Computational bipolar fuzzy soft matrices with applications in decision making problems

Abstract

A fuzzy soft matrix is a type of mathematical matrix that combines the principles of fuzzy set theory and soft set theory. It is used to handle uncertainty and vagueness in decision-making problems. Fuzzy soft matrix theory cannot handle negative information. To overcome this difficulty, we define the notion of bipolar fuzzy soft (BFS) matrices and study their fundamental properties. We define products of BFS matrices and investigate some useful properties and results. We also give an application of bipolar fuzzy soft matrices to decision-making problems. We propose a decision-making algorithm based on computer programs under the environment of the bipolar fuzzy soft sets.

Keywords

Soft sets fuzzy soft matrices bipolar fuzzy soft matrices BFS decision-makings

1 Introduction

The applications of fuzzy sets [26] in applied sciences, engineering, and computer science is well-known. A fuzzy set is similar to a probability set in that it has values in the closed interval [0,1]. In fact, because there are no constraints on choices, this concept is better suited to modeling in almost all domains of science. Real-world problem models frequently contain ambiguities that cannot be resolved with crisp sets. Fuzzy sets can also be used in business, medicine, and related health sciences. But the lack of an adequate parameterization tool is the fundamental issue affecting the FS model.

To overcome this difficulty, Molodtsov adopted the notion of soft sets [19]. The basic properties of the soft set (SS) theory are presented in [16, 19]. Cagman et al., [11] introduced a fuzzy soft aggregation operator and applied it successfully to many decision-making problems. Zhan and Alcantud introduced a novel type of soft rough covering and discussed its applications to multicriteria group decision-making problems [27]. Moreover, they developed some different algorithms of parameter reduction based on some types of (fuzzy) soft sets [28]. A method of soft max-min decision-making has been developed In [10]. Zhan and Wang developed five new different types of soft coverings based on rough sets [29]. They proposed two special algorithms based on the first two types of soft coverings and utilized them to solve an actual problem.

Majumdar et al., [18] defined generalized fuzzy soft sets and discussed their applications in decision-making problems and medical diagnosis problems. Basu et al., [8] introduced the concept of the interval-valued fuzzy soft matrix (IVFS-matrix) together with some different types of matrices in interval-valued fuzzy soft set theory. They discussed their applications in decision-making problems. In [12], the authors investigated some fruitful results and introduced a uni-int decision-making method. Maji et al., [17] generalized FSSs to intuitionistic FSSs which are based on the combination of the IFSs [5 –7] and SSs. Fuzzy soft β-covering based on fuzzy rough sets was proposed by Zhang and Zhan in [30]. They discussed their applications to decision-makingproblems.

In 2000, Lee [15] extended the idea of an FS and introduced a bipolar fuzzy set (BF-set). The range of the membership degree in the BF-set is extended from the interval [0, 1] to [-1, 1]. In a BF-set, the membership degree 0 indicates that the elements are irrelevant to the associated property, the elements whose functional values lie in (0, 1] show that they obey the property and the set of those elements whose range is [-1, 0) gives that they satisfy the reverse property [15]. In 1994, Zhang [31, 32] investigated bipolar fuzzy sets. Poulik and Ghorai discussed the Applications of the graph’s complete degree with bipolar fuzzy information [22]. Abughazala et al. [2] introduced the concepts of bipolar fuzzy (closed) BCI-positive implicative ideals and bipolar fuzzy (closed) BCI-commutative ideals of BCI-algebras. Sing proposed the concept of bipolar fuzzy attribute implications for the multi-decision process [24]. Akram et al. [3] suggested a new approach for determining a patient’s health status and evaluating the factors influencing that status in a bipolar fuzzy environment.

Abdullah et al. [1] combined the concept of a bipolar fuzzy set and a soft set and introduced the notion of a bipolar fuzzy soft set. They discussed its applications to decision-making problems. Shabir and Naz [25] in 2013, initiated the concept of bipolar soft sets and its set-theoretic operations such as union, intersection, and complement and discuss its application to decision-making problems. Gull et al., [13] studied a novel approach toward the roughness of bipolar soft sets and discussed their applications in multicriteria group decision-making problems. Riaz et al. [23] introduced distance and similarity measures for bipolar fuzzy soft sets and presented applications to pharmaceutical logistics and supply chain management. A decision-making method based on bipolar soft sets was proposed by Mahmood in [20]. Ali et al. [4] proposed many reduction techniques under the environment of bipolar fuzzy soft sets. They developed a decision-making method based on these reduction techniques. Naz and Shabir gave the notion of fuzzy bipolar soft sets and discussed their applications in decision-making problems [21]. Karaaslan and Karatas [14] redefine the concept of bipolar soft set and bipolar soft set operations as more functional than the former. They studied their applications in decision-making problems.

Thus in this paper, we define the notion of bipolar fuzzy soft (BFS) matrices and study their fundamental properties. We define products of BFS matrices and investigate some useful properties and results. We also give an application of bipolar fuzzy soft matrices to decision-making problems. We give a general algorithm based on computer programs to solve decision-making problems under the environment of the bipolar fuzzy soft set.

2 Preliminaries

Let $R$ be a universe of discourse, and $P (R)$ denotes the power set of $R$ . Let ℵ represent the set of all parameters and ℑ ⊆ ℵ . A soft set (ϝ _ℑ, ℵ) on the universe $R$ is introduced by the set of order pairs $\begin{matrix} (ϝ_{ℑ}, ℵ) = {(ϰ, ϝ_{ℑ} (ϰ)) : ϰ \in ℵ, ϝ_{ℑ} (ϰ) \in P (R)}, \end{matrix}$ where $ϝ_{ℑ} : ℵ \to P (R)$ such that ϝ_ℑ (ϰ) = φ if ϰ∉ ℑ. Here ϝ_ℑ is called an approximate function of the SS (ϝ _ℑ, ℵ). The set ϝ_ℑ (ϰ) is said to be an ϰ-approximate value set or ϰ-approximate set which consists of related objects of the parameterϰ∈ ℵ.

Example 1. Let $R = {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}}$ be a set of six shirts, and $\begin{matrix} ℵ = {ϰ_{1} (blue), ϰ_{2} (white), ϰ_{3} (skin), ϰ_{4} (red)}, \end{matrix}$ be a set of parameters. If ℑ = { ϰ₁, ϰ₂, ϰ₃ } ⊆ ℵ .

Let $\begin{matrix} ϝ_{ℑ} (ϰ_{1}) = {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}}, \\ ϝ_{ℑ} (ϰ_{2}) = {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}}, \end{matrix}$ and ϝ_ℑ (ϰ₃) = { ℓ ₃, ℓ ₄, ℓ ₅, ℓ ₆ } , then we write the soft set: $\begin{matrix} (ϝ_{ℑ}, ℵ) = {\begin{matrix} [c] c (ϰ_{1}, {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}}), (ϰ_{2}, {ℓ_{1}, ℓ_{2}, ℓ_{3}, ℓ_{4}, ℓ_{5}}), \\ (ϰ_{3}, {ℓ_{3}, ℓ_{4}, ℓ_{5}, ℓ_{6}}) \end{matrix}}, \end{matrix}$ over U which describes the “colour of the shirts” that Mr. ℑ is going to buy.

