3}}{(0.3, 0.2, 0.4)}, \\ \frac{h_{4}}{(0.5, 0.2, 0.3)}, \frac{h_{5}}{(0.4, 0.3, 0.2)}} \end{matrix}$ $\begin{matrix} F (e_{3}) \\ = {\frac{h_{1}}{(0.5, 0.3, 0.2)}, \frac{h_{2}}{(0.4, 0.5, 0.1)}, \frac{h_{3}}{(0.3, 0.4, 0.3)}, \\ \frac{h_{4}}{(0.3, 0.3, 0.3)}, \frac{h_{5}}{(0.2, 0.3, 0.2)}} \end{matrix}$ $\begin{matrix} G (\neg e_{3}) \\ = {\frac{h_{1}}{(0.5, 0.2, 0.3)}, \frac{h_{2}}{(0.5, 0.4, 0.1)}, \frac{h_{3}}{(0.4, 0.3, 0.3)}, \\ \frac{h_{4}}{(0.4, 0.3, 0.3)}, \frac{h_{5}}{(0.1, 0.4, 0.2)}} \end{matrix}$

In matrix form, the above multi-fuzzy bipolar soft set can be expressed as follows: $F = [\begin{matrix} (0.4, 0.2, 0.3) & (0.2, 0.1, 0.6) & (0.1, 0.3, 0.4) \\ (0.3, 0.1, 0.5) & (0.7, 0.1, 0.2) \\ (0.5, 0.2, 0.3) & (0.4, 0.0, 0.6) & (0.3, 0.4, 0.2) \\ (0.4, 0.1, 0.4) & (0.6, 0.2, 0.2) \\ (0.5, 0.3, 0.2) & (0.4, 0.5, 0.1) & (0.3, 0.4, 0.3) \\ (0.3, 0.3, 0.3) & (0.2, 0.3, 0.2) \end{matrix}]$ $G = [\begin{matrix} (0.3, 0.1, 0.5) & (0.7, 0.1, 0.2) & (0.4, 0.5, 0.1) \\ (0.5, 0.3, 0.1) & (0.6, 0.3, 0.1) \\ (0.4, 0.3, 0.4) & (0.5, 0.1, 0.4) & (0.3, 0.2, 0.4) \\ (0.5, 0.2, 0.3) & (0.4, 0.3, 0.2) \\ (0.5, 0.2, 0.3) & (0.5, 0.4, 0.1) & (0.4, 0.3, 0.3) \\ (0.4, 0.3, 0.3) & (0.1, 0.4, 0.2) \end{matrix}]$ 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.

Definition 12. Let A, B ⊆ E. Let (F, G, A) and (F₁, G₁, 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 (F₁, G₁, B) if:

A ⊆ B, and

F (e) ⊑ F₁ (e) and G₁ (e) ⊑ G (e), for all e ∈ A.

Here, we write

(F, G, A) \tilde{⊑} (F_{1}, G_{1}, B)

Definition 13. Two multi-fuzzy bipolar soft sets (F, G, A) and (F₁, G₁, 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 (F₁, G₁, B) and (F₁, G₁, B) is a multi-fuzzy bipolar soft subset of (F, G, A).

Definition 14. (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.

Definition 15. (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.

4 Generalised multi-fuzzy bipolar soft sets

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

4.1 The concept of a generalised multi-fuzzy bipolar soft set

Definition 16. Let U ={ x₁, x₂, . . . , x_n } be the universal set of elements and E ={ e₁, e₂, . . . , e_n } be the universal set of parameters and A ⊆ E. Let (F, G, A) be a multi-fuzzy bipolar soft set of dimension k over U, where F and G are mappings given by F : A ⟶ M^kFP (U) and G : ¬ A ⟶ M^kFP (U) . Additionally, let μ and υ be a fuzzy subsets of A i.e., μ : A ⟶ [0, 1] and υ : ¬ A ⟶ [0, 1] . Next the triplet (F_μ, G_υ, A) is called a generalised multi-fuzzy bipolar soft set dimension k over U, where F_μ and G_υ, are mappings given by F_μ : A ⟶ M^kFP (U) × I and G_υ : ¬ A ⟶ M^kFP (U) × I with F_μ (e) = (F (e) , μ (e)), G_υ (¬ e) = (G (¬ e) , υ (¬ e)) and F (e) ∈ M^kFP (U) , G (¬ e) ∈ M^kFP (U) and μ (e) , υ (¬ e) ∈ [0, 1] .

Here for each parameter e_i, F_μ (e_i) = (F (e_i) , μ (e_i)), indicates the degrees of belongingness of the elements of U in F (e_i) and of the possibility of such belongingness. This parameter is represented by μ (e_i) . While G_υ (¬ e_i) = (G (¬ e_i) , υ (¬ e_i)) indicates the degrees of non-belongingness of the elements of U in G (¬ e_i) and of the possibility of such non-belongingness. This parameter is represented by υ (¬ e_i) .

Example 17. 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, and 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” which consists of 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. Let μ, υ : A ⟶ [0, 1] be defined by μ (e₁) =0.2, μ (e₂) =0.3, μ (e₃) =0.5 and υ (¬ e₁) =0.2, υ (¬ e₂) =0.3, υ (¬ e₃) =0.4 . We define a generalised multi-fuzzy bipolar soft set of dimension 3 over U as follows: $\begin{matrix} F_{μ} (e_{1}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.2, 0.3), h_{2} / (0.2, 0.1, 0.6), \\ h_{3} / (0.1, 0.3, 0.4), \\ h_{4} / (0.3, 0.1, 0.5), h_{5} / (0.7, 0.1, 0.2) \end{matrix}}, 0.2) \\ G_{υ} (\neg e_{1}) \\ = ({\begin{matrix} h_{1} / (0.3, 0.1, 0.5), h_{2} / (0.7, 0.1, 0.2), \\ h_{3} / (0.4, 0.5, 0.1), \\ h_{4} / (0.5, 0.3, 0.1), h_{5} / (0.6, 0.3, 0.1) \end{matrix}}, 0.1) \\ F_{μ} (e_{2}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.2, 0.3), h_{2} / (0.4, 0.0, 0.6), \\ h_{3} / (0.3, 0.4, 0.2), \\ h_{4} / (0.4, 0.1, 0.4), h_{5} / (0.6, 0.2, 0.2) \end{matrix}}, 0.3) \\ G_{υ} (\neg e_{2}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.3, 0.4), h_{2} / (0.5, 0.1, 0.4), \\ h_{3} / (0.3, 0.2, 0.4), \\ h_{4} / (0.5, 0.2, 0.3), h_{5} / (0.4, 0.3, 0.2) \end{matrix}}, 0.2) \\ F_{μ} (e_{3}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.3, 0.2), h_{2} / (0.4, 0.5, 0.1), \\ h_{3} / (0.3, 0.4, 0.3), \\ h_{4} / (0.3, 0.3, 0.3), h_{5} / (0.2, 0.3, 0.2) \end{matrix}}, 0.5) \\ G_{υ} (\neg e_{3}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.2, 0.3), h_{2} / (0.5, 0.4, 0.1), \\ h_{3} / (0.4, 0.3, 0.3), \\ h_{4} / (0.4, 0.3, 0.3), h_{5} / (0.1, 0.4, 0.2) \end{matrix}}, 0.4) . \end{matrix}$

