Mehar and Keerat method for solving system of fuzzy complex linear equations

Abstract

In this paper, it is pointed out that in the existing methods [Sarangam Majumdar, Numerical solutions of fuzzy complex system of linear equations, German Journal of Advanced Mathematical Sciences 1 (2013), 20–26] for solving fuzzy complex system of linear equations in which all the parameters except variables are represented by triangular/trapezoidal fuzzy complex numbers as well as the fuzzy complex system of linear equations in which all the parameters except coefficients are represented by triangular/trapezoidal fuzzy complex numbers, Majumdar has assumed that on subtracting a fuzzy number from itself, the obtained value is a zero fuzzy number as well as on dividing a fuzzy number by itself, the obtained value is a unit fuzzy number. To resolve these flaws of the existing methods, two new methods (named as Mehar method and Keerat method) are proposed for solving the same systems of fuzzy complex linear equations.

Keywords

Fuzzy number fuzzy complex number system of fuzzy complex linear equations

1 Introduction

Systems of linear equations have been studied by various authors. There are many methods for solving system of linear equations with real parameters. However, when only some vague and imprecise information about the system’s parameters is given, then some or all the parameters are represented by fuzzy numbers [19]. Moreover, in many applications such as circuit analysis, wave function in quantum mechanics etc., physical quantities are in the form of complex numbers and have fuzzy nature, so these are represented by fuzzy complex numbers [10].

Several authors have proposed direct and iterative methods for solving system of fuzzy linear equations with real fuzzy parameters [1–6 , 17], while, very few authors have proposed some direct and iterative methods for solving system of fuzzy complex linear equations [7–9 , 18].

Majumdar [16] proposed iterative methods (Gauss Jacobi method and Gauss Seidel method) for solving fuzzy complex system of linear equations with fuzzy complex coefficients as well as for solving fuzzy complex system of linear equations with fuzzy complex variables.

In this paper, flaws in the existing methods [16] for solving fuzzy complex system of linear equations are pointed out. Also, to overcome the flaws, two new methods (named as Mehar method and Keerat method) are proposed for solving system of fuzzy complex linear equations with fuzzy complex coefficients and system of fuzzy complex linear equations with fuzzy complex variables.

This paper is organized as follows. In Section 2, some basic definitions are presented. In Section 3, flaws in the existing methods [16] are pointed out. In Section 4, two new methods (named as Mehar method and Keerat method) are proposed for solving system of fuzzy complex linear equations with fuzzy complex coefficients and system of fuzzy complex linear equations with fuzzy complex variables. In Section 5, to illustrate the proposed methods, two example problems are solved. Section 6 concludes thepaper.

2 Preliminaries

In this section, some basic definitions are presented.

Definition 2.1. [12] A fuzzy set $\tilde{A}$ , defined on universal set of real numbers R, is said to be a fuzzy number if it has the following characteristics:

$\tilde{A}$ is convex, i.e., $\begin{matrix} μ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) & \geq & minimum (μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2})) \\ \forall x_{1}, x_{2} \in R, \forall λ \in [0, 1] . \end{matrix}$

$\tilde{A}$ is normal, i.e., ∃ x₀ ∈ R such that $μ_{\tilde{A}} (x_{0}) = 1 .$

$μ_{\tilde{A}}$ is piecewise continuous.

Definition 2.2. [12] A fuzzy number $\tilde{U}$ is said to be triangular fuzzy number if its membership function $μ_{\tilde{U}}$ is given by $μ_{\tilde{U}} (x) = {\begin{matrix} \frac{x - a}{b - a}, & a \leq x < b \\ 1, & x = b \\ \frac{x - c}{b - c}, & b < x \leq c \\ 0, & otherwise \end{matrix}$

Definition 2.3. [12] A fuzzy number $\tilde{U}$ is said to be trapezoidal fuzzy number if its membership function $μ_{\tilde{U}}$ is given by $μ_{\tilde{U}} (x) = {\begin{matrix} \frac{x - a}{b - a}, & a \leq x < b \\ 1, & b \leq x \leq c \\ \frac{x - d}{c - d}, & c < x \leq d \\ 0, & otherwise \end{matrix}$

Definition 2.4. [10] If $\tilde{X}$ and $\tilde{Y}$ are real fuzzy numbers with membership functions $μ (x | \tilde{X})$ and $μ (y | \tilde{Y})$ respectively, then $\tilde{Z} = \tilde{X} + i \tilde{Y}$ will be a fuzzy complex number with membership function $μ (z | \tilde{Z}) = \min (μ (x | \tilde{X}), μ (y | \tilde{Y})), where z = x + iy .$

Definition 2.5. [10] A triangular fuzzy complex number (a₁, b₁, c₁) + i (a₂, b₂, c₂) represents the fuzzy set, ${\begin{matrix} ((x + iy, μ (x + iy)) : x \in [a_{1}, c_{1}], y \in [a_{2}, c_{2}], \\ μ (x + iy) = \min {μ (x), μ (y)} \end{matrix}} .$

Definition 2.6. [10] A trapezoidal fuzzy complex number (a₁, b₁, c₁, d₁) + i (a₂, b₂, c₂, d₂) represents the fuzzy set, ${\begin{matrix} ((x + iy, μ (x + iy)) : x \in [a_{1}, d_{1}], y \in [a_{2}, d_{2}], \\ μ (x + iy) = \min {μ (x), μ (y)} \end{matrix}} .$

3 Flaws in the existing methods

In this section, flaws in the existing methods [16] are pointed out.

