Linear system of equations in m -polar fuzzy environment

Abstract

In this paper, we present the notion of m-polar fuzzy number(m-PFN) along with some properties. Further, we describe the m-polar fuzzy linear system of equations (m-PFLSEs) along with weak and strong solutions. Moreover, we characterize a new technique to find the solution of fully m-PFLSE using one-cut extension. By applying this technique, we calculate the minimal and maximal symmetric solutions (SSs) of the fully m-PFLSEs which based on a controllable solution set (CSS) and a tolerable solution set (TSS), respectively. We consider an example to find the solutions of fully m-PFLSEs. In the end, we prove some elementary results on the base of our proposed method.

Keywords

minimal and maximal symmetric solution

1 Introduction

The fuzzy set (FS) was introduced by Zadeh [26] which led to fuzzy number (FN) and its utilization in estimated logic and fuzzy control issues. The concept of trapezoidal FNs was developed by Moghadam et al. [20]. Zhang [25] studied the notion of bipolar fuzzy sets (BFSs) by describing a truthness degree and falseness degree, which is an extension of fuzzy sets [26]. This notion has been used in various domains, including comparatively for appliance in argumentation, preference modeling, cooperative games, multi-criteria decision inquiry and knowledge representation. The truthness degree and falseness degree of BFSs is lying in the interval [-1, 1]. In spite of the fact that BFSs and intuitionistic fuzzy sets seem to be like one another, these are basically different notions as identified by Lee [19]. Akram [3] proposed the concept of bipolar fuzzy graphs. Chen et al. [17] proposed m-polar fuzzy sets (m-PFSs) as the extension of BFSs. He exhibited that BFSs and 2-PFSs are cryptomorphic scientific concepts and that we can acquire briefly one from the relating one. The concept back of this is “multipolar information" (not simply bipolar data which compare to two-valued data) exists on the grounds that the information for this present reality issues is some of the time taken from n operators (n ≥ 2). Furthermore, Akram et al. [4 , 7] proposed different notions on m-PFSs. Wei et al. [22 –24] discussed the decision analysis criteria in different environments. Akram et al. [8 –10] proposed the new concept in group-decision making and solve the linear system with iterative technique in bipolar fuzzy environment.

The concept of linear systems having a vital role in several fields including mathematical, physical sciences and engineering such as fluid flow, traffic flow, circuit analysis and so on. In by far most of the issue, generally, work with rough information. To beat this mistake, we may represent the given information as a more general m-PFN as opposed to a crisp, FN and bipolar FN. In a couple of utilizations, at any rate, several parameters of the system are communicated by fuzzy instead of the classical numbers. Friedman et al. [18] introduced a general model to get ride of a fuzzy linear system (FLS) by using the inserted methodology. For solving a FLS Allahviranloo et al. [11 , 13–15] introduced some important numerical methods. Moreover, Abbasbandy et al. [1, 2] introduced LU-decomposition strategy and Steepest descent technique to solve FLS. Moreover, Allahviranloo and Salahshour [12] built up a novel strategy to explain a FLS in light of the 1-cut expansion. Allahviranloo et al. [16] introduced another plan to illuminate the fully FLS. They built up another strategy to solve the fully FLS. Recently, Akram et al. [6] proposed the concept of bipolar fuzzy linear system of equations. In this article, we present certain notions, including m-polar fuzzy number (m-PFN), the distance between two m-PFNs and some basic operations on m-PFNs. We illustrate these concepts with examples. We develop an m-PFLSEs and obtain weak and strong solutions of the system. Moreover, we investigate another way to deal with discover the solutions of fully m-PFLSEs which depends on 1-cut extension. Further, by applying this technique, we conclude the minimal and maximal SSs of the fully m-PFLSEs which depends on a CSS and a TSS, respectively. We also describe some examples of fully m-PFLSEs.

For further terminologies and applications, readers are referred to [20 , 27–32].

2 Parametric m-polar fuzzy numbers

The concept of m-polar fuzzy set was introduced in [17]. First we develop some basic definitions and concepts of m-polar fuzzy numbers that are necessary for this work.

Definition 2.1. An m-PFN in parametric form is an m-tuple $≺ [\underline{k^{(1)}} (δ), \bar{k^{(1)}} (δ)], [\underline{k^{(2)}} (δ), \bar{k^{(2)}} (δ)], . . ., [\underline{k^{(m)}} (δ), \bar{k^{(m)}} (δ)] ≻ = ≺ [\underline{k^{(i)}} (δ), \bar{k^{(i)}} (δ)] ≻$ of functions $\underline{k^{(i)}} (δ), \bar{k^{(i)}} (δ), 0 \leq δ \leq 1, 1 \leq i \leq m,$ which satisfy the following properties:

$\underline{k^{(i)}} (δ)$ is a bounded non-decreasing right continuous function at the point 0 and left continuous over the interval (0,1],

$\bar{k^{(i)}} (δ)$ is a bounded non-increasing right continuous function at the point 0 and left continuous over the interval (0,1],

$\underline{k^{(i)}} (δ) \leq \bar{k^{(i)}} (δ) .$

Throughout the paper i = 1, 2, 3, ⋯ , m .

Definition 2.2. For arbitrary $k = ≺ [{\underline{k}}^{(i)} (δ), {\bar{k}}^{(i)}$ ( δ )]≻, $l = ≺ [{\underline{l}}^{(i)} (δ), {\bar{l}}^{(i)} (δ)] ≻$ and a > 0, we define (k + l), (k . l) and scalar multiplication by a as follows:

$(\underline{k^{(i)} + l^{(i)}}) (δ) = {\underline{k}}^{(i)} (δ) + {\underline{l}}^{(i)} (δ)$ , $(\bar{k^{(i)} + l^{(i)}})$ $(δ) = {\bar{k}}^{(i)} (δ) + {\bar{l}}^{(i)} (δ)$ ,

$(\underline{k^{(i)} l^{(i)}}) (δ) = min {{\underline{k}}^{(i)} (δ) {\underline{l}}^{(i)} (δ), {\underline{k}}^{(i)} (δ) {\bar{l}}^{(i)}$ $(δ), {\bar{k}}^{(i)} (δ) {\underline{l}}^{(i)} (δ), {\bar{k}}^{(i)} (δ) {\bar{l}}^{(i)} (δ)}$ ,

$(\bar{k^{(i)} l^{(i)}}) (δ) = max {{\underline{k}}^{(i)} (δ) {\underline{l}}^{(i)} (δ), {\underline{k}}^{(i)} (δ) {\bar{l}}^{(i)}$ $(δ), {\bar{k}}^{(i)} (δ) {\underline{l}}^{(i)} (δ), {\bar{k}}^{(i)} (δ) {\bar{l}}^{(i)} (δ)}$ ,

$({\underline{a k}}^{(i)}) (δ) = a ({\underline{k}}^{(i)}) (δ)$ , $({\bar{a k}}^{(i)}) (δ) = a ({\bar{k}}^{(i)})$ ( δ ) .

The family of all m-PFNs is denoted by Ψ.

We currently characterize the metric D on Ψ.

Definition 2.3. Let m-PFNs $k = ≺ [{\underline{k}}^{(i)} (δ), {\bar{k}}^{(i)}$ ( δ )]≻ and $l = ≺ [{\underline{l}}^{(i)} (δ), {\bar{l}}^{(i)} (δ)] ≻$ , the quantity $\begin{matrix} D^{(i)} (k, l) & = & sup_{0 \leq δ \leq 1} {max [| {\underline{k}}^{(i)} (δ) \\ - {\underline{l}}^{(i)} (δ) |, | {\bar{k}}^{(i)} (δ) - {\bar{l}}^{(i)} (δ) |]} \end{matrix}$ (1) is called the distances between k and l, where 1 ≤ i ≤ m.

Definition 2.4. For arbitrary m-PFN $k = ≺ [{\underline{k}}^{(i)} (δ), {\bar{k}}^{(i)} (δ)] ≻$ , the numbers $k_{0}^{(i)} = \frac{1}{2} ({\underline{k}}^{(i)} (1) + {\bar{k}}^{(i)} (1))$ (2) are called location index numbers (LINs) of k, two non-increasing left continuous functions $k_{*}^{(i)} (δ) = k_{0}^{(i)} - {\underline{k}}^{(i)} (δ), k^{(i) *} (δ) = {\bar{k}}^{(i)} (δ) - k_{0}^{(i)}$ (3) are called the left m-polar fuzzy index functions and the right m-polar fuzzy index functions (m-PFIFs), respectively.

According to this definition, every m-PFN can be described as

$k = ≺ [{\underline{k}}^{(i)} (δ), {\bar{k}}^{(i)} (δ)] ≻ : = ≺ [k_{0}^{(i)}, k_{*}^{(i)} (δ), k^{(i) *}$ ( δ )]≻.

Example 2.5. Let $k = ≺ [{\underline{k}}^{(1)} (δ), {\bar{k}}^{(1)} (δ)],$ $[{\underline{k}}^{(2)} (δ), {\bar{k}}^{(2)} (δ)], [{\underline{k}}^{(3)} (δ), {\bar{k}}^{(3)} (δ)] ≻ = ≺ [- 4 + 2 δ, 7 - 9 δ],$ [-9+ 3 δ , - 2 -8 δ ] , [-3 + δ , 4 - 6 δ ] ≻ be a 3-polar fuzzy number, where $\begin{matrix} {\underline{k}}^{(1)} (δ) & = - 4 + 2 δ, {\bar{k}}^{(1)} (δ) = 7 - 9 δ, \\ {\underline{k}}^{(2)} (δ) & = - 9 + 3 δ, {\bar{k}}^{(2)} (δ) = - 2 - 8 δ, \\ {\underline{k}}^{(3)} (δ) & = - 3 + δ, {\bar{k}}^{(3)} (δ) = 4 - 6 δ . \end{matrix}$ Then the location index number $k_{0}^{(i)}$ are $\begin{matrix} k_{0}^{(1)} = \frac{1}{2} ({\underline{k}}^{(1)} (1) + {\bar{k}}^{(1)} (1)) = \frac{1}{2} (- 2 - 2) = - 2, \end{matrix}$ $\begin{matrix} k_{0}^{(2)} = \frac{1}{2} ({\underline{k}}^{(2)} (1) + {\bar{k}}^{(2)} (1)) = \frac{1}{2} (- 6 - 6) = - 6, \end{matrix}$ $\begin{matrix} k_{0}^{(3)} = \frac{1}{2} ({\underline{k}}^{(3)} (1) + {\bar{k}}^{(3)} (1)) = \frac{1}{2} (- 2 - 2) = - 2, \end{matrix}$ and 3-polar fuzzy index functions are $\begin{matrix} k_{*}^{(1)} (δ) & = k_{0}^{(1)} - {\underline{k}}^{(1)} (δ) = 2 (1 - δ), \\ k^{(1) *} (δ) & = {\bar{k}}^{(1)} (δ) - k_{0}^{(1)} = 9 (1 - δ), \\ k_{*}^{(2)} (δ) & = k_{0}^{(2)} - {\underline{k}}^{(2)} (δ) = 3 (1 - δ), \\ k^{(2) *} (δ) & = {\bar{k}}^{(2)} (δ) - k_{0}^{(2)} = 8 (1 - δ), \\ k_{*}^{(3)} (δ) & = k_{0}^{(3)} - {\underline{k}}^{(3)} (δ) = (1 - δ), \\ k^{(3) *} (δ) & = {\bar{k}}^{(3)} (δ) - k_{0}^{(3)} = 6 (1 - δ) . \end{matrix}$