In matrix form, the above multi-fuzzy bipolar soft set can be expressed as follows: $\begin{matrix} F_{μ} = \\ [\begin{matrix} (0.4, 0.2, 0.3) & (0.2, 0.1, 0.6) & (0.1, 0.3, 0.4) \\ (0.3, 0.1, 0.5) & (0.7, 0.1, 0.2) \\ (0.5, 0.2, 0.3) & (0.4, 0.0, 0.6) & (0.3, 0.4, 0.2) \\ (0.4, 0.1, 0.4) & (0.6, 0.2, 0.2) \\ (0.5, 0.3, 0.2) & (0.4, 0.5, 0.1) & (0.3, 0.4, 0.3) \\ (0.3, 0.3, 0.3) & (0.2, 0.3, 0.2) \end{matrix} \begin{matrix} 0.2 \\ 0.3 \\ 0.5 \end{matrix}] \end{matrix}$ $\begin{matrix} G_{υ} = \\ [\begin{matrix} (0.3, 0.1, 0.5) & (0.7, 0.1, 0.2) & (0.4, 0.5, 0.1) \\ (0.5, 0.3, 0.1) & (0.6, 0.3, 0.1) \\ (0.4, 0.3, 0.4) & (0.5, 0.1, 0.4) & (0.3, 0.2, 0.4) \\ (0.5, 0.2, 0.3) & (0.4, 0.3, 0.2) \\ (0.5, 0.2, 0.3) & (0.5, 0.4, 0.1) & (0.4, 0.3, 0.3) \\ (0.4, 0.3, 0.3) & (0.1, 0.4, 0.2) \end{matrix} \begin{matrix} 0.1 \\ 0.2 \\ 0.4 \end{matrix}] \end{matrix}$ where the ith row vector represents F (e_i) and G (e_i) , the ith column vector represents h_i, the last column represents the values of μ and υ and are called the membership matrices of F (e_i) and G (¬ e_i) , respectively.

Definition 18. Let A, B ⊆ E. Let (F_μ, G_υ, A) and (F_1,λ, G_1,ν, B) be two generalised multi-fuzzy bipolar soft set of dimension k over U. Then (F_μ, G_υ, A) is called a generalised multi-fuzzy bipolar soft subset of (F_1,λ, G_1,ν, B) if:

A ⊆ B,

$(F_{μ}, G_{υ}, A) \tilde{⊑} (F_{1, λ}, G_{1, ν}, B)$ and

μ (e) ≤ λ (e) and ν (¬ e) ≤ υ (¬ e), for all e ∈ A.

Here, we write

(F_{μ}, G_{υ}, A) \tilde{⊑} (F_{1, λ}, G_{1, ν}, B)

Example 19. Let U = { h₁, h₂, h₃, h₄ } , A = {e₁, e₂} and B = {e₁, e₂, e₃} . Define fuzzy subsets μ, λ, υ, ν as follows: $μ (x) = {\begin{matrix} 0.4, if x = e_{1}, \\ 0.3, if x = e_{2}, \end{matrix}, λ (x) = {\begin{matrix} 0.6, if x = e_{1}, \\ 0.3, if x = e_{2}, \\ 0.1, if x = e_{3}, \end{matrix}$ and $ν (x) = {\begin{matrix} 0.3, if x = \neg e_{1}, \\ 0.2, if x = \neg e_{2}, \end{matrix}$ $υ (x) = {\begin{matrix} 0.5, if x = \neg e_{1}, \\ 0.4, if x = \neg e_{2} . \end{matrix}$ Clearly, A ⊆ B. Suppose that (F_μ, G_υ, A) and (F_1,λ, G_1,ν, B) are two generalised multi-fuzzy bipolar soft set of dimension 3 over U, defined as follows. $\begin{matrix} F_{μ} (e_{1}) \\ = ({\begin{matrix} h_{1} / (0.3, 0.3, 0.1), h_{2} / (0.4, 0.3, 0.2), \\ h_{3} / (0.5, 0.2, 0.1), \\ h_{4} / (0.4, 0.2, 0.2), h_{5} / (0.4, 0.2, 0.3) \end{matrix}}, 0.4) \end{matrix}$ $\begin{matrix} G_{υ} (\neg e_{1}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.4, 0.1), h_{2} / (0.4, 0.4, 0.2), \\ h_{3} / (0.5, 0.2, 0.3), \\ h_{4} / (0.4, 0.5, 0.1), h_{5} / (0.4, 0.3, 0.3) \end{matrix}}, 0.5) \\ F_{1, λ} (e_{1}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.3, 0.2), h_{2} / (0.5, 0.3, 0.2), \\ h_{3} / (0.5, 0.3, 0.2), \\ h_{4} / (0.4, 0.3, 0.3), h_{5} / (0.4, 0.3, 0.3) \end{matrix}}, 0.6) \\ G_{1, ν} (\neg e_{1}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.3, 0.1), h_{2} / (0.3, 0.3, 0.2), \\ h_{3} / (0.4, 0.1, 0.3), \\ h_{4} / (0.3, 0.4, 0.1), h_{5} / (0.3, 0.3, 0.3) \end{matrix}}, 0.3) \\ F_{μ} (e_{2}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.4, 0.1), h_{2} / (0.5, 0.2, 0.2), \\ h_{3} / (0.4, 0.1, 0.1), \\ h_{4} / (0.3, 0.1, 0.2), h_{5} / (0.4, 0.3, 0.3) \end{matrix}}, 0.3) \\ G_{υ} (\neg e_{2}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.3, 0.2), h_{2} / (0.4, 0.5, 0.1), \\ h_{3} / (0.4, 0.3, 0.3), \\ h_{4} / (0.3, 0.4, 0.2), h_{5} / (0.4, 0.2, 0.3) \end{matrix}}, 0.4) \\ F_{1, λ} (e_{2}) \\ = ({\begin{matrix} h_{1} / (0.5, 0.4, 0.1), h_{2} / (0.5, 0.3, 0.2), \\ h_{3} / (0.4, 0.2, 0.2), \\ h_{4} / (0.3, 0.3, 0.3), h_{5} / (0.4, 0.4, 0.3) \end{matrix}}, 0.4) \\ G_{1, ν} (\neg e_{2}) \\ = ({\begin{matrix} h_{1} / (0.4, 0.3, 0.2), h_{2} / (0.4, 0.4, 0.1), \\ h_{3} / (0.3, 0.3, 0.3), \\ h_{4} / (0.2, 0.3, 0.2), h_{5} / (0.3, 0.2, 0.2) \end{matrix}}, 0.2) . \end{matrix}$

From above generalised multi-fuzzy sets, we observe that $(F, G, A) \tilde{⊑} (F_{1}, G_{1}, B)$ and μ (e) ≤ λ (e) and ν (¬ e) ≤ υ (¬ e). Therefore, $(F_{μ}, G_{υ}, A) \tilde{⊑} (F_{1, λ}, G_{1, ν}, B)$ .

