Least Squares Method for Solving Fuzzy LR Interval Algebraic Linear Systems

Abstract

We first investigate the solvability conditions of fuzzy LR interval algebraic linear systems with fuzzy LR interval coefficient matrix and fuzzy LR interval hand-right vector. Next, we compute the solution of FLRIALS (fuzzy LR interval algebraic linear systems) using the least squares approach. We show that solving the least squares problem is equivalent to solving a quadratic programming problem. Indeed, instead of the FLRIALS, this quadratic programming has a non-convex constraint. Thus, we relax this constraint to have a convex search space. Then, from the solution obtained from quadratic programming on convex search space, we give an approximate solution for FLRIALS.

Keywords:

• Fuzzy LR interval number
• fuzzy LR interval algebraic linear system
• least squares method
• non-convex quadratic programming

1. Introduction

Linear systems are one of the most important and usable problems in the field of engineering, economy, and other sciences. Since the parameters of systems are expressed by non-exact and fuzzy quantities in real problems, exhibition and offering methods are very important in solving such systems. Including of fuzzy LR linear systems can be cited to the application of the neural network in [1], financial mathematics, and economics in [2] and so on (see other applications in [3,4]).

There are many numerical methods for solving fuzzy linear systems. Ezzati [5] replaced an original system with two crisp linear systems using S-Matrices. Mansouri and Asady [6], using the Gaussian elimination, solved a fuzzy linear system with fuzzy variables. Karthik and Chandrasekaran [7], using the singular value decomposition, and Radhakrishnan et al. [8], using the LU Decomposition method, proposed a method to solve fully fuzzy linear systems.

Also, there are non-numerical methods for solving fuzzy linear systems. Mishmast et al. [9], with the introduction of three parameters fuzzy center, fuzzy support, and α-cut and Khezerloo et al. [10], using α-cut, solved fuzzy linear systems with fuzzy variables. Allahviranloo and Mikaeilvand [11], using multiobjective linear programming, and Pandit [12], using α-cut, solved fully fuzzy linear systems. Abbasbandy and Ezzati [13], with the parametrization of fuzzy LR trapezoidal numbers, proposed a new method to solve fuzzy algebraic linear systems. Also, Amirfakharian [14] considered a fuzzy algebraic linear system using a normal system to obtain an approximate solution of fuzzy algebraic linear systems. See other works [15–17].

Recently, Ghanbari and Mahdavi-Amiri [18] showed the efficiency of using the least squares method for solving other types of fuzzy LR linear systems. Also, for solving fuzzy LR linear systems, Ghanbari and Mahdavi-Amiri [19] proposed a method by using the ranking function and the ABS algorithm. Ghanbari et al. [20], by offering a new algorithm and using a least squares method and ABS approach, solved fuzzy LR linear systems with fuzzy triangular variables.

Ghanbari [21] gave the necessary and sufficient conditions for the solvability of fuzzy LR algebraic linear systems with fuzzy LR triangular parameters. He proposed an approximate solution for fuzzy LR algebraic linear systems (considering selecting the entering variable) using a special pivoting in simplex algorithm and solving a linear programming problem.

The rest of the paper is organized as follows. In Section 2, we briefly review some basic definitions. In Section 3, we provide some necessary and sufficient conditions for the solvability of fuzzy LR interval algebraic linear system (FLRIALS). We show that solving a least squares problem is equivalent to solving a quadratic programming problem. Indeed, instead of FLRIALS, this quadratic programming has a non-convex constraint. Thus, we relax this constraint to have a convex search space. Then, from the solution obtained from quadratic programming on the convex search space, we give an approximate solution for FLRIALS. In Section 4, by the numerical experiments, we show the appropriateness of our approximate solution in comparison with other approximate solutions. We conclude in Section 5.

2. Preliminaries

Here, some concepts, which are used in our paper, are described.

Definition 2.1

[22]

Fuzzy LR interval number $\tilde{A}=(a,b,\alpha ,\beta {)}_{LR}$ with ${\mu }_{\tilde{a}}(x)=\{\begin{array}{ll}\text{L}\left({\displaystyle \frac{a_l-x}{\alpha }}\right),& a_l-\alpha \le x\le a_l,\\ 1,& a_l\le x\le a_r,\\ \text{R}\left({\displaystyle \frac{x-a_r}{\beta }}\right),& a_r\le x\le a_r+\beta ,\\ 0,& o.w.\end{array}$ is called a fuzzy LR interval number, where $\alpha ,\beta \ge 0$, $a\le b$, and $L,R$ are generating functions nondecreasing and nonincreasing of ${\mathbb{R}}^{+}$ into $[0,1]$, respectively.