Theorem 2.6. For arbitrary $k = ≺ [k_{0}^{(i)}, k_{*}^{(i)} (δ), k^{(i) *} (δ)] ≻$ and $l = ≺ [l_{0}^{(i)}, l_{*}^{(i)} (δ), l^{(i) *} (δ)] ≻$ , the distance metric is defined as $\begin{matrix} D_{*}^{(i)} (k, l) = & | k_{0}^{(i)} - l_{0}^{(i)} | + sup_{0 \leq δ \leq 1} {| k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) |, \\ | k^{(i) *} (δ) - l^{(i) *} (δ) |} . \end{matrix}$ Then $D_{*}^{(i)}$ are comparable to the metric D and $D^{(i)} (k, l) \leq D_{*}^{(i)} (k, l) \leq 3 D^{(i)} (k, l) .$ (4)

Proof. For arbitrary k, l ∈ Ψ, equation (3) gives us $\begin{matrix} | {\underline{k}}^{(i)} (δ) - {\underline{l}}^{(i)} (δ) | & = | k_{0}^{(i)} - l_{0}^{(i)} - (k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ)) | \\ \leq | k_{0}^{(i)} - l_{0}^{(i)} | + | k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) | \end{matrix}$ $\begin{matrix} | {\bar{k}}^{(i)} (δ) - {\bar{l}}^{(i)} (δ) | & = | k_{0}^{(i)} - l_{0}^{(i)} + (k^{(i) *} (δ) - l^{(i) *} (δ)) | \\ \leq | k_{0}^{(i)} - l_{0}^{(i)} | + | k^{(i) *} (δ) - l^{(i) *} (δ) | . \end{matrix}$

Therefore,

$\begin{matrix} D^{(i)} (k, & l) = sup_{0 \leq δ \leq 1} max {| {\underline{k}}^{(i)} (δ) - {\underline{l}}^{(i)} (δ) |, | {\bar{k}}^{(i)} (δ) - {\bar{l}}^{(i)} (δ) |} \\ \leq sup_{0 \leq δ \leq 1} max {| k_{0}^{(i)} - l_{0}^{(i)} | + | k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) |, \\ | k_{0}^{(i)} - l_{0}^{(i)} | + | k^{(i) *} (δ) - l^{(i) *} (δ) |} \\ = sup_{0 \leq δ \leq 1} {| k_{0}^{(i)} - l_{0}^{(i)} | + max {| k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) |, \\ | k^{(i) *} (δ) - l^{(i) *} (δ) |}} \\ = | k_{0}^{(i)} - l_{0}^{(i)} | + \leq sup_{0 \leq δ \leq 1} max {| k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) |, \\ | k^{(i) *} (δ) - l^{(i) *} (δ) |} \\ = D_{*}^{(i)} (k, l) . \end{matrix}$

Moreover, $\begin{matrix} k_{*}^{(i)} (δ) = k_{0}^{(i)} - {\underline{k}}^{(i)} (δ), k^{(i) *} (δ) = {\bar{k}}^{(i)} (δ) - k_{0}^{(i)}, \end{matrix}$ We emphasize that $\begin{matrix} | k_{*}^{(i)} (δ) - l_{*}^{(i)} (δ) | \leq | k_{0}^{(i)} - l_{0}^{(i)} | + | {\underline{k}}^{(i)} (δ) - {\underline{l}}^{(i)} (δ) |, \\ | k^{(i) *} (δ) - l^{(i) *} (δ) | \leq | k_{0}^{(i)} - l_{0}^{(i)} | + | {\bar{k}}^{(i)} (δ) - {\bar{l}}^{(i)} (δ) |, \end{matrix}$ so that $\begin{matrix} D_{*}^{(i)} (k, & l) = | k_{0}^{(i)} - l_{0}^{(i)} | + sup_{0 \leq δ \leq 1} max {| k_{*}^{(i)} (δ) - \\ l_{*}^{(i)} (δ) |, | k^{(i) *} (δ) - l^{(i) *} (δ) |} \\ \leq | k_{0}^{(i)} - l_{0}^{(i)} | + sup_{0 \leq δ \leq 1} max {| k_{0}^{(i)} - l_{0}^{(i)} | + \\ | {\underline{k}}^{(i)} (δ) - {\underline{l}}^{(i)} (δ) |, | k_{0}^{(i)} - l_{0}^{(i)} | + \\ | {\bar{l}}^{(i)} (δ) - {\bar{k}}^{(i)} (δ) |} \\ \leq 2 | k_{0}^{(i)} - l_{0}^{(i)} | + sup_{0 \leq δ \leq 1} max {| {\underline{k}}^{(i)} (δ) - \\ {\underline{l}}^{(i)} (δ) |, | {\bar{k}}^{(i)} (δ) - {\bar{l}}^{(i)} (δ) |} \\ \leq 2 | k_{0}^{(i)} - l_{0}^{(i)} | + D^{(i)} (k, l) . \end{matrix}$

Notice that

$\begin{matrix} | k_{0}^{(i)} - l_{0}^{(i)} | & = | \frac{1}{2} [{\underline{k}}^{(i)} (1) + {\bar{k}}^{(i)} (1)] - \frac{1}{2} [{\underline{l}}^{(i)} (1) + {\bar{l}}^{(i)} (1)] | \\ \leq \frac{1}{2} | {\underline{k}}^{(i)} (1) - {\underline{l}}^{(i)} (1) | + \frac{1}{2} | {\bar{k}}^{(i)} (1) - {\bar{l}}^{(i)} (1) | \\ \leq D^{(i)} (k, l) . \end{matrix}$

Thus, we have $D^{(i)} (k, l) \leq D_{*}^{(i)} (k, l) \leq 3 D^{(i)} (k, l) .$ □

Definition 2.7. For any two m-PFNs $k = ≺ [k_{0}^{(i)},$ $k_{*}^{(i)} (δ), k^{(i) *} (δ)] ≻$ and $l = ≺ [l_{0}^{(i)}, l_{*}^{(i)} (δ), l^{(i) *} (δ)] ≻$ , the basic operations describe as follows $\begin{matrix} k * l = & ≺ [k_{0}^{(i)} * l_{0}^{(i)}, k_{*}^{(i)} (δ) \lor l_{*}^{(i)} (δ), \\ k^{(i) *} (δ) \lor l^{(i) *} (δ)] ≻, \end{matrix}$ where k ∗ l is either k + l, k \ l, k · l and k/l.

The primary thought of the above definition is that one can determine arithmetic, by the operations on both LIN and m-PFIFs. The LIN is taken in the conventional math, while the m-PFIFs are considered to pursue the cross-section rule which is least upper bound in the Lattice S.

Example 2.8. $\begin{matrix} Let k = & ≺ [- 2, 2 (1 - δ), 9 (1 - δ), - 6, 3 (1 - δ), \\ 8 (1 - δ), - 2, (1 - δ), 6 (1 - δ)] ≻ and \end{matrix}$ $\begin{matrix} l = & ≺ [- 7, 4 (1 - δ), 15 (1 - δ), - 5, 2 (1 - δ), \\ 10 (1 - δ), - 4, 3 (1 - δ), 5 (1 - δ)] ≻ . \end{matrix}$ Then

k+ l = ≺ [-9, 4 (1 - δ ) , 15 (1 - δ ) , -11, 3 (1 - δ ) , 10 (1 - δ ) , -6, 3 (1 - δ ) , 6 (1 - δ )] ≻,

k\ l = ≺ [5, 4 (1 - δ ) , 15 (1 - δ ) , -1, 3 (1 - δ ) , 10 (1 - δ ) , 2, 3 (1 - δ ) , 6 (1 - δ )] ≻,

k· l = ≺ [14, 4 (1 - δ ) , 15 (1 - δ ) , 30, 3 (1 - δ ) , 10 (1 - δ ) , 8, 3 (1 - δ ) , 6 (1 - δ )] ≻,

k/l =≺ [0.28, 4 (1 - δ ) , 15 (1 - δ ) , 1.2, 3 (1 - δ ) , 10 (1 - δ ) , 0.5, 3 (1 - δ ) , 6 (1 - δ )] ≻.

Theorem 2.9. For arbitrary three m-PFNs k,l and w, we have

k + l = l + k,

(k + l) + w = k + (l + w),

kl = lk,

(kl) w = k (lw),

There exists 1 ∈ Ω such that k · 1 = k, ∀ k ∈ Ω,

k (l + w) = kl + kw.

Proof. The proof follows from Definition 2.7□

3 m-Polar fuzzy linear system of equations

Definition 3.1. The n × n linear system $\begin{matrix} d_{11} k_{1} + d_{12} k_{2} + \dots + d_{1 n} k_{n} & = w_{1}^{(i)} \\ d_{21} k_{1} + d_{22} k_{2} + \dots + d_{2 n} k_{n} & = w_{2}^{(i)} \\ ⋮ \end{matrix}$ (5) $d_{n 1} k_{1} + d_{n 2} k_{2} + \dots + d_{nn} k_{n} = w_{n}^{(i)}$ where the coefficient matrix D = (d_pq) , 1 ≤ p, q ≤ n is a crisp n × n matrix and $w_{p}^{(i)}, 1 \leq p \leq n, 1 \leq i \leq m$ are known m-PFNs and k_q, 1 ≤ q ≤ n are unknowns which may or may not be m-PFNs, is called an m-PFLS.