We may write the SS in the following form:

ℜ	blue(ϰ1)	white(ϰ2)	skin(ϰ3)	red(ϰ4)
l ₁	1	1	0	0
l ₂	1	1	0	0
l ³	1	1	1	0
l ₄	1	1	1	0
l ₅	1	1	1	0
l ₆	1	0	1	0

Definition 1. Let $R$ be a universe of discourse, ℵ denotes the set of all parameters and ℑ⊆ ℵ. A pair (ϝ , ℑ) is called a fuzzy soft set (FSS) over $R$ where ϝ :ℑ → $P (R)$ is a mapping from ℑ into $P (R)$ , where $P (R)$ represents the collection of all subsets of $R$ .

Example 2. Consider the values in Example 1, here we can not describe the values with only two real numbers 0 and 1 . The values of the truth grade lie in [0, 1] instead of crisp numbers 0 and 1 . Then, $\begin{matrix} (ϝ_{ℑ}, ℵ) = {\begin{matrix} [c] c ϝ_{ℑ} (ϰ_{1}) = {(ℓ_{1}, 0.5), (ℓ_{2}, 0.7), (ℓ_{3}, 0.2), (ℓ_{4}, 0.6), (ℓ_{5}, 0.1), (ℓ_{6}, 0.8)}, \\ ϝ_{ℑ} (ϰ_{2}) = {(ℓ_{1}, 0.7), (ℓ_{2}, 0.2), (ℓ_{3}, 0.5), (ℓ_{4}, 0.9), (ℓ_{5}, 0.5)} \\ and ϝ_{ℑ} (ϰ_{3}) = {(ℓ_{3}, 0.4), (ℓ_{4}, 0.3), (ℓ_{5}, 0.7), (ℓ_{6}, 0.1)} \end{matrix}}, \end{matrix}$ is the FSS representing the “colour of the shirts” which Mr. ℑ is going to buy.

We may express the FSS in the following form:

ℜ	blue(ϰ1)	white(ϰ2)	skin(ϰ3)	red(ϰ4)
l ₁	0.5	0.7	0	0
l ₂	0.7	0.2	0	0
l ₃	0.2	0.5	0.4	0
l ₄	0.6	0.9	0.3	0
l ₅	0.1	0.5	0.7	0
l ₆	0.8	0	0.1	0

Definition 2. [15] A BF-set ℑ in a universe $R$ is an object having the form, $ℑ = {(ℓ, θ_{ℑ}^{+} (ℓ), θ_{ℑ}^{-} (ℓ)) : ℓ \in R}$ where $θ_{ℑ}^{+} : R \to [0, 1]$ , $θ_{ℑ}^{-} : R \to [- 1, 0]$ . So $θ_{ℑ}^{+}$ describes positive information and $θ_{ℑ}^{-}$ describes negative information.

Definition 3. Fuzzy Soft Matrices (FSM)

Let (ϝ _ℑ, ℵ) be FSS over $R$ . Then a subset of $R \times ℵ$ is uniquely defined by $\begin{matrix} R_{ℑ} = {(\overset{⌣}{ℓ} . ϰ) : ϰ \in ℑ, \overset{⌣}{ℓ} \in ϝ_{ℑ} (ϰ)}, \end{matrix}$ which is said to be a relation form of (ϝ _ℑ, ℵ). The characteristic function of R_ℑ is written by $\begin{matrix} F_{R_{ℑ}} : R \times ℵ \to [0, 1], where F_{R_{ℑ}} (\overset{⌣}{ℓ}, ϰ) \in [0, 1], \end{matrix}$ is the membership value of $\overset{⌣}{ℓ} \in R$ for each $ϰ \in R$ . If $F_{ij} =$ $F_{R_{ℑ}} (ℓ_{i}^{⌣}, ϰ_{i}),$ we can define a matrix

${[F_{i j}]}_{m \times n} = [\begin{matrix} F_{11} & F_{12} & .. & F_{1 n} \\ F_{21} & F_{22} & .. & F_{2 n} \\ : & : & : & : \\ F_{m 1} & F_{m 2} & .. & F_{m n} \end{matrix}],$

which is called an m × n soft matrix of the SS (ϝ _ℑ, ℵ) over $R$ .

Therefore we can say that an FSS (ϝ _ℑ, ℵ) is uniquely characterized by the matrix ${[F_{ij}]}_{m \times n},$ and both concepts are interchangeable.

Example 3. Assume that $R = {ℓ_{1}^{⌣}, ℓ_{2}^{⌣}, ℓ_{3}^{⌣}, ℓ_{4}^{⌣}, ℓ_{5}^{⌣}}$ is a universal set, and ℵ ={ ϰ₁, ϰ₂, ϰ₃, ϰ₄ } is a set of parameters. If ℑ ⊆ ℵ { ϰ₁, ϰ₂, ϰ₃ } , and $\begin{matrix} ϝ_{ℑ} (ϰ_{1}) & = {(ℓ_{1}^{⌣}, 0.5), (ℓ_{2}^{⌣}, 0.7), (ℓ_{3}^{⌣}, 0.2), \\ (ℓ_{4}^{⌣}, 0.8), (ℓ_{5}^{⌣}, 0.1)}, \\ ϝ_{ℑ} (ϰ_{2}) & = {(ℓ_{1}^{⌣}, 0.2), (ℓ_{2}^{⌣}, 0.9), (ℓ_{3}^{⌣}, 0.1), \\ (ℓ_{4}^{⌣}, 0.7), (ℓ_{5}^{⌣}, 0.3)}, \\ ϝ_{ℑ} (ϰ_{3}) & = {(ℓ_{1}^{⌣}, 0.3), (ℓ_{2}^{⌣}, 0.1), (ℓ_{3}^{⌣}, 0.6), \\ (ℓ_{4}^{⌣}, 0.3), (ℓ_{5}^{⌣}, 0.9)} . \end{matrix}$ Then, the FSS (ϝ _ℑ, ℵ) is a parameterized family {ϝ _ℑ (ϰ₁) , ϝ _ℑ (ϰ₂) , ϝ _ℑ (ϰ₃) } of all FSs over $R$ .

Hence the fuzzy soft matrix $[F_{ij}]$ can be written as $\begin{matrix} [F_{ij}] = [\begin{matrix} 0.5 & 0.2 & 0.3 & 0 \\ 0.7 & 0.9 & 0.1 & 0 \\ 0.2 & 0.1 & 0.6 & 0 \\ 0.8 & 0.7 & 0.3 & 0 \\ 0.1 & 0.3 & 0.9 & 0 \end{matrix}] . \end{matrix}$

3 Bipolar fuzzy soft (BFS) matrix theory

Bipolar Fuzzy Soft Matrix (BFSM)

Let $R$ be an initial universe, ℵ be the set of parameters, and ℑ⊆ ℵ. Let (ϝ _ℑ, ℵ) be a bipolar FSS (BFSS) over $R$ . Then a subset of $R \times ℵ$ is uniquely defined by $\begin{matrix} R_{ℑ} = {(\overset{⌣}{ℓ} . ϰ) : ϰ \in ℑ, \overset{⌣}{ℓ} \in ϝ_{ℑ} (ϰ)}, \end{matrix}$

which is called the relation form of (ϝ _ℑ, ℵ). The membership and non-membership functions are written by $\begin{matrix} F_{R_{ℑ}}^{+} : R \times ℵ \to [0, 1] \\ and F_{R_{ℑ}}^{-} : R \times ℵ \to [- 1, 0], \end{matrix}$ where $\begin{matrix} F_{R_{ℑ}}^{+} : (\overset{⌣}{ℓ}, ϰ) \in [0, 1] \\ and F_{R_{ℑ}}^{-} : (\overset{⌣}{ℓ}, ϰ) \in [- 1, 0], \end{matrix}$ are the membership value and nonmembership value of $\overset{⌣}{ℓ} \in R$ for each ϰ ∈ ℵ .