Remark 2.1

[22]

Let $\tilde{a}=(a^l,a^r,a^{\alpha },a^{\beta }{)}_{LR}$, let $\tilde{b}=(b^l,b^r,b^{\theta },b^{\gamma }{)}_{LR}$ and let $\delta \in \mathbb{R}$. Then

1. $\delta \tilde{a}=\{\begin{array}{ll}(\delta a^l,\delta a^r,\delta a^{\alpha },\delta a^{\beta }{)}_{LR},& \delta \ge 0,\\ (\delta a^r,\delta a^l,-\delta a^{\beta },-\delta a^{\alpha }{)}_{LR},& \delta \le 0.\end{array}$

2. $\tilde{a}⨁\tilde{b}=(a^l+b^l,a^r+b^r,a^{\alpha }+b^{\theta },a^{\beta }+b^{\gamma }{)}_{LR}$.

Remark 2.2

Here, we show the set of the fuzzy LR interval numbers by $\mathbb{I}({\mathbb{R}}^1{)}_{LR}$.

Definition 2.2

The system (1) $\tilde{A}x=\tilde{b},$(1) where ${\tilde{A}}_{m\times n}\in \mathbb{I}({\mathbb{R}}^{m\times n}{)}_{LR}$, $\tilde{b}\in \mathbb{I}({\mathbb{R}}^m{)}_{LR}$, and $x\in {\mathbb{R}}^n$. System (1(1) $\tilde{A}x=\tilde{b},$(1) ) is called FLRIALS (fuzzy LR interval algebraic linear systems).

Definition 2.3

Corresponding to $x=x^{+}-x^{-}(x^{+},x^{-}\ge 0)$ in system (1(1) $\tilde{A}x=\tilde{b},$(1) ), we define $x^{+}=max\{0,x\}$ and $x^{-}=x^{+}-x$. Corresponding to $\tilde{A}=({\tilde{a}}_{ij}{)}_{m\times n}$ and $\tilde{b}=({\tilde{b}}_i{)}_{m\times 1}$, we have ${\tilde{a}}_{ij}=(a_{ij}^l,a_{ij}^r,a_{ij}^{\alpha },a_{ij}^{\beta }{)}_{LR}$ and ${\tilde{b}}_i=(b_i^l,b_i^r,b_i^{\alpha },b_i^{\beta }{)}_{LR}$ for $i=1,\dots ,m$, $j=1,\dots ,n$.

3. Solvability FLRIALS

In this section, first, we conclude the proven results in [18,21] for FLRIALS. Then, the proofs of the results are done inspired by the proofs of the results in [18,21] and linear programming.

Theorem 3.1

Fundamental theorem FLRIALS

Let $\tilde{A}\in \mathbb{I}({\mathbb{R}}^{m\times n}{)}_{LR}$ and let $\tilde{b}\in \mathbb{I}({\mathbb{R}}^m{)}_{LR}$. Then $x\in {\mathbb{R}}^n$ is a solution to (1(1) $\tilde{A}x=\tilde{b},$(1) ) if and only if $({x^{+}}^T,{x^{-}}^T{)}^T\in {\mathbb{R}}^{2n}$ is a solution to the following problem: (2) $\begin{array}{rl}& \left[\begin{array}{cc}{A}^l& -A^r\\ A^r& -A^l\end{array}\right]\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]=\left[\begin{array}{c}{b}^l\\ b^r\end{array}\right],\end{array}$(2) (3) $\begin{array}{rl}& \left[\begin{array}{cc}{A}^{\alpha }& A^{\beta }\\ A^{\beta }& A^{\alpha }\end{array}\right]\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]=\left[\begin{array}{c}{b}^{\alpha }\\ b^{\beta }\end{array}\right],\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(3)

Theorem 3.2

If the following system has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) does not have a solution: $\{\begin{array}{l}{b^l}^{T}w+{b^r}^{T}u+{b^{\alpha }}^{T}s+{b^{\beta }}^{T}t<0,\\ {A^l}^{T}w+{A^r}^{T}u+{A^{\alpha }}^{T}s+{A^{\beta }}^{T}t\ge 0,\\ {-A^r}^{T}w-{A^l}^{T}u+{A^{\beta }}^{T}s+{A^{\alpha }}^{T}t\ge 0.\end{array}$

Corollary 3.1

If the system $\{\begin{array}{l}{b^{\alpha }}^{T}s+{b^{\beta }}^{T}t<0,\\ {A^{\alpha }}^{T}s+{A^{\beta }}^{T}t\ge 0,\\ {A^{\beta }}^{T}s+{A^{\alpha }}^{T}t\ge 0,\end{array}$ has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is not solvable.