Theorem 3.2. An m-PFN vector $(k_{1}^{(i)}, k_{2}^{(i)}, . . ., k_{n}^{(i)})^{T}$ given by $(K_{q}^{(i)})_{δ} = k_{q}^{(i)} = ≺ [{\underline{k}}_{q}^{(i)} (δ), {\bar{k}}_{q}^{(i)} (δ)] ≻, 1 \leq q \leq n, 0 \leq δ \leq 1, 1 \leq i \leq m$ is called a solution of the m-PFLS (5) if $\begin{matrix} \underline{\sum_{1 \leq q \leq n} d_{pq} k_{q}} & = \sum_{1 \leq q \leq n} \underline{d_{pq} k_{q}} = {\underline{w}}_{p}^{(i)} \end{matrix}$ (6) $\bar{\sum_{1 \leq q \leq n} d_{pq} k_{q}} = \sum_{1 \leq q \leq n} \bar{d_{pq} k_{q}} = {\bar{w}}_{p}^{(i)} .$ For a particular p, d_pq > 0, 1 ≤ p ≤ n, we get $\sum_{1 \leq q \leq n} d_{pq} {\underline{k}}_{q} = {\underline{w}}_{p}^{(i)}, \sum_{1 \leq q \leq n} d_{pq} {\bar{k}}_{q} = {\bar{w}}_{p}^{(i)} .$ (7)

4 Solution procedure for n × nm-PFLS

Consider the n × n linear system $\begin{matrix} d_{11} k_{1} + d_{12} k_{2} + \dots + d_{1 n} k_{n} & = w_{1}^{(i)} \\ d_{21} k_{1} + d_{22} k_{2} + \dots + d_{2 n} k_{n} & = w_{2}^{(i)} \\ ⋮ \end{matrix}$ (8) $d_{n 1} k_{1} + d_{n 2} k_{2} + \dots + d_{nn} k_{n} = w_{n}^{(i)}$ where the coefficient matrix D = (d_pq) , 1 ≤ p, q ≤ n is a crisp n × n matrix and $w_{p}^{(i)}, 1 \leq p \leq n$ are known m-PFNs and k_q, 1 ≤ q ≤ n are unknowns which may or may not be m-PFNs.

In order to solve the above m-PFLS we must solve m (2n) × (2n) crisp linear systems where the right hand side columns are the function vector $({\underline{w}}_{1}^{(i)}, {\underline{w}}_{2}^{(i)}, . . ., {\underline{w}}_{n}^{(i)}, {\bar{w}}_{1}^{(i)}, {\bar{w}}_{2}^{(i)}, . . ., {\bar{w}}_{n}^{(i)})^{T} .$

Let us now rearrange the linear system so that the unknowns are ${\underline{k}}_{q}^{(i)}, (- {\bar{k}}_{q}^{(i)}), 1 \leq q \leq n, 1 \leq i \leq m$ and the right hand side columns are $({\underline{w}}_{1}^{(i)}, {\underline{w}}_{2}^{(i)}, . . ., {\underline{w}}_{n}^{(i)}, {\bar{w}}_{1}^{(i)}, {\bar{w}}_{2}^{(i)}, . . ., {\bar{w}}_{n}^{(i)})^{T} .$

We obtain (2n) × (2n) crisp linear system

$\begin{matrix} N_{1, 1} {\underline{k}}_{1}^{(i)} + N_{1, 2} {\underline{k}}_{2}^{(i)} + \dots + N_{1, n} {\underline{k}}_{n}^{(i)} + N_{1, n + 1} (- {\bar{k}}_{1}^{(i)}) \\ + N_{1, n + 2} (- {\bar{k}}_{2}^{(i)}) + \dots + N_{1, 2 n} (- {\bar{k}}_{n}^{(i)}) = {\underline{w}}_{1}^{(i)} \\ ⋮ \\ N_{n, 1} {\underline{k}}_{1}^{(i)} + N_{n, 2} {\underline{k}}_{2}^{(i)} + \dots + N_{n, n} {\underline{k}}_{n}^{(i)} + N_{n, n + 1} (- {\bar{k}}_{1}^{(i)}) \\ + N_{n, n + 2} (- {\bar{k}}_{2}^{(i)}) + \dots + N_{n, 2 n} (- {\bar{k}}_{n}^{(i)}) = {\underline{w}}_{n}^{(i)} \\ N_{n + 1, 1} {\underline{k}}_{1}^{(i)} + N_{n + 1, 2} {\underline{k}}_{2}^{(i)} + \dots + N_{n + 1, n} {\underline{k}}_{n}^{(i)} \\ + N_{n + 1, n + 1} (- {\bar{k}}_{1}^{(i)}) + N_{n + 1, n + 2} (- {\bar{k}}_{2}^{(i)}) \\ + \dots + N_{n + 1, 2 n} (- {\bar{k}}_{n}^{(i)}) = - {\bar{w}}_{1}^{(i)} \\ ⋮ \\ N_{2 n, 1} {\underline{k}}_{1}^{(i)} + N_{2 n, 2} {\underline{k}}_{2}^{(i)} + \dots + N_{2 n, n} {\underline{k}}_{n}^{(i)} + N_{2 n, n + 1} (- {\bar{k}}_{1}^{(i)}) \\ + N_{2 n, n + 2} (- {\bar{k}}_{2}^{(i)}) + \dots + N_{2 n, 2 n} (- {\bar{k}}_{n}^{(i)}) = - {\bar{w}}_{n}^{(i)}, \end{matrix}$

where N_p,q are determined as follows:

$\begin{matrix} d_{pq} \geq 0 & \Rightarrow N_{p, q} = d_{pq}, N_{p, q + n} = 0, N_{p + n, q} = 0, \\ N_{p + n, q + n} = d_{pq}, \\ d_{pq} < 0 & \Rightarrow N_{p, q} = 0, N_{p, q + n} = - d_{pq}, N_{p + n, q} = - d_{pq}, \\ N_{p + n, q + n} = 0, 1 \leq p \leq n, 1 \leq q \leq n, 1 \leq i \leq m . \end{matrix}$

Using matrix notation, we have

$N k^{(i)} = w^{(i)}$ (9) where N = (N_p,q) , 1 ≤ p ≤ 2n, 1 ≤ q ≤ 2n, k⁽ⁱ⁾ $= ({\underline{k}}_{1}^{(i)}, {\underline{k}}_{2}^{(i)}, . . ., {\underline{k}}_{n}^{(i)}, (- {\bar{k}}_{1}^{(i)}), (- {\bar{k}}_{2}^{(i)}), . . ., (- {\bar{k}}_{n}^{(i)}))^{T}$ and $w^{(i)} = ({\underline{w}}_{1}^{(i)}, {\underline{w}}_{2}^{(i)}, . . ., {\underline{w}}_{n}^{(i)}, (- {\bar{w}}_{1}^{(i)}), (- {\bar{w}}_{2}^{(i)}),$ $. . ., (- {\bar{w}}_{n}^{(i)}))^{T} .$

Hence, by solving the equation (9) as k⁽ⁱ⁾ = N^-1w⁽ⁱ⁾, we get solutions of m-PFLS, i.e., $k_{q} = ≺ [{\underline{k}}_{q}^{(i)}, {\bar{k}}_{q}^{(i)}] ≻, 1 \leq q \leq n, 1 \leq i \leq m .$

Definition 4.1. If $S = ≺ ({\underline{k}}_{1}^{(i)}, {\underline{k}}_{2}^{(i)}, . . ., {\underline{k}}_{n}^{(i)}, {\bar{k}}_{1}^{(i)},$ ${\bar{k}}_{2}^{(i)}, . . ., {\bar{k}}_{n}^{(i)})^{T} ≻$ is a solution set of equations (8) and for each q ∈ [1, n] the inequality ${\underline{k}}_{q}^{(i)} \leq {\bar{k}}_{q}^{(i)}$ hold, then the solution k_q is called a strong solution of the system of equations (8).

Definition 4.2. If $S = ≺ ({\underline{k}}_{1}^{(i)}, {\underline{k}}_{2}^{(i)}, . . ., {\underline{k}}_{n}^{(i)}, {\bar{k}}_{1}^{(i)},$ ${\bar{k}}_{2}^{(i)}, . . ., {\bar{k}}_{n}^{(i)})^{T} ≻$ is a solution set of equations (8) and for some q ∈ [1, n] the inequality ${\underline{k}}_{q}^{(i)} > {\bar{k}}_{q}^{(i)}$ hold, then the solution k_q is called a weak solution of the system of equations (8).

Example 4.3. Consider 2 × 2 3-PFLS $\begin{matrix} 3 k_{1} + 5 k_{2} = & ≺ [2 + 3 δ, 8 - 3 δ], [5 + 4 δ, 13 - 4 δ], \\ [8 + 2 δ, 12 - 2 δ] ≻ \\ 4 k_{1} + 6 k_{2} = & ≺ [3 + 6 δ, 15 - 6 δ], [6 + 5 δ, 16 - 5 δ], \\ [10 + 4 δ, 18 - 4 δ] ≻ . \end{matrix}$ Then the first 4 × 4 system is $\begin{matrix} 3 \cdot {\underline{k}}_{1}^{(1)} + 5 \cdot {\underline{k}}_{2}^{(1)} + 0 \cdot (- {\bar{k}}_{1}^{(1)}) + 0 \cdot (- {\bar{k}}_{2}^{(1)}) = 2 + 3 δ, \\ 4 \cdot {\underline{k}}_{1}^{(1)} + 6 \cdot {\underline{k}}_{2}^{(1)} + 0 \cdot (- {\bar{k}}_{1}^{(1)}) + 0 \cdot (- {\bar{k}}_{2}^{(1)}) = 3 + 6 δ, \\ 0 \cdot {\underline{k}}_{1}^{(1)} + 0 \cdot {\underline{k}}_{2}^{(1)} + 3 \cdot (- {\bar{k}}_{1}^{(1)}) + 5 \cdot (- {\bar{k}}_{2}^{(1)}) \\ = - (8 - 3 δ), \\ 0 \cdot {\underline{k}}_{1}^{(1)} + 0 \cdot {\underline{k}}_{2}^{(1)} + 4 \cdot (- {\bar{k}}_{1}^{(1)}) + 6 \cdot (- {\bar{k}}_{2}^{(1)}) \\ = - (15 - 6 δ), \end{matrix}$ or $(\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}) (\begin{matrix} {\underline{k}}_{1}^{(1)} \\ {\underline{k}}_{2}^{(1)} \\ - {\bar{k}}_{1}^{(1)} \\ - {\bar{k}}_{2}^{(1)} \end{matrix}) = (\begin{matrix} 2 + 3 δ \\ 3 + 6 δ \\ 3 δ - 8 \\ 6 δ - 15 \end{matrix}),$ here $N = (\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}),$ $k^{(1)} = (\begin{matrix} {\underline{k}}_{1}^{(1)} \\ {\underline{k}}_{2}^{(1)} \\ - {\bar{k}}_{1}^{(1)} \\ - {\bar{k}}_{2}^{(1)} \end{matrix}), w^{(1)} = (\begin{matrix} 2 + 3 δ \\ 3 + 6 δ \\ 3 δ - 8 \\ 6 δ - 15 \end{matrix}) .$

