Bipolar fuzzy system of linear equations with polynomial parametric form

Abstract

The notions of center and distance between two Parametric Bipolar Fuzzy Numbers (PBFNs) by giving the preference of right dominance as compared to the left dominance are presented. The solution of Bipolar Fuzzy System of Linear Equations (BFSLEs) is discussed with polynomial parametric bipolar fuzzy number coefficients matrix having crisp real variables and the right-hand side is polynomial parametric bipolar fuzzy numbers. Some of their related properties are investigated. It is proved that if the real coefficient matrix considered as crisp in an original system, while the unknown variable vectors and Right Hand Side (RHS) column vector functions are treated as PBFNs, then initially, the general BFSLEs in polynomial parametric form is solved by the addition and subtraction of the vectors of the lower and upper bound, respectively. The solution procedure is computationally efficient in a bipolar fuzzy environment, and our proposed method is simple as well as efficient to handle the BFSLEs.

Keywords

Polynomial PBFNs center distance BFSLEs

1 Introduction

Zadeh [38, 39] proposed the concept of the fuzzy sets. Dubois and Prade [22, 23], Tanaka and Mizumoto [31, 32] and Nahmias [33] discussed fundamental arithmetic operations and methods of fuzzy numbers. Fuzzy set theory is a mathematical tool for dealing with objects that are complicated or uncertain in nature. Zhang [40] introduced the concept of bipolar fuzzy sets as an extension of fuzzy sets. A bipolar fuzzy set is a mapping from universe X to [-1, 0] × [0, 1]. The positive membership of this mapping indicates that an object satisfies the certain property and the negative membership of this mapping indicate that an object satisfies the counter property. A wide verity of the human decision based on the bipolar judgment that is the positive and negative side. For example, likelihood and unlikelihood, feed-forward and feed backward, effects and side effects, etc. Bipolarity seems to pervade human understanding of information and preference, and bipolar representations look very useful in the development of intelligent technologies. Bipolarity refers to explicit handling of positive and negative sides of information. Basic notions and background on bipolar representations are provided in [21].

Linear system plays a significant role in the field of science and engineering. In the wide majority of the problems, sometimes work with approximate data. In several applications, a few parameters are expressed fuzzy and more general bipolar fuzzy instead of the crisp number. It is extremely essential to propose the numerical techniques that would suitably treat Bipolar Fuzzy System of Linear Equations (BFSLEs) and solve them. There is a vast literature for solving Fuzzy System of Linear Equations (FSLEs). The notion of FSLEs with real crisp coefficient entries of matrix and vector on the Right Hand Side (RHS) is Parametric Fuzzy Numbers (PFNs) arises in many domains of technology and engineering sciences such as telecommunications, statistics, economics, social sciences, image processing and even in physics. Friedman [26] studied the general solution to the FSLEs with crisp coefficient entries of the matrix and the RHS column vector function is the PFN. Friedman studied the embedding technique and replace original n × n FSLEs to the extended 2n × 2n FSLEs. There are several reported studies through which many authors attract our attention to solve the FSLEs numerically. Various procedures for solving fuzzy system would be used by many authors. These methods depend on the fuzzy variables, fuzzy matrix and fuzzy vectors to the RHS of the system, etc. Because of the arithmetical complexity, these methods sometimes provide a non-unique solution. This procedure is lengthy and also not so powerful and efficient. The problem for the system having crisp real coefficients and fuzzy variables is considered in most of these studies. Some new research article has been done with fuzzy coefficients and fuzzy variables which is called fully FSLEs. Few kinds of literature are examined in this paragraph to make the problem complete. Although the open literature has other papers, only corresponding papers are studied and discussed here. Resolution of fuzzy polynomial equations using eigenvalue and an upper bound on the number of solutions were developed in [24, 25]. Parameter reductions of bipolar fuzzy soft sets with their decision-making algorithms was introduced in [7]. Cong-Xin and Ming [18] used the idea of embedding technique in a fuzzy environment. The iterative approach for solving the linear system in the form of an equation X = AX + U was discussed in [37]. Asady et al. [12] considered a general fuzzy system and used the embedding approach to develop different methods. Horcik [28] investigated the solution of FSLEs. To find the solution of the general FSLEs, Vroman et al. [36] used a parametric form of fuzzy numbers. Li et al. [30] introduced the new algorithm for solving FSLEs. The new scheme for an interval and fuzzy systems was proposed in [34]. In [27], the numerical approach was used for solving FSLEs using Gaussian fuzzy weight (membership) function. Behera and Chakraarm developed a new method for handling the complex as well as real FSLEs in [15]. Allahviranloo et al. [9, 10] proposed some notable numerical techniques for solving FSLEs. Allahviranloo and Salahshour [8] considered a novel method for solving the fuzzy linear system of equations by using the 1-cut extension. The new trapezoidal bipolar fuzzy technique for an order of preference by similarity to ideal solution (TOPSIS) scheme for group decision-making was introduced to Akram and Arshad [4]. Akram et al. [2] considered the bipolar fuzzy linear system of equations and also solve it for fully bipolar fuzzy linear system with (-1, 1)-cut position. Akram et al. [3] developed the concept of iterative methods for solving a system of linear equations in a bipolar fuzzy environment. Behera and Chakraverty [13, 14] proposed the solution of FSLEs with crisp coefficients and polynomial parametric form. The numerical solution of a fuzzy system of linear equations with polynomial parametric form is introduced in [11]. In the literature, several traditional techniques have been discussed to find the solutions of linear systems in fuzzy environment (see [8, 26]). In present work, the solutions of linear system of equations in bipolar fuzzy environment is discussed.

In this paper, certain notions including center and distance between two BFNs by giving the preference of right dominance as compared to the left dominance are developed. This property of distance holds good in Hausdorff distance. The solution procedure of the BFSLEs having either crisp or bipolar fuzzy number coefficients matrix and crisp variables using Polynomial Parametric Bipolar Fuzzy Numbers (PPBFNs) on the RHS is discussed. The new method to solve a BFSLEs with the RHS is l-degree of PPBFNs column vector functions and real crisp variables by using the least-squares method is developed. Moreover, we extend this technique for solving the BFSLEs with PPBFN coefficients and crisp variables. Related theorems are stated and prove it by giving true reasoning related to this work. If the coefficient entries of the matrix are treated as crisp in an original system, while unknown variable vectors and RHS column vector functions are considered as BFNs. Then initially, the general BFSLEs in polynomial parametric form is solved by the addition and subtraction of the vectors of the lower and upper bound,respectively.