Definition 20. Two generalised multi-fuzzy bipolar soft sets (F_μ, G_υ, A) and (F_1,λ, G_1,ν, B) of dimension k over a common universe U are said to be generalised multi-fuzzy bipolar soft equal if (F_μ, G_υ, A) is a multi-fuzzy bipolar soft subset of (F_1,λ, G_1,ν, B) and (F_1,λ, G_1,ν, B) is a multi-fuzzy bipolar soft subset of (F_μ, G_υ, A).

Definition 21. (Null generalised multi-fuzzy bipolar soft set) A generalised multi-fuzzy bipolar soft set (F_μ, G_υ, A) of dimension k over U is said to be null generalised multi-fuzzy bipolar soft set, denoted by ${\tilde{Φ}}_{(μ, λ, A)}^{k}$ if $F_{μ} (e_{i}) = (F (e_{i}), μ (e_{i})) = ({\tilde{Φ}}_{k}, 0)$ and $G_{λ} (\neg e_{i}) = (G (\neg e_{i}), λ (\neg e_{i})) = ({\tilde{1}}_{k}, 1)$ for all e_i ∈ A.

Definition 22. (Absolute generalised multi-fuzzy bipolar soft set) A generalised multi-fuzzy bipolar soft set (F_μ, G_υ, A) of dimension k over U is said to be absolute generalised multi-fuzzy soft set, denoted by ${\tilde{U}}_{(μ, λ, A)}^{k}$ if $F_{μ} (e_{i}) = (F (e_{i}), μ (e_{i})) = ({\tilde{1}}_{k}, 1)$ and $G (\neg e_{i}) = (G (\neg e_{i}), λ (\neg e_{i})) = ({\tilde{Φ}}_{k}, 0)$ for all e_i ∈ A.

4.2 Operations on generalised multi-fuzzy bipolar soft sets

Definition 23. The complement of a generalised multi-fuzzy soft bipolar soft set $(F_{μ}, G_{λ}, A)$ of dimension k over U is denoted by (F_μ, G_υ, A) ^c and is defined by ${(F_{μ}, G_{υ}, A)}^{c} = (F_{μ^{c}}^{c}, G_{λ^{c}}^{c}, A)$ where $F_{μ^{c}}^{c}$ and $G_{λ^{c}}^{c}$ are mappings given by $F_{μ^{c}}^{c} (e_{i}) = ({(F (e_{i}))}^{c}, μ^{c} (e))$ and $G_{λ^{c}}^{c} (\neg e_{i}) = ({(G (\neg e_{i}))}^{c}, λ^{c} (\neg e_{i}))$ for all e_i ∈ A. Clearly, ((F_μ, G_λ, A) ^c) ^c is the same as (F_μ, G_λ, A) , i.e., complement of a generalised multi-fuzzy bipolar soft set is involute in nature. Suppose ${\tilde{Φ}}_{(μ, λ, A)}^{k}$ and ${\tilde{U}}_{(μ, λ, A)}^{k}$ are generalised multi-fuzzy bipolar soft sets of dimension k over U, then ${({\tilde{Φ}}_{(μ, λ, A)}^{k})}^{c} = {\tilde{U}}_{(μ, λ, A)}^{k}$ and ${({\tilde{U}}_{(μ, λ, A)}^{k})}^{c} = {\tilde{Φ}}_{(μ, λ, A)}^{k}$ .

Example 24. Reconsider Example 16, we have ${(F_{μ}, G_{λ}, A)}^{c} = (F_{μ^{c}}^{c}, G_{λ^{c}}^{c}, A)$ as follows: $\begin{matrix} F_{μ^{c}}^{c} (e_{1}) = ({h_{1} / (0.7, 0.7, 0.9), h_{2} / (0.6, 0.7, 0.8), \\ h_{3} / (0.5, 0.8, 0.9), h_{4} / (0.6, 0.8, 0.8), \\ h_{5} / (0.6, 0.8, 0.7)}, 0.6) \end{matrix}$ $\begin{matrix} G_{λ^{c}}^{c} (\neg e_{1}) = ({h_{1} / (0.5, 0.6, 0.9), h_{2} / (0.6, 0.6, 0.8), \\ h_{3} / (0.5, 0.8, 0.7), h_{4} / (0.6, 0.5, 0.9), \\ h_{5} / (0.6, 0.7, 0.7)}, 0.5) \end{matrix}$ $\begin{matrix} F_{μ^{c}}^{c} (e_{2}) = ({h_{1} / (0.6, 0.6, 0.9), h_{2} / (0.5, 0.8, 0.8), \\ h_{3} / (0.6, 0.9, 0.9), h_{4} / (0.7, 0.9, 0.8), \\ h_{5} / (0.6, 0.7, 0.7)}, 0.7) \end{matrix}$ $\begin{matrix} G_{λ^{c}}^{c} (\neg e_{2}) = ({h_{1} / (0.5, 0.7, 0.8), h_{2} / (0.6, 0.5, 0.9), \\ h_{3} / (0.6, 0.7, 0.7), h_{4} / (0.7, 0.6, 0.8), \\ h_{5} / (0.6, 0.8, 0.7)}, 0.6) . \end{matrix}$

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

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

Definition 25. If $(F_{μ}, G_{υ}, A)$ and $(F_{1, λ}, G_{1, ν}, B)$ are two generalised multi-fuzzy bipolar soft sets of dimension k over U, then “ $(F_{μ}, G_{υ}, A)$ AND $(F_{1, λ}, G_{1, ν}, B)$ ” denoted by $(F_{μ}, G_{υ}, A) \land (F_{1, λ}, G_{1, ν}, B)$ is defined by $(F_{μ}, G_{υ}, A) \land (F_{1, λ}, G_{1, ν}, B) = (H_{η}, I_{δ}, A \times B),$ where $H (α, β) = F (α) ⊓ F_{1} (β),$ η (α, β) = μ (α) ∧ λ (β) and $I (\neg α, \neg β) = G (\neg α) ⊔ G_{1} (\neg β), δ (\neg α, \neg β) = υ (\neg α) \lor ν (\neg β)$ for all (α, β) ∈ A × B .

Definition 26. If $(F_{μ}, G_{υ}, A)$ and $(F_{1, λ}, G_{1, ν}, B)$ are two generalised multi-fuzzy bipolar soft sets of dimension k over U, then “ $(F_{μ}, G_{υ}, A)$ OR $(F_{1, λ}, G_{1, ν}, B)$ ” denoted by $(F_{μ}, G_{υ}, A) \lor (F_{1, λ}, G_{1, ν}, B)$ is defined by $(F_{μ}, G_{υ}, A) \lor (F_{1, λ}, G_{1, ν}, B) = (O_{η}, J_{δ}, A \times B),$ where $O (α, β) = F (α) ⊔ F_{1} (β), η (α, β) = μ (α) \lor λ (β)$ and $J (\neg α, \neg β) = G (\neg α) ⊓ G_{1} (\neg β), δ (\neg α, \neg β) = υ (\neg α) \land ν (\neg β)$ for all (α, β) ∈ A × B .