Similarly, the second 4 × 4 system is

$\begin{matrix} 3 \cdot {\underline{k}}_{1}^{(2)} + 5 \cdot {\underline{k}}_{2}^{(2)} + 0 \cdot (- {\bar{k}}_{1}^{(2)}) + 0 \cdot (- {\bar{k}}_{2}^{(2)}) & = 5 + 4 δ, \\ 4 \cdot {\underline{k}}_{1}^{(2)} + 6 \cdot {\underline{k}}_{2}^{(2)} + 0 \cdot (- {\bar{k}}_{1}^{(2)}) + 0 \cdot (- {\bar{k}}_{2}^{(2)}) & = 6 + 5 δ, \\ 0 \cdot {\underline{k}}_{1}^{(2)} + 0 \cdot {\underline{k}}_{2}^{(2)} + 3 \cdot (- {\bar{k}}_{1}^{(2)}) + 5 \cdot (- {\bar{k}}_{2}^{(2)}) & = - (13 - 4 δ), \\ 0 \cdot {\underline{k}}_{1}^{(2)} + 0 \cdot {\underline{k}}_{2}^{(2)} + 4 \cdot (- {\bar{k}}_{1}^{(2)}) + 6 \cdot (- {\bar{k}}_{2}^{(2)}) & = - (16 - 5 δ), \end{matrix}$

or $(\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}) (\begin{matrix} {\underline{k}}_{1}^{(2)} \\ {\underline{k}}_{2}^{(2)} \\ - {\bar{k}}_{1}^{(2)} \\ - {\bar{k}}_{2}^{(2)} \end{matrix}) = (\begin{matrix} 5 + 4 δ \\ 6 + 5 δ \\ 4 δ - 13 \\ 5 δ - 16 \end{matrix}),$

here $N = (\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}),$ $k^{(2)} = (\begin{matrix} {\underline{k}}_{1}^{(2)} \\ {\underline{k}}_{2}^{(2)} \\ - {\bar{k}}_{1}^{(2)} \\ - {\bar{k}}_{2}^{(2)} \end{matrix}), w^{(2)} = (\begin{matrix} 5 + 4 δ \\ 6 + 5 δ \\ 4 δ - 13 \\ 5 δ - 16 \end{matrix}) .$

Similarly, the third 4 × 4 system is

$\begin{matrix} 3 \cdot {\underline{k}}_{1}^{(3)} + 5 \cdot {\underline{k}}_{2}^{(3)} + 0 \cdot (- {\bar{k}}_{1}^{(3)}) + 0 \cdot (- {\bar{k}}_{2}^{(3)}) & = 8 + 2 δ, \\ 4 \cdot {\underline{k}}_{1}^{(3)} + 6 \cdot {\underline{k}}_{2}^{(3)} + 0 \cdot (- {\bar{k}}_{1}^{(3)}) + 0 \cdot (- {\bar{k}}_{2}^{(3)}) & = 10 + 4 δ, \\ 0 \cdot {\underline{k}}_{1}^{(3)} + 0 \cdot {\underline{k}}_{2}^{(3)} + 3 \cdot (- {\bar{k}}_{1}^{(3)}) + 5 \cdot (- {\bar{k}}_{2}^{(3)}) & = - (12 - 2 δ), \\ 0 \cdot {\underline{k}}_{1}^{(3)} + 0 \cdot {\underline{k}}_{2}^{(3)} + 4 \cdot (- {\bar{k}}_{1}^{(3)}) + 6 \cdot (- {\bar{k}}_{2}^{(3)}) & = - (18 - 4 δ), \end{matrix}$

$(\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}) (\begin{matrix} {\underline{k}}_{1}^{(3)} \\ {\underline{k}}_{2}^{(3)} \\ - {\bar{k}}_{1}^{(3)} \\ - {\bar{k}}_{2}^{(3)} \end{matrix}) = (\begin{matrix} 8 + 2 δ \\ 10 + 4 δ \\ 2 δ - 12 \\ 4 δ - 18 \end{matrix}),$

here $N = (\begin{matrix} 3 & 5 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 0 & 0 & 3 & 5 \\ 0 & 0 & 4 & 6 \end{matrix}),$ $k^{(3)} = (\begin{matrix} {\underline{k}}_{1}^{(3)} \\ {\underline{k}}_{2}^{(3)} \\ - {\bar{k}}_{1}^{(3)} \\ - {\bar{k}}_{2}^{(3)} \end{matrix}), w^{(3)} = (\begin{matrix} 8 + 2 δ \\ 10 + 4 δ \\ 2 δ - 12 \\ 4 δ - 18 \end{matrix}) .$

Therefore, we obtain the solution from k⁽¹⁾ = N^-1w⁽¹⁾, k⁽²⁾ = N^-1w⁽²⁾ and k⁽³⁾ = N^-1w⁽³⁾ .

Now,

k⁽¹⁾ = N^-1w⁽¹⁾ ⇒ $(\begin{matrix} {\underline{k}}_{1}^{(1)} \\ {\underline{k}}_{2}^{(1)} \\ - {\bar{k}}_{1}^{(1)} \\ - {\bar{k}}_{2}^{(1)} \end{matrix})$ = $(\begin{matrix} - 3.0000 & 2.5000 & 0 & 0 \\ 2.0000 & - 1.5000 & 0 & 0 \\ 0 & 0 & - 3.0000 & 2.5000 \\ 0 & 0 & 2.0000 & - 1.5000 \end{matrix})$ $(\begin{matrix} 2 + 3 δ \\ 3 + 6 δ \\ 3 δ - 8 \\ 6 δ - 15 \end{matrix}),$

k⁽²⁾ = N^-1w⁽²⁾ ⇒ $(\begin{matrix} {\underline{k}}_{1}^{(2)} \\ {\underline{k}}_{2}^{(2)} \\ - {\bar{k}}_{1}^{(2)} \\ - {\bar{k}}_{2}^{(2)} \end{matrix})$ = $(\begin{matrix} - 3.0000 & 2.5000 & 0 & 0 \\ 2.0000 & - 1.5000 & 0 & 0 \\ 0 & 0 & - 3.0000 & 2.5000 \\ 0 & 0 & 2.0000 & - 1.5000 \end{matrix})$ $(\begin{matrix} 5 + 4 δ \\ 6 + 5 δ \\ 4 δ - 13 \\ 5 δ - 16 \end{matrix}),$

k⁽³⁾ = N^-1w⁽³⁾ ⇒ $(\begin{matrix} {\underline{k}}_{1}^{(3)} \\ {\underline{k}}_{2}^{(3)} \\ - {\bar{k}}_{1}^{(3)} \\ - {\bar{k}}_{2}^{(3)} \end{matrix})$ = $(\begin{matrix} - 3.0000 & 2.5000 & 0 & 0 \\ 2.0000 & - 1.5000 & 0 & 0 \\ 0 & 0 & - 3.0000 & 2.5000 \\ 0 & 0 & 2.0000 & - 1.5000 \end{matrix})$ $(\begin{matrix} 8 + 2 δ \\ 10 + 4 δ \\ 2 δ - 12 \\ 4 δ - 18 \end{matrix}) .$

Thus, ${\underline{k}}_{1}^{(1)} = 1.5000 + 6.0000 δ, {\underline{k}}_{2}^{(1)} = - 0.5000 - 3.0000 δ, {\bar{k}}_{1}^{(1)} = 13.5000 - 6.0000 δ, {\bar{k}}_{2}^{(1)} = - 6.5000 + 3.0000 δ,$ ${\underline{k}}_{1}^{(2)} = 0.0000 + 0.5000 δ, {\underline{k}}_{2}^{(2)} = 1.0000 + 0.5000 δ, {\bar{k}}_{1}^{(2)} = 1.0000 - 0.5000 δ, {\bar{k}}_{2}^{(2)} = 2.0000 - 0.5000 δ,$ and ${\underline{k}}_{1}^{(3)} = 1.0000 + 4.0000 δ, {\underline{k}}_{2}^{(3)} = 1.0000 - 2.0000 δ, {\bar{k}}_{1}^{(3)} = 9.0000 - 4.0000 δ, {\bar{k}}_{2}^{(3)} = - 3.0000 + 2.0000 δ .$

Thus here we see strong solution in Fig. 1 and weak solution in Fig. 2. $\begin{matrix} k_{1} = & ≺ [{\underline{k}}_{1}^{(1)}, {\bar{k}}_{1}^{(1)}], [{\underline{k}}_{1}^{(2)}, {\bar{k}}_{1}^{(2)}], [{\underline{k}}_{1}^{(3)}, {\bar{k}}_{1}^{(3)}] ≻ \\ = & ≺ [1.5000 + 6.0000 δ, 13.5000 - 6.0000 δ], \\ [0.0000 + 0.5000 δ, 1.0000 - 0.5000 δ], \\ [1.0000 + 4.0000 δ, 9.0000 - 4.0000 δ] ≻ . \end{matrix}$ where $(a) = [{\underline{k}}_{1}^{(1)}, {\bar{k}}_{1}^{(1)}]$ , $(b) = [{\underline{k}}_{1}^{(2)}, {\bar{k}}_{1}^{(2)}]$ , $(c) = [{\underline{k}}_{1}^{(3)}, {\bar{k}}_{1}^{(3)}]$ , is a strong solution by Definition 4.1 and $\begin{matrix} k_{2} = & ≺ [{\underline{k}}_{2}^{(1)}, {\bar{k}}_{2}^{(1)}], [{\underline{k}}_{2}^{(2)}, {\bar{k}}_{2}^{(2)}], [{\underline{k}}_{2}^{(3)}, {\bar{k}}_{2}^{(3)}] ≻ \\ = & ≺ [- 0.5000 - 3.0000 δ, - 6.5000 + 3.0000 δ], \\ [1.0000 + 0.5000 δ, 2.0000 - 0.5000 δ], \\ [1.0000 - 2.0000 δ, - 3.0000 + 2.0000 δ] ≻ . \end{matrix}$

where $(a) = [{\underline{k}}_{2}^{(1)}, {\bar{k}}_{2}^{(1)}]$ , $(b) = [{\underline{k}}_{2}^{(2)}, {\bar{k}}_{2}^{(2)}]$ , $(c) = [{\underline{k}}_{2}^{(3)}, {\bar{k}}_{2}^{(3)}]$ , is a weak solution by Definition 4.2.