2 Polynomial parametric bipolar fuzzy numbers

Definition 2.1. [4]A Bipolar Fuzzy Number (BFN) $A$ with satisfaction and dissatisfaction degree ξ and η, respectively can be defined as $ξ_{A} (t) = {\begin{matrix} L_{A}^{P} (t), & if t \in [a, b], \\ 1, & if t \in [b, c], \\ R_{A}^{P} (t), & if t \in [c, d], \\ 0, & Otherwise, \end{matrix}$

$η_{A} (t) = {\begin{matrix} L_{A}^{N} (t), & if t \in [e, f], \\ - 1, & if t \in [f, g], \\ R_{A}^{N} (t), & if t \in [g, h], \\ 0, & Otherwise, \end{matrix}$ where the function $L_{A}^{P} (t)$ is left satisfaction and is non-decreasing over the interval [a, b] and the function $R_{A}^{P} (t)$ is right satisfaction and is non-increasing over the interval [c, d], such that $\begin{matrix} L_{A}^{P} (a) = R_{A}^{P} (d) = 0 and L_{A}^{P} (b) = R_{A}^{P} (c) = 1 . \end{matrix}$ The function $L_{A}^{N} (t)$ is left dissatisfaction and is non-increasing over the interval [e, f] and the function $R_{A}^{N} (t)$ is right dissatisfaction and is non-decreasing over the interval [g, h] such that $\begin{matrix} L_{A}^{N} (e) = R_{A}^{N} (h) = 0 \end{matrix}$ and $\begin{matrix} L_{A}^{N} (f) = R_{A}^{N} (g) = - 1 . \end{matrix}$ Moreover, if the satisfaction functions $L_{A}^{P} (a), R_{A}^{P} (d)$ and dissatisfaction functions $L_{A}^{N} (a), R_{A}^{N} (d)$ are linear, then we say that $A$ is the trapezoidal BFN and is denoted by 〈 [a, b, c, d] , [e, f, g, h] 〉. If b = c and f = g, then we can write as 〈 [a, b, d] , [e, f, h] 〉 or 〈 [a, c, d] , [e, g, h] 〉 which is triangular BFN. Now we write the proper definition of parametric bipolar fuzzy number base on (r, s)-cut of BFN.

A Parametric Bipolar Fuzzy Number (PBFN) $j$ is a quadruple $〈 [\underline{j} (r), \bar{j} (r)], [\underline{j} (s), \bar{j} (s)] 〉$ of the functions $\underline{j} (r), \bar{j} (r), \underline{j} (s), \bar{j} (s); r \in [0, 1], s \in [- 1, 0]$ and satisfying the given conditions:

$\underline{j} (r)$ is bounded, non-decreasing (monotonically increasing), right continuous at point 0 and left continuous function on set (0, 1],

$\bar{j} (r)$ is bounded, non-increasing (monotonically decreasing), right continuous at point 0 and left continuous functions on set (0, 1],

$\underline{j} (s)$ is bounded, non-increasing (monotonically decreasing), right continuous at point -1 and left continuous functions on set (-1, 0],

$\bar{j} (s)$ is bounded, non-decreasing (monotonically increasing), right continuous at point -1 and left continuous functions on set (-1, 0],

$\bar{j} (r) \geq \underline{j} (r)$ ,

$\bar{j} (s) \geq \underline{j} (s)$ .

The class of all parametric BFNs with scalar multiplication and addition, denoted by I E, is convex and concave cone.

Definition 2.2. Center for an arbitrary parametric BFN $j = 〈 [\underline{j} (r), \bar{j} (r)], [\underline{j} (s), \bar{j} (s)] 〉$ is defined as $\begin{matrix} u^{c} = [\frac{\underline{j} (r) + \bar{j} (r)}{2}, \frac{\underline{j} (s) + \bar{j} (s)}{2}], \\ \forall r \in [0, 1], s \in [- 1, 0] . \end{matrix}$

Definition 2.3. For any two parametric BFNs $j$ and $k$ with (r, s)-cut representations $\begin{matrix} 〈 [\underline{j} (r), \bar{j} (r)], [\underline{j} (s), \bar{j} (s)] 〉 and 〈 [\underline{l} (r), \bar{l} (r)], [\underline{l} (s), \bar{l} (s)] 〉, \end{matrix}$ respectively. Then the distance between $j$ and $l$ is defined as

$\begin{matrix} D (j, l) & = | \int_{0}^{\frac{1}{2}} [r (\underline{j} (r) - \underline{l} (r)) + (1 - r) (\bar{j} (r) - \bar{l} (r))] dr \\ + \int_{\frac{1}{2}}^{1} [(1 - r) (\underline{j} (r) - \underline{l} (r)) + r (\bar{j} (r) - \bar{l} (r))] dr | \\ + | \int_{- 1}^{- \frac{1}{2}} [s (\bar{j} (s) - \bar{l} (s)) + (1 - s) (\underline{j} (s) - \underline{l} (s))] ds \\ + \int_{- \frac{1}{2}}^{0} [(1 - s) (\bar{j} (s) - \bar{l} (s)) + s (\underline{j} (s) - \underline{l} (s))] ds | . \end{matrix}$ (1) On the other hands, right dominance has a preference as compared with the left dominance in the bipolar fuzzy environment.

Theorem 2.1. For any arbitrary three bipolar fuzzy numbers $j$ , $l$ and $q$ , the properties below are satisfied.

$D (j, l) \geq 0$ we have non-negativity and $D (j, l) = 0$ if and only if $j = l$ ,

$D (j, l) = D (l, j)$ symmetry,

$D (j, l) \leq D (j, q) + D (q, l)$ triangle inequality.

Proof. The proof is straightforward and is left as an exercise to the reader.□

Since, the above distance is developed on the base of dominance. So, we can easily prove these properties hold in Hausdorff distance

$D (j + q, l + q) = D (j, l), for all j, l, q \in I E$ ,

$D (j, \tilde{0}) + D (l, \tilde{0}) \geq D (j + l, \tilde{0}), for all j, l \in I E$ ,

$D (κ j, κ l) = | κ | D (j, l), for all j, l \in I E$ and κ ∈ I R,

$D (j, q) + D (l, y) \geq D (j + l, q + y), for all j,$ $l, q and y \in I E$ .

In above expression,

\tilde{0}

is a zero BFN. For zero trapezoidal and triangular BFN,

\tilde{0}

can be represented as 〈[0, 0, 0, 0] , [0, 0, 0, 0] 〉 and 〈[0, 0, 0] , [0, 0, 0] 〉, respectively.

Now we define suprimum and infimum for (r, s)-cut of bipolar fuzzy numbers.