Corollary 3.2

If the system $\{\begin{array}{l}{b^{\alpha }}^{T}s<0,\\ {A^{\alpha }}^{T}s\ge 0,\\ {A^{\beta }}^{T}s\ge 0,\end{array}$ has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is not solvable.

Corollary 3.3

If the system $\{\begin{array}{l}{b^{\beta }}^{T}t<0,\\ {A^{\beta }}^{T}t\ge 0,\\ {A^{\alpha }}^{T}t\ge 0,\end{array}$ has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is not solvable.

Corollary 3.4

If the system $\{\begin{array}{l}{b^{\alpha }}^{T}s<0,\\ {A^{\alpha }}^{T}s\ge 0,\\ {A^{\beta }}^{T}s\ge 0,\end{array}$ has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is not solvable.

Corollary 3.5

If the system $\{\begin{array}{l}{b^{\beta }}^{T}t<0,\\ {A^{\beta }}^{T}t\ge 0,\\ {A^{\alpha }}^{T}t\ge 0,\end{array}$ has a solution, then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is not solvable.

Theorem 3.3

Let $\begin{array}{rl}S& =\{(w^T,u^T,s^T,t^T{)}^T|{A^l}^{T}w+{A^r}^{T}u+{A^{\alpha }}^{T}s+{A^{\beta }}^{T}t\ge 0,\\ & {-A^r}^{T}w-{A^l}^{T}u+{A^{\beta }}^{T}s+{A^{\alpha }}^{T}t\ge 0,w,u,s,t\in {\mathbb{R}}^m\}.\end{array}$ Suppose that, for each $(w^T,u^T,s^T,t^T{)}^T\in S$, we have ${b^l}^{T}w+{b^r}^{T}u+{b^{\alpha }}^{T}s+{b^{\beta }}^{T}t\ge 0$ and there exists a vector $({\overline{w}}^T,{\overline{u}}^T,{\overline{s}}^T,{\overline{t}}^T{)}^T\in S$ such that the following properties hold:

(1)	${b^l}^T\overline{w}+{b^r}^T\overline{u}+{b^{\alpha }}^T\overline{s}+{b^{\beta }}^T\overline{t}=0$,
(2)	For each $j=1,\dots ,n$, if $({A^l}^T\overline{w}+{A^r}^T\overline{u}+{A^{\alpha }}^T\overline{s}+{A^{\beta }}^T\overline{t}{)}_j=0$, then $({-A^r}^T\overline{w}-{A^l}^T\overline{u}+{A^{\beta }}^T\overline{s}+{A^{\alpha }}^T\overline{t}{)}_j>0.$
(3)	For each $j=1,\dots ,n$, if $(-{A^r}^T\overline{w}-{A^l}^T\overline{u}+{A^{\beta }}^T\overline{s}+{A^{\alpha }}^T\overline{t}{)}_j=0$, then $({A^l}^T\overline{w}+{A^r}^T\overline{u}+{A^{\alpha }}^T\overline{s}+{A^{\beta }}^T\overline{t}{)}_j>0.$

Then system (1(1) $\tilde{A}x=\tilde{b},$(1) ) is solvable.

3.1. Least Squares Solution

Here, we propose a method to compute exact and approximate solutions of FLRIALS. Ghanbari and Mahdavi-Amiri [18] showed that least squares methods are more efficient than other methods. Therefore, we want to propose a least squares method to solve FLRILS.