It is well known fact that if $\tilde{a}, \tilde{b} and \tilde{c}$ are three fuzzy numbers, then $\tilde{a} + \tilde{b} = \tilde{c}$ does not imply $\tilde{a} = \tilde{c} - \tilde{b} .$ However, Majumdar [16] has used this mathematical incorrect assumption to transform the system of fuzzy complex linear equations $\sum_{j = 1}^{n} {\tilde{c}}_{ij} x_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots, n$ into system of fuzzy complex linear equations

$x_{i} = \frac{{\tilde{w}}_{i} - \sum_{j = 1 j \neq i}^{n} {\tilde{c}}_{ij} x_{j}}{{\tilde{c}}_{ii}}, i = 1, 2, \dots, n,$ as well as to transform the system of fuzzy complex linear equations $\sum_{j = 1}^{n} c_{ij} {\tilde{x}}_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots, n$ into system of fuzzy complex linear equations ${\tilde{x}}_{i} = \frac{{\tilde{w}}_{i} - \sum_{j = 1 j \neq i}^{n} c_{ij} {\tilde{x}}_{j}}{c_{ii}}, i = 1, 2, \dots, n .$

Since, the system of fuzzy complex linear equations $\sum_{j = 1}^{n} {\tilde{c}}_{ij} x_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots, n,$ cannot be transformed into system of fuzzy complex linear equations $x_{i} = \frac{{\tilde{w}}_{i} - \sum_{j = 1 j \neq i}^{n} {\tilde{c}}_{ij} x_{j}}{{\tilde{c}}_{ii}}, i = 1, 2, \dots, n$ and the system of fuzzy complex linear equations $\sum_{j = 1}^{n} c_{ij} {\tilde{x}}_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots n,$ cannot be transformed into system of fuzzy complex linear equations ${\tilde{x}}_{i} = \frac{{\tilde{w}}_{i} - \sum_{j = 1 j \neq i}^{n} c_{ij} {\tilde{x}}_{j}}{c_{ii}}, i = 1, 2, \dots, n .$

Therefore, the method, proposed by Majumdar [16], is not valid.

4 Proposed methods

In this section, two methods are proposed for solving system of fuzzy complex linear equations. Mehar method for solving system of fuzzy complex linear equations with fuzzy complex coefficients and Keerat method for solving system of fuzzy complex linear equations with fuzzy complex variables.

4.1 Mehar method for solving system of fuzzy complex linear equations with fuzzy complex coefficients

In this section, Mehar method for solving system of fuzzy complex linear Equation (1) with fuzzy complex coefficients is proposed.

$\begin{matrix} {\tilde{c}}_{11} z_{1} + {\tilde{c}}_{12} z_{2} + \dots + {\tilde{c}}_{1 n} z_{n} = {\tilde{w}}_{1} \\ {\tilde{c}}_{21} z_{1} + {\tilde{c}}_{22} z_{2} + \dots + {\tilde{c}}_{2 n} z_{n} = {\tilde{w}}_{2} \\ ⋮ \\ {\tilde{c}}_{n 1} z_{1} + {\tilde{c}}_{n 2} z_{2} + \dots + {\tilde{c}}_{nn} z_{n} = {\tilde{w}}_{n}, \\ i . e . \sum_{j = 1}^{n} {\tilde{c}}_{ij} z_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots, n . \end{matrix}$ (1) where ${\tilde{c}}_{ij}, {\tilde{w}}_{i}$ are triangular/trapezoidal fuzzy complex numbers and z_j is a crisp complex number.

The steps of the proposed Mehar method are as follows:

Step 1: Assuming $\begin{matrix} {\tilde{c}}_{ij} = (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4}) + i (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4}), \\ z_{j} = (x_{j} + {iy}_{j}), \\ {\tilde{w}}_{i} = (u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4}), \end{matrix}$ the system of fuzzy complex linear Equation (1) can be transformed into system of fuzzy complex linear Equation (2).

$\begin{matrix} \sum_{j = 1}^{n} {[(a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4}) \\ + i (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4})] (x_{j} + {iy}_{j})} \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (2)

Step 2: Using the existing product [18, Section IV, pp. 537] $\begin{matrix} [(x + iy)] [(a_{1}, a_{2}, a_{3}, a_{4}) + i (b_{1}, b_{2}, b_{3}, b_{4})] \\ = [x (a_{1}, a_{2}, a_{3}, a_{4}) - y (b_{1}, b_{2}, b_{3}, b_{4})] \\ + i [x (b_{1}, b_{2}, b_{3}, b_{4}) \\ + y (a_{1}, a_{2}, a_{3}, a_{4})], \end{matrix}$ the system of fuzzy complex linear Equation (2) can be transformed into system of fuzzy complex linear Equation (3).

$\begin{matrix} \sum_{j = 1}^{n} {[x_{j} (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4}) \\ - y_{j} (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4})] \\ + i [x_{j} (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4}) \\ + y_{j} (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4})]} \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (3)

Step 3: Assuming [15, Section 3.4, pp. 33]

$\begin{matrix} x_{j} (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4}) = (e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) \\ = {\begin{matrix} (x_{j} a_{ij 1}, x_{j} a_{ij 2}, x_{j} a_{ij 3}, x_{j} a_{ij 4}) if x_{j} \geq 0, \\ (x_{j} a_{ij 4}, x_{j} a_{ij 3}, x_{j} a_{ij 2}, x_{j} a_{ij 1}) if x_{j} < 0, \end{matrix} \\ y_{j} (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4}) = (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4}) \\ = {\begin{matrix} (y_{j} b_{ij 1}, y_{j} b_{ij 2}, y_{j} b_{ij 3}, y_{j} b_{ij 4}) if y_{j} \geq 0, \\ (y_{j} b_{ij 4}, y_{j} b_{ij 3}, y_{j} b_{ij 2}, y_{j} b_{ij 1}) if y_{j} < 0, \end{matrix} \\ x_{j} (b_{ij 1}, b_{ij 2}, b_{ij 3}, b_{ij 4}) = (e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) \\ = {\begin{matrix} (x_{j} b_{ij 1}, x_{j} b_{ij 2}, x_{j} b_{ij 3}, x_{j} b_{ij 4}) if x_{j} \geq 0, \\ (x_{j} b_{ij 4}, x_{j} b_{ij 3}, x_{j} b_{ij 2}, x_{j} b_{ij 1}) if x_{j} < 0, \end{matrix} \\ y_{j} (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4}) = (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'}) \\ = {\begin{matrix} (y_{j} a_{ij 1}, y_{j} a_{ij 2}, y_{j} a_{ij 3}, y_{j} a_{ij 4}) if y_{j} \geq 0, \\ (y_{j} a_{ij 4}, y_{j} a_{ij 3}, y_{j} a_{ij 2}, y_{j} a_{ij 1}) if y_{j} < 0, \end{matrix} \end{matrix}$ the system of fuzzy complex linear Equation (3) can be transformed into system of fuzzy complex linear Equation (4).