Definition 2.4. For any two parametric BFNs $u$ and $v$ , we have $\begin{matrix} u ≾ v \Leftrightarrow sup [u]_{{r, s}} \leq sup [v]_{{r, s}} \end{matrix}$ and $\begin{matrix} inf [u]_{{r, s}} \leq inf [v]_{{r, s}}, \forall r \in [0, 1], s \in [- 1, 0] . \end{matrix}$ In general, $\begin{matrix} u_{1} ≾ u_{2}, \dots, ≾ u_{m} \\ \Leftrightarrow sup [u_{1}]_{{r, s}} \leq sup [u_{2}]_{{r, s}}, \dots, \leq sup [u_{m}]_{{r, s}} \end{matrix}$ and $\begin{matrix} inf [u_{1}]_{{r, s}} \leq inf [u_{2}]_{{r, s}}, . . ., \leq inf [u_{m}]_{{r, s}}, \\ \forall r \in [0, 1], s \in [- 1, 0] . \end{matrix}$ We now define the relation of inequality ≺ for ⪯ as follows:

Definition 2.5. For any two parametric BFNs $u$ and $v$ , we have $\begin{matrix} u ≺ v \Leftrightarrow u ⪯ v \end{matrix}$ and ∃ r ∈ [0, 1], s ∈ [-1, 0] holding either $\begin{matrix} sup [u]_{{r, s}} < sup [v]_{{r, s}} or inf [u]_{{r, s}} < inf [v]_{{r, s}} . \end{matrix}$ In general, $\begin{matrix} u_{1} ≺ u_{2}, \dots, ≺ u_{m} \Leftrightarrow u_{1} ≾ u_{2}, \dots, ≾ u_{m} \end{matrix}$ holding either $\begin{matrix} sup [u_{1}]_{{r, s}} < sup [u_{2}]_{{r, s}}, \dots, < sup [u_{m}]_{{r, s}}, \end{matrix}$ or $\begin{matrix} inf [u_{1}]_{{r, s}} < inf [u_{2}]_{{r, s}}, \dots, < inf [u_{m}]_{{r, s}}, \\ \forall r \in [0, 1], s \in [- 1, 0] . \end{matrix}$

Definition 2.6. A parametric BFN $j$ has l-degree of polynomial and if there exist four polynomials ${\underline{j}}_{l} (r), {\bar{j}}_{l} (r), {\underline{j}}_{l} (s)$ and ${\bar{j}}_{l} (s)$ having degree at most l, such that $j = 〈 [{\underline{j}}_{l} (r), {\bar{j}}_{l} (r)], [{\underline{j}}_{l} (s), {\bar{j}}_{l} (s)] 〉$ is called polynomial parametric bipolar fuzzy number.

Definition 2.7. The n × n linear system having l-degree of polynomial in parametric form can be defined as follows ${\begin{matrix} a_{11} y_{1} + a_{12} y_{2} + a_{13} y_{3} + \dots + a_{1 n} y_{n} = z_{1}, \\ a_{21} y_{1} + a_{22} y_{2} + a_{23} y_{3} + \dots + a_{2 n} y_{n} = z_{2}, \\ ⋮ \\ a_{n 1} y_{1} + a_{n 2} y_{2} + a_{n 3} y_{3} + \dots + a_{nn} y_{n} = z_{n} . \end{matrix}$ In matrix notation, the Definition 2.7 can be setting in the following

$[A] {Y} = {Z},$ (2) where the coefficient matrix $[A]$ of the elements (a_uv) either crisp or polynomial parametric bipolar fuzzy numbers $\begin{matrix} (a_{uv}) = 〈 [{\underline{a}}_{uv}^{P} (r), {\bar{a}}_{uv}^{P} (r)], [{\underline{a}}_{uv}^{N} (s), {\bar{a}}_{uv}^{N} (s)] 〉, \\ 1 \leq u, v \leq n \end{matrix}$ is a bipolar fuzzy n × n matrix, $Z = z_{u} = 〈 [{\underline{z}}_{u}^{P} (r), {\bar{z}}_{u}^{P} (r)], [{\underline{z}}_{u}^{N} (s), {\bar{z}}_{u}^{N} (s)] 〉$ , 1 ≤ u ≤ n is column vector bipolar fuzzy number functions and $Y = y_{v} = 〈 [{\underline{y}}_{u}^{P} (r), {\bar{y}}_{u}^{P} (r)], [{\underline{y}}_{u}^{N} (s), {\bar{y}}_{u}^{N} (s)] 〉$ is the unknown vectors. For a positive integer l, all $\begin{matrix} 〈 [{\underline{a}}_{uv}^{P} (r), {\bar{a}}_{uv}^{P} (r)], [{\underline{a}}_{uv}^{N} (s), {\bar{a}}_{uv}^{N} (s)] 〉 \end{matrix}$ and $\begin{matrix} 〈 [{\underline{z}}_{u}^{P} (r), {\bar{z}}_{u}^{P} (r)], [{\underline{z}}_{u}^{N} (s), {\bar{z}}_{u}^{N} (s)] 〉 \end{matrix}$ are bipolar fuzzy numbers with l-degree of polynomial form. The system (2) can also be written as

$\sum_{v = 1}^{n} a_{uv} y_{v} = z_{u}, \forall u = 1, 2, \dots, n .$ (3) We can write equation (3) in parametric form as.

$\begin{matrix} \sum_{v = 1}^{n} 〈 [{\underline{a}}_{uv}^{P} (r), {\bar{a}}_{uv}^{P} (r)], [{\underline{a}}_{uv}^{N} (s), {\bar{a}}_{uv}^{N} (s)] 〉 y_{v} \\ = 〈 [{\underline{z}}_{u}^{P} (r), {\bar{z}}_{u}^{P} (r)], [{\underline{z}}_{u}^{N} (s), {\bar{z}}_{u}^{N} (s)] 〉, \forall u = 1, 2, \dots, n . \end{matrix}$ (4) Equation (4) is equivalent as the following equations (5) and (6)

$\begin{matrix} \sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{0 > y_{y}} {\bar{a}}_{uv}^{P} (r) y_{v}^{P} = {\underline{z}}_{u}^{P} (r), \\ \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{0 > y_{y}} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} = {\bar{z}}_{u}^{P} (r) \end{matrix}$ (5) and

$\begin{matrix} \sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{0 > y_{y}} {\bar{a}}_{uv}^{N} (s) y_{v}^{N} = {\underline{z}}_{u}^{N} (s), \\ \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{0 > y_{y}} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} = {\bar{z}}_{u}^{N} (s) . \end{matrix}$ (6)