If $\begin{matrix} 〈 F_{R_{ℑ}}^{+}, F_{R_{ℑ}}^{-} 〉 = 〈 F_{R_{ℑ}}^{+} (ℓ_{i}^{⌣}, ϰ_{j}), F_{R_{ℑ}}^{-} (ℓ_{i}^{⌣}, ϰ_{j}) 〉, \end{matrix}$ we can define a matrix $\begin{array}{l} {[F_{R 3}^{+}, F_{R 3}^{-}]}_{m \times}_{n} \\ = [\begin{matrix} (F_{11}^{+}, F_{11}^{-}) & (F_{12}^{+}, F_{12}^{-}) & ... & (F_{1 n}^{+}, F_{1 n}^{-}) \\ (F_{21}^{+}, F_{21}^{-}) & (F_{22}^{+}, F_{22}^{-}) & ... & (F_{2 n}^{+}, F_{2 n}^{-}) \\ : & : & : \\ (F_{m 1}^{+}, F_{m 1}^{-}) & (F_{m 2}^{+}, F_{m 2}^{-}) & ... & (F_{m n}^{+}, F_{m n}^{-}) \end{matrix}] \end{array}$ which is called an m × n BFSM of the BFSS (ϝ _ℑ, ℵ) over $R$ . Therefore, we can say that BFSS (ϝ _ℑ, ℵ) is uniquely characterized by the matrix ${[F_{R_{ℑ}}^{+}, F_{R_{ℑ}}^{-}]}_{m \times n}$ and both concepts are interchangeable. The set of all m × n BFS matrices will be denoted by BFSM_m×n.

Example 4. Let $R = {ℓ_{1}^{⌣}, ℓ_{2}^{⌣}, ℓ_{3}^{⌣}, ℓ_{4}^{⌣}, ℓ_{5}^{⌣}, ℓ_{6}^{⌣}, ℓ_{7}^{⌣}, ℓ_{8}^{⌣}}$ is a universal set and ℵ ={ ϰ₁, ϰ₂, ϰ₃, ϰ₄, ϰ₅, ϰ₆ } is a set of parameters. If ℑ = { ϰ₁, ϰ₂, ϰ₃, ϰ₄ } ⊆ ℵ , and $\begin{matrix} ϝ_{ℑ} (ϰ_{1}) & = {\begin{matrix} [c] c (ℓ_{1}^{⌣}, 0.4, - 0.5), (ℓ_{2}^{⌣}, 0.6, - 0.3), (ℓ_{3}^{⌣}, 0.7, - 0.2), (ℓ_{4}^{⌣}, 0.8, - 0.5), \\ (ℓ_{5}^{⌣}, 0.1, - 0.3), (ℓ_{6}^{⌣}, 0.5, - 0.2), (ℓ_{7}^{⌣}, 0.3, - 0.1), (ℓ_{8}^{⌣}, 0.6, - 0.3) . \end{matrix}}, \\ ϝ_{ℑ} (ϰ_{2}) & = {\begin{matrix} [c] c (ℓ_{1}^{⌣}, 0.3, - 0.4), (ℓ_{2}^{⌣}, 0.4, - 0.7), (ℓ_{3}^{⌣}, 0.7, - 0.5), (ℓ_{4}^{⌣}, 0.5, - 0.4), \\ (ℓ_{5}^{⌣}, 0.7, - 0.3), (ℓ_{6}^{⌣}, 0.3, - 0.6), (ℓ_{7}^{⌣}, 0.5, - 0.1), (ℓ_{8}^{⌣}, 0.1, - 0.7) . \end{matrix}}, \end{matrix}$ $\begin{matrix} ϝ_{ℑ} (ϰ_{3}) & = {\begin{matrix} [c] c (ℓ_{1}^{⌣}, 0.5, - 0.4), (ℓ_{2}^{⌣}, 0.4, - 0.3), (ℓ_{3}^{⌣}, 0.8, - 0.1), (ℓ_{4}^{⌣}, 0.3, - 0.5), \\ (ℓ_{5}^{⌣}, 0.1, - 0.5), (ℓ_{6}^{⌣}, 0.1, - 0.8), (ℓ_{7}^{⌣}, 0.2, - 0.7), (ℓ_{8}^{⌣}, 0.5, - 0.2) . \end{matrix}}, \\ ϝ_{ℑ} (ϰ_{4}) & = {\begin{matrix} [c] c (ℓ_{1}^{⌣}, 0.4, - 0.6), (ℓ_{2}^{⌣}, 0.1, - 0.6), (ℓ_{3}^{⌣}, 0.3, - 0.5), (ℓ_{4}^{⌣}, 0.2, - 0.6), \\ (ℓ_{5}^{⌣}, 0.2, - 0.7), (ℓ_{6}^{⌣}, 0.1, - 0.3), (ℓ_{7}^{⌣}, 0.6, - 0.5), (ℓ_{8}^{⌣}, 0.4, - 0.3) . \end{matrix}} . \end{matrix}$ Then, the BFS set (ϝ _ℑ, ℵ) is a parameterized family {ϝ _ℑ (ϰ₁) , ϝ _ℑ (ϰ₂) , ϝ _ℑ (ϰ₃) , ϝ _ℑ (ϰ₄) } of all BFS sets over $R$ . Hence $[{\tilde{μ}}_{R_{ℑ}}^{+}, {\tilde{μ}}_{R_{ℑ}}^{-}]$ can be written as; $\begin{matrix} [{\tilde{μ}}_{R_{ℑ}}^{+}, {\tilde{μ}}_{R_{ℑ}}^{-}] = [\begin{matrix} (0.4, - 0.5) & (0.3, - 0.4) & (0.5, - 0.4) & (0.4, - 0.6) & (0, 0) \\ (0.6, - 0.3) & (0.4, - 0.7) & (0.4, - 0.3) & (0.1, - 0.6) & (0, 0) \\ (0.7, - 0.2) & (0.7, - 0.5) & (0.8, - 0.1) & (0.3, - 0.5) & (0, 0) \\ (0.8, - 0.5) & (0.5, - 0.4) & (0.3, - 0.5) & (0.2, - 0.6) & (0, 0) \\ (0.1, - 0.3) & (0.7, - 0.3) & (0.1, - 0.5) & (0.2, - 0.7) & (0, 0) \\ (0.5, - 0.2) & (0.3, - 0.6) & (0.1, - 0.8) & (0.1, - 0.3) & (0, 0) \\ (0.3, - 0.1) & (0.5, - 0.1), & (0.2, - 0.7) & (0.6, - 0.5) & (0, 0) \\ (0.6, - 0.3) & (0.1, - 0.7) & (0.5, - 0.2) & (0.4, - 0.3) & (0, 0) \end{matrix}] . \end{matrix}$