Example 27. Let U = {u₁, u₂, u₃}, A = {e₁, e₂} and B = {e₂, e₃, e₄} . Let μ, λ, υ, ν : A ⟶ [0, 1] be fuzzy subsets such that:

$μ (x) = {\begin{matrix} 0.4, if x = e_{1}, \\ 0.3, if x = e_{2}, \end{matrix}, λ (x) = {\begin{matrix} 0.6, if x = e_{2}, \\ 0.2, if x = e_{3}, \\ 0.5, if x = e_{4}, \end{matrix}, υ (x) = {\begin{matrix} 0.6, if x = e_{1}, \\ 0.5, if x = e_{2}, \end{matrix},$ $ν (x) = {\begin{matrix} 0.4, if x = e_{2}, \\ 0.5, if x = e_{3}, \\ 0.3, if x = e_{4} . \end{matrix}$

Suppose that $(F_{μ}, G_{υ}, A)$ and $(F_{1, λ}, G_{1, ν}, B)$ are two generalised multi-fuzzy bipolar soft sets of dimension 2 over U, defined as follows: $\begin{matrix} F_{μ} (e_{1}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), \\ u_{3} / (0.7, 0.3)}, 0.4) \\ G_{λ} (\neg e_{1}) = ({u_{1} / (0.7, 0.2), u_{2} / (0.8, 0.1), \\ u_{3} / (0.6, 0.3)}, 0.6) \\ F_{μ} (e_{2}) = ({u_{1} / (0.6, 0.4), u_{2} / (0.5, 0.4), \\ u_{3} / (0.8, 0.2)}, 0.3) \\ G_{λ} (\neg e_{2}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.2), \\ u_{3} / (0.8, 0.2)}, 0.5) \\ F_{1, υ} (e_{2}) = ({u_{1} / (0.8, 0.2), u_{2} / (0.6, 0.4), \\ u_{3} / (0.5, 0.3)}, 0.6) \\ G_{1, ν} (\neg e_{2}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.3, 0.7), \\ u_{3} / (0.5, 0.5)}, 0.4) \\ F_{1, υ} (e_{3}) = ({u_{1} / (0.4, 0.5), u_{2} / (0.2, 0.7), \\ u_{3} / (0.3, 0.5)}, 0.2) \\ G_{1, ν} (\neg e_{3}) = ({u_{1} / (0.5, 0.3), u_{2} / (0.3, 0.6), \\ u_{3} / (0.4, 0.6)}, 0.5) \\ F_{1, υ} (e_{4}) = ({u_{1} / (0.1, 0.8), u_{2} / (0.3, 0.7), \\ u_{3} / (0.4, 0.5)}, 0.5) \\ G_{1, ν} (\neg e_{4}) = ({u_{1} / (0.2, 0.6), u_{2} / (0.4, 0.5), \\ u_{3} / (0.5, 0.3)}, 0.3) . \end{matrix}$

Then we have $(F_{μ}, G_{λ}, A) \land (F_{1, υ}, G_{1, ν}, B) = (H_{η}, I_{δ}, A \times B)$ and $(F_{μ}, G_{λ}, A) \lor (F_{1, υ}, G_{1, ν}$ , $B) = (O_{η}, J_{δ}, A \times B)$ as follows: $\begin{matrix} H_{η} (e_{1}, e_{2}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), \\ u_{3} / (0.7, 0.2)}, 0.6) \\ I_{δ} (\neg e_{1}, \neg e_{2}) = ({u_{1} / (0.7, 0.4), u_{2} / (0.8, 0.2), \\ u_{3} / (0.8, 0.3)}, 0.4) \\ H_{η} (e_{1}, e_{3}) = ({u_{1} / (0.4, 0.4), u_{2} / (0.2, 0.4), \\ u_{3} / (0.3, 0.3)}, 0.4) \end{matrix}$ $\begin{matrix} I_{δ} (\neg e_{1}, \neg e_{3}) = ({u_{1} / (0.7, 0.3), u_{2} / (0.8, 0.6), \\ u_{3} / (0.6, 0.6)}, 0.5) \\ H_{η} (e_{1}, e_{4}) = ({u_{1} / (0.1, 0.4), u_{2} / (0.3, 0.4), \\ u_{3} / (0.4, 0.3)}, 0.5) \\ I_{δ} (\neg e_{1}, \neg e_{4}) = ({u_{1} / (0.5, 0.6), u_{2} / (0.7, 0.5), \\ u_{3} / (0.8, 0.3)}, 0.6) \\ H_{η} (e_{2}, e_{2}) = ({u_{1} / (0.6, 0.2), u_{2} / (0.5, 0.4), \\ u_{3} / (0.5, 0.2)}, 0.3) \\ I_{δ} (\neg e_{2}, \neg e_{2}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.7), \\ u_{3} / (0.8, 0.5)}, 0.5) \\ H_{η} (e_{2}, e_{3}) = ({u_{1} / (0.4, 0.4), u_{2} / (0.2, 0.4), \\ u_{3} / (0.3, 0.2)}, 0.3) \\ I_{δ} (\neg e_{2}, \neg e_{3}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.3, 0.7), \\ u_{3} / (0.5, 0.6)}, 0.3) \\ H_{η} (e_{2}, e_{4}) = ({u_{1} / (0.1, 0.4), u_{2} / (0.3, 0.4), \\ u_{3} / (0.4, 0.2)}, 0.5) \\ I_{δ} (\neg e_{2}, \neg e_{4}) = ({u_{1} / (0.8, 0.8), u_{2} / (0.6, 0.7), \\ u_{3} / (0.5, 0.5)}, 0.3) . \end{matrix}$

Similarly, we can find $(F_{μ}, G_{λ}, A) \lor (F_{1, υ}, G_{1, ν}, B) = (O_{η}, J_{δ}, A \times B)$ .

Theorem 28. Let $(F_{μ}, G_{λ}, A)$ and $(F_{1, υ}, G_{1, ν}, B)$ be two generalised multi-fuzzy bipolar soft sets of dimension k over U. Then

$((F_{μ}, G_{λ}, A) \land (F_{1, υ}, G_{1, ν}, B))^{c} = (F_{μ}, G_{λ}, A)^{c} \lor (F_{1, υ}, G_{1, ν}, B)^{c}$ .

$((F_{μ}, G_{λ}, A) \lor (F_{1, υ}, G_{1, ν}, B))^{c} = (F_{μ}, G_{λ}, A)^{c} \land (F_{1, υ}, G_{1, ν}, B)^{c}$ .