Fig.1

k₁,

Fig.2

k₂,

Definition 4.4. The CSS, the TSS and the unites solution set (USS) for system of equations (5), respectively as follows

$\begin{matrix} K_{\exists \forall} & = {k^{(i)^{'}} \in I R^{n} : (\forall w^{(i)^{'}} \in w^{(i)}) (\exists N' \in N s . t \\ N' k^{(i)^{'}} = w^{(i)^{'}})} = {k^{(i)^{'}} \in I R^{n} : N k^{(i)^{'}} \supseteq w^{(i)}}, \\ K_{\forall \exists} & = {k^{(i)^{'}} \in I R^{n} : (\forall w^{(i)'} \in w) (\exists N' \in N s . t \\ N' k^{(i)^{'}} = w^{(i)^{'}})} = {k^{(i)^{'}} \in I R^{n} : N k^{(i)^{'}} \subseteq w^{(i)}}, \\ K_{\exists \exists} & = {k^{(i)^{'}} \in I R^{n} : (\exists w^{(i)' \in w) (\exists N' \in N s . t} \\ N' k^{(i)^{'}} = w^{(i)^{'}})} = {k^{(i)^{'}} \in I R^{n} : N k^{(i)^{'}} \cap w^{(i)^{'}} \neq φ} . \end{matrix}$

Definition 4.5. An m-PFN vector (k₁, k₂, ⋯ , k_n) ^T given by $\begin{matrix} (K_{q})_{(δ)} = & k_{q} = ≺ [{\underline{k}}_{q}^{(i)} (δ), {\bar{k}}_{q}^{(i)} (δ)] ≻, 1 \leq q \leq n; \\ 0 \leq δ \leq 1, \end{matrix}$ is said to be minimal SS of m-polar fuzzy linear system of equations (5) which is put in CSS if for a subjective SS (t₁, t₂, ⋯ , t_n) ^T which put in CSS, i,e., T (1) = K (1), we have (t₁, t₂, ⋯ , t_n) ^T ⊇ (k₁, k₂, ⋯ , k_n) ^T, that is, (t_q ⊇ k_q) , i . e . , (φ_{t
_q} ≥ φ_{k
_q}) ∀ q = 1, 2, 3, ⋯ , n, where φ_{t
_q} and φ_{k
_q} are the symmetric spreads of t_q and k_q, respectively.

Definition 4.6. An m-PFN vector (k₁, k₂, ⋯ , k_n) ^T given by $\begin{matrix} (K_{q})_{(δ)} = & k_{q} = ≺ [{\underline{k}}_{q}^{(i)} (δ), {\bar{k}}_{q}^{(i)} (δ)] ≻, 1 \leq q \leq n; \\ 0 \leq δ \leq 1, \end{matrix}$ is said to be maximal SS of m-polar fuzzy linear system of equations (5) which is put in TSS if for a subjective SS (u₁, u₂, ⋯ , u_n) ^T which put in TSS, i,e., U (1) = K (1), we have (u₁, u₂, ⋯ , u_n) ^T ⊇ (k₁, k₂, ⋯ , k_n) ^T, that is, (u_q ⊇ k_q) , i . e . , (φ_{u
_q} ≥ φ_{k
_q}) ∀ q = 1, 2, 3, ⋯ , n, where φ_{u
_q} and φ_{k
_q} are the symmetric spreads of u_q and k_q, respectively.

Note. If the coefficient components, $c_{pq}^{(i)}$ of the matrix N, 1 ≤ p, q ≤ n and the components $w_{p}^{(i)}$ of the vector w⁽ⁱ⁾ are m-PFNs are called fully m-PFLSEs.

We presently describe the practical technique to solve the fully m-PFLSEs. In any case, we solve the 1-cut of the fully m-PFLSEs to calculate a classical solution of the fully m-PFLSEs. In this way, we should tackle the following classical system:

$\sum_{1 \leq q \leq n} c_{pq}^{(i)} (1) k_{q}^{(i)} = w_{p}^{(i)} (1), p = 1, 2, 3, \dots, n .$ (10) $c_{pq}^{(i)} (1), w_{p}^{(i)} (1) \in I R$ and $k_{q}^{(i)}$ is an unknown classical variable which will be obtained by solving system (10). After calculating the solution of the classical system (10), we fuzzify this solution by assigning some unknown symmetric spreads to each row of the system (10). All things considered, finding the solution of the first fully m-PFLSEs is analogous to calculate the solution of 2n linear equations.

Presently, the classical system of equations (10) is transformed to the following system:

$\begin{matrix} ≺ [{\underline{c}}_{11}^{(i)} (δ), {\bar{c}}_{11}^{(i)} (δ)] [k_{1}^{(i)} - γ_{1}^{(i)} (δ), k_{1}^{(i)} + γ_{1}^{(i)} (δ)] ≻ \\ + \dots + \\ ≺ [{\underline{c}}_{1 n}^{(i)} (δ), {\bar{c}}_{1 n}^{(i)} (δ)] [k_{n}^{(i)} - γ_{1}^{(i)} (δ), k_{n}^{(i)} + γ_{1}^{(i)} (δ)] ≻ \\ = ≺ [{\underline{w}}_{1}^{(i)} (δ), {\bar{w}}_{1}^{(i)} (δ)] ≻ \\ ≺ [{\underline{c}}_{21}^{(i)} (δ), {\bar{c}}_{21}^{(i)} (δ)] [k_{1}^{(i)} - γ_{2}^{(i)} (δ), k_{1}^{(i)} + γ_{2}^{(i)} (δ)] ≻ \\ + \dots + \\ ≺ [{\underline{c}}_{2 n}^{(i)} (δ), {\bar{c}}_{2 n}^{(i)} (δ)] [k_{n}^{(i)} - γ_{2}^{(i)} (δ), k_{n}^{(i)} + γ_{2}^{(i)} (δ)] \\ = ≺ [{\underline{w}}_{2}^{(i)} (δ), {\bar{w}}_{2}^{(i)} (δ)] ≻ \\ ⋮ \\ ≺ [{\underline{c}}_{n 1}^{(i)} (δ), {\bar{c}}_{n 1}^{(i)} (δ)] [k_{1}^{(i)} - γ_{n}^{(i)} (δ), k_{1}^{(i)} + γ_{n}^{(i)} (δ)] ≻ \\ + \dots + \\ ≺ [{\underline{c}}_{nn}^{(i)} (δ), {\bar{c}}_{nn}^{(i)} (δ)] [k_{n}^{(i)} - γ_{n}^{(i)} (δ), k_{n}^{(i)} + γ_{n}^{(i)} (δ)] \\ = ≺ [{\underline{w}}_{n}^{(i)} (δ), {\bar{w}}_{n}^{(i)} (δ)] ≻, \end{matrix}$ (11) where, $γ_{p}^{(i)}, p = 1, 2, \dots, n$ are unknown spreads which will be calculated by solving above 2n system of equations and $k_{q}^{(i)}, q = 1, \dots, n$ are components of the acquired vector solution of system (10). There exist three conceivable cases for the components of the matrix N follows as:

I P₁ = {(p, q) ∈ I M_n × I M_n| c_pq > 0},

I P₂ = {(p, q) ∈ I M_n × I M_n| c_pq < 0},

I P₃ = {(p, q)∈ I M_n × I M_n| c_pq > 0} ∪{(p, q) ∈ I M_n × I M_n| c_pq < 0},

where, I M_n = 1, 2, 3, ⋯ , n.

Case I: I P₁ = {(p, q) ∈ I M_n × I M_n| c_pq > 0} = |I P₁| = n²

Here, m-polar fuzzy matrix N is thought to be positive. In this way, the p-th row of the system (11) is exhibited as follows: $\begin{matrix} ≺ [{\underline{c}}_{p 1}^{(i)} (δ), {\bar{c}}_{p 1}^{(i)} (δ)] [k_{1}^{(i)} - γ_{p}^{(i)} (δ), k_{1}^{(i)} + γ_{p}^{(i)} (δ)] ≻ \\ + \dots + \\ [{\underline{c}}_{pn}^{(i)} (δ), {\bar{c}}_{pn}^{(i)} (δ)] [k_{n}^{(i)} - γ_{p}^{(i)} (δ), k_{n}^{(i)} + γ_{p}^{(i)} (δ)] ≻ \\ = [{\underline{w}}_{n}^{(i)} (δ), {\bar{w}}_{n}^{(i)} (δ)] ≻ . \end{matrix}$ By supposition of the positiveness of N,compact kind of the previously mentioned equation is given as: $\begin{matrix} \sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} - γ_{p}^{(i)} (δ)) = {\underline{w}}_{p}^{(i)} (δ) \Rightarrow \\ γ_{p}^{(i)} (δ) = k_{1} (k_{1}^{(i)}, k_{2}^{(i)}, k_{3}^{(i)}, \dots, k_{n}^{(i)}, \\ {\underline{k}}_{p 1}^{(i)} (δ), {\underline{k}}_{p 2}^{(i)} (δ) \dots, {\underline{k}}_{pn}^{(i)} (δ), {\underline{w}}_{p}^{(i)} (δ)) \end{matrix}$ $\begin{matrix} \sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} + γ_{p}^{(i)} (δ)) = {\bar{w}}_{p}^{(i)} (δ) \Rightarrow \\ γ_{p}^{(i)} (δ) = k_{2} (k_{1}^{(i)}, k_{2}^{(i)}, k_{3}^{(i)}, \dots, k_{n}^{(i)}, \\ {\bar{k}}_{p 1}^{(i)} (δ), {\bar{k}}_{p 2}^{(i)} (δ) \dots, {\bar{k}}_{pn}^{(i)} (δ), {\bar{w}}_{p}^{(i)} (δ)) \end{matrix}$ From now on, we replace $γ_{p 1}^{(i)} (δ)$ and $γ_{p 2}^{(i)} (δ)$ with $γ_{p}^{(i)} (δ)$ . Thus, by such documentations, we get for p = 1, 2, ⋯ , n: $γ_{p 1}^{(i)} (δ) = k_{1} (k_{1}^{(i)}, k_{2}^{(i)}, k_{3}^{(i)}, \dots, k_{n}^{(i)}, {\underline{k}}_{p 1}^{(i)} (δ),$ (12) ${\underline{k}}_{p 2}^{(i)} (δ) \dots, {\underline{k}}_{pn}^{(i)} (δ), {\underline{w}}_{p}^{(i)} (δ)),$ $γ_{p 2}^{(i)} (δ) = k_{2} (k_{1}^{(i)}, k_{2}^{(i)}, k_{3}^{(i)}, \dots, k_{n}^{(i)}, {\bar{k}}_{p 1}^{(i)} (δ),$ (13) ${\bar{k}}_{p 2}^{(i)} (δ) \dots, {\bar{k}}_{pn}^{(i)} (δ), {\bar{w}}_{p}^{(i)} (δ)),$ We propose two total methods to find the unknown symmetric spreads of solutions (SSSs) of the fully m-PFLS, which are indicated by: $\begin{array}{l} {(γ^{(i)})}_{m^{'}}^{-} (δ) = \min_{δ \in [0, 1]} {| γ_{p 1}^{(i)} (δ) |, | γ_{p 2}^{(i)} (δ) |}, \\ p = 1, 2, 3, \dots, n, \end{array}$ (14) $\begin{array}{l} {(γ^{(i)})}_{m^{'}}^{+} (δ) = \max_{δ \in [0, 1]} {| γ_{p 1}^{(i)} (δ) |, | γ_{p 2}^{(i)} (δ) |}, \\ p = 1, 2, 3, \dots, n, \end{array}$ (15) So, $K = (k_{1}^{(i)} (δ), k_{2}^{(i)} (δ), \dots, k_{n}^{(i)} (δ))^{T}$ is computed by applying (14) and (15) respectively, as follows:

$\begin{array}{l} k_{p}^{(i)} (δ) = [k_{p}^{(i)} - {(γ^{(i)})}_{m^{'}}^{-} (δ), k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{-} (δ)], \\ k_{p}^{(i)} (δ) = [k_{p}^{(i)} - {(γ^{(i)})}_{m^{'}}^{+} (δ), k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{+} (δ)] . \end{array}$ (16) Some of the time, find the base of different functions on the interval [0, 1] is not easy work, even we do it. So as to beat this trouble, we actualize a couple of adjustments in the structure of the gained spreads equation (14)-(15). By utilizing equations (12)-(13), the linear form of the spreads of m-polar fuzzy symmetric solutions are acquired as follows: $\begin{matrix} γ_{p 1}^{(i)} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) + ρ_{p}^{(i)} - ρ_{p}^{(i)} δ}{\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)}, \\ γ_{p 2}^{(i)} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) - π_{p}^{(i)} + π_{p}^{(i)} δ}{\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ)} . \end{matrix}$

Let ${\underline{k}}_{pq}^{(i)} (δ)$ in $[{\underline{k}}_{pq}^{(i)} (0), {\underline{k}}_{pq}^{(i)} (1)]$ , then $\begin{matrix} min_{δ \in [0, 1]} [\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)] = \sum_{1 \leq q \leq n} min_{δ \in [0, 1]} [{\underline{k}}_{pq}^{(i)} (δ)] \\ = \sum_{1 \leq q \leq n} [{\underline{k}}_{pq}^{(i)} (0)], p = 1, 2, 3, \dots, n, \\ max_{δ \in [0, 1]} [\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)] = \sum_{1 \leq q \leq n} max_{δ \in [0, 1]} [{\underline{k}}_{pq}^{(i)} (δ)] \\ = \sum_{1 \leq q \leq n} [{\underline{k}}_{pq}^{(i)} (1)], p = 1, 2, 3, \dots, n, \end{matrix}$ Similarly, consider ${\bar{k}}_{pq}^{(i)} (δ)$ in $[{\bar{k}}_{pq}^{(i)} (1), {\bar{k}}_{pq}^{(i)} (0)]$ , then $\begin{matrix} min_{δ \in [0, 1]} [\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ)] = \sum_{1 \leq q \leq n} min_{δ \in [0, 1]} [{\bar{k}}_{pq}^{(i)} (δ)] \\ = \sum_{1 \leq q \leq n} [{\bar{k}}_{pq}^{(i)} (1)], p = 1, 2, 3, \dots, n, \\ max_{δ \in [0, 1]} [\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ)] = \sum_{1 \leq q \leq n} max_{δ \in [0, 1]} [{\bar{k}}_{pq}^{(i)} (δ)] \\ = \sum_{1 \leq q \leq n} [{\bar{k}}_{pq}^{(i)} (0)], p = 1, 2, 3, \dots, n . \end{matrix}$ thus, we have

$\begin{array}{l} {(γ^{(i)})}_{p 1}^{j^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) + ρ_{p} - ρ_{p} δ}{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (1)}, \\ {(γ^{(i)})}_{p 1}^{k^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) + ρ_{p} - ρ_{p} δ}{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (0)}, \\ {(γ^{(i)})}_{p 2}^{j^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) - π_{p} + π_{p} δ}{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (0)}, \\ {(γ^{(i)})}_{p 2}^{k^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - w_{p}^{(i)} (1) - π_{p} + π_{p} δ}{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (1)} . \end{array}$

Presently, a few notations are utilized for execute linear spreads of solutions of the fully m-PFLS as follows:

${(γ^{(i)})}_{m^{'}}^{-, j^{'}} (δ) = \min_{δ \in [0, 1]} {| {(γ^{(i)})}_{p 1}^{j^{'}} (δ) |, | {(γ^{(i)})}_{p 2}^{j^{'}} (δ) |},$ (17)

${(γ^{(i)})}_{m^{'}}^{-, k^{'}} (δ) = \min_{δ \in [0, 1]} {| {(γ^{(i)})}_{p 1}^{k^{'}} (δ) |, | {(γ^{(i)})}_{p 2}^{k^{'}} (δ) |},$ (18)

${(γ^{(i)})}_{m^{'}}^{+, j^{'}} (δ) = \min_{δ \in [0, 1]} {| {(γ^{(i)})}_{p 1}^{j^{'}} (δ) |, | {(γ^{(i)})}_{p 2}^{j^{'}} (δ) |},$ (19)

${(γ^{(i)})}_{m^{'}}^{+, k^{'}} (δ) = \min_{δ \in [0, 1]} {| {(γ^{(i)})}_{p 1}^{k^{'}} (δ) |, | {(γ^{(i)})}_{p 2}^{k^{'}} (δ) |},$ (20) In this manner, by applying different spreads, the relating solutions are determined as:

${(K^{(i)})}^{-, j^{'}} (δ) = {({(K^{(i)})}_{1}^{^{-, j^{'}}} (δ), {(K^{(i)})}_{2}^{^{-, j^{'}}} (δ), ..., {(K^{(i)})}_{n}^{^{-, j^{'}}} n (δ))}^{t} s . t {(K^{(i)})}_{p}^{^{-, j^{'}}} (δ) = [k_{p}^{(i)} - {(γ^{(i)})}_{m^{'}}^{^{-, j^{'}}} (δ), k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{-, j^{'}}} (δ)],$

${(K^{(i)})}^{-, k^{'}} (δ) = {({(K^{(i)})}_{1}^{^{-, k^{'}}} (δ), {(K^{(i)})}_{2}^{^{-, k^{'}}} (δ), ..., {(K^{(i)})}_{n}^{^{-, k^{'}}} n (δ))}^{t} s . t {(K^{(i)})}_{p}^{^{-, k^{'}}} (δ) = [k_{p}^{(i)} - {(γ^{(i)})}_{m^{'}}^{^{-, k^{'}}} (δ), k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{-, k^{'}}} (δ)],$

${(K^{(i)})}^{+, j^{'}} (δ) = {({(K^{(i)})}_{1}^{^{+, j^{'}}} (δ), {(K^{(i)})}_{2}^{^{+, j^{'}}} (δ), ..., {(K^{(i)})}_{n}^{^{+, j^{'}}} n (δ))}^{t} s . t {(K^{(i)})}_{p}^{^{+, j^{'}}} (δ) = [j_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{+, j^{'}}} (δ), j_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{+, j^{'}}} (δ)],$

${(K^{(i)})}^{+, k^{'}} (δ) = {({(K^{(i)})}_{1}^{^{+, k^{'}}} (δ), {(K^{(i)})}_{2}^{^{+, k^{'}}} (δ), ..., {(K^{(i)})}_{n}^{^{+, k^{'}}} n (δ))}^{t} s . t {(K^{(i)})}_{p}^{^{+, k^{'}}} (δ) = [k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{+, k^{'}}} (δ), k_{p}^{(i)} + {(γ^{(i)})}_{m^{'}}^{^{+, k^{'}}} (δ)],$ Case (II): ${I P}_{1} = {(p, q) \in I M_{n} \times I M_{n} | c_{pq}^{(i)} < 0} = | {I P}_{1} | = n^{2}$

Here, m-polar fuzzy matrix N is thought to be negative. In this way, the p-th row of the system (11) is exhibited as follows: $\begin{matrix} \sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} + γ_{p 1}^{(i)} (δ)) = {\underline{w}}_{p}^{(i)} (δ) \Rightarrow \\ γ_{p 1}^{(i)} (δ) = \frac{{\underline{w}}_{p}^{(i)} (δ) - \sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) k_{q}^{(i)}}{\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)} \end{matrix}$ $\begin{matrix} \sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} - γ_{p 2}^{(i)} (δ)) = {\bar{w}}_{p}^{(i)} (δ) \Rightarrow \\ γ_{p 2}^{(i)} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ)} . \end{matrix}$ Analogous to Case (I), we altered the got spreads. Thus, different spreads of the solution of the fully m-PFLS for every row are computed as follows:

$\begin{array}{l} {(γ^{(i)})}_{p 1}^{j^{'}} (δ) = \frac{{\underline{w}}_{p}^{(i)} (δ) - \sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)}}{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (1)}, \\ {(γ^{(i)})}_{p 1}^{k^{'}} (δ) = \frac{{\underline{w}}_{p}^{(i)} (δ) - \sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)}}{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (0)}, \end{array}$ $\begin{array}{l} {(γ^{(i)})}_{p 2}^{j^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (0)}, \\ {(γ^{(i)})}_{p 2}^{k^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (1)}, \end{array}$ analogously to use of equations (17)-(20), different kinds of spreads will also be computed.