$\begin{matrix} \sum_{j = 1}^{n} {[(e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) - (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4})] \\ + i [(e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) + (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'})]} \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (4)

Step 4: Using the property $\begin{matrix} \sum_{i = 1}^{n} [(a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) + i (a_{i 5}, a_{i 6}, a_{i 7}, a_{i 8})] \\ = \sum_{i = 1}^{n} (a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) + i \sum_{i = 1}^{n} (a_{i 5}, a_{i 6}, a_{i 7}, a_{i 8}), \end{matrix}$ the system of fuzzy complex linear Equation (4) can be transformed into system of fuzzy complex linear Equation (5).

$\begin{matrix} \sum_{j = 1}^{n} [(e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) - (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4})] \\ + i \sum_{j = 1}^{n} [(e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) + (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'})] \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (5)

Step 5: Equating real and imaginary parts of the system of fuzzy complex linear Equation (5), it can be transformed into system of fuzzy linear Equation (6).

$\begin{matrix} \sum_{j = 1}^{n} [(e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) - (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 3})] \\ = (u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}), i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} [(e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) + (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'})] \\ = (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4}), i = 1, 2, \dots, n . \end{matrix}$ (6)

Step 6: Using the property

$\begin{matrix} \sum_{i = 1}^{n} (a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) \\ = (\sum_{j = 1}^{n} a_{i 1}, \sum_{j = 1}^{n} a_{i 2}, \sum_{j = 1}^{n} a_{i 3}, \sum_{j = 1}^{n} a_{i 4}), \end{matrix}$ the system of fuzzy linear Equation (6) can be transformed into system of fuzzy linear Equation (7).

$\begin{matrix} (\sum_{j = 1}^{n} e_{ij 1}, \sum_{j = 1}^{n} e_{ij 2}, \sum_{j = 1}^{n} e_{ij 3}, \sum_{j = 1}^{n} e_{ij 4}) \\ - (\sum_{j = 1}^{n} f_{ij 1}, \sum_{j = 1}^{n} f_{ij 2}, \sum_{j = 1}^{n} f_{ij 3}, \sum_{j = 1}^{n} f_{ij 4}) \\ = (u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}), i = 1, 2, \dots, n; \\ (\sum_{j = 1}^{n} e_{ij 1}^{'}, \sum_{j = 1}^{n} e_{ij 2}^{'}, \sum_{j = 1}^{n} e_{ij 3}^{'}, \sum_{j = 1}^{n} e_{ij 4}^{'}) \\ + (\sum_{j = 1}^{n} f_{ij 1}^{'}, \sum_{j = 1}^{n} f_{ij 2}^{'}, \sum_{j = 1}^{n} f_{ij 3}^{'}, \sum_{j = 1}^{n} f_{ij 4}^{'}) \\ = (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4}), i = 1, 2, \dots, n . \end{matrix}$ (7)

Step 7: Using the properties

$\begin{matrix} (\sum_{i = 1}^{n} a_{i 1}, \sum_{i = 1}^{n} a_{i 2}, \sum_{i = 1}^{n} a_{i 3}, \sum_{i = 1}^{n} a_{i 4}) \\ + (\sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} b_{i 4}) \\ = (\sum_{i = 1}^{n} a_{i 1} + \sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} a_{i 2} + \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} a_{i 3} \\ + \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} a_{i 4} + \sum_{i = 1}^{n} b_{i 4}) \end{matrix}$ and

$\begin{matrix} (\sum_{i = 1}^{n} a_{i 1}, \sum_{i = 1}^{n} a_{i 2}, \sum_{i = 1}^{n} a_{i 3}, \sum_{i = 1}^{n} a_{i 4}) \\ - (\sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} b_{i 4}) \\ = (\sum_{i = 1}^{n} a_{i 1} - \sum_{i = 1}^{n} b_{i 4}, \sum_{i = 1}^{n} a_{i 2} - \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} a_{i 3} \\ - \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} a_{i 4} - \sum_{i = 1}^{n} b_{i 1}), \end{matrix}$ the system of fuzzy linear Equation (7) can be transformed into system of fuzzy linear Equation (8).

$\begin{matrix} (\sum_{j = 1}^{n} e_{ij 1} - \sum_{j = 1}^{n} f_{ij 4}, \sum_{j = 1}^{n} e_{ij 2} - \sum_{j = 1}^{n} f_{ij 3}, \sum_{j = 1}^{n} e_{ij 3} \\ - \sum_{j = 1}^{n} f_{ij 2}, \sum_{j = 1}^{n} e_{ij 4} - \sum_{j = 1}^{n} f_{ij 1}) \\ = (u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}), i = 1, 2, \dots, n; \\ (\sum_{j = 1}^{n} e_{ij 1}^{'} + \sum_{j = 1}^{n} f_{ij 1}^{'}, \sum_{j = 1}^{n} e_{ij 2}^{'} + \sum_{j = 1}^{n} f_{ij 2}^{'}, \sum_{j = 1}^{n} e_{ij 3}^{'} \\ + \sum_{j = 1}^{n} f_{ij 3}^{'}, \sum_{j = 1}^{n} e_{ij 4}^{'} + \sum_{j = 1}^{n} f_{ij 4}^{'}) \\ = (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4}), i = 1, 2, \dots, n . \end{matrix}$ (8)

Step 8: Using the property $\begin{matrix} (a_{1}, a_{2}, a_{3}, a_{4}) \\ = (b_{1}, b_{2}, b_{3}, b_{4}) \Rightarrow a_{1} = b_{1}, a_{2} = b_{2}, \\ a_{3} = b_{3}, a_{4} = b_{4}, \end{matrix}$ the system of fuzzy linear Equation (8) can be transformed into system of crisp linear Equation (9). $\begin{matrix} \sum_{j = 1}^{n} e_{ij 1} - \sum_{j = 1}^{n} f_{ij 4} = u_{i 1}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 2} - \sum_{j = 1}^{n} f_{ij 3} = u_{i 2}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 3} - \sum_{j = 1}^{n} f_{ij 2} = u_{i 3}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 4} - \sum_{j = 1}^{n} f_{ij 1} = u_{i 4}, i = 1, 2, \dots, n; \end{matrix} \begin{matrix} \sum_{j = 1}^{n} e_{ij 1}^{'} + \sum_{j = 1}^{n} f_{ij 1}^{'} = v_{i 1}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 2}^{'} + \sum_{j = 1}^{n} f_{ij 2}^{'} = v_{i 2}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 3}^{'} + \sum_{j = 1}^{n} f_{ij 3}^{'} = v_{i 3}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 4}^{'} + \sum_{j = 1}^{n} f_{ij 4}^{'} = v_{i 4}, i = 1, 2, \dots, n . \end{matrix}$ (9)

Step 9: Find the solution {x_j, y_j} of system of crisp linear Equation (9).