Example 5. Suppose that there are four houses under consideration, namely the universes $R = {h_{1}, h_{2}, h_{3}, h_{4}}$ , and the parameter set ℵ = {ϰ₁, ϰ₂, ϰ₃, ϰ₄} , where ϰ_i stands for “beautiful”, “large”, “cheap” and “in green surroundings” respectively. Consider the set ℑ = {ϰ₁, ϰ₂}⊂ ℵ. Then, $\begin{matrix} F_{R 3}^{+} (X 1) = {(h_{1}, 0.65), (h_{2}, 0.85), (h_{3}, 0.65), (h_{4}, 0.55)}, \\ F_{R 3}^{+} (X 2) = {(h_{1}, 0.57), (h_{2}, 0.56), (h_{3}, 0.45), (h_{4}, 0.78)}, \\ a n d F_{R 3}^{-} (X 1) = {(h_{1}, - 0.66), (h_{2}, - 0.68), (h_{3}, - 0.66), (h_{4}, - 0.65)}, \\ F_{R 3}^{-} (X 2) = {(h_{1}, - 0.87), (h_{2}, - 0.86), (h_{3}, - 0.85), (h_{4}, - 0.88,)} . \end{matrix}$

We would represent this bipolar soft set in matrix form as:

$\begin{matrix} [{\tilde{μ}}_{R_{ℑ}}^{+}, {\tilde{μ}}_{R_{ℑ}}^{-}] \\ = [\begin{matrix} (0.65, - 0.66) & (0.57, - 0.87) & (0, 0) & (0, 0) \\ (0.85, - 0.68) & (0.56, - 0.86) & (0, 0) & (0, 0) \\ (0.65, - 0.66) & (0.45, - 0.85) & (0, 0) & (0, 0) \\ (0.55, - 0.65) & (0.78, - 0.88) & (0, 0) & (0, 0) \end{matrix}] . \end{matrix}$

Bipolar Fuzzy Soft Zero Matrix

Let $\begin{matrix} [{\hat{a}}_{ij}] = [({\tilde{μ}}_{ij}^{+}, {\tilde{μ}}_{ij}^{-})] \in {BFSM}_{m \times n} . \end{matrix}$ Then ${\hat{a}}_{ij}$ is a zero BFSM denoted by 0 = [(0, 0)] , if ${\tilde{μ}}_{ij}^{+} = 0$ and ${\tilde{μ}}_{ij}^{-} = 0$ for all i and j.

Bipolar Fuzzy Soft $\tilde{μ}$ -Universal Matrix

A BFS matrix of order m × n is said to be a BFS $\tilde{μ}$ -Universal Matrix if ${\tilde{μ}}_{ij}^{+}$ = 1 and ${\tilde{μ}}_{ij}^{-}$ = -1 for all i and j. It is denoted by $I^{\tilde{μ}} .$

Bipolar Fuzzy Soft $\tilde{ν}$ -Universal Matrix

A BFS matrix of order m × n is said to be a BFS $\tilde{ν}$ -Universal Matrix if ${\tilde{μ}}_{ij}^{+}$ = -1 and ${\tilde{μ}}_{ij}^{-}$ = 1 for all i and j. It is denoted by $I^{\tilde{ν}} .$

Definition 4. Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then $[{\hat{a}}_{ij}]$ is said to be BFS sub matrix of $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] \subseteq [{\hat{b}}_{ij}]$ if ${\tilde{μ}}_{ij}^{{\hat{a}}^{+}} \leq {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}$ and ${\tilde{μ}}_{ij}^{{\hat{a}}^{-}} \geq {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}$ for all i and j.

Bipolar Fuzzy Soft Super Matrix

Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then $[{\hat{a}}_{ij}]$ is said to be BFS super matrix of $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] \supseteq [{\hat{b}}_{ij}]$ if ${\tilde{μ}}_{ij}^{{\hat{a}}^{+}} \geq {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}$ and ${\tilde{μ}}_{ij}^{{\hat{a}}^{-}} \leq {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}$ for all i and j.

Bipolar Fuzzy Soft Equal Matrix

Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then $[{\hat{a}}_{ij}]$ is said to be equal $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] = [{\hat{b}}_{ij}]$ if ${\tilde{μ}}_{ij}^{{\hat{a}}^{+}} = {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}$ and ${\tilde{μ}}_{ij}^{{\hat{a}}^{-}} = {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}$ for all i and j.

Union of Bipolar Fuzzy Soft Matrix

Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then, the union $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}],$ and is defined as $\begin{matrix} [{\dot{G}}_{ij}] = [〈 max {{\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}}, min {{\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}} 〉], \end{matrix}$ for all i and j.

Intersection of Bipolar Fuzzy Soft Matrix

Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then, the intersection $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] \cap [{\hat{b}}_{ij}],$ and is defined as $\begin{matrix} [{\dot{G}}_{ij}] = [〈 min {{\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}}, max {{\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}} 〉], \end{matrix}$ for all i and j.

Complement of Bipolar Fuzzy Soft Matrix

Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then, the complement of A denoted by $[{\hat{a}}_{ij}]^{0},$ and is defined as $\begin{matrix} [{\hat{a}}_{ij}]^{o} = [〈 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, - 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \end{matrix}$ for all i and j.

Example 6. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \in {BFSM}_{2 \times 2},$ then $\begin{matrix} [{\hat{a}}_{ij}] & = [\begin{matrix} (0.5 . - 0.2) & (0.3, - 0.7) \\ (0.2, - 0.3) & (0.6, - 0.1) \end{matrix}], \\ {[{\hat{a}}_{ij}]}^{0} & = [\begin{matrix} (0.5, - 0.8) & (0.7, - 0.3) \\ (0.8, - 0.7) & (0.4, - 0.9) \end{matrix}] . \end{matrix}$

Proposition 1.Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then, the De Morgan’s type results are true which can be written as $\begin{matrix} 1) {([{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}])}^{o} & = [{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o}, \\ 2) {([{\hat{a}}_{ij}] \cap [{\hat{b}}_{ij}])}^{o} & = [{\hat{a}}_{ij}]^{o} \cup [{\hat{b}}_{ij}]^{o} . \end{matrix}$

Proof. For all i and j we have, $\begin{matrix} {([{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}])}^{o} \\ = {([〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \cup [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉])}^{o} \\ = {[〈 max {{\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}}, min {{\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}} 〉]}^{o} \\ = [〈 \begin{matrix} [c] c (min {1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}}), \\ (max {- 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, - 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}}) \end{matrix} 〉] \\ = [〈 1 - max ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), - 1 - min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}) 〉] \\ = [{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o} . \end{matrix}$ The result (2) can be proved similar way. □

Proposition 2. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then

i) ${([{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o})}^{o} = [{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}],$

ii) ${([{\hat{a}}_{ij}]^{o} \cup [{\hat{b}}_{ij}]^{o})}^{o} = [{\hat{a}}_{ij}] \cap [{\hat{b}}_{ij}] .$

Proof. i)