Assume that $y_{v}^{P} = y_{v}^{' P} - y_{v}^{″ P}$ and $y_{v}^{N} = y_{v}^{' N} - y_{v}^{″ N}$ , where $y_{v}^{' P} = {\begin{matrix} y_{v}^{P}, & if y_{v}^{P} \geq 0, \\ 0, & if y_{v}^{P} < 0, \end{matrix}$ $y_{v}^{″ P} = {\begin{matrix} 0, & if y_{v}^{P} \geq 0, \\ - y_{v}^{P}, & if y_{v}^{P} < 0, \end{matrix}$ and $y_{v}^{' N} = {\begin{matrix} y_{v}^{N}, & if y_{v}^{N} \geq 0, \\ 0, & if y_{v}^{N} < 0, \end{matrix}$ $y_{v}^{″ N} = {\begin{matrix} 0, & if y_{v}^{N} \geq 0, \\ - y_{v}^{N}, & if y_{v}^{N} < 0 . \end{matrix}$ As $y_{v}^{' P}, y_{v}^{″ P}, y_{v}^{' N}, y_{v}^{″ N} \geq 0$ , so equations (5-6) implies that $\begin{matrix} \sum_{v = 0}^{n} {\underline{a}}_{uv}^{P} (r) y_{v}^{' P} - \sum_{v = 0}^{n} {\bar{a}}_{uv}^{P} (r) y_{v}^{″ P} = {\underline{z}}_{u}^{P} (r), \\ \sum_{v = 0}^{n} {\bar{a}}_{uv}^{P} (r) y_{v}^{' P} - \sum_{v = 0}^{n} {\underline{a}}_{uv}^{P} (r) y_{v}^{″ P} = {\bar{z}}_{u}^{P} (r) \end{matrix}$ (7) and $\sum_{v = 0}^{n} {\underline{a}}_{uv}^{N} (s) y_{v}^{' N} - \sum_{v = 0}^{n} {\bar{a}}_{uv}^{N} (s) y_{v}^{″ N} = {\underline{z}}_{u}^{N} (s),$ (8) $\sum_{v = 0}^{n} {\bar{a}}_{uv}^{N} (s) y_{v}^{' N} - \sum_{v = 0}^{n} {\underline{a}}_{uv}^{N} (s) y_{v}^{″ N} = {\bar{z}}_{u}^{N} (s) .$ (9) Since a_uv and z_u are BFNs having l-degree of polynomial form, therefore ${\begin{matrix} {\underline{a}}_{uv}^{P} (r) = \sum_{i = 0}^{l} c_{uvi} r^{i}, & u, v = 1, 2, \dots, n, \\ {\bar{a}}_{uv}^{P} (r) = \sum_{i = 0}^{l} d_{uvi} r^{i}, & u, v = 1, 2, \dots, n, \\ {\underline{a}}_{uv}^{N} (s) = \sum_{i = 0}^{l} e_{uvi} s^{i}, & u, v = 1, 2, \dots, n, \\ {\bar{a}}_{uv}^{N} (s) = \sum_{i = 0}^{l} f_{uvi} s^{i}, & u, v = 1, 2, \dots, n, \\ {\underline{z}}_{u}^{P} (r) = \sum_{i = 0}^{l} g_{ui} r^{i}, & u = 1, 2, \dots, n, \\ {\bar{z}}_{u}^{P} (r) = \sum_{i = 0}^{l} h_{ui} r^{i}, & u = 1, 2, \dots, n, \\ {\underline{z}}_{u}^{N} (s) = \sum_{i = 0}^{l} k_{ui} s^{i}, & u = 1, 2, \dots, n, \\ {\bar{z}}_{u}^{N} (s) = \sum_{i = 0}^{l} t_{ui} s^{i}, & u = 1, 2, \dots, n . \end{matrix}$ By using above expression, equations (7-8) become as

$\begin{matrix} \sum_{v = 0}^{n} \sum_{i = 0}^{l} c_{uvi} r^{i} y_{v}^{' P} - \sum_{v = 0}^{n} \sum_{i = 0}^{l} d_{uvi} r^{i} y_{v}^{″ P} = \sum_{i = 0}^{l} g_{ui} r^{i}, \\ \sum_{v = 0}^{n} \sum_{i = 0}^{l} d_{uvi} r^{i} y_{v}^{' P} - \sum_{v = 0}^{n} \sum_{i = 0}^{l} c_{uvi} r^{i} y_{v}^{″ P} = \sum_{i = 0}^{l} h_{ui} r^{i}, \end{matrix}$ (10) and

$\begin{matrix} \sum_{v = 0}^{n} \sum_{i = 0}^{l} e_{uvi} s^{i} y_{v}^{' N} - \sum_{v = 0}^{n} \sum_{i = 0}^{l} f_{uvi} s^{i} y_{v}^{″ N} = \sum_{i = 0}^{l} k_{ui} s^{i}, \\ \sum_{v = 0}^{n} \sum_{i = 0}^{l} f_{uvi} s^{i} y_{v}^{' N} - \sum_{v = 0}^{n} \sum_{i = 0}^{l} e_{uvi} s^{i} y_{v}^{″ N} = \sum_{i = 0}^{l} t_{ui} s^{i} . \end{matrix}$ (11) Consequently, for every u = 1, 2, ⋯ , n and i = 1, 2, ⋯ , l, we have

$\begin{matrix} \sum_{v = 0}^{n} c_{uvi} y_{v}^{' P} - \sum_{v = 0}^{n} d_{uvi} y_{v}^{″ P} = g_{ui}, \\ \sum_{v = 0}^{n} d_{uvi} y_{v}^{' P} - \sum_{v = 0}^{n} c_{uvi} y_{v}^{″ P} = h_{ui}, u = 1, 2, \dots, n, \end{matrix}$ (12) and