For two fuzzy LR interval vectors $\tilde{x}$ and $\tilde{y}$, we compute the Ming et al.'s distance function [23] ${\tilde{x}}_j=(x_j^l,x_j^r,x_j^{\alpha },x_j^{\beta }{)}_{LR}$ and ${\tilde{y}}_j=(y_j^l,y_j^r,y_j^{\alpha },y_j^{\beta }{)}_{LR}$, for each $j=1,\dots ,n$, as follows: (4) $\begin{array}{rl}2D_n^2(\tilde{x},\tilde{y})& =2(x^l-y^l{)}^T(x^l-y^l)+2(x^r-y^r{)}^T(x^r-y^r)\\ & -2(x^l-y^l{)}^T(x^{\alpha }-y^{\alpha })+2(x^r-y^r{)}^T(x^{\beta }-y^{\beta })\\ & +(x^{\alpha }-y^{\alpha }{)}^T(x^{\alpha }-y^{\alpha })+(x^{\beta }-y^{\beta }{)}^T(x^{\beta }-y^{\beta }),\end{array}$(4) where $x^l=(x_1^l,\dots ,x_n^l{)}^T$, $y^l=(y_1^l,\dots ,y_n^l{)}^T$, $x^r=(x_1^r,\dots ,x_n^r{)}^T$, $y^r=(y_1^r,\dots ,y_n^r{)}^T$, $x^{\alpha }=(x_1^{\alpha },\dots ,x_n^{\alpha }{)}^T$, $y^{\alpha }=(y_1^{\alpha },\dots ,y_n^{\alpha }{)}^T$, $x^{\beta }=(x_1^{\beta },\dots ,x_n^{\beta }{)}^T$, and $y^{\beta }=(y_1^{\beta },\dots ,y_n^{\beta }{)}^T$. Now, we define the following residual at x, for each $x=x^{+}-x^{-}\in {\mathbb{R}}^n$: $r(x^{+},x^{-})=2D_n^2(\tilde{A}(x^{+}-x^{-}),\tilde{b}).$

Remark 3.1

The solution x is called exact for system (1(1) $\tilde{A}x=\tilde{b},$(1) ) if and only if $r(x^{+},x^{-})$ is equal to zero; otherwise, x is an approximate solution.

Thus, to compute a solution to (1(1) $\tilde{A}x=\tilde{b},$(1) ), the following optimization problem must be solved: (5) $\{\begin{array}{l}minr(x^{+},x^{-})=2D_n^2(\tilde{A}(x^{+}-x^{-}),\tilde{b})\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(5) Based on Definition 2.3 and by computing $\tilde{A}x$ of Remark 2.1, problem (5(5) $\{\begin{array}{l}minr(x^{+},x^{-})=2D_n^2(\tilde{A}(x^{+}-x^{-}),\tilde{b})\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(5) ) can be rewritten as (6) $\{\begin{array}{ll}minr(x^{+},x^{-})& =2(A^l{x}^{+}-A^r{x}^{-}-b^l{)}^T(A^l{x}^{+}-A^r{x}^{-}-b^l)\\ & +2(A^r{x}^{+}-A^l{x}^{-}-b^r{)}^T(A^r{x}^{+}-A^l{x}^{-}-b^r)\\ & +2(A^l{x}^{+}-A^r{x}^{-}-b^l{)}^T(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha })\\ & -2(A^r{x}^{+}-A^l{x}^{-}-b^r{)}^T(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta })\\ & +(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha }{)}^T(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha })\\ & +(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta }{)}^T(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta })\\ s.t.\\ x^{+},x^{-}\ge 0,& {x^{+}}^T{x}^{-}=0.\end{array}$(6) Let (7) $\begin{array}{rl}S& =4{A^l}^T{A}^l+4{A^r}^T{A}^r+2{A^{\alpha }}^T{A}^{\alpha }+2{A^{\beta }}^T{A}^{\beta }-2{A^l}^T{A}^{\alpha }+2{A^r}^T{A}^{\beta }\\ & -2{A^{\alpha }}^T{A}^l+2{A^{\beta }}^T{A}^r,\\ R& =-4{A^l}^T{A}^r-4{A^r}^T{A}^l+2{A^{\alpha }}^T{A}^{\beta }+2{A^{\beta }}^T{A}^{\alpha }-2{A^l}^T{A}^{\beta }+2{A^r}^T{A}^{\alpha }\\ & -2{A^{\beta }}^T{A}^l+2{A^{\alpha }}^T{A}^r,\\ T& =-4{A^l}^T{b}^l-4{A^r}^T{b}^r-2{A^{\alpha }}^T{b}^{\alpha }-2{A^{\beta }}^T{b}^{\beta }+2{A^l}^T{b}^{\alpha }+2{A^{\alpha }}^T{b}^l\\ & -2{A^r}^T{b}^{\beta }-2{A^{\beta }}^T{b}^r,\\ K& =4{A^r}^T{b}^l+4{A^l}^T{b}^r-2{A^{\alpha }}^T{b}^{\beta }-2{A^{\beta }}^T{b}^{\alpha }-2{A^r}^T{b}^{\alpha }+2{A^{\beta }}^T{b}^l\\ & +2{A^l}^T{b}^{\beta }-2{A^{\alpha }}^T{b}^r,\\ Q& ={\left[\begin{array}{cc}S& R\\ R& S\end{array}\right]}_{2n\times 2n},\end{array}$(7) (8) $\begin{array}{rl}f& ={\left[\begin{array}{c}T\\ K\end{array}\right]}_{2n\times 1},\end{array}$(8) and $c={2b^l}^T{b}^l+{2b^r}^T{b}^r+{b^{\alpha }}^T{b}^{\alpha }+{b^{\beta }}^T{b}^{\beta }-{2b^l}^T{b}^{\alpha }+2{b^r}^{T}b\beta .$ Then $r(x^{+},x^{-})$ in problem (6(6) $\{\begin{array}{ll}minr(x^{+},x^{-})& =2(A^l{x}^{+}-A^r{x}^{-}-b^l{)}^T(A^l{x}^{+}-A^r{x}^{-}-b^l)\\ & +2(A^r{x}^{+}-A^l{x}^{-}-b^r{)}^T(A^r{x}^{+}-A^l{x}^{-}-b^r)\\ & +2(A^l{x}^{+}-A^r{x}^{-}-b^l{)}^T(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha })\\ & -2(A^r{x}^{+}-A^l{x}^{-}-b^r{)}^T(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta })\\ & +(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha }{)}^T(A^{\beta }{x}^{-}+A^{\alpha }{x}^{+}-b^{\alpha })\\ & +(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta }{)}^T(A^{\alpha }{x}^{-}+A^{\beta }{x}^{+}-b^{\beta })\\ s.t.\\ x^{+},x^{-}\ge 0,& {x^{+}}^T{x}^{-}=0.\end{array}$(6) ) is equal to $r(x^{+},x^{-})=\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+c,$ where c is a constant number. Therefore, to compute the solution for FLRIALS, we must solve the following problem: (9) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ s.t.\\ x^{+},x^{-}\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(9) Since problem (9(9) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ s.t.\\ x^{+},x^{-}\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(9) ) contains ${x^{+}}^T{x}^{-}=0$, thus (9(9) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ s.t.\\ x^{+},x^{-}\ge 0,{x^{+}}^T{x}^{-}=0.\end{array}$(9) ) is not a convex programming problem, and it may solve difficulty [24]. Then, we consider the problem without the condition ${x^{+}}^T{x}^{-}=0$, as follows: (10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) For the obtained solution to problem (10(10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) ), if ${x^{+}}^T{x}^{-}=0$ is correct, then $x=x^{+}-x^{-}$ is an exact or approximate solution to (1(1) $\tilde{A}x=\tilde{b},$(1) ). Otherwise, we construct a new solution to (1(1) $\tilde{A}x=\tilde{b},$(1) ) by the following definition.