$\begin{matrix} {([{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o})}^{o} \\ = {({[〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉]}^{o} \cap {[〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉]}^{o})}^{o} \\ = {([〈 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, - 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \cap [〈 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, - 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉])}^{o} \\ = {[〈 min (1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), max (- 1 - {\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, - 1 - {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}) 〉]}^{o} \\ = {[〈 1 - max ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), - 1 - min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}) 〉]}^{o} \\ = [〈 \begin{matrix} [c] c 1 - (1 - max ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}})), \\ - 1 - (- 1 - min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}})) \end{matrix} 〉] \\ = [〈 max ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}}) 〉] \\ = [{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}] . \end{matrix}$

So, $\begin{matrix} {([{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o})}^{o} = [{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}] . \end{matrix}$ The result (ii) can be proved similar way. □

Example 7. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{3 \times 3},$ where $\begin{matrix} [{\hat{a}}_{ij}] = [\begin{matrix} (0.3, - 0.4) & (0.1, - 0.2) & (0.9, - 0.1) \\ (0.4, - 0.5) & (0.6, - 0.3) & (0.7, - 0.3) \\ (0.7, - 0.3) & (0.5, - 0.4) & (0.4, - 0.2) \end{matrix}], \end{matrix}$

and $\begin{matrix} [{\hat{b}}_{ij}] = [\begin{matrix} (0.5, - 0.4) & (0.4, - 0.6) & (0.3, - 0.5) \\ (0.3, - 0.4) & (0.7, - 0.4) & (0.8, - 0.3) \\ (0.9, - 0.2) & (0.1, - 0.3) & (0.5, - 0.4) \end{matrix}] . \end{matrix}$

Now,

i) ${([{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o})}^{o} = [{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}],$ $\begin{matrix} {([{\hat{a}}_{ij}]^{o} \cap [{\hat{b}}_{ij}]^{o})}^{o} \\ = [\begin{matrix} (0.5, - 0.4) & (0.4, - 0.6) & (0.9, - 0.5) \\ (0.4, - 0.5) & (0.7, - 0.4) & (0.8, - 0.3) \\ (0.9, - 0.3) & (0.5, - 0.4) & (0.5, - 0.4) \end{matrix}] \\ = [{\hat{a}}_{ij}] \cup [{\hat{b}}_{ij}] . \end{matrix}$

Proposition 3. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then,

i) ${([{\hat{a}}_{ij}]^{o} + [{\hat{b}}_{ij}]^{o})}^{o} \neq [{\hat{a}}_{ij}] . [{\hat{b}}_{ij}],$

ii) ${([{\hat{a}}_{ij}]^{o} . [{\hat{b}}_{ij}]^{o})}^{o} \neq [{\hat{a}}_{ij}] + [{\hat{b}}_{ij}] .$

Example 8. i) ${([{\hat{a}}_{ij}]^{o} + [{\hat{b}}_{ij}]^{o})}^{o} \neq [{\hat{a}}_{ij}] . [{\hat{b}}_{ij}]$ $\begin{matrix} {([{\hat{a}}_{ij}]^{o} + [{\hat{b}}_{ij}]^{o})}^{o} \\ = [\begin{matrix} (0.15, - 0.64) & (0.04, - 0.68) & (0.27, - 0.55) \\ (0.12, - 0.70) & (0.42, - 0.58) & (0.56, - 0.51) \\ (0.63, - 0.44) & (0.05, - 0.58) & (0.20, - 0.52) \end{matrix}], \\ [{\hat{a}}_{ij}] . [{\hat{b}}_{ij}] \\ = [\begin{matrix} (0.65, - 0.16) & (0.46, - 0.12) & (0.93, - 0.05) \\ (0.58, - 0.20) & (0.88, - 0.12) & (0.94, - 0.09) \\ (0.97, - 0.06) & (0.55, - 0.12) & (0.70, - 0.08) \end{matrix}] . \end{matrix}$

So, $\begin{matrix} {([{\hat{a}}_{ij}]^{o} + [{\hat{b}}_{ij}]^{o})}^{o} \neq [{\hat{a}}_{ij}] . [{\hat{b}}_{ij}] . \end{matrix}$

The result (ii) can be proved similar way.

Proposition 4. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then

i) ${([{\hat{a}}_{ij}]^{o})}^{o} = [{\hat{a}}_{ij}],$

ii) $[{\hat{a}}_{ij}] \cup [{\hat{a}}_{ij}] = [{\hat{a}}_{ij}],$

iii) $[{\hat{a}}_{ij}] \cap [{\hat{a}}_{ij}] = [{\hat{a}}_{ij}] .$

Max-Min Product of Bipolar Fuzzy Soft Matrices

Let $\begin{matrix} [{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \in {BFSM}_{m \times n}, \\ [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} . \end{matrix}$ Then, the Max - Min Product of $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ij}]$ denoted by $[{\hat{a}}_{ij}] * [{\hat{b}}_{ij}],$ and is defined as $\begin{matrix} [{\hat{a}}_{ij}] * [{\hat{b}}_{ij}] = [〈 max - min ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), \\ max - min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}) 〉], \end{matrix}$ for all i and j.

Operators of Bipolar Fuzzy Soft Matrices

a) the “.” (product) operation of $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ij}]$ , denoted by $\overset{\circ}{C} = [{\hat{a}}_{ij}] . [{\hat{b}}_{ij}]$ if $\begin{matrix} [〈 μ_{ij}^{{\overset{\circ}{C}}^{+}}, μ_{ij}^{{\overset{\circ}{C}}^{-}} 〉] \\ = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}} . {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} + {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} - | {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} | | {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} | 〉], \end{matrix}$ for all i and j.

b) the “+” (Probabilisticsum) operation of $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ij}]$ , denoted by $\overset{\circ}{C} = [{\hat{a}}_{ij}] + [{\hat{b}}_{ij}]$ if $\begin{matrix} [〈 μ_{ij}^{{\overset{\circ}{C}}^{+}}, μ_{ij}^{{\overset{\circ}{C}}^{-}} 〉] \\ = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}} + {\tilde{μ}}_{ij}^{{\hat{b}}^{+}} - {\tilde{μ}}_{ij}^{{\hat{a}}^{+}} . {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, - | {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} | . | {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} | 〉], \end{matrix}$ for all i and j.

Product of BFS-Matrices

In this section, we define special products of BFS-matrices to construct BFS-decision-making methods.

Definition 5. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \in {BFSM}_{m \times n}, [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then, And-product of $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ik}]$ is defined by

$\overset{˘}{\land} :$ ${BFSM}_{m \times n} \times {BFSM}_{m \times n} \to {BFSM}_{m \times n^{2}}, [{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}] = [{\ddot{c}}_{ip}],$

where $\begin{matrix} {\ddot{c}}_{ip} = [min ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), max ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}})], \end{matrix}$ such that p = n (j - 1) + k .

Definition 6. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉] \in {BFSM}_{m \times n}, [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then, Or-product of $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ik}]$ is defined by

$\lor$ : ${BFSM}_{m \times n} \times {BFSM}_{m \times n} \to {BFSM}_{m \times n^{2}}, [{\hat{a}}_{ij}] \lor [{\hat{b}}_{ik}] = [{\ddot{c}}_{ip}],$

where $\begin{matrix} {\ddot{c}}_{ip} = [max ({\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}), min ({\tilde{μ}}_{ij}^{{\hat{a}}^{-}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}})], \end{matrix}$ such that p = n (j - 1) + k .