$\begin{matrix} \sum_{v = 0}^{n} e_{uvi} y_{v}^{' N} - \sum_{v = 0}^{n} f_{uvi} y_{v}^{″ N} = k_{ui}, \\ \sum_{v = 0}^{n} f_{uvi} y_{v}^{' N} - \sum_{v = 0}^{n} e_{uvi} y_{v}^{″ N} = t_{ui} u = 1, 2, \dots, n . \end{matrix}$ (13) Consider in matrix form $\begin{matrix} C = [\begin{matrix} c_{110} & \dots & c_{1 n 0} \\ ⋮ & ⋮ & ⋮ \\ c_{11 l} & \dots & c_{1 nl} \\ ⋮ & ⋮ & ⋮ \\ c_{n 10} & \dots & c_{nn 0} \\ ⋮ & ⋮ & ⋮ \\ c_{n 1 l} & \dots & c_{nnl} \end{matrix}], D = [\begin{matrix} d_{110} & \dots & d_{1 n 0} \\ ⋮ & ⋮ & ⋮ \\ d_{11 l} & \dots & d_{1 nl} \\ ⋮ & ⋮ & ⋮ \\ d_{n 10} & \dots & d_{nn 0} \\ ⋮ & ⋮ & ⋮ \\ d_{n 1 l} & \dots & d_{nnl} \end{matrix}], \\ E = [\begin{matrix} e_{110} & \dots & e_{1 n 0} \\ ⋮ & ⋮ & ⋮ \\ e_{11 l} & \dots & e_{1 nl} \\ ⋮ & ⋮ & ⋮ \\ e_{n 10} & \dots & e_{nn 0} \\ ⋮ & ⋮ & ⋮ \\ e_{n 1 l} & \dots & e_{nnl} \end{matrix}], F = [\begin{matrix} f_{110} & \dots & f_{1 n 0} \\ ⋮ & ⋮ & ⋮ \\ f_{11 l} & \dots & f_{1 nl} \\ ⋮ & ⋮ & ⋮ \\ f_{n 10} & \dots & f_{nn 0} \\ ⋮ & ⋮ & ⋮ \\ f_{n 1 l} & \dots & f_{nnl} \end{matrix}], \\ G = [\begin{matrix} g_{10} \\ ⋮ \\ g_{1 l} \\ ⋮ \\ g_{n 0} \\ ⋮ \\ g_{nl} \end{matrix}], H = [\begin{matrix} h_{10} \\ ⋮ \\ h_{1 l} \\ ⋮ \\ h_{n 0} \\ ⋮ \\ h_{nl} \end{matrix}], K = [\begin{matrix} k_{10} \\ ⋮ \\ k_{1 l} \\ ⋮ \\ k_{n 0} \\ ⋮ \\ k_{nl} \end{matrix}], T = [\begin{matrix} t_{10} \\ ⋮ \\ t_{1 l} \\ ⋮ \\ t_{n 0} \\ ⋮ \\ t_{nl} \end{matrix}] \end{matrix}$ and $\begin{array}{l} Y^{' P} = [\begin{matrix} y_{1}^{' P} \\ ⋮ \\ y_{n}^{' P} \end{matrix}], Y^{" P} = [\begin{matrix} y_{1}^{" P} \\ ⋮ \\ y_{n}^{" P} \end{matrix}], \\ Y^{' N} = [\begin{matrix} y_{1}^{' N} \\ ⋮ \\ y_{n}^{' N} \end{matrix}], Y^{" N} = [\begin{matrix} y_{1}^{" N} \\ ⋮ \\ y_{n}^{" N} \end{matrix}] . \end{array}$ Now we will solve the following linear system

$Q Y = Z$ (14) where $\begin{matrix} Q = [\begin{matrix} C & - D & O & O \\ O & O & E & - F \\ D & - C & O & O \\ O & O & F & - E \end{matrix}], Y = [\begin{matrix} y^{'} P \\ y^{'} N \\ y^{″} P \\ y^{″} N \end{matrix}], Z = [\begin{matrix} z^{'} P \\ z' N \\ z ″ P \\ z ″ N \end{matrix}] . \end{matrix}$ As 𝒬 is a 4n (l + 1) ×4n matrix, O is the null matrix having order n × n and also l > 0, so we will solve the following system

$min_{Y \in ℝ^{4 n}} ∥ Q Y - Z ∥_{4} .$ (15) The solution of equation (15) is equivalent to the following system

$Q^{T} Q Y = Q^{T} Z .$ (16)

Theorem 2.2. Let ${\hat{Y}}$ be the least squares solution of the system (2), and also if its corresponding 𝒴 satisfies the general system 𝒬𝒴 = 𝒵. Then it is an exact solution of this system (2).

Proof. Suppose that 𝒬^T𝒬 is nonsingular. Then the solution of corresponding systems (14) and (16) are identical and 𝒴^∗ = (𝒬^T𝒬) ^-1𝒬^T𝒵. For a simple calculation, it tends to be appeared $\begin{matrix} Q^{T} Q = [\begin{matrix} K & - L & O & O \\ - L & K & O & O \\ O & O & M & - N \\ O & O & - N & M \end{matrix}], \end{matrix}$ where $\begin{matrix} K = C^{T} C + D^{T} D, L = C^{T} D + D^{T} C, \\ M = E^{T} E + F^{T} F, N = E^{T} F + F^{T} E . \end{matrix}$ So $\begin{matrix} Q^{T} Q \to [\begin{matrix} K - L & - L + K & O & O \\ - L & K & O & O \\ O & O & M - N & - N + M \\ O & O & - N & M \end{matrix}] \end{matrix}$ $\begin{matrix} \to [\begin{matrix} K - L & O & O & O \\ - L & K + L & O & O \\ O & O & M - N & O \\ O & O & - N & M + N \end{matrix}] . \end{matrix}$ Therefore $\begin{matrix} | Q^{T} Q | = | K - L | . | K + L | . | M - N | . | M + N | . \end{matrix}$ Thus we have $\begin{matrix} K - L = (C - D)^{T} (C - D), \\ K + L = (C + D)^{T} (C + D), \\ M - N = (E - F)^{T} (E - F), \\ M + N = (E + F)^{T} (E + F) . \end{matrix}$ If $C - D, C + D, E - F$ and $E + F$ are full rank matrices then 𝒦 - ℒ, 𝒦 + ℒ, ℳ - 𝒩 and ℳ + 𝒩 are nonsingular, respectively, $(if rank (| C \pm D | . | E \pm F |) = 2 n)$ then 𝒬^T𝒬 is nonsingular. This completes the proof.□

Theorem 2.3. Suppose that the vector {𝒴} is the solution of BFSLEs $[A] {Y} = {Z}$ , then the vector {𝒴} is solution of the crisp linear system $[\underline{A} + \bar{A}] {Y} = {\underline{Z} + \bar{Z}}$ .

Proof. First we assume that the left side of this system $[\underline{A} + \bar{A}] {Y} = {\underline{Z} + \bar{Z}}$ . We can write it $[\underline{A} + \bar{A}] {Y}$ as $\begin{matrix} \sum_{v = 1}^{n} [{\underline{a}}_{uv}^{P} (r) + {\bar{a}}_{uv}^{P} (r), {\underline{a}}_{uv}^{N} (s) + {\bar{a}}_{uv}^{N} (s)] {y_{v}}, \end{matrix}$ which implies that

$\begin{matrix} [\sum_{v = 1}^{n} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{v = 1}^{n} {\bar{a}}_{uv}^{P} (r) y_{v}^{P}, \sum_{v = 1}^{n} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{v = 1}^{n} {\bar{a}}_{uv}^{N} (s) y_{v}^{N}], \\ u = 1, 2, \dots, n . \end{matrix}$

which is equivalent to

$\begin{matrix} [\sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{y_{v} < 0} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{y_{v} < 0} {\bar{a}}_{uv}^{P} (r) y_{v}^{P}, \\ \sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{y_{v} < 0} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{y_{v} < 0} {\bar{a}}_{uv}^{N} (s) y_{v}^{N}] . \end{matrix}$ (17)