Step 10: Using the solution, obtained in Step 9, the solution of system of fuzzy complex linear Equation (1) is z_j = x_j + iy_j .

4.2 Keerat method for solving system of fuzzy complex linear equations with fuzzy complex variables

In this section, Keerat method for solving system of fuzzy complex linear Equation (10) with fuzzy complex variables is proposed. $\begin{matrix} c_{11} {\tilde{z}}_{1} + c_{12} {\tilde{z}}_{2} + \dots + c_{1 n} {\tilde{z}}_{n} = {\tilde{w}}_{1} \\ c_{21} {\tilde{z}}_{1} + c_{22} {\tilde{z}}_{2} + \dots + c_{2 n} {\tilde{z}}_{n} = {\tilde{w}}_{2} \\ ⋮ \\ c_{n 1} {\tilde{z}}_{1} + c_{n 2} {\tilde{z}}_{2} + \dots + c_{nn} {\tilde{z}}_{n} = {\tilde{w}}_{n} \\ i . e . \sum_{j = 1}^{n} c_{ij} {\tilde{z}}_{j} = {\tilde{w}}_{i}, i = 1, 2, \dots, n . \end{matrix}$ (10) where ${\tilde{z}}_{j}, {\tilde{w}}_{i}$ are triangular/trapezoidal fuzzy complex numbers and c_ij is a crisp complex number.

The steps of the proposed Keerat method are as follows:

Step 1: Assuming $\begin{matrix} c_{ij} = (a_{ij} + {ib}_{ij}), \\ {\tilde{z}}_{j} = (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) + i (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4}), \\ {\tilde{w}}_{i} = (u_{i 1}, u_{i 2}, u_{i 3}, u_{j 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{j 4}), \end{matrix}$ the system of fuzzy complex linear Equation (10) can be transformed into system of fuzzy complex linear Equation (11).

$\begin{matrix} \sum_{j = 1}^{n} {(a_{ij} + {ib}_{ij}) [(x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) \\ + i (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4})]} \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{j 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{j 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (11)

Step 2: Using the existing product [18, Section IV, pp. 537] $\begin{matrix} [(x + iy)] [(a_{1}, a_{2}, a_{3}, a_{4}) + i (b_{1}, b_{2}, b_{3}, b_{4})] \\ = [x (a_{1}, a_{2}, a_{3}, a_{4}) - y (b_{1}, b_{2}, b_{3}, b_{4})] \\ + i [x (b_{1}, b_{2}, b_{3}, b_{4}) + y (a_{1}, a_{2}, a_{3}, a_{4})], \end{matrix}$ the system of fuzzy complex linear Equation (11) can be transformed into system of fuzzy complex linear Equation (12).

$\begin{matrix} \sum_{j = 1}^{n} {[a_{ij} (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) \\ - b_{ij} (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4})] \\ + i [a_{ij} (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4}) \\ + b_{ij} (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4})]} \\ = [(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})]; \\ i = 1, 2, \dots, n . \end{matrix}$ (12)

Step 3: Assuming [15, Section 3.4, pp. 33] $\begin{matrix} a_{ij} (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) = (e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) \\ = {\begin{matrix} (a_{ij} x_{j 1}, a_{ij} x_{j 2}, a_{ij} x_{j 3}, a_{ij} x_{j 4}) if a_{ij} \geq 0, \\ (a_{ij} x_{j 4}, a_{ij} x_{j 3}, a_{ij} x_{j 2}, a_{ij} x_{j 1}) if a_{ij} < 0, \end{matrix} \end{matrix}$ $\begin{matrix} b_{ij} (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4}) = (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4}) \\ = {\begin{matrix} (b_{ij} y_{j 1}, b_{ij} y_{j 2}, b_{ij} y_{j 3}, b_{ij} y_{j 4}) if b_{ij} \geq 0, \\ (b_{ij} y_{j 4}, b_{ij} y_{j 3}, b_{ij} y_{j 2}, b_{ij} y_{j 1}) if b_{ij} < 0, \end{matrix} \end{matrix}$ $\begin{matrix} a_{ij} (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4}) = (e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) \\ = {\begin{matrix} (a_{ij} y_{j 1}, a_{ij} y_{j 2}, a_{ij} y_{j 3}, a_{ij} y_{j 4}) if a_{ij} \geq 0, \\ (a_{ij} y_{j 4}, a_{ij} y_{j 3}, a_{ij} y_{j 2}, a_{ij} y_{j 1}) if a_{ij} < 0, \end{matrix} \end{matrix}$ $\begin{matrix} b_{ij} (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) = (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'}) \\ = {\begin{matrix} (b_{ij} x_{j 1}, b_{ij} x_{j 2}, b_{ij} x_{j 3}, b_{ij} x_{j 4}) if b_{ij} \geq 0, \\ (b_{ij} x_{j 4}, b_{ij} x_{j 3}, b_{ij} x_{j 2}, b_{ij} x_{j 1}) if b_{ij} < 0, \end{matrix} \end{matrix}$ the system of fuzzy complex linear Equation (12) can be transformed into system of fuzzy complex linear Equation (13).

$\begin{matrix} \sum_{j = 1}^{n} {[(e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) - (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4})] \\ + i [(e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) + (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'})]} \\ = {(u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}) + i (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4})}; \\ i = 1, 2, \dots, n . \end{matrix}$ (13)

Step 4: Using the property $\begin{matrix} \sum_{i = 1}^{n} [(a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) + i (a_{i 5}, a_{i 6}, a_{i 7}, a_{i 8})] \\ = \sum_{i = 1}^{n} (a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) + i \sum_{i = 1}^{n} (a_{i 5}, a_{i 6}, a_{i 7}, a_{i 8}), \end{matrix}$ the system of fuzzy complex linear Equation (13) can be transformed into system of fuzzy complex linear Equation (14).