Proof. (1) Suppose that $(F_{μ}, G_{λ}, A) \land (F_{1, υ}$ , $G_{1, ν}, B) = (H_{η}, I_{δ}, A \times B)$ . Then $((F_{μ}, G_{λ}, A) \land (F_{1, υ}, G_{1, ν}, B))^{c} = (H_{η}, I_{δ}, A \times B)^{c} = (H_{η^{c}}^{c}, I_{δ^{c}}^{c}, A$ ×B). Where η^c (α, β) = (μ ∧ υ) ^c (α, β) = μ^c (α) ∨υ^c (β) and δ^c (¬ α, ¬ β) = (λ ∨ ν) ^c (¬ α, ¬ β) = λ^c (¬ α) ∧ ν^c (¬ β) for all (α, β) ∈ A × B . Now, $(F_{μ}, G_{λ}, A)^{c} \lor (F_{1, υ}, G_{1, ν}, B)^{c} = (F_{μ^{c}}^{c}, G_{λ^{c}}^{c}, A) \lor (F_{1, υ^{c}}^{c}, G_{1, ν^{c}}^{c}, B) = (O_{τ}, J_{κ}, A \times B)$ . Where τ (α, β) = μ^c (α) ∨ υ^c (β) = η^c (α, β) and κ (¬ α, ¬ β) = λ^c (¬ α) ∧ ν^c (¬ β) = δ^c (¬ α, ¬ β) for all (α, β) ∈ A ×B . Therefore, $\begin{matrix} H_{η c}^{c} (α, β) & = & (H^{c} (α, β), η^{c} (α, β))^{c} \\ = & ((F (α) ⊓ F_{1} (β))^{c}, η^{c} (α, β)) \\ = & (F^{c} (α) ⊔ F_{1}^{c} (β), η^{c} (α, β)) \\ = & (O (α, β), τ (α, β)) \\ = & (O_{τ}, A \times B), \end{matrix}$ and $\begin{matrix} I_{δ^{c}}^{c} (\neg α, \neg β) & = & (I^{c} (\neg α, \neg β), δ^{c} (\neg α, \neg β)) \\ = & (F^{c} (\neg α) ⊓ F_{1}^{c} (\neg β), δ^{c} (\neg α, \neg β)) \\ = & (J (\neg α, \neg β), δ^{c} (\neg α, \neg β)) \\ = & (J_{κ}, A \times B) . \end{matrix}$ Hence, $((F_{μ}, G_{λ}, A) \land (F_{1, υ}, G_{1, ν}, B))^{c} = (F_{μ}, G_{λ}$ , $A)^{c} \lor (F_{1, υ}, G_{1, ν}, B)^{c} .$

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

Definition 29. The union of two generalised multi-fuzzy bipolar soft sets $(F_{μ}, G_{λ}, A)$ and $(F_{1, υ}, G_{1, ν}, B)$ of dimension k over U is the generalised multi-fuzzy bipolar soft set $(H_{η}, I_{δ}, C),$ where C = A ∪ B and for all e ∈ C, $H (e) = {\begin{matrix} F (e), if e \in A - B, \\ F_{1} (e), if e \in B - A, \\ F (e) ⊔ F_{1} (e), if e \in A \cap B \end{matrix},$ $η (e) = {\begin{matrix} μ (e), if e \in A - B, \\ υ (e), if e \in B - A, \\ μ (e) \lor υ (e), if e \in A \cap B \end{matrix}$ and $\begin{array}{l} ℐ (\neg e) = {\begin{matrix} G (\neg e), if \neg e \in (\neg A) - (\neg B), \\ G_{1} (\neg e), if \neg e \in (\neg B) - (\neg A), \\ G (e) ⊓ G_{1} (e), if \neg e \in (\neg A) \cap (\neg B) % \end{matrix}, \\ δ (\neg e) = {\begin{matrix} λ (\neg e), if \neg e \in (\neg A) - (\neg B), \\ ν (\neg e), if \neg e \in (\neg B) - (\neg A), \\ λ (\neg e) \land ν (\neg e), if % \neg e \in (\neg A) \cap (\neg B) \end{matrix} \end{array}$ We write $(F_{μ}, G_{λ}, A) \tilde{⊔} (F_{1, υ}, G_{1, ν}, B) = (H_{η}, I_{δ}$ ,C).

Example 30. Reconsider Example 18, we have $(F_{μ}, G_{λ}, A) \tilde{⊔} (F_{1, υ}, G_{1, ν}, B) = (H_{η}, I_{δ}, C),$ where C = A ∪ B = {e₁, e₂, e₄, e₅} , $\begin{matrix} H_{η} (e_{1}) = F_{μ} (e_{1}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.4, 0.4), \\ u_{3} / (0.7, 0.3)}, 0.4) \\ H_{η} (e_{4}) = F_{1, υ} (e_{4}) = ({u_{1} / (0.4, 0.5), \\ u_{2} / (0.2, 0.7), u_{3} / (0.3, 0.5)}, 0.5) \end{matrix}$ $\begin{matrix} H_{η} (e_{5}) = F_{1, υ} (e_{5}) = ({u_{1} / (0.1, 0.8), \\ u_{2} / (0.3, 0.7), u_{3} / (0.4, 0.5)}, 0.3) \\ H_{η} (e_{2}) = F_{μ} (e_{2}) ⊔ F_{1, υ} (e_{2}) = ({u_{1} / (0.8, 0.4), \\ u_{2} / (0.6, 0.4), u_{3} / (0.8, 0.3)}, 0.4) \\ I_{δ} (\neg e_{1}) = G_{λ} (\neg e_{1}) = ({u_{1} / (0.7, 0.2), \\ u_{2} / (0.8, 0.1), u_{3} / (0.6, 0.3)}, 0.5) \\ I_{δ} (\neg e_{4}) = G_{1, ν} (\neg e_{4}) = ({u_{1} / (0.5, 0.3), \\ u_{2} / (0.3, 0.6), u_{3} / (0.4, 0.6)}, 0.8) \\ I_{δ} (\neg e_{5}) = G_{1, ν} (\neg e_{5}) = ({u_{1} / (0.2, 0.6), \\ u_{2} / (0.4, 0.5), u_{3} / (0.5, 0.3)}, 0.6) \\ I_{δ} (\neg e_{2}) = G_{λ} (\neg e_{2}) ⊓ G_{1, ν} (\neg e_{2}) = ({u_{1} / (0.5, 0.4), \\ u_{2} / (0.3, 0.2), u_{3} / (0.5, 0.2)}, 0.5) . \end{matrix}$ Definition 31. The intersection of two generalised multi-fuzzy bipolar soft sets $(F_{μ}, G_{λ}, A)$ and $(F_{1, υ}, G_{1, ν}, B)$ of dimension k with A∩ B ≠ ∅ over U is the generalised multi-fuzzy bipolar soft set $(H_{η}, I_{δ}, C),$ where C = A ∩ B and for all e ∈ C, $H (e) = F (e) ⊓ F_{1} (e), η (e) = μ (e) \land λ (e)$ and $I (\neg e) = G (\neg e) ⊔ G_{1} (\neg e), δ (\neg e) = υ (\neg e)$ ∨ν (¬ e).