Using equations (5-6), the above expression can be written as in equation (17) we get $\begin{matrix} [{\underline{z}}_{u}^{P} (r) + {\bar{z}}_{u}^{P} (r), {\underline{z}}^{N} u (s) + {\bar{z}}_{u}^{N} (s)] \\ = & [{\underline{z}}^{P} (r) + {\bar{z}}^{P} (r), {\underline{z}}^{N} (s) + {\bar{z}}^{N} (s)] = {\underline{Z} + \bar{Z}} . \end{matrix}$ Thus we have $\begin{matrix} [\underline{A} + \bar{A}] {Y} = {\underline{Z} + \bar{Z}} . \end{matrix}$ This proves that the vector {𝒴} is the solution of the system $\begin{matrix} [\underline{A} + \bar{A}] {Y} = {\underline{Z} + \bar{Z}} . \end{matrix}$ □

Theorem 2.4. Assume that the vector {𝒴} is the solution of BFSLEs $[A] {Y} = {Z}$ , then the vector {𝒴} is the solution of the crisp linear system $[A^{c}] {Y} = {Z^{c}}$ , where $\begin{matrix} [A^{c}] = \frac{〈 [{\underline{a}}_{uv}^{P} + {\bar{a}}_{uv}^{P}], [{\underline{a}}_{uv}^{N} + {\bar{a}}_{uv}^{N}] 〉}{2} \end{matrix}$ and $\begin{matrix} {Z^{c}} = \frac{〈 [{\underline{z}}_{u}^{P} + {\bar{z}}_{uv}^{P}], [{\underline{z}}_{u}^{N} + {\bar{z}}_{uv}^{N}] 〉}{2} . \end{matrix}$

Proof. The proof is obvious as in Theorem 2.3.□ Now we state some useful results of linear system when the matrix $[A]$ is crisp coefficient in bipolar fuzzy environment.

Theorem 2.5. If $\begin{matrix} [A] {Y} = {Z}, \end{matrix}$ then $\begin{matrix} {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} \end{matrix}$ is the solution vector of BFSLEs

$[A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}},$ (18) where $[A^{*}] = (a_{uv}^{*}) \geq 0 \in [A]$ .

Proof. First we assume the left side of this system $[A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}},$ which can be written as $\begin{matrix} (\sum_{v = 1}^{n} [{\underline{a}}_{uv}^{P} (r) - {\bar{a}}_{uv}^{P} (r), {\underline{a}}_{uv}^{N} (s) - {\bar{a}}_{uv}^{N} (s)] \\ {{\underline{y}}_{v}^{P} (r) - {\bar{y}}_{v}^{P} (r), {\underline{y}}_{v}^{N} (s) - {\bar{y}}_{v}^{N} (s)}), \end{matrix}$ which implies that $\begin{matrix} [ & \sum_{v = 1}^{n} {\underline{a}}_{uv}^{P} (r) {\underline{y}}_{v}^{P} - \sum_{v = 1}^{n} {\bar{a}}_{uv}^{P} (r) {\bar{y}}_{v}^{P}, \\ \sum_{v = 1}^{n} {\underline{a}}_{uv}^{N} (s) {\underline{y}}_{v}^{N} - \sum_{v = 1}^{n} {\bar{a}}_{uv}^{N} (s) {\bar{y}}_{v}^{N}], u = 1, 2, \dots, n . \end{matrix}$ which is equivalent to

$\begin{matrix} [\sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} + \sum_{y_{v} < 0} {\underline{a}}_{uv}^{P} (r) y_{v}^{P} - \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{P} (r) y_{v}^{P} - \sum_{y_{v} < 0} {\bar{a}}_{uv}^{P} (r) y_{v}^{P}, \\ \sum_{0 \leq y_{v}} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} + \sum_{y_{v} < 0} {\underline{a}}_{uv}^{N} (s) y_{v}^{N} - \sum_{0 \leq y_{v}} {\bar{a}}_{uv}^{N} (s) y_{v}^{N} - \sum_{y_{v} < 0} {\bar{a}}_{uv}^{N} (s) y_{v}^{N}] . \end{matrix}$ (19)

Using equations (5-6), the above expression can be written as in equation (19), we get $\begin{matrix} {{\underline{z}}_{u}^{P} (r) - {\bar{z}}_{u}^{P} (r), {\underline{z}}^{N} u (s) - {\bar{z}}_{u}^{N} (s)} \\ = {{\underline{z}}^{P} (r) - {\bar{z}}^{P} (r), {\underline{z}}^{N} (s) - {\bar{z}}^{N} (s)} = {\underline{Z} - \bar{Z}} . \end{matrix}$ Thus we have $\begin{matrix} [A^{*}] {\underline{Y} - \bar{Y}} = {\underline{Z} - \bar{Z}} . \end{matrix}$ This shows that the vector ${\underline{Y} - \bar{Y}}$ is solution of the linear system in bipolar fuzzy environment.□

Theorem 2.6. If $\begin{matrix} [A] {Y} = {Z}, then {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} \end{matrix}$ is the solution vector of BFSLEs

$[A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}} .$ (20)

Proof. The proof is straightforward as in Theorem 2.5.□

Theorem 2.7. BFSLEs $[A] {Y} = {Z}$ (21) are inconsistent if and only if $\begin{matrix} [A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}} \end{matrix}$ and $\begin{matrix} [A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}} \end{matrix}$ are inconsistent, where $[A^{*}] = (a_{uv}^{*}) \geq 0 \in [A]$ .

Proof. Suppose that the system (21) is inconsistent. Then the system (21) can be put in parametric form as

$\begin{matrix} [A] {[{\underline{y}}^{P} (r), {\bar{y}}^{P} (r), {\underline{y}}^{N} (s), {\bar{y}}^{N} (s)]} \\ = {[{\underline{z}}^{P} (r), {\bar{z}}^{P} (r), {\underline{z}}^{N} (s), {\bar{z}}^{N} (s)]} . \end{matrix}$ (22) One can write the system (22) in term of center in bipolar fuzzy environment as

$[A] {Y^{c}} = {Z^{c}},$ (23) where $\begin{matrix} {Y^{c}} = {\frac{{\underline{y}}^{P} (r) + {\bar{y}}^{P} (r)}{2}, \frac{{\underline{y}}^{N} (s) + {\bar{y}}^{N} (s)}{2}} \\ and \\ {Z^{c}} = {\frac{{\underline{z}}^{P} (r) + {\bar{z}}^{P} (r)}{2}, \frac{{\underline{z}}^{N} (s) + {\bar{z}}^{N} (s)}{2}} . \end{matrix}$ So, equation (23) implies that