Step 5: Equating real and imaginary parts of the system of fuzzy complex linear Equation (14), it can be transformed into system of fuzzy linear Equation (15). $\begin{matrix} \sum_{j = 1}^{n} [(e_{ij 1}, e_{ij 2}, e_{ij 3}, e_{ij 4}) - (f_{ij 1}, f_{ij 2}, f_{ij 3}, f_{ij 4})] \\ = (u_{i 1}, u_{i 2}, u_{i 3}, u_{i 4}), i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} [(e_{ij 1}^{'}, e_{ij 2}^{'}, e_{ij 3}^{'}, e_{ij 4}^{'}) + (f_{ij 1}^{'}, f_{ij 2}^{'}, f_{ij 3}^{'}, f_{ij 4}^{'})] \\ = (v_{i 1}, v_{i 2}, v_{i 3}, v_{i 4}), i = 1, 2, \dots, n . \end{matrix}$ (15)

Step 6: Using the property $\begin{matrix} \sum_{i = 1}^{n} (a_{i 1}, a_{i 2}, a_{i 3}, a_{i 4}) \\ = (\sum_{j = 1}^{n} a_{i 1}, \sum_{j = 1}^{n} a_{i 2}, \sum_{j = 1}^{n} a_{i 3}, \sum_{j = 1}^{n} a_{i 4}), \end{matrix}$ the system of fuzzy linear Equation (15) can be transformed into system of fuzzy linear Equation (16).

Step 7: Using the properties $\begin{matrix} (\sum_{i = 1}^{n} a_{i 1}, \sum_{i = 1}^{n} a_{i 2}, \sum_{i = 1}^{n} a_{i 3}, \sum_{i = 1}^{n} a_{i 4}) \\ + (\sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} b_{i 4}) \\ = (\sum_{i = 1}^{n} a_{i 1} + \sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} a_{i 2} + \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} a_{i 3} \\ + \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} a_{i 4} + \sum_{i = 1}^{n} b_{i 4}) \end{matrix}$ and $\begin{matrix} (\sum_{i = 1}^{n} a_{i 1}, \sum_{i = 1}^{n} a_{i 2}, \sum_{i = 1}^{n} a_{i 3}, \sum_{i = 1}^{n} a_{i 4}) \\ - (\sum_{i = 1}^{n} b_{i 1}, \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} b_{i 4}) \\ = (\sum_{i = 1}^{n} a_{i 1} - \sum_{i = 1}^{n} b_{i 4}, \sum_{i = 1}^{n} a_{i 2} - \sum_{i = 1}^{n} b_{i 3}, \sum_{i = 1}^{n} a_{i 3} \\ - \sum_{i = 1}^{n} b_{i 2}, \sum_{i = 1}^{n} a_{i 4} - \sum_{i = 1}^{n} b_{i 1}), \end{matrix}$ the system of fuzzy linear Equation (16) can be transformed into system of fuzzy linear Equation (17).

Step 8: Using the property (a₁, a₂, a₃, a₄) = (b₁, b₂, b₃, b₄) ⇒ a₁ = b₁, a₂ = b₂, a₃ = b₃, a₄ = b₄, the system of fuzzy linear Equation (17) can be transformed into system of crisp linear Equation (18). $\begin{matrix} \sum_{j = 1}^{n} e_{ij 1} - \sum_{j = 1}^{n} f_{ij 4} = u_{i 1}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 2} - \sum_{j = 1}^{n} f_{ij 3} = u_{i 2}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 3} - \sum_{j = 1}^{n} f_{ij 2} = u_{i 3}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 4} - \sum_{j = 1}^{n} f_{ij 1} = u_{i 4}, i = 1, 2, \dots, n; \end{matrix} \begin{matrix} \sum_{j = 1}^{n} e_{ij 1}^{'} + \sum_{j = 1}^{n} f_{ij 1}^{'} = v_{i 1}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 2}^{'} + \sum_{j = 1}^{n} f_{ij 2}^{'} = v_{i 2}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 3}^{'} + \sum_{j = 1}^{n} f_{ij 3}^{'} = v_{i 3}, i = 1, 2, \dots, n; \\ \sum_{j = 1}^{n} e_{ij 4}^{'} + \sum_{j = 1}^{n} f_{ij 4}^{'} = v_{i 4}, i = 1, 2, \dots, n . \end{matrix}$ (18)

Step 9: Find the solution {x_j1, x_j2, x_j3, x_j4, y_j1, y_j2, y_j3, y_j4} of system of crisp linear Equation (18) with the restrictions x_j1 ≤ x_j2 ≤ x_j3 ≤ x_j4 and y_j1 ≤ y_j2 ≤ y_j3 ≤ y_j4 .

Step 10: Using the solution {x_j1, x_j2, x_j3, x_j4, y_j1, y_j2, y_j3, y_j4} obtained in Step 9, the fuzzy complex solution ${\tilde{z}}_{j}$ of system of fuzzy complex linear Equation (10) is ${\tilde{z}}_{j} = (x_{j 1}, x_{j 2}, x_{j 3}, x_{j 4}) + i (y_{j 1}, y_{j 2}, y_{j 3}, y_{j 4}) .$

Remark 1. The solution, attained by proposed Mehar method, will be a real complex number and the solution, attained by proposed Keerat method, will always be a fuzzy complex number.

Remark 2. The restrictions x_j1 ≤ x_j2 ≤ x_j3 ≤ x_j4 and y_j1 ≤ y_j2 ≤ y_j3 ≤ y_j4, in Step 9 in Section 4.2, guarantee that the solutions are always fuzzy number solutions.

Remark 3. The system of fuzzy complex linear Equation (1) will be consistent if the equivalent system of crisp linear Equation (9) is consistent.

Remark 4. The system of fuzzy complex linear Equation (10) will be consistent if the equivalent system of crisp linear Equation (18) is consistent.

5 Illustrative examples

In this section, to illustrate the proposed methods, system of fuzzy complex linear equations with fuzzy complex coefficients and system of fuzzy complex linear equations with fuzzy complex variables are solved.