Proposition 5. Let $[{\hat{a}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{a}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{a}}^{-}} 〉], [{\hat{b}}_{ij}] = [〈 {\tilde{μ}}_{ij}^{{\hat{b}}^{+}}, {\tilde{μ}}_{ij}^{{\hat{b}}^{-}} 〉] \in {BFSM}_{m \times n} .$ Then, the following De Morgan’s types of results are true.

(1) ${([{\hat{a}}_{ij}] \lor [{\hat{b}}_{ik}])}^{o} = [{\hat{a}}_{ij}]^{o} \lor [{\hat{b}}_{ik}]^{o},$

(2) ${([{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}])}^{o} = [{\hat{a}}_{ij}]^{o} \overset{˘}{\land} [{\hat{b}}_{ik}]^{o} .$

4 Bipolar fuzzy soft max-min decision- making with computer Pseudocode

In this section, we construct a BFS max-min decision-making (BFSMmDM) method by using BFS max-min decision function which is also defined here. The method selects optimum alternatives from the set of alternatives.

Definition 7. Let $[{\ddot{c}}_{ip}] \in {BFSM}_{m \times n^{2}}, I_{k} = {p : \exists i, {\ddot{c}}_{ip} \neq 0, (k - 1) < p \leq kn}, \forall k \in I = {1, 2, . . ., n}$ . Then BFS max-min decision function, denoted Mm, is defined as follows

$Mm : {BFSM}_{m \times n^{2}} \to {BFSM}_{m \times 1}, Mm [{\ddot{c}}_{ip}] = [d_{i 1}] = [max_{k} {t_{ik}}],$

where $\begin{matrix} t_{ik} & = min p \in I_{k} {{\ddot{c}}_{ip}}, if I_{k} \neq Φ, \\ 0, if I_{k} & \neq Φ . \end{matrix}$ The one column BFS-matrix $Mm [{\ddot{c}}_{ip}]$ is called max - min decision BFS-martix.

Definition 8. Let $R = {ℓ_{1}^{⌣}, ℓ_{2}^{⌣}, . . ., ℓ_{m}^{⌣}}$ be a universe of discourse and $Mm [{\ddot{c}}_{ip}] = [d_{i 1}] .$ Then, a subset of $R$ can be defined by utilizing [d_i1] in the following way $\begin{matrix} {opt}_{[d_{i 1}]} (R) = {(ℓ_{i}^{⌣}, d_{i 1}) : ℓ_{i}^{⌣} \in R, d_{i 1} \neq o}, \end{matrix}$ which is called an optimum fuzzy set on U.

We can construct the following BFSMmDM method using the proposed definitions.

Algorithm 1. Step 1: Select feasible subsets of the set of parameters.

Step 2: For each set of parameters, propose the BFS matrix.

Step 3: Determine a convenient product of the BFS matrices.

Step 4: Compute max - min decision BFS matrix.

Step 5: Find an optimum fuzzy set on U.

Similarly, we can define BFS min-max, BFS min-min, and BFSs max-max decision-making methods which may be denoted by (BFSmMDM), (BFSmmDM), (BFSMMDM), respectively. One of them may be more useful than the others according to the type of the problems.

Computer Program

Pseudocode. A computer program is developed for BFSMm decision-making, its Pseudocode is given below.

input: Read input.txt for

m : = rows

n : = columns

${\hat{a}}_{ij} : = m \times n$ BFS matrix

${\hat{b}}_{ij} : = m \times n$ BFS matrix

Define $\ddot{C}$ an m × n² matrix

for : i = 1 to m

for : j = 1 to n

for : k = 1 to n

p = n × (j - 1) + k

${\ddot{c}}_{ip}^{+} : = min (a_{ij}^{+}, b_{ij}^{+})$

${\ddot{c}}_{ip}^{-} : = max (a_{ij}^{-}, b_{ij}^{-})$

end for

define D for m × n matrix

for : i = 1 to m

for : j = 1 to n

$d_{ij}^{+} : = min_{n (j - 1) < k \leq nj} c_{ik}^{+}$

$d_{ij}^{-} : = min_{n (j - 1) < k \leq nj} c_{ik}^{-}$

end for

Define ℵ an m - vector

for : i = 1 to m

$e_{i}^{+} : = max_{1 \leq j \leq n} d_{ij}^{+}$

$e_{i}^{-} : = max_{1 \leq j \leq n} d_{ij}^{-}$

end for

optimum $value : = max_{1 \leq i \leq n} e_{i}^{+}$

out put:

Numerical Example

Example 9. Assume that a real estate agent has a set of different types of houses $R = {ℓ_{1}^{⌣}, ℓ_{2}^{⌣}, ℓ_{3}^{⌣}, ℓ_{4}^{⌣}}$ which may be characterized by a set of parameters ℵ = { x₁, x₂, x₃ } . For j = 1, 2, 3, the parameters x_j stand for “in good location”, “cheap”, and “modern”, respectively. Then, we can give the following examples. Suppose that Mr. X. comes to the real estate agent to buy a house. If each partner has to consider their own set of parameters, then we select a house on the basis of the sets of partners’ parameters by using the BFSMmDM as follows.

Assume that $R = {ℓ_{1}^{⌣}, ℓ_{2}^{⌣}, ℓ_{3}^{⌣}, ℓ_{4}^{⌣}}$ is a universal set and ℵ ={ x₁, x₂, x₃ } is a set of all parameters.

Step 1: First, Mr. X has to choose the sets of their parameters, $[{\hat{a}}_{ij}] = {x_{1}, x_{2}, x_{3}}$ and $[{\hat{b}}_{ij}] = {x_{1}, x_{2}, x_{3}}$ , respectively.

Step 2: Then, we can write the following BFS matrices which are constructed

according to their parameters. $\begin{matrix} [{\hat{a}}_{ij}] = [\begin{matrix} (0.3, - 0.5) & (0.7, - 0.4) & (0.3, - 0.1) \\ (0.5, - 0.4) & (0.8, - 0.6) & (0.4, - 0.3) \\ (0.9, - 0.4) & (0.4, - 0.5) & (0.6, - 0.4) \\ (0.5, - 0.4) & (0.3, - 0.2) & (0.9, - 0.6) \end{matrix}], \end{matrix}$

and $\begin{matrix} [{\hat{b}}_{ij}] = [\begin{matrix} (0.8, - 0.4) & (0.1, - 0.3) & (0.2, - 0.6) \\ (0.4, - 0.3) & (0.8, - 0.7) & (0.7, - 0.4) \\ (0.9, - 0.5) & (0.7, - 0.8) & (0.6, - 0.8) \\ (0.3, - 0.5) & (0.6, - 0.3) & (0.8, - 0.4) \end{matrix}] . \end{matrix}$

Step 3: Now, we compute a product of the BFS-matrices $[{\hat{a}}_{ij}]$ and $[{\hat{b}}_{ik}]$ by using And - product as follows;