$\begin{matrix} [A] {\frac{{\underline{y}}^{P} (r) + {\bar{y}}^{P} (r)}{2}, \frac{{\underline{y}}^{N} (s) + {\bar{y}}^{N} (s)}{2}} \\ = {\frac{{\underline{z}}^{P} (r) + {\bar{z}}^{P} (r)}{2}, \frac{{\underline{z}}^{N} (s) + {\bar{z}}^{N} (s)}{2}} . \end{matrix}$ (24) Now we can conclude that equation (24) is inconsistent, because ${Y^{c}}$ is the middle point of ${{\underline{y}}^{P} (r), {\bar{y}}^{P} (r), {\underline{y}}^{N} (s), {\bar{y}}^{N} (s)} .$ So from equation (24), we conclude that $[A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}}$ is inconsistent. Moreover, the system (2) may be represented by width of bipolar fuzzy solution as $\begin{matrix} [A^{*}] {△ Y} = {△ Z}, \end{matrix}$ where $\begin{matrix} {△ Y} = {\frac{{\underline{y}}^{P} (r) - {\bar{y}}^{P} (r)}{2}, \frac{{\underline{y}}^{N} (s) - {\bar{y}}^{N} (s)}{2}} \\ and \\ {△ Z} = {\frac{{\underline{z}}^{P} (r) - {\bar{z}}^{P} (r)}{2}, \frac{{\underline{z}}^{N} (s) - {\bar{z}}^{N} (s)}{2}} . \end{matrix}$ So $\begin{matrix} [A^{*}] {\frac{{\underline{y}}^{P} (r) - {\bar{y}}^{P} (r)}{2}, \frac{{\underline{y}}^{N} (s) - {\bar{y}}^{N} (s)}{2}} \\ = {\frac{{\underline{z}}^{P} (r) - {\bar{z}}^{P} (r)}{2}, \frac{{\underline{z}}^{N} (s) - {\bar{z}}^{N} (s)}{2}} \end{matrix}$ is inconsistent, because ${△ Y}$ is the left and right spread of the solution in bipolar fuzzy environment. From this, we conclud that $[A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}}$ is inconsistent. Hence the necessary case of this theorem holds good.

Conversely, now we will prove the sufficient condition of this theorem. Suppose that $[A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}}$ and $[A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}}$ are inconsistent. Now we have to show that $[A] {Y} = {Z}$ that is $\begin{matrix} [A] {[{\underline{y}}^{P} (r), {\bar{y}}^{P} (r), {\underline{y}}^{N} (s), {\bar{y}}^{N} (s)]} \\ = {[{\underline{z}}^{P} (r), {\bar{z}}^{P} (r), {\underline{z}}^{N} (s), {\bar{z}}^{N} (s)]} \end{matrix}$ to be inconsistent. $\begin{matrix} [A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}} \end{matrix}$ can be equivalently written as $\begin{matrix} [A] {\frac{{\underline{y}}^{P} + {\bar{y}}^{P}}{2}, \frac{{\underline{y}}^{N} + {\bar{y}}^{N}}{2}} = {\frac{{\underline{z}}^{P} + {\bar{z}}^{P}}{2}, \frac{{\underline{z}}^{N} + {\bar{z}}^{N}}{2}} . \end{matrix}$ So, we have $[A] {Y^{c}} = {Z^{c}} .$ Thus the center solution of equation (21) is inconsistent and is equivalent to $[A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}} .$ Based on [16], the bipolar fuzzy solution vector $[{\underline{y}}^{P}, {\bar{y}}^{P}, {\underline{y}}^{N}, {\bar{y}}^{N}]$ can be represented as $[Y^{c} - △ Y, Y^{c} + △ Y, Y^{c} - △ Y, Y^{c} + △ Y] .$ As the solution center is inconsistent, hence the system (21) is inconsistent, because we can not proceed for [Y^c - △ Y, Y^c + △ Y, Y^c - △ Y, Y^c + △ Y]. So the sufficient part of this theorem is proved.□

Theorem 2.8. The system (21) is consistent if and only if $[A] {{\underline{y}}^{P} + {\bar{y}}^{P}, {\underline{y}}^{N} + {\bar{y}}^{N}} = {{\underline{z}}^{P} + {\bar{z}}^{P}, {\underline{z}}^{N} + {\bar{z}}^{N}}$ and $[A^{*}] {{\underline{y}}^{P} - {\bar{y}}^{P}, {\underline{y}}^{N} - {\bar{y}}^{N}} = {{\underline{z}}^{P} - {\bar{z}}^{P}, {\underline{z}}^{N} - {\bar{z}}^{N}}$ are consistent, where $[A^{*}] = (a_{uv}^{*}) \geq 0 \in [A]$ .

Proof. As a theorem 2.7, the proof is straight-forward.□

3 Numerical examples

Example 3.1. Consider 2 × 2 fully bipolar fuzzy system of linear equations with (Definition 2.6) l = 1 as

$\begin{matrix} 〈 [3 r + 2, 8 - 3 r], [4 - 3 s, 10 + 3 s] 〉 y_{1} + 〈 [3 r + 1, 7 - 3 r], \\ 〈 [2 - 3 s, 8 + 3 s] 〉 y_{2} = 〈 [3 r + 3, 9 - 3 r], [2 - 2 s, 2 s + 6] 〉, \\ 〈 [2 r + 2, 6 - 2 r], [3 - 2 s, 8 + 3 s] 〉 y_{1} + 〈 [4 r + 4, 12 - 4 r], \\ 〈 [2 - 4 s, 10 + 4 s] 〉 y_{2} = 〈 [2 r + 3, 7 - 2 r], [4 - 3 s, 3 s + 10] 〉 . \end{matrix}$

The above linear system has now been transformed into the following system using the procedure discussed in Theorem 2.3

$\begin{matrix} 〈 [3 r + 2 + 8 - 3 r], [4 - 3 s + 10 + 3 s] 〉 y_{1} + 〈 [3 r + 1 + 7 - 3 r], \\ 〈 [2 - 3 s + 8 + 3 s] 〉 y_{2} = 〈 [3 r + 3 + 9 - 3 r], [2 - 2 s + 2 s + 6] 〉, \\ 〈 [2 r + 2 + 6 - 2 r], [3 - 2 s + 8 + 3 s] 〉 y_{1} + 〈 [4 r + 4 + 12 - 4 r], \\ 〈 [2 - 4 s + 10 + 4 s] 〉 y_{2} = 〈 [2 r + 3 + 7 - 2 r], [4 - 3 s + 3 s + 10] 〉 . \end{matrix}$

The above system is the same as the following system $\begin{matrix} 10 y_{1}^{P} + 8 y_{2}^{P} = 12, 8 y_{1}^{P} + 16 y_{2}^{P} = 10, \end{matrix}$ $\begin{matrix} 14 y_{1}^{N} + 10 y_{2}^{N} = 8, (11 + s) y_{1}^{N} + 12 y_{2}^{N} = 14 . \end{matrix}$ By solving the above system, we have $y_{1}^{P} = \frac{7}{6}, y_{2}^{P} = \frac{1}{24}, y_{1}^{N} = \frac{- 11}{17}$ and $y_{2}^{N} = \frac{- 69}{17}$ . This is also the solution to the main system. Likewise, by using Theorem 2.4, we can also find the same solution.