Example 5.1. Solve the following system of fuzzy complex linear equations by the Mehar method proposed in Section 4.1.

$\begin{matrix} [(1, 2, 3, 4) + i (2, 3, 4, 5)] z_{1} \\ + [(0, 1, 2, 3) + i (6, 7, 8, 9)] z_{2} \\ = [(- 23, - 15, - 7, 1) + i (23, 31, 39, 47)], \\ [(- 2, - 1, 0, 1) + i (3, 4, 5, 6)] z_{1} \\ + [(0, 2, 4, 6) + i (- 3, - 2, - 1, 0)] z_{2} \\ = [(- 20, - 9, 2, 13) + i (- 12, - 3, 6, 15)] . \end{matrix}$ (19) where, z₁ and z₂ are crisp complex numbers.

Solution: Using the Mehar method, proposed in Section 4.1, the solution of system of fuzzy complex linear Equation (19) can be obtained as follows:

Step 1: Assuming z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, the system of fuzzy complex linear Equation (19) can be transformed into system of fuzzy complex linear Equation (20).

$\begin{matrix} [(1, 2, 3, 4) + i (2, 3, 4, 5)] (x_{1} + {iy}_{1}) \\ + [(0, 1, 2, 3) + i (6, 7, 8, 9)] (x_{2} + {iy}_{2}) \\ = [(- 23, - 15, - 7, 1) + i (23, 31, 39, 47)], \\ [(- 2, - 1, 0, 1) + i (3, 4, 5, 6)] (x_{1} + {iy}_{1}) \\ + [(0, 2, 4, 6) + i (- 3, - 2, - 1, 0)] (x_{2} + {iy}_{2}) \\ = [(- 20, - 9, 2, 13) + i (- 12, - 3, 6, 15)] . \end{matrix}$ (20)

Step 2: Using Step 2 of the Mehar method, proposed in Section 4.1, the system of fuzzy complex linear Equation (20) can be transformed into system of fuzzy complex linear Equation (21).

$\begin{matrix} [(1, 2, 3, 4) x_{1} - (2, 3, 4, 5) y_{1} \\ + (0, 1, 2, 3) x_{2} - (6, 7, 8, 9) y_{2}] \\ + i [(1, 2, 3, 4) y_{1} + (2, 3, 4, 5) x_{1} \\ + (0, 1, 2, 3) y_{2} + (6, 7, 8, 9) x_{2}] \\ = [(- 23, - 15, - 7, 1) + i (23, 31, 39, 47)], \\ [(- 2, - 1, 0, 1) x_{1} - (3, 4, 5, 6) y_{1} \\ + (0, 2, 4, 6) x_{2} - (- 3, - 2, - 1, 0) y_{2}] \\ + i [(- 2, - 1, 0, 1) y_{1} + (3, 4, 5, 6) x_{1} \\ + (0, 2, 4, 6) y_{2} + (- 3, - 2, - 1, 0) x_{2}] \\ = [(- 20, - 9, 2, 13) + i (- 12, - 3, 6, 15)] . \end{matrix}$ (21)

Step 3: Using Step 3 of the Mehar method, proposed in Section 4.1, the system of fuzzy complex linear Equation (21) can be transformed into system of fuzzy complex linear Equation (22).

$\begin{matrix} [(1 x_{1}, 2 x_{1}, 3 x_{1}, 4 x_{1}) + (- 5 y_{1}, - 4 y_{1}, - 3 y_{1}, - 2 y_{1}) \\ + (0, 1 x_{2}, 2 x_{2}, 3 x_{2}) + (- 9 y_{2}, - 8 y_{2}, - 7 y_{2}, - 6 y_{2})] \\ + i [(1 y_{1}, 2 y_{1}, 3 y_{1}, 4 y_{1}) + (2 x_{1}, 3 x_{1}, 4 x_{1}, 5 x_{1}) \\ + (0, 1 y_{2}, 2 y_{2}, 3 y_{2}) + (6 x_{2}, 7 x_{2}, 8 x_{2}, 9 x_{2})] \\ = [(- 23, - 15, - 7, 1) + i (23, 31, 39, 47)], \\ [(- 2 x_{1}, - 1 x_{1}, 0, 1 x_{1}) + (- 6 y_{1}, - 5 y_{1}, - 4 y_{1}, - 3 y_{1}) \\ + (0, 2 x_{2}, 4 x_{2}, 6 x_{2}) + (0, 1 y_{2}, 2 y_{2}, 3 y_{2})] \\ + i [(- 2 y_{1}, - 1 y_{1}, 0, 1 y_{1}) \\ + (3 x_{1}, 4 x_{1}, 5 x_{1}, 6 x_{1}) + (0, 2 y_{2}, 4 y_{2}, 6 y_{2}) \\ + (- 3 x_{2}, - 2 x_{2}, - 1 x_{2}, 0)] \\ = [(- 20, - 9, 2, 13) + i (- 12, - 3, 6, 15)] . \end{matrix}$ (22)

Step 4: Using Step 4 to Step 8 of the Mehar method, proposed in Section 4.1, the system of fuzzy complex linear Equation (22) can be transformed into system of crisp linear Equation (23). $\begin{matrix} x_{1} - 5 y_{1} - 9 y_{2} = - 23, \\ 2 x_{1} + x_{2} - 4 y_{1} - 8 y_{2} = - 15, \\ 3 x_{1} + 2 x_{2} - 3 y_{1} - 7 y_{2} = - 7, \\ 4 x_{1} + 3 x_{2} - 2 y_{1} - 6 y_{2} = 1, \\ - 2 x_{1} - 6 y_{1} = - 20, \\ - x_{1} + 2 x_{2} - 5 y_{1} + y_{2} = - 9, \\ 4 x_{2} - 4 y_{1} + 2 y_{2} = 2, \\ x_{1} + 6 x_{2} - 3 y_{1} + 3 y_{2} = 13, \\ 2 x_{1} + 6 x_{2} + y_{1} = 23, \\ 3 x_{1} + 7 x_{2} + 2 y_{1} + y_{2} = 31, \\ 4 x_{1} + 8 x_{2} + 3 y_{1} + 2 y_{2} = 39, \\ 5 x_{1} + 9 x_{2} + 4 y_{1} + 3 y_{2} = 47, \\ 3 x_{1} - 3 x_{2} - 2 y_{1} = - 12, \\ 4 x_{1} - 2 x_{2} - y_{1} + 2 y_{2} = - 3, \\ 5 x_{1} - x_{2} + 4 y_{2} = 6, \\ 6 x_{1} + y_{1} + 6 y_{2} = 15 . \end{matrix}$ (23)

Step 5: The solution of the system of crisp linear Equation (23) is x₁ = 1, x₂ = 3, y₁ = 3 and y₂ = 1 .