Proposition 4.7. Let $K = ([k_{1}^{(i)}, k_{2}^{(i)}, \dots, k_{n}^{(i)}])^{T}$ and the SSS of fully m-PFLS calculated by equations (17)-(20) s.t |I K_r| = n², k = 1, 2, then

${(γ^{(i)})}_{m^{'}}^{-, j^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{-, k^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{+, j^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{+, k^{'}} (1) = 0,$

${(K^{(i)})}_{m^{'}}^{-, j^{'}} (1) = {(K^{(i)})}_{m^{'}}^{-, k^{'}} (1) = {(K^{(i)})}_{m^{'}}^{+, j^{'}} (1) = {(K^{(i)})}_{m^{'}}^{-, j^{'}} (1) = K^{c} .$

Case (III): I P₃ = I P₁ ∪ I P₂, |I P₁| = t₁, |I P₂| = t₂ such that |I P₃| = t₁ + t₂ = n²

Without any loss of all inclusive statement, let the p-th row of the system (11): $c_{pq}^{(i)} (δ) > 0, \forall q \in {I M}_{t}, c_{pq}^{(i)} (δ) < 0, \forall q \in {I M}_{n} \ {I M}_{t}, c_{pq}^{(i)} (δ) > 0$ . So, we have:

$\begin{matrix} \sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} - γ_{p}^{(i)} (δ)) + \\ \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} + γ_{p}^{(i)} (δ)) = {\underline{w}}_{p}^{(i)} (δ), \\ p = 1, 2, 3, \dots, n \end{matrix}$ (21)

$\begin{matrix} \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} + γ_{p}^{(i)} (δ)) + \\ \sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) (k_{q}^{(i)} - γ_{p}^{(i)} (δ)) = {\bar{w}}_{p}^{(i)} (δ), \\ p = 1, 2, 3, \dots, n \end{matrix}$ (22) From equations (21) and (22), we get the following respectively:

$\begin{matrix} γ_{p 1}^{(i)} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ) k_{q}^{(i)} - {\underline{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (δ) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)}, \\ γ_{p 2}^{(i)} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) - \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (δ)}, \\ p = 1, 2, 3, \dots, n . \end{matrix}$ (23) In addition, to find the least and greatest denominators of the equations (23), we have for p = 1, 2, 3, ⋯ , n:

$\begin{matrix} min_{δ \in [0, 1]} {\sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (δ) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)} \\ = \sum_{1 \leq q \leq t} min_{δ \in [0, 1]} {{\underline{k}}_{pq}^{(i)} (δ)} - \sum_{(1 + t) \leq q \leq n} max_{δ \in [0, 1]} {{\underline{k}}_{pq}^{(i)} (δ)} \\ = \sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (0) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (1), \end{matrix}$ $\begin{matrix} max_{δ \in [0, 1]} {\sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (δ) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (δ)} \\ = \sum_{1 \leq q \leq t} max_{δ \in [0, 1]} {{\underline{k}}_{pq}^{(i)} (δ)} - \sum_{(1 + t) \leq q \leq n} min_{δ \in [0, 1]} {{\underline{k}}_{pq}^{(i)} (δ)} \\ = \sum_{1 \leq q \leq t} {\underline{k}}_{pq}^{(i)} (1) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{pq}^{(i)} (0), \end{matrix}$ $\begin{matrix} min_{δ \in [0, 1]} {\sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) - \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (δ)} \\ = \sum_{(1 + t) \leq q \leq n} min_{δ \in [0, 1]} {{\bar{k}}_{pq}^{(i)} (δ)} - \sum_{1 \leq q \leq t} max_{δ \in [0, 1]} {{\bar{k}}_{pq}^{(i)} (δ)} \\ = \sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (1) - \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (0), \end{matrix}$ $\begin{matrix} max_{δ \in [0, 1]} {\sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (δ) - \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (δ)} \\ = \sum_{(1 + t) \leq q \leq n} max_{δ \in [0, 1]} {{\bar{k}}_{pq}^{(i)} (δ)} - \sum_{1 \leq q \leq t} min_{δ \in [0, 1]} {{\bar{k}}_{pq}^{(i)} (δ)} \\ = \sum_{(1 + t) \leq q \leq n} {\bar{k}}_{pq}^{(i)} (0) - \sum_{1 \leq q \leq t} {\bar{k}}_{pq}^{(i)} (1) . \end{matrix}$

Thus, different kinds of linear spreads of solutions are follows as:

$\begin{array}{l} {(γ^{(i)})}_{p 1}^{j^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\underline{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq t} {\underline{k}}_{p q}^{(i)} (1) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{p q}^{(i)} (0)}, \\ {(γ^{(i)})}_{p 1}^{k^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\underline{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\underline{w}}_{p}^{(i)} (δ)}{\sum_{1 \leq q \leq t} {\underline{k}}_{p q}^{(i)} (0) - \sum_{(1 + t) \leq q \leq n} {\underline{k}}_{p q}^{(i)} (1)}, \\ p = 1, 2, 3, \dots, n, \end{array}$ (24)

$\begin{array}{l} {(γ^{(i)})}_{p 2}^{j^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{(1 + t) \leq q \leq n} {\bar{k}}_{p q}^{(i)} (0) - \sum_{1 \leq q \leq t} {\bar{k}}_{p q}^{(i)} (1)}, \\ {(γ^{(i)})}_{p 2}^{k^{'}} (δ) = \frac{\sum_{1 \leq q \leq n} {\bar{k}}_{p q}^{(i)} (δ) k_{q}^{(i)} - {\bar{w}}_{p}^{(i)} (δ)}{\sum_{(1 + t) \leq q \leq n} {\bar{k}}_{p q}^{(i)} (1) - \sum_{1 \leq q \leq t} {\bar{k}}_{p q}^{(i)} (0)}, \\ p = 1, 2, 3, \dots, n . \end{array}$ (25) Finally, different unknown spreads of the solution of the fully m-PFLS can be calculated by utilizing equations (17)-(20).

Proposition 4.8. Let $K^{c} = ([k_{1}^{(i)}, k_{2}^{(i)}, \dots, k_{n}^{(i)}])^{T}$ and the SSS of fully m-PFLS calculated by equations (17)-(20) s.t |I P₃| = |I P₁| ∪ |I P₂|, |I P₁| = t₁, |I P₂| = t₂, |I P₃| = t₁ + t₂ then

${(γ^{(i)})}_{m^{'}}^{-, j^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{-, k^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{+, j^{'}} (1) = {(γ^{(i)})}_{m^{'}}^{+, k^{'}} (1) = 0,$

${(K^{(i)})}_{m^{'}}^{-, j^{'}} (1) = {(K^{(i)})}_{m^{'}}^{-, k^{'}} (1) = {(K^{(i)})}_{m^{'}}^{+, j^{'}} (1) = {(K^{(i)})}_{m^{'}}^{+, k^{'}} (1) = K^{c}$ .

Theorem 4.9. Suppose that ${(γ^{(i)})}_{m^{'}}^{K^{'}} (δ) = \max {{(γ^{(i)})}_{m^{'}}^{-, †} (δ), {(γ^{(i)})}_{m^{'}}^{+, †} (δ)}, {(γ^{(i)})}_{m^{'}}^{j^{'}} (δ) = \min {{(γ^{(i)})}_{m^{'}}^{-, †} (δ), {(γ^{(i)})}_{m^{'}}^{+, †} (δ)} f o r δ \in [0, 1] a n d † \in {j^{'}, k^{'}}$ . Then, we have

(K^{(i⁾)^K′} ∈ CSS and (K^{(i⁾)^J′} ∈ TSS,

where

K^{(i), J^{'}} = (K_{1}^{(i), J^{'}}, K_{2}^{(i), J^{'}}, K_{3}^{(i), J^{'}}, \dots,

K_{n}^{(i), J^{'}})^{T}

and

K_{p}^{(i), J^{'}} (δ) = [k_{p}^{(i)} - γ_{m^{'}}^{j^{'}} (δ), k_{p}^{(i)} + γ_{m^{'}}^{j^{'}}

( δ )] and so on for K^(i),K′ .

Proof. The proof follows from Definition (4.4).□

Example 4.10. Consider fully m-PFLS:

$N = (\begin{matrix} ≺ [2 + δ, 4 - δ], [4 + δ, 6 - δ], [2 + 2 δ, 6 - 2 δ] ≻ & ≺ [4 + δ, 6 - δ], [5 + δ, 8 - 2 δ], [1 + δ, 5 - 3 δ] ≻ \\ ≺ [1 + 2 δ, 5 - 2 δ], [6 + δ, 7], [1 + δ, 6 - 4 δ] ≻ & ≺ [8 + 2 δ, 12 - 2 δ], [4, 5 - δ], [2 + δ, 4 - δ] ≻ \end{matrix})$

$w = (\begin{matrix} ≺ [35 + 15 δ, 65 - 15 δ], [40 + 10 δ, 67 - 17 δ], [9 + 11 δ, 31 - 11 δ] ≻ \\ ≺ [40 + 30 δ, 100 - 30 δ], [43 + 5 δ, 55 - 7 δ], [3 + 7 δ, 32 - 22 δ] ≻ \end{matrix})$ .

The classical solution is calculated as $(K)_{(δ)}^{c} = ({\underline{k}}^{(1)} (δ), {\bar{k}}^{(1)} (δ), {\underline{k}}^{(2)} (δ), {\bar{k}}^{(2)} (δ), {\underline{k}}^{(3)} (δ), {\bar{k}}^{(3)} (δ)) = (10, 4, 4, 5, 5, 0)^{T}$ . By applying the above method, different solutions of the fully m-PFLS depend on the calculated spreads are follows: ${(K^{(1)})}^{-, j^{'}} (δ) = {([10 - \frac{1 - δ}{10}, 10 + \frac{1 - δ}{10}], [4 - \frac{1 - δ}{10}, 4 + \frac{1 - δ}{10}])}^{T},$ (26) ${(K^{(1)})}^{-, k^{'}} (δ) = {([10 - \frac{1 - δ}{8}, 10 + \frac{1 - δ}{8}], [4 - \frac{1 - δ}{8}, 4 + \frac{1 - δ}{8}])}^{T},$ (27) ${(K^{(1)})}^{+, j^{'}} (δ) = {([10 - \frac{2 - 2 δ}{13}, 4 + \frac{2 - 2 δ}{13}], [4 - \frac{2 - 2 δ}{13}, 4 + \frac{2 - 2 δ}{13}])}^{T},$ (28) ${(K^{(1)})}^{+, k^{'}} (δ) = {([10 - \frac{2 - 2 δ}{9}, 10 + \frac{2 - 2 δ}{9}], [4 - \frac{2 - 2 δ}{9}, 4 + \frac{2 - 2 δ}{9}])}^{T},$ (29) ${(K^{(2)})}^{-, j^{'}} (δ) = {([4 - \frac{1 - δ}{11}, 4 + \frac{1 - δ}{11}], [5 - \frac{1 - δ}{11}, 5 + \frac{1 - δ}{11}])}^{T},$ (30)

$\begin{matrix} {(K^{(2)})}^{-, k^{'}} (δ) = ([4 - \frac{1 - δ}{10}, 4 + \frac{1 - δ}{10}], \\ [5 - \frac{1 - δ}{10}, 5 + \frac{1 - δ}{10}])^{T}, \end{matrix}$ (31)

$\begin{matrix} {(K^{(2)})}^{+, j^{'}} (δ) = ([4 - \frac{3 - 3 δ}{14}, 4 + \frac{3 - 3 δ}{14}], \\ [5 - \frac{3 - 3 δ}{14}, 5 + \frac{3 - 3 δ}{14}])^{T}, \end{matrix}$ (32)