Example 3.2. Consider 2 × 2 BFSLEs with (Definition 2.6) l = 2 $\begin{matrix} A_{11} y_{1} + A_{12} y_{2} & = 〈 [5 + 2 r + r^{2}, 12 - 4 r + r^{2}], \\ [9 - 2 s + s^{2}, 15 + 2 s + s^{2}] 〉, \\ A_{21} y_{1} + A_{22} y_{2} & = 〈 [4 + r + r^{2}, 9 - 2 r + r^{2}], \\ [7 - 3 s + s^{2}, 14 + 4 s + s^{2}] 〉 . \\ A_{11} = a_{11} & = 〈 [3 + 2 r + r^{2}, 10 - 3 r + r^{2}], \\ [4 - 2 s + s^{2}, 10 + 5 s + s^{2}] 〉, \\ A_{12} = a_{12} & = 〈 [2 + r + r^{2}, 8 - 2 r + r^{2}], \\ [7 - 2 s + s^{2}, 13 + 4 s + 2 s^{2}] 〉, \\ A_{21} = a_{21} & = 〈 [7 + 6 r + r^{2}, 13 - 5 r + r^{2}], \\ [8 - 5 s + s^{2}, 14 + 7 s + s^{2}] 〉, \\ A_{22} = a_{22} & = 〈 [1 + 3 r + r^{2}, 8 - 2 r + r^{2}], \\ [5 - 2 s + s^{2}, 11 + 6 s + s^{2}] 〉 . \end{matrix}$ Using the procedure of Theorem 2.3 and 2.4. We have $\begin{matrix} y_{1}^{P} = \frac{- 10}{17}, y_{2}^{P} = \frac{39}{17}, y_{1}^{N} = \frac{7}{20}, y_{2}^{N} = \frac{143}{140} . \end{matrix}$

Example 3.3. Consider a crisp coefficient 2 × 2 BFSLEs with l = 1 and RHS column vector is BFNs as $\begin{matrix} 2 y_{1} + y_{2} & = 〈 [3 + 3 r, 9 - 3 r], [- 2 s + 2, 2 s + 6] 〉, \\ y_{1} + 3 y_{2} & = 〈 [5 + 4 r, 13 - 4 r], [- 5 s + 2, 4 s + 11] 〉 . \end{matrix}$ Now by solving, the corresponding vector elements are $\begin{matrix} y_{1} & = [{\underline{y}}_{1}^{P}, {\bar{y}}_{1}^{P}, {\underline{y}}_{1}^{N}, {\bar{y}}_{1}^{N}] \\ = [\frac{4}{5} + r, \frac{14}{5} - r, - \frac{1 s}{5} + \frac{4}{5}, \frac{7}{5} + \frac{2 s}{5}], \\ y_{2} & = [{\underline{y}}_{2}^{P}, {\bar{y}}_{2}^{P}, {\underline{y}}_{2}^{N}, {\bar{y}}_{2}^{N}] \\ = [\frac{7}{5} + r, \frac{17}{5} - r, - \frac{8 s}{5} + \frac{2}{5}, \frac{16}{5} + \frac{6 s}{5}] . \end{matrix}$ According to the Theorem 2.5 and 2.6 $\begin{matrix} [\begin{matrix} {\underline{y}}_{1}^{P} - {\bar{y}}_{1}^{P}, {\underline{y}}_{1}^{N} - {\bar{y}}_{1}^{N} \\ {\underline{y}}_{2}^{P} - {\bar{y}}_{2}^{P}, {\underline{y}}_{2}^{N} - {\bar{y}}_{2}^{N} \end{matrix}] & = [\begin{matrix} - 2 + 2 r, - \frac{3 s}{5} - \frac{3}{5} \\ - 2 + 2 r, - \frac{14 s}{5} - \frac{14}{5} \end{matrix}] \\ and \\ [\begin{matrix} {\underline{y}}_{1}^{P} + {\bar{y}}_{1}^{P}, {\underline{y}}_{1}^{N} + {\bar{y}}_{1}^{N} \\ {\underline{y}}_{2}^{P} + {\bar{y}}_{2}^{P}, {\underline{y}}_{2}^{N} + {\bar{y}}_{2}^{N} \end{matrix}] & = [\begin{matrix} \frac{18}{5}, \frac{1 s}{5} + \frac{11}{5} \\ \frac{24}{5}, - \frac{2 s}{5} + \frac{18}{5} \end{matrix}], \end{matrix}$ are the solution of the equation (18) and (20), respectively. The graphical representation of the exact solution is shown in Figures 1, 2, 3 and 4

Fig.1

Exact solution of positive membership function of 2 × 2 BFSLEs.

Fig.2

Exact solution of positive membership function of 2 × 2 BFSLEs.

Fig.3

Exact solution of negative membership function of 2 × 2 BFSLEs.

Fig.4

Exact solution of negative membership function of 2 × 2 BFSLEs.

4 Conclusions

The simple procedures are used to solve FSLEs with fuzzy’s coefficient and real crisp variables using polynomial PFNs. These simple procedures are now used to solve BFSLEs with bipolar fuzzy’s coefficient and real crisp variables using PPBFNs. The system may have an exact solution if all coefficients are PPBFNs otherwise, we can find the approximate solution of the system by setting l-degree polynomial bipolar fuzzy number coefficients. The selection of l depends on the shape and order of derivation of the left and right parametric bipolar fuzzy number functions. In the proposed method, solution of the linear system in bipolar fuzzy environment can be found by adding or subtracting of polynomial parametric bipolar fuzzy numbers to the left and right side of the original system. In this way, the crisp solution of the original system can be found by using (r, s)-cut expansion. But in general, the proposed embedding technique which studied by Friedman in [26], if the original system is n × n then the extend system must be 2n × 2n in fuzzy environment. By following [3], this technique is extended to 4n × 4n in bipolar fuzzy environment. Thus the current procedure has less effort to solve the system and are therefore computationally efficient. For future research, we are working to extend our research to m-polar metric spaces and some fixed point theorems.

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this researcharticle.

Footnotes

Acknowledgments

The authors are very thankful to the Editor and referees for their valuable comments and suggestions for improving the paper.

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