We write $(F_{μ}, G_{λ}, A) \tilde{⊓} (F_{1, υ}, G_{1, ν}, B) = (H_{η}, I_{δ}$ ,C).

Example 32. Reconsider Example 18, we have A ∩ B = {e₂} and $(F_{μ}, G_{λ}, A) \tilde{⊓} (F_{1, υ}, G_{1, ν}, B) = (H_{η}, I_{δ}, C)$ implies

$H_{η} (e_{2}) = F_{μ} (e_{2}) ⊓ F_{1, υ} (e_{2}) = ({u_{1} / (0.6, 0.2)$ , u₂/(0.5, 0.4) , u₃/(0.8, 0.2)} , 0.2) and $I_{δ} (\neg e_{2}) = G_{λ}$ $(\neg e_{2}) ⊔ G_{1, ν} (\neg e_{2}) = ({u_{1} / (0.5, 0.4), u_{2} / (0.7, 0.7)$ , u₃/(0.8, 0.5)} , 0.7).

The following results are obvious.

Theorem 33. Let $(F_{μ}, G_{λ}, A)$ and $(F_{1, υ}, G_{1, ν}, B)$ be two generalised multi-fuzzy bipolar soft sets of dimension k over U. Then

$(F_{μ}, G_{λ}, A) \tilde{⊔} (F_{μ}, G_{λ}, A) = (F_{μ}, G_{λ}, A)$ .

$(F_{μ}, G_{λ}, A) \tilde{⊓} (F_{μ}, G_{λ}, A) = (F_{μ}, G_{λ}, A)$ .

$(F_{μ}, G_{λ}, A) \tilde{⊔} {\tilde{Φ}}_{(μ, λ, A)}^{k} = (F_{μ}, G_{λ}, A)$ .

$(F_{μ}, G_{λ}, A) \tilde{⊓} {\tilde{Φ}}_{(μ, λ, A)}^{k} = {\tilde{Φ}}_{(μ, λ, A)}^{k}$ .

$(F_{μ}, G_{λ}, A) \tilde{⊔} {\tilde{U}}_{(μ, λ, A)}^{k} = {\tilde{U}}_{(μ, λ, A)}^{k}$ .

$(F_{μ}, G_{λ}, A) \tilde{⊓} {\tilde{U}}_{(μ, λ, A)}^{k} = (F_{μ}, G_{λ}, A)$ .

$(F_{μ}, G_{λ}, A) \tilde{⊔} (F_{1, υ}, G_{1, ν}, B) = (F_{1, υ}, G_{1, ν}, B) \tilde{⊔} (F_{μ}, G_{λ}, A)$ .

$(F_{μ}, G_{λ}, A) \tilde{⊓} (F_{1, υ}, G_{1, ν}, B) = (F_{1, υ}, G_{1, ν}, B) \tilde{⊓} (F_{μ}, G_{λ}, A)$ .

5 Application of generalised 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.

6 Application of generalised multi-fuzzy bipolar soft sets in decision making

Let U = {x₁, x₂, . . . , x_n} be the universal set of objects. E = {e₁, e₂, . . . , e_n} be the universal set of parameters and A ⊆ E. Let (F_{δ
₁}, G_{β
₁}, A) be a generalised multi-fuzzy bipolar soft set of dimention k over U. For each e ∈ A, $\begin{matrix} F_{δ_{1}} (e) = \\ ({\begin{matrix} x_{1} / (μ_{1} (x_{1}), μ_{2} (x_{1}), . . ., μ_{k} (x_{1})), \\ x_{2} / (μ_{1} (x_{2}), μ_{2} (x_{2}), . . ., μ_{k} (x_{2})), . . ., \\ x_{n} / (μ_{1} (x_{n}), μ_{2} (x_{n}), . . ., μ_{k} (x_{n})) \end{matrix}}, \\ δ_{1} (e)) = (F (e), δ_{1} (e)), \end{matrix}$ $\begin{matrix} G_{β_{1}} (\neg e) = \\ ({\begin{matrix} x_{1} / (λ_{1} (x_{1}), λ_{2} (x_{1}), . . ., λ_{k} (x_{1})), \\ x_{2} / (λ_{1} (x_{2}), λ_{2} (x_{2}), . . ., λ_{k} (x_{2})), . . ., \\ x_{n} / (λ_{1} (x_{n}), λ_{2} (x_{n}), . . ., λ_{k} (x_{n})) \end{matrix}}, \\ β_{1} (\neg e)) = (G (\neg e), β_{1} (\neg e)), \end{matrix}$ where F (e) and G (¬ e) are normalized multi-fuzzy softs of dimention k over U.

Now, $F (e) = (\begin{matrix} μ_{1} (x_{1}), μ_{2} (x_{1}), . . ., μ_{k} (x_{1}) \\ μ_{1} (x_{2}), μ_{2} (x_{2}), . . ., μ_{k} (x_{2}) \\ . . . \\ μ_{1} (x_{n}), μ_{2} (x_{n}), . . ., μ_{k} (x_{n}) \end{matrix})$ and $G (\neg e) = (\begin{matrix} λ_{1} (x_{1}), λ_{2} (x_{1}), . . ., λ_{k} (x_{1}) \\ λ_{1} (x_{2}), λ_{2} (x_{2}), . . ., λ_{k} (x_{2}) \\ . . . \\ λ_{1} (x_{n}), λ_{2} (x_{n}), . . ., λ_{k} (x_{n}) \end{matrix}) .$

Then $\begin{matrix} F_{δ_{1}} (e) = ((\begin{matrix} μ_{1} (x_{1}), μ_{2} (x_{1}), . . ., μ_{k} (x_{1}) \\ μ_{1} (x_{2}), μ_{2} (x_{2}), . . ., μ_{k} (x_{2}) \\ . . . \\ μ_{1} (x_{n}), μ_{2} (x_{n}), . . ., μ_{k} (x_{n}) \end{matrix}), \\ δ_{1} (e)) \end{matrix}$ and $\begin{matrix} G_{β_{1}} (\neg e) = ((\begin{matrix} λ_{1} (x_{1}), λ_{2} (x_{1}), . . ., λ_{k} (x_{1}) \\ λ_{1} (x_{2}), λ_{2} (x_{2}), . . ., λ_{k} (x_{2}) \\ . . . \\ λ_{1} (x_{n}), λ_{2} (x_{n}), . . ., λ_{k} (x_{n}) \end{matrix}), \\ β_{1} (\neg e)) . \end{matrix}$

Suppose that ω_{δ
₂} (e) = ((ω₁, ω₂, . . . , ω_k) ^T, δ₂ (e)) and ϖ_{β
₂} (¬ e) = ((ϖ₁, ϖ₂, . . . , ϖ_k) ^T, β₂ (¬ e)) $(\sum_{i = 1}^{k}$ $ω_{i} = 1 and \sum_{i = 1}^{k} ϖ_{i} = 1),$ be the relative weights of the parameters e and ¬e respectively.