$\begin{matrix} ℏ_{ip} = [\begin{matrix} (0.3, - 0.4) & (0.1, - 0.3) & (0.2, - 0.5) & (0.7, - 0.4) & (0.1, - 0.3) & (0.2, - 0.4) & (0.3, - 0.1) & (0.1, - 0.1) & (0.2, - 0.1) \\ (0.4, - 0.3) & (0.5, - 0.4) & (0.5, - 0.4) & (0.4, - 0.3) & (0.8, - 0.6) & (0.7, - 0.4) & (0.4, - 0.3) & (0.4, - 0.3) & (0.4, - 0.3) \\ (0.9, - 0.4) & (0.7, - 0.4) & (0.6, - 0.4) & (0.4, - 0.5) & (0.4, - 0.5) & (0.4, - 0.5) & (0.6, - 0.4) & (0.6, - 0.4) & (0.6, - 0.4) \\ (0.3, - 0.4) & (0.5, - 0.3) & (0.5, - 0.4) & (0.3, - 0.2) & (0.3, - 0.2) & (0.3, - 0.2) & (0.3, - 0.5) & (0.6, - 0.3) & (0.8, - 0.4) \end{matrix}] \end{matrix}$

Here, we use And-product since Mr. X′s choices have to be considered.

Step 4: To calculate $Mm ([{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}]) = [d_{i 1}],$ we have to find d_i1 ∀i ∈ { 1, 2, 3, 4 } . To demonstrate, let us find $\begin{matrix} I_{k} & = {p : \exists i, {\ddot{c}}_{ip} \neq 0, (k - 1) < p \leq kn}, \\ I_{1} & = {p : {\ddot{c}}_{ip} \neq 0, 0 < p \leq 3} = {1, 2, 3}, \\ I_{2} & = {p : {\ddot{c}}_{ip} \neq 0, 3 < p \leq 6} = {4, 5, 6}, \\ I_{3} & = {p : {\ddot{c}}_{ip} \neq 0, 6 < p \leq 9} = {7, 8, 9} . \end{matrix}$ Hence, $\begin{matrix} t_{11} & = min {{\ddot{c}}_{11}, {\ddot{c}}_{12}, {\ddot{c}}_{13}} \\ = min {(0.3, - 0.4), (0.1, - 0.3), (0.2, - 0.5)} \\ = (0.1, - 0.5) . \end{matrix}$ Similarly, $\begin{matrix} t_{12} = (0.1, - 0.4) and t_{13} = (0.1, - 0.1) . \end{matrix}$ Since i = 1 and k ∈ { 1, 2, 3, 4 } , $\begin{matrix} d_{11} & = max_{k} {t_{1 k}} \\ = max {(0.1, - 0.5), (0.1, - 0.4), (0.1, - 0.1)} \\ = (0.1, - 0.1) . \end{matrix}$ Similarly we can find $\begin{matrix} d_{21} = (0.4, - 0.3), d_{31} = (0.6, - 0.4) \\ and d_{41} = (0.3, - 0.2) . \end{matrix}$ Finally, we can obtain the BFS matrix as $\begin{matrix} Mm ([{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}]) = [d_{i 1}] = [\begin{matrix} (0.1, - 0.1) \\ (0.4, - 0.3) \\ (0.6, - 0.4) \\ (0.3, - 0.2) \end{matrix}], \end{matrix}$ Step 5: Finally we construct an optimum fuzzy set on $R$ according to $Mm ([{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}])$ $\begin{matrix} {opt}_{Mm ([{\hat{a}}_{ij}] \overset{˘}{\land} [{\hat{b}}_{ik}])} (R) \\ = {\begin{matrix} [c] c (ℓ_{1}^{⌣}, 0.1, - 0.1), (ℓ_{2}^{⌣}, 0.4, - 0.3), \\ (ℓ_{3}^{⌣}, 0.6, - 0.4), (ℓ_{4}^{⌣}, 0.3, - 0.2) \end{matrix}}, \end{matrix}$ where $ℓ_{3}^{⌣}$ is an optimum house to buy for Mr. X.

5 Bipolar fuzzy soft mean decision-making with computer Pseudocode

Arithmetic Mean. The sum of all of the numbers of a group, when divided by the number of items in that list is known as the Arithmetic Mean or Mean of the group. $\begin{matrix} \bar{X} = \frac{\sum_{i = 1}^{n} X_{i}}{n}, where n is total number of observations . \end{matrix}$

Product of Bipolar Soft Matrices: If $\tilde{A} = {[({\tilde{a}}_{ij}, a_{ij})]}_{m \times n}$ and $\tilde{B} = {[({\tilde{b}}_{ij}, b_{ij})]}_{n \times p}$ are bipolar fuzzy soft matrices, then we define multiplication of $\tilde{A}$ and $\tilde{B}$ as: $\begin{matrix} \tilde{A} * \tilde{B} = \tilde{C} = {[({\tilde{c}}_{ij}, c_{ij})]}_{m \times p}, where {\tilde{c}}_{ij} = {\tilde{a}}_{ij} . {\tilde{b}}_{ij} \\ for all i, j and c_{ij} = a_{ij} . b_{ij} for all i, j . \end{matrix}$

Now using the arithmetic mean and BFS matrix, we construct the following algorithm for decision-making.

Algorithm 2. Step 1: Choose a feasible subset of the set of parameters.

Step 2: Construct the BFSM for each set of parameters.

Step 3: Find the product of the BFSM using the usual product of matrices.

Step 4: Find the BFS matrix arithmetic mean (BFSMam) by calculating the arithmetic mean of values representing positive and negative information column-wise.

Step 5: Find the optimum bipolar FSS on a universal set.

Computer Program

Pseudocode. Computer Pseudocode developed for BFSMam decision-making is given below.

input: Read input.txt for

m : = rows

n : = columns

${\hat{a}}_{ij} : = m \times n$ BFS matrix

${\hat{b}}_{ij} : = m \times n$ BFS matrix

Define $\ddot{C}$ an m × n² matrix

for : i = 1 to m

for : j = 1 to n

for : k = 1 to n

p = n × (j - 1) + k

${\ddot{c}}_{ip}^{+} : = mean (a_{ij}^{+}, b_{ij}^{+})$

${\ddot{c}}_{ip}^{-} : = mean (a_{ij}^{-}, b_{ij}^{-})$

end for

define D for m × n matrix

for : i = 1 to m

for : j = 1 to n

$d_{ij}^{+} : = \underset{n (j - 1) < k \leq nj}{mean} c_{ik}^{+}$

$d_{ij}^{-} : = \underset{n (j - 1) < k \leq nj}{mean} c_{ik}^{-}$

end for

optimum $value : = \underset{1 \leq i \leq n}{mean} e_{i}^{+}$

out put:

Numerical Example

Example 10. Assume that an investor wants to invest his/her money in opening a shop and choose a city among three different cities $R = {x_{1}, x_{2}, x_{3}}$ . Let the set of parameters ℵ ={ ϰ₁, ϰ₂, ϰ₃ } represent his/her choices for selecting the best city to invest in, where the parameters ϰ_j (j ∈ {1, 2, 3}) stand for “near to down town”, “low land price”, “good roads”, respectively. Suppose that the investor is ready to choose the city. Thus selection for the best choice will be made according to his/her choices using the BFSMam as follows.