$\begin{matrix} {(K^{(2)})}^{+, k^{'}} (δ) = ([4 - \frac{3 - 3 δ}{11}, 4 + \frac{3 - 3 δ}{11}], \\ [5 - \frac{3 - 3 δ}{11}, 5 + \frac{3 - 3 δ}{11}])^{T}, \end{matrix}$ (33) $\begin{matrix} {(K^{(3)})}^{-, j^{'}} (δ) = ([5 - \frac{1 - δ}{11}, 5 + \frac{1 - δ}{11}], \\ [0 - \frac{1 - δ}{11}, 0 + \frac{1 - δ}{11}])^{T}, \end{matrix}$ (34) $\begin{matrix} {(K^{(3)})}^{-, k^{'}} (δ) = ([5 - \frac{1 - δ}{6}, 5 + \frac{1 - δ}{6}], \\ [0 - \frac{1 - δ}{6}, 0 + \frac{1 - δ}{6}])^{T}, \end{matrix}$ (35) $\begin{matrix} {(K^{(3)})}^{+, j^{'}} (δ) = ([5 - \frac{2 - 2 δ}{5}, 5 + \frac{2 - 2 δ}{5}], \\ [0 - \frac{2 - 2 δ}{5}, 0 + \frac{2 - 2 δ}{5}])^{T}, \end{matrix}$ (36) $\begin{matrix} {(K^{(3)})}^{+, k^{'}} (δ) = ([5 - \frac{2 - 2 δ}{3}, 5 + \frac{2 - 2 δ}{3}], \\ [0 - \frac{2 - 2 δ}{3}, 0 + \frac{2 - 2 δ}{3}])^{T}, \end{matrix}$ (37)

Moreover, we put the solutions of fully m-PFLSEs of equations (26-37) into the original fully m-PFLSEs to analyze the distinction between the primary component of $w_{1}^{(i)} i = 1, 2, 3$ and the estimation of the 1st row and the distinction between the second component of $w_{2}^{(i)} i = 1, 2, 3,$ and the estimation of the 2nd row for distinct acquired SS in equation (26-37). For more feature see Figures 3, 4, 5, 6, 7 and 8.

Fig.3

Compare $w_{1}^{(1)} (-)$ and the 1^st row for 4 different solutions, ${(K^{(1)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(1)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(1)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(1)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ ,

Fig.4

Compare $w_{1}^{(2)} (-)$ and the 1^st row for 4 different solutions, ${(K^{(2)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(2)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(2)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(2)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ ,

Fig.5

Compare $w_{1}^{(3)} (-)$ and the 1^st row for 4 distinct solutions, ${(K^{(3)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(3)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(3)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(3)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ ,

Fig.6

Compare $w_{2}^{(1)} (-)$ and the 2nd row for 4 distinct solutions, ${(K^{(1)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(1)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(1)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(1)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ ,

Fig.7

Compare $w_{2}^{(2)} (-)$ and the 2nd row for 4 distinct solutions, ${(K^{(2)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(2)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(2)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(2)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ ,

Fig.8

Compare $w_{2}^{(3)} (-)$ and the 2nd row for 4 distinct solutions, ${(K^{(3)})}_{m^{'}}^{-, j^{'}} (-), {(K^{(3)})}_{m^{'}}^{-, k^{'}} (- .), {(K^{(3)})}_{m^{'}}^{+, j^{'}} (- °)$ and ${(K^{(3)})}_{m^{'}}^{+, k^{'}} (- ⋆)$ .

Theorem 4.11. If the solution exists of the fully m-PFLSEs, at that point the following constraints are satisfied:

$\sum_{1 \leq q \leq n} (c_{pq})_{0} k_{q}^{(i)} = (w_{u}^{(i)})_{0}$ has solution as a classical SLE,

$max_{1 \leq q \leq n} {(c_{pq})_{*}} \leq (w_{u}^{(i)})_{*}$ , q = 1, 2, ⋯ , n,

$max_{1 \leq q \leq n} {(c_{pq})^{*}} \leq (w_{u}^{(i)})^{*}$ , q = 1, 2, ⋯ , n.

Proof. Let $k_{q}^{(i)}$ be the solution of the SLE. That is $\sum_{1 \leq q \leq n} c_{pq} k_{q}^{(i)} = w_{u}^{(i)}$ , q = 1, 2, ⋯ , n. Then $\begin{matrix} \sum_{1 \leq q \leq n} ((c_{pq})_{0}, (c_{pq})_{*}, (c_{pq})^{*}) . ((k_{q}^{(i)})_{0}, (k_{q}^{(i)})_{*}, \\ (k_{q}^{(i)})^{*}) = ((w_{u}^{(i)})_{0}, (w_{u}^{(i)})_{*}, (w_{u}^{(i)})^{*}) \end{matrix}$ This implies $\begin{matrix} \sum_{1 \leq q \leq n} (c_{pq})_{0} (k_{q}^{(i)})_{0}, max_{1 \leq q \leq n} {(c_{pq})_{*}, (k_{q}^{(i)})_{*}}, \\ max_{1 \leq q \leq n} {(c_{pq})^{*}, (k^{i})^{*}}, \\ = ((w_{u}^{(i)})_{0}, (w_{u}^{(i)})_{*}, (w_{u}^{(i)})^{*}) . \end{matrix}$ It pursues that $\begin{matrix} \sum_{1 \leq q \leq n} (c_{pq})_{0} (k_{q}^{(i)})_{0} = (w_{u}^{(i)})_{0}, \\ max_{1 \leq q \leq n} {(c_{pq})_{*}, (k_{q}^{(i)})_{*}} = (w_{u}^{(i)})_{*}, \\ max_{1 \leq q \leq n} {(c_{pq})^{*}, (k_{q}^{(i)})^{*}} = (w_{u}^{(i)})^{*} . \end{matrix}$ Subsequently conditions (i-iii) are fulfilled.□

Theorem 4.12. If

$\sum_{1 \leq q \leq n} (c_{pq})_{0} k_{q}^{(i)} = (w_{p}^{(i)})_{0}$ has crisp solutions $(k_{q}^{(i)})_{0}$ .

$max_{1 \leq q \leq n} {(c_{pq})_{*}} = (w_{p}^{(i)})_{*}$ , $max_{1 \leq q \leq n} {(c_{pq})^{*}} = (w_{p}^{(i)})^{*}$ , q = 1, 2, ⋯ , n.

Then for every

k_{q}^{(i)} = ((k_{q}^{(i)})_{0}, (k_{q}^{(i)})_{*}, (k_{q}^{(i)})^{*})

with

\begin{matrix} (k_{q}^{(i)})_{*} \leq min_{q \in [1, n]} (w_{p}^{(i)})_{*}, (k_{q}^{(i)})^{*} \leq min_{q \in [1, n]} (w_{p}^{(i)})^{*}, \end{matrix}

is a solution of

\sum_{1 \leq q \leq n} c_{pq} k_{q}^{(i)} = w_{p}^{(i)}

, q = 1, 2, ⋯ , n.

Proof. Let $(k_{q}^{(i)})_{0}$ be the crisp solution of the equation $\sum_{1 \leq q \leq n} (c_{pq})_{0} (k_{q}^{(i)}) = (w_{p}^{(i)})_{0}$ , q = 2, 3, ⋯ , n. Then for every $k_{q}^{(i)} = ((k_{q}^{(i)})_{0}, (k_{q}^{(i)})_{*}, (k_{q}^{(i)})^{*})$ with $\begin{matrix} (k_{q}^{(i)})_{*} \leq min_{q \in [1, n]} (w_{p}^{(i)})_{*}, & (k_{q}^{(i)})^{*} \leq min_{q \in [1, n]} (w_{p}^{(i)})^{*}, \end{matrix}$ is a solution of $\sum_{1 \leq q \leq n} c_{pq} k_{q}^{(i)} = w_{p}^{(i)}$ , q = 1, 2, ⋯ , n. we have $\begin{matrix} \sum_{1 \leq q \leq n} c_{pq} k_{q}^{(i)} \\ = \sum_{1 \leq q \leq n} ((c_{pq})_{0}, (c_{pq})_{*}, (c_{pq})^{*}) . ((k_{q}^{(i)})_{0}, (k_{q}^{(i)})_{*}, \\ (k_{q}^{(i)})^{*}) \\ = (\sum_{1 \leq q \leq n} (c_{pq})_{0} (k_{q}^{(i)})_{0}, max_{1 \leq q \leq n} {(c_{pq})_{*}, (k_{q}^{(i)})_{*}}), \\ max_{1 \leq q \leq n} {(c_{pq})^{*}, (k_{q}^{(i)})^{*}}, \\ = ((w_{p}^{(i)})_{0}, max_{1 \leq q \leq n} {(c_{pq})_{*}, (k_{q}^{(i)})_{*}}, max_{1 \leq q \leq n} {(c_{pq})^{*}, \\ (k_{q}^{(i)})^{*}}) . \end{matrix}$ By requirements $\begin{matrix} max_{1 \leq q \leq n} {(c_{pq})_{*}, (k_{q}^{(i)})_{*}} \\ = max {max_{1 \leq q \leq n} {(c_{pq})_{*}, max_{1 \leq q \leq n} (k_{q}^{(i)})_{*}} \\ = max {(w_{p}^{(i)})_{*}, max_{1 \leq q \leq n} (k_{q}^{(i)})_{*}} \\ = (w_{p}^{(i)})_{*} \end{matrix}$ Similarly, $\begin{matrix} max_{1 \leq q \leq n} {(c_{pq})^{*}, (k_{q}^{(i)})^{*}} & = (w_{p}^{(i)})^{*} . \end{matrix}$ □

5 Conclusions

An m-PFS on an underlying set Z is a mapping M : Z → [0, 1] ^m. The truthness degree of each element z ∈ Z is defined as M (z) = (P₁∘M (z) , P₂ ∘ M (z) , ⋯ , P_m ∘ M (z)), where P_i ∘ M :[0, 1] ^m → [0, 1] is the i-th projection mapping. We have proposed the concept of m-polar fuzzy number along with some properties. We have developed the system of linear equation in m-polar fuzzy environment and obtained weak and strong solutions. We have developed a new and practical method to solve fully m-PFLSEs. To end this method, we have solved this system using 1-cut expansion which is the crisp system. Then, this crisp system is m-polar fuzzified to locate the unknown symmetric spreads of the solutions of fully m-PFLSEs. Our method has shown the bounded solution of fully m-PFLSEs that can help the decision-makers to choose some special symmetric solutions which are placed in TSS or CSS. We have noticed that the minimal and maximal symmetric solutions are the approximated solution but not an exact solution. To show the validity and accuracy of this technique, an example of having an approximate solution is illustrated. In the end, some basic results are proven on the base of this work. For the future directions, we will extend our work to (i) LU-factorization method in m-polar fuzzy environment, (ii) iterative schemes in m-polar fuzzy environment, (iii) differential equations under m-polar fuzzy environment and (iv) iterative schemes in differential equations under m-polar fuzzy environment.

Conflicts of interest

The authors declare no conflict of interest.

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