Step 6: Using Step 10 of the Mehar method, proposed in Section 4.1, the solution of system of fuzzy complex linear Equation (19) is $\begin{matrix} z_{1} = x_{1} + {iy}_{1} = 1 + 3 i, \\ z_{2} = x_{2} + {iy}_{2} = 3 + i . \end{matrix}$

Example 5.2. Solve the following system of fuzzy complex linear equations by the Keerat method proposed in Section 4.2.

$\begin{matrix} (10 - 7.5 i) {\tilde{z}}_{1} - (6 - 5 i) {\tilde{z}}_{2} \\ = [(4, 5, 5, 6) + i (- 1, 0, 0, 1)], \\ - (6 - 5 i) {\tilde{z}}_{1} + (16 + 3 i) {\tilde{z}}_{2} \\ = (- 2, - 1, - 1, 0) + i (- 3, - 2, - 2, - 1) . \end{matrix}$ (24) where, ${\tilde{z}}_{1} and {\tilde{z}}_{2}$ are trapezoidal fuzzy complex numbers.

Solution: Using the Keerat method, proposed in Section 4.2, the solution of system of fuzzy complex linear Equation (24) can be obtained as follows:

Step 1: Assuming $\begin{matrix} {\tilde{z}}_{1} = (x_{1}, x_{2}, x_{3}, x_{4}) + i (y_{1}, y_{2}, y_{3}, y_{4}), \\ {\tilde{z}}_{2} = (x_{5}, x_{6}, x_{7}, x_{8}) + i (y_{5}, y_{6}, y_{7}, y_{8}), \end{matrix}$ the system of fuzzy complex linear Equation (24) can be transformed into system of fuzzy complex linear Equation (25).

$\begin{matrix} [(10 - 7.5 i) (x_{1}, x_{2}, x_{3}, x_{4}) + i (y_{1}, y_{2}, y_{3}, y_{4})] \\ - (6 - 5 i) [(x_{5}, x_{6}, x_{7}, x_{8}) \\ + i (y_{5}, y_{6}, y_{7}, y_{8})] \\ = [(4, 5, 5, 6) + i (- 1, 0, 0, 1)], \\ - (6 - 5 i) [(x_{1}, x_{2}, x_{3}, x_{4}) + i (y_{1}, y_{2}, y_{3}, y_{4})] \\ + (16 + 3 i) [(x_{5}, x_{6}, x_{7}, x_{8}) + i (y_{5}, y_{6}, y_{7}, y_{8})] \\ = [(- 2, - 1, - 1, 0) + i (- 3, - 2, - 2, - 1)] . \end{matrix}$ (25)

Step 2: Using Step 2 of the Keerat method, proposed in Section 4.2, the system of fuzzy complex linear Equation (25) can be transformed into system of fuzzy complex linear Equation (26).

$\begin{matrix} [10 (x_{1}, x_{2}, x_{3}, x_{4}) + 7.5 (y_{1}, y_{2}, y_{3}, y_{4}) \\ - 6 (x_{5}, x_{6}, x_{7}, x_{8}) - 5 (y_{5}, y_{6}, y_{7}, y_{8})] \\ + i [10 (y_{1}, y_{2}, y_{3}, y_{4}) - 7.5 (x_{1}, x_{2}, x_{3}, x_{4}) \\ - 6 (y_{5}, y_{6}, y_{7}, y_{8}) + 5 (x_{5}, x_{6}, x_{7}, x_{8})] \\ = [(4, 5, 5, 6) + i (- 1, 0, 0, 1)], \\ [- 6 (x_{1}, x_{2}, x_{3}, x_{4}) - 5 (y_{1}, y_{2}, y_{3}, y_{4}) \\ + 16 (x_{5}, x_{6}, x_{7}, x_{8}) - 3 (y_{5}, y_{6}, y_{7}, y_{8})] \\ + i [- 6 (y_{1}, y_{2}, y_{3}, y_{4}) + 5 (x_{1}, x_{2}, x_{3}, x_{4}) \\ + 16 (y_{5}, y_{6}, y_{7}, y_{8}) + 3 (x_{5}, x_{6}, x_{7}, x_{8})] \\ = [(- 2, - 1, - 1, 0) + i (- 3, - 2, - 2, - 1)] . \end{matrix}$ (26)

Step 3: Using Step 3 of the Keerat method, proposed in Section 4.2, the system of fuzzy complex linear Equation (26) can be transformed into system of fuzzy complex linear Equation (27).

$\begin{matrix} [(10 x_{1}, 10 x_{2}, 10 x_{3}, 10 x_{4}) \\ + (7.5 y_{1}, 7.5 y_{2}, 7.5 y_{3}, 7.5 y_{4}) \\ + (- 6 x_{8}, - 6 x_{7}, - 6 x_{6}, - 6 x_{5}) \\ + (- 5 y_{8}, - 5 y_{7}, - 5 y_{6}, - 5 y_{5})] \\ + i [(10 y_{1}, 10 y_{2}, 10 y_{3}, 10 y_{4}) \\ + (- 7.5 x_{4}, - 7.5 x_{3}, - 7.5 x_{2}, - 7.5 x_{1}) \\ + (- 6 y_{8}, - 6 y_{7}, - 6 y_{6}, - 6 y_{5}) \\ + (5 x_{5}, 5 x_{6}, 5 x_{7}, 5 x_{8})] \\ = [(4, 5, 5, 6) + i (- 1, 0, 0, 1)], \\ [(- 6 x_{4}, - 6 x_{3}, - 6 x_{2}, - 6 x_{1}) \\ + (- 5 y_{4}, - 5 y_{3}, - 5 y_{2}, - 5 y_{1}) \\ + (16 x_{5}, 16 x_{6}, 16 x_{7}, 16 x_{8}) \\ + (- 3 y_{8}, - 3 y_{7}, - 3 y_{6}, - 3 y_{5})] \\ + i [(- 6 y_{4}, - 6 y_{3}, - 6 y_{2}, - 6 y_{1}) \\ + (5 x_{1}, 5 x_{2}, 5 x_{3}, 5 x_{4}) \\ + (16 y_{5}, 16 y_{6}, 16 y_{7}, 16 y_{8}) \\ + (3 x_{5}, 3 x_{6}, 3 x_{7}, 3 x_{8})] \\ = [(- 2, - 1, - 1, 0) + i (- 3, - 2, - 2, - 1)] . \end{matrix}$ (27)