Step 1: First of all, select the parameters, $[{\hat{a}}_{ij}] = {x_{1}, x_{2}, x_{3}}$ and $[{\hat{b}}_{ij}] = {x_{1}, x_{2}, x_{3}}$ , respectively.

Step 2: Then, we can write the following BFS matrices which are constructed

according to the parameters. $\begin{matrix} [{\hat{a}}_{ij}] = [\begin{matrix} (0.7, - 0.4) & (0.6, - 0.7) & (0.8, - 1) \\ (0.8, - 0.5) & (0.2, 0) & (0.5, - 0.8) \\ (0.4, 0) & (0.8, - 0.4) & (0, - 0.3) \end{matrix}], \end{matrix}$ $\begin{matrix} [{\hat{b}}_{ik}] = [\begin{matrix} (0.4, - 0.4) & (0.6, - 0.7) & (0.1, - 0.3) \\ (0.3, - 0.7) & (0.4, - 0.9) & (0.1, - 0.6) \\ (0.2, - 0.4) & (0.4, - 0.5) & (0.8, - 0.5) \end{matrix}] . \end{matrix}$

Step 3: Now we will calculate the product of above defined matrices by the usual multiplication of matrices. $\begin{matrix} [{\hat{a}}_{ij}] [{\hat{b}}_{ik}] \\ = [\begin{matrix} (0.62, - 0.69) & (0.98, - 0.96) & (0.77, - 0.59) \\ (0.48, - 0.52) & (0.76, - 0.75) & (0.5, - 0.55) \\ (0.4, - 0.4) & (0.56, - 0.51) & (0.12, - 0.39) \end{matrix}] . \end{matrix}$

Step 4: $\begin{matrix} MAMBFSM ([{\hat{a}}_{ij}] [{\hat{b}}_{ik}]) \\ = [\begin{matrix} (0.5, - . 54) & (0.76, - 0.74) & (0.46, - 0.51) \end{matrix}], \end{matrix}$

$\begin{matrix} {opt}_{MAMBFSM ([{\hat{a}}_{ij}] [{\hat{b}}_{ik}])} (R) \\ = [\begin{matrix} (x_{1}, 0.5, - . 54) & (x_{2}, 0.76, - 0.74) & (x_{3}, 0.46, - 0.51) \end{matrix}] . \end{matrix}$

Therefore city x₂ is an optimum city to be chosen for investing money by the investor according to his choices.

6 Comparison analysis

The BFSSs have many applications, particularly in decision-making problems. Here we presented the applications of BFS matrices in decision-making problems. In this practical application, one of the main issues is how to choose a suitable model. We examined this idea in-depth and used the BFS matrices and computer programs for calculations, and developed an algorithm. In this application, we choose an optimum city for investing money by the investor according to his choices.

Borah et al. in [9] introduced a decision-making method based on a fuzzy soft matrix by using a fuzzy soft T-product. The T-product of fuzzy soft matrices is given by the following.

Definition 9. [9] Let $A_{k} = [{\hat{a}}_{ij}^{k}]$ be complex fuzzy soft matrices. Then, the T-product of fuzzy soft matrices, denoted by $\overset{l}{\underset{k = 1}{Π}} A_{k} = A_{1} * A_{2} * . . . * A_{l},$ is defined by $\overset{l}{\underset{k = 1}{Π}} A_{k} = {[c_{i}]}_{m \times 1},$ where $\begin{matrix} c_{i} = \underset{j = 1}{\sum^{n}} \overset{l}{\underset{k = 1}{Π}} {\hat{a}}_{ij}^{k} . \end{matrix}$

The example is demonstrated as follows:

Step 1. Choose the set of parameters, that is; Suppose U = {c₁, c₂, . . . , c₅} be the five candidates appearing in an interview for an appointment at the managerial level in a company, and E = {e₁, e₂, e₃} be the set of parameters.

Step 2. Suppose three experts, Mr. A, Mr. B and Mr. C take interview the five candidates. Construct the fuzzy soft matrices for each set of parameters. $\begin{matrix} A & = [\begin{matrix} 0.3 & 0.2 & 0.1 \\ 0.5 & 0.4 & 0.2 \\ 0.6 & 0.5 & 0.7 \\ 0.4 & 0.6 & 0.8 \\ 0.8 & 0.6 & 0.3 \end{matrix}], B = [\begin{matrix} 0.7 & 0.2 & 0.5 \\ 0.6 & 0.4 & 0.9 \\ 0.7 & 0.8 & 0.6 \\ 0.5 & 0.6 & 0.1 \\ 0.4 & 0.5 & 0.7 \end{matrix}], \\ C & = [\begin{matrix} 0.5 & 0.4 & 0.6 \\ 0.4 & 0.7 & 0.6 \\ 0.6 & 0.5 & 0.5 \\ 0.8 & 0.6 & 0.4 \\ 0.5 & 0.6 & 0.5 \end{matrix}] . \end{matrix}$

Step 3. Compute T-product of the fuzzy soft matrices;

$\begin{matrix} A * B * C \\ = [\begin{matrix} 0.3 * 0.7 * 0.5 + 0.2 * 0.2 * 0.4 + 0.1 * 0.5 * 0.6 \\ 0.5 * 0.6 * 0.4 + 0.4 * 0.4 * 0.7 + 0.2 * 0.9 * 0.6 \\ 0.6 * 0.7 * 0.6 + 0.5 * 0.8 * 0.5 + 0.7 * 0.6 * 0.5 \\ 0.4 * 0.5 * 0.8 + 0.6 * 0.6 * 0.6 + 0.8 * 1.0 * 0.4 \\ 0.8 * 0.4 * 0.5 + 0.6 * 0.5 * 0.6 + 0.3 * 0.7 * 0.5 \end{matrix}], \\ = [\begin{matrix} 0.151 \\ 0.340 \\ 0.662 \\ 0.696 \\ 0.445 \end{matrix}] . \end{matrix}$

It is clear that the maximum score is 0.696, scored by c₄ and the decision is in favor of selecting c₄ .

The main idea of this paper is similar to the decision-making method given in [9], which only depends on SSs. We extended the idea of bipolar fuzzy soft sets and introduced bipolar fuzzy soft matrices. Moreover, we generalized the concept of the fuzzy soft product to bipolar fuzzy soft product. We introduced and developed algorithms using soft matrices and apply these techniques to different decision-making problems. We developed computer programs for calculations and gave them Pseudocodes. The programs make the calculations easier and faster than regular manual calculations which saves time and avoids fatigue.

So our proposed algorithm is more significant than the Borah [9] algorithm.

7 Conclusion

The BFS theory is being applied to many fields varying from theoretical to practical. In this paper, we defined BFS matrices which are matrix representations of the BFFs. We defined the set-theoretic operations of BFS matrices which are more functional to improve several new results. Afterward, we constructed BFS decision-making models based on the BFS matrix theory. These new decision-making methods depend on the ideas of bipolar FSSs and can be used for making decisions in social sciences, we gave some particular decision-making applications for a real estate agent to choose an optimal house and for investors to choose the best city to invest in. We hope that this work will give some new directions for applications for decision-making problems.

Acknowledgments

This work is financially supported by the Higher Education Commission of Pakistan (Grant No: 7750/Federal/ NRPU/R&D/HEC/ 2017).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.

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