In the following we define induced generalised multi-fuzzy bipolar soft sets: $\begin{matrix} F_{δ_{1} δ_{2}} (e) = ((\begin{matrix} μ_{1} (x_{1}), μ_{2} (x_{1}), . . ., μ_{k} (x_{1}) \\ μ_{1} (x_{2}), μ_{2} (x_{2}), . . ., μ_{k} (x_{2}) \\ . . . \\ μ_{1} (x_{n}), μ_{2} (x_{n}), . . ., μ_{k} (x_{n}) \end{matrix}) \\ (\begin{matrix} ω_{1} \\ ω_{2} \\ . \\ . \\ . \\ ω_{k} \end{matrix}), δ_{1} (e) δ_{2} (e)) = (\begin{matrix} \sum_{i = 1}^{k} ω_{i} μ_{i} (x_{1}) \\ \sum_{i = 1}^{k} ω_{i} μ_{i} (x_{2}) \\ . \\ . \\ . \\ \sum_{i = 1}^{k} ω_{i} μ_{i} (x_{n}) \end{matrix}) \end{matrix}$ and $\begin{matrix} G_{β_{1} β_{2}} (\neg e) = ((\begin{matrix} λ_{1} (x_{1}), λ_{2} (x_{1}), . . ., λ_{k} (x_{1}) \\ λ_{1} (x_{2}), λ_{2} (x_{2}), . . ., λ_{k} (x_{2}) \\ . . . \\ λ_{1} (x_{n}), λ_{2} (x_{n}), . . ., λ_{k} (x_{n}) \end{matrix}) \\ (\begin{matrix} ϖ_{1} \\ ϖ_{2} \\ . \\ . \\ . \\ ϖ_{k} \end{matrix}), β_{1} (\neg e) β_{2} (\neg e)) = (\begin{matrix} \sum_{i = 1}^{k} ϖ_{i} λ_{i} (x_{1}) \\ \sum_{i = 1}^{k} ϖ_{i} λ_{i} (x_{2}) \\ . \\ . \\ . \\ \sum_{i = 1}^{k} ϖ_{i} λ_{i} (x_{n}) \end{matrix}) . \end{matrix}$

Thus, if we know the respective waights ω_{δ
₂} (e) and ϖ_{β
₂} (¬ e) , we can change the generalised multi-fuzzy bipolar soft sets F_{δ
₁} (e) and G_{β
₁} (¬ e) into induced generalised multi-fuzzy bipolar soft sets F_{δ
₁
δ
₂} (e) and G_{β
₁
β
₂} (¬ e) respectively.

We now use induced generalised multi-fuzzy bipolar soft sets and present an algorithm to select an optimal decision.

Algorithm

Step 1. Input a generalised multi-fuzzy bipolar soft set (F_{δ
₁}, G_{β
₁}, A).

Step 2. Find the generalised multi-fuzzy soft sets (F_{δ
₁}, A) and (G_{β
₁}, A).

Step 3. Change (F_{δ
₁}, A) and (G_{β
₁}, A) into normalized multi-fuzzy soft set, respectively.

Step 4. Input the relative weights (ω (e_i) , δ₂ (e_i)) and (ϖ (¬ e_i) , δ₂ (¬ e_i)) of parameter e_i.

Step 5. Compute the induced generalised fuzzy soft sets ▵_{F
_{δ
₁
δ
₂}} = (F_{δ
₁
δ
₂}, A) and ▵_{G
_{β
₁
β
₂}} = (G_{β
₁
β
₂}, A).

Step 6. Choose the mid-level decision rule for decision making.

Step 7. Compute the mid-level soft sets U (▵ _{F
_{δ
₁
δ
₂}} ; mid) and L (G_{β
₁
β
₂} ; mid) with respect to the mid-level decision rule for decision making.

Step 8. Present the mid-level soft sets U (▵ _{F
_{δ
₁
δ
₂}} ; mid) and L (G_{β
₁
β
₂} ; mid) in tabular forms and compute the choice values $c_{i}^{+}$ and $c_{i}^{-}$ of x_i ∀i, respectively.

Step 9. 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_j =max_iξ_i .

Step 10. If j has more than one value then any one of x_j may be chosen.

6.1 Experimental analysis

Let U ={ m₁, m₂, m₃, m₄, m₅ } be the universe consisting of five types of cell phones. Let A ={ e₁, e₂, e₃ } be the set of all parameters. Where e₁ stands for the parameter colour, which consists of red, green and blue; e₂ stands for the parameter ingredient, which consists of liquid crystal, inflexible plastic and metallic; e₃ stands for the parameter price, which consists of very high, very medium and very low. Let ¬A ={ ¬ e₁, ¬ e₂, ¬ e₃ } be the ’not’ set of parameter set A, where ¬e₁ stands for the parameter opposite of colour parameter “e₁”, which consists of green, red and orange; ¬e₂ stands for the parameter opposite of ingredient parameter “e₂”, which consists of thermotropic crystal, hard plastic and nonmetallic; ¬e₃ stands for the parameter opposite of price parameter “e₃”, which consists of low, medium and high. Suppose that the generalised multi-fuzzy bipolar soft set $(F_{δ_{1}}, G_{β_{1}}, A)$ is given as follows:

$\begin{matrix} \begin{matrix} F_{δ_{1}} (e_{1}) = \\ ({\begin{matrix} m_{1} / (0.22, 0.42, 0.36), m_{2} / (0.52, 0.31, 0.16), \\ m_{3} / (0.13, 0.31, 0.53), \\ m_{4} / (0.57, 0.22, 0.13), m_{5} / (0.27, 0.62, 0.03) \end{matrix}}, \\ 0.45) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G_{β_{1}} (\neg e_{1}) = \\ ({\begin{matrix} m_{1} / (0.33, 0.37, 0.24), m_{2} / (0.13, 0.35, 0.48), \\ m_{3} / (0.46, 0.27, 0.18), \\ m_{4} / (0.39, 0.26, 0.32), m_{5} / (0.34, 0.46, 0.16) \end{matrix}}, \\ 0.55) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} F_{δ_{1}} (e_{2}) = \\ ({\begin{matrix} m_{1} / (0.31, 0.24, 0.45), m_{2} / (0.30, 0.42, 0.24), \\ m_{3} / (0.15, 0.38, 0.41), \\ m_{4} / (0.36, 0.28, 0.30), m_{5} / (0.47, 0.20, 0.31) \end{matrix}}, \\ 0.43) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G_{β_{1}} (\neg e_{2}) = \\ ({\begin{matrix} m_{1} / (0.47, 0.14, 0.37), m_{2} / (0.38, 0.25, 0.33), \\ m_{3} / (0.44, 0.16, 0.39), \\ m_{4} / (0.50, 0.07, 0.39), m_{5} / (0.31, 0.56, 0.11) \end{matrix}}, \\ 0.53) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} F_{δ_{1}} (e_{3}) = \\ ({\begin{matrix} m_{1} / (0.30, 0.45, 0.25), m_{2} / (0.14, 0.61, 0.24), \\ m_{3} / (0.35, 0.20, 0.37), \\ m_{4} / (0.47, 0.34, 0.18), m_{5} / (0.17, 0.29, 0.53) \end{matrix}}, \\ 0.46) \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} G_{β_{1}} (\neg e_{3}) = \\ ({\begin{matrix} m_{1} / (0.27, 0.35, 0.38), m_{2} / (0.57, 0.16, 0.24), \\ m_{3} / (0.46, 0.29, 0.17), \\ m_{4} / (0.16, 0.50, 0.27), m_{5} / (0.32, 0.56, 0.06) \end{matrix}}, \\ 0.56) . \end{matrix} \end{matrix}$