Step 4: Using Step 4 to Step 8 of the Keerat method, proposed in Section 4.2, the system of fuzzy complex linear Equation (27) can be transformed into system of crisp linear Equation (28). $\begin{matrix} 10 x_{1} - 6 x_{8} + 7.5 y_{1} - 5 y_{8} = 4, \\ 10 x_{2} - 6 x_{7} + 7.5 y_{2} - 5 y_{7} = 5, \\ 10 x_{3} - 6 x_{6} + 7.5 y_{3} - 5 y_{6} = 5, \\ 10 x_{4} - 6 x_{5} + 7.5 y_{4} - 5 y_{5} = 6, \\ - 6 x_{4} + 16 x_{5} - 5 y_{4} - 3 y_{8} = - 2, \\ - 6 x_{3} + 16 x_{6} - 5 y_{3} - 3 y_{7} = - 1, \\ - 6 x_{2} + 16 x_{7} - 5 y_{2} - 3 y_{6} = - 1, \\ - 6 x_{1} + 16 x_{8} - 5 y_{1} - 3 y_{5} = 0, \end{matrix} \begin{matrix} - 7.5 x_{4} + 5 x_{5} + 10 y_{1} - 6 y_{8} = - 1, \\ - 7.5 x_{3} + 5 x_{6} + 10 y_{2} - 6 y_{7} = 0, \\ - 7.5 x_{2} + 5 x_{7} + 10 y_{3} - 6 y_{6} = 0, \\ - 7.5 x_{1} + 5 x_{8} + 10 y_{4} - 6 y_{5} = 1, \\ 5 x_{1} + 3 x_{5} - 6 y_{4} + 16 y_{5} = - 3, \\ 5 x_{2} + 3 x_{6} - 6 y_{3} + 16 y_{6} = - 2, \\ 5 x_{3} + 3 x_{7} - 6 y_{2} + 16 y_{7} = - 2, \\ 5 x_{4} + 3 x_{8} - 6 y_{1} + 16 y_{8} = - 1 . \end{matrix}$ (28)

Step 5: The solution {x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈, y₁, y₂, y₃, y₄, y₅, y₆, y₇, y₈} of the system of crisp linear Equation (28) with the restrictions x₁ ≤ x₂ ≤ x₃ ≤ x₄, x₅ ≤ x₆ ≤ x₇ ≤ x₈ and y₁ ≤ y₂ ≤ y₃ ≤ y₄, y₅ ≤ y₆ ≤ y₇ ≤ y₈, is $\begin{matrix} x_{1} = \frac{16225328}{51288327}, x_{2} = \frac{42944}{121249}, x_{3} = \frac{42944}{121249}, \\ x_{4} = \frac{20105296}{51288327}, x_{5} = \frac{1778153}{51288327}, x_{6} = \frac{7930}{121249}, \\ x_{7} = \frac{7930}{121249}, x_{8} = \frac{4930627}{51288327}, y_{1} = \frac{3630080}{51288327}, \\ y_{2} = \frac{13168}{121249}, y_{3} = \frac{13168}{121249}, y_{4} = \frac{7510048}{51288327}, \\ y_{5} = - \frac{12204112}{51288327}, y_{6} = - \frac{25125}{121249}, y_{7} = - \frac{25125}{121249}, \\ y_{8} = - \frac{9051638}{51288327} . \end{matrix}$

Step 6: Using Step 10 of the Keerat method, proposed in Section 4.2, the solution of system of fuzzy complex linear Equation (24) is $\begin{matrix} {\tilde{z}}_{1} = (x_{1}, x_{2}, x_{3}, x_{4}) + i (y_{1}, y_{2}, y_{3}, y_{4}) \\ = (\frac{16225328}{51288327}, \frac{42944}{121249}, \frac{42944}{121249}, \frac{20105296}{51288327}) \\ + i (\frac{3630080}{51288327}, \frac{13168}{121249}, \frac{13168}{121249}, \frac{7510048}{51288327}), \end{matrix}$ $\begin{matrix} {\tilde{z}}_{2} = (x_{5}, x_{6}, x_{7}, x_{8}) + i (y_{5}, y_{6}, y_{7}, y_{8}) \\ = (\frac{1778153}{51288327}, \frac{7930}{121249}, \frac{7930}{121249}, \frac{4930627}{51288327}) \\ + i (- \frac{12204112}{51288327}, - \frac{25125}{121249}, - \frac{25125}{121249}, - \frac{9051638}{51288327}) . \end{matrix}$

6 Conclusion

On the basis of the present study, it can be concluded that some mathematical incorrect assumptions are used in the existing methods [16]. Therefore, it is not genuine to use the existing methods [16]. Further, two new methods, named as Mehar method and Keerat method, for solving different types of system of fuzzy complex linear equations are proposed. In future, by combining both the methods, a new method can be proposed for solving such system of linear equations, in which all the parameters and the decision variables are represented by fuzzy complex numbers.

Footnotes

Acknowledgments

The authors would like to thank the anonymous referees as well as Associate Editor “Prof. Tofigh Allahviranloo” for their valuable comments which have led to an improvement in both quality and clarity of the paper. Dr. Amit Kumar would like to acknowledge the adolescent inner blessings of Mehar (lovely daughter of his cousin sister Dr. Parmpreet Kaur). He believes that MATA VAISHNO DEVI has appeared on earth in the form of Mehar and without her blessings it would not be possible to think the ideas presented in this paper. The first author would also like to acknowledge Department of Science and Technology, Government of India for financial support vide reference no. SR/WOS-A/PM-1025/2015 under Women Scientist Scheme to carry out this work.

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