Suppose that a customer wants to select a cell phone depending on its performance and the customer has imposed the following weights for the parameters e and ¬e respectively, as following:

ω_{δ
₂} (e₁) = ((0.3, 0.4, 0.3) ^T, 0.38) , ω_{δ
₂} (e₂) = ((0.5, 0.4, 0.1) ^T, 0.37) , ω_{δ
₂} (e₃) = ((0.2, 0.3, 0.5) ^T, 0.39),

ϖ_{β
₂} (¬ e₁) = ((0.4, 0.5, 0.1) ^T, 0.48) , ϖ_{β
₂} (¬ e₂) = ((0.4, 0.3, 0.1) ^T, 0.47) , ω_{β
₂} (¬ e₃) = ((0.5, 0.2, 0.3) ^T, 0.49).

Thus we have the induced generalised fuzzy soft sets ▵_{F
_{δ
₁
δ
₂}} = (F_{δ
₁
δ
₂}, A) and ▵_{G
_{β
₁
β
₂}} = (G_{β
₁
β
₂}, A) with their tabular representations as given in Tables 1 and 2

Table 1

Tabular representation of induced generalised fuzzy soft set ▵_{F
_{δ
₁
δ
₂}} = (F_{δ
₁
δ
₂}, A)

U	e₁ω_{δ ₂} (e₁) = ((0.3, 0.4, 0.3) ^T, 0.38)	e₂ω_{δ ₂} (e₂) = ((0.5, 0.4, 0.1) ^T, 0.37)	e₃ω_{δ ₂} (e₃) = ((0.2, 0.30, 0.5) ^T, 0.39)
m ₁	(0.291, 0.1710)	(0.295, 0.1591)	(0.299, 0.1794)
m ₂	(0.327, 0.1710)	(0.381, 0.1591)	(0.263, 0.1794)
m ₃	(0.271, 0.1710)	(0.358, 0.1591)	(0.234, 0.1794)
m ₄	(0.322, 0.1710)	(0.361, 0.1591)	(0.330, 0.1794)
m ₅	(0.302, 0.1710)	(0.153, 0.1591)	(0.345, 0.1794)

Table 2

Tabular representation of induced generalised fuzzy soft set ▵_{G
_{β
₁
β
₂}} = (G_{β
₁
β
₂}, ¬ A)

U	¬e₁ϖ_{β ₂} (¬ e₁) = ((0.3, 0.4, 0.3) ^T, 0.55)	¬e₂ω_{δ ₂} (¬ e₂) = ((0.5, 0.4, 0.1) ^T, 0.53)	¬e₃ω_{δ ₂} (¬ e₃) = ((0.2, 0.30, 0.5) ^T, 0.56)
m ₁	(0.341, 0.264)	(0.267, 0.2491)	(0.319, 0.2744)
m ₂	(0.251, 0.264)	(0.935, 0.2491)	(0.405, 0.2744)
m ₃	(0.367, 0.264)	(0.263, 0.2491)	(0.339, 0.2744)
m ₄	(0.318, 0.264)	(0.260, 0.2491)	(0.261, 0.2744)
m ₅	(0.337, 0.264)	(0.303, 0.2491)	(0.290, 0.2744)

Now we find mid-level decision rule. Therefore, mid▵_{F
_{δ
₁
δ
₂}} = {(e₁, 0.3026) , (e₂, 0.3096) , (e₃, 0.2942)} and mid▵_{G
_{β
₁
β
₂}} = {(¬ e₁, 0.3228), (¬ e₂, 0.4056), (¬ e₃, 0.3228)}. Thus, we shall obtain the mid-level soft sets U (▵ _{F
_{δ
₁
δ
₂}} ; mid) of ▵_{F
_{δ
₁
δ
₂}} and L (▵ _{G
_{β
₁
β
₂}} ; mid) of ▵_{G
_{β
₁
β
₂}}, with choice values in tabular representation as given in Tables 3 and 4.

Table 3

Tabular representation of the mid-level soft set U (▵ _{F
_{δ
₁
δ
₂}} ; mid) with choice value

U	e₁δ₁ (e₁) δ₂ (e₁) = 0.1710	e₂δ₁ (e₂) δ₂ (e₂) = 0.1591	e₃δ₁ (e₃) δ₂ (e₃) = 0.1794	Choice value $c_{i}^{+}$
m ₁	0	0	1	$c_{1}^{+} = 1$
m ₂	1	1	0	$c_{2}^{+} = 2$
m ₃	0	1	0	$c_{3}^{+} = 1$
m ₄	1	1	1	$c_{4}^{+} = 2$
m ₅	0	0	1	$c_{5}^{+} = 1$

Table 4

Tabular representation of the mid-level soft set L (▵ _{G
_{β
₁
β
₂}} ; mid) with choice value

U	¬e₁δ₁ (¬ e₁) δ₂ (¬ e₁) =	e₂δ₁ (e₂) δ₂ (e₂) =	e₃δ₁ (e₃) δ₂ (e₃) =	Choice value $c_{i}^{+}$
0.1710	0.1591	0.1794
m ₁	1	1	1	$c_{1}^{-} = 3$
m ₂	1	0	0	$c_{2}^{-} = 1$
m ₃	0	1	0	$c_{3}^{-} = 1$
m ₄	1	1	1	$c_{4}^{-} = 3$
m ₅	0	1	1	$c_{5}^{-} = 2$

From Tables 3 and 4, we have the bipolar choice values of alternatives as given below $\begin{matrix} ξ_{1} = c_{1}^{+} - c_{1}^{-} = - 2 & ξ_{2} = c_{2}^{+} - c_{2}^{-} = 1 \\ ξ_{3} = c_{3}^{+} - c_{3}^{-} = 0 & ξ_{4} = c_{4}^{+} - c_{4}^{-} = - 1 \\ ξ_{5} = c_{5}^{+} - c_{5}^{-} = - 2 . \end{matrix}$

From above bipolar choice values we see that the maximum bipolar value is 1 and clearly the optimal solution is to select m₂. Therefore, the customer should select m₂ as the best cell phone after specifying weights for different parameters.

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