Definition 3.1

Let $y=x^{+}-x^{-}$. We define $y^{+}$ and $y^{-}$ as follows: (11) $y^{+}=max\{0,y\},y^{-}=y^{+}-y.$(11) Then, the new vector $y=y^{+}-y^{-}$ is an exact or approximate solution to system (1(1) $\tilde{A}x=\tilde{b},$(1) ).

3.2. Proposed Algorithm

Ghanbari and Mahdavi-Amiri [18], using an interior-point algorithm, proposed three methods to find initial feasible interior points, and they solved a quadratic programming problem similar to the quadratic programming problem (10(10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) ). Here, we want to solve FLRIALS inspired by the given approach in [18]. Thus, we define initial feasible interior points as follows:

1. Simple Initial Point (SIP). Let $x_{SIP}=(e_{n\times 1}^{+},e_{n\times 1}^{-}{)}^T$, where $e_{n\times 1}^{-}=(1,\dots ,1)$ and $e_{n\times 1}^{+}=(1,\dots ,1)$.

2. Karush–Kuhn–Tucker Initial Point (KKTIP). Let $x_{KKTIP}=arg\underset{\begin{array}{c}{x}^{+}\ge 0\\ x^{-}\ge 0\end{array}}{min}\Vert Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f{\Vert }_2,$ where $x_{KKTIP}=({x^{+}}^T,{x^{-}}^T{)}^T$, $x^{+}=({x^{+}}_1,\dots ,{x^{+}}_n{)}^T$, and $x^{-}=({x^{-}}_1,\dots ,{x^{-}}_n{)}^T$.

3. Least Squares Initial Point (LSIP). Let $x_{LSIP}=arg\underset{\begin{array}{c}{x}^{+}\ge 0\\ x^{-}\ge 0\end{array}}{min}\Vert A_{LS}\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]-b_{LS}{\Vert }_2,$ where $x_{LSIP}=({x^{+}}^T,{x^{-}}^T{)}^T,x^{+}=({x^{+}}_1,\dots ,{x^{+}}_n{)}^T$, $x^{-}=({x^{-}}_1,\dots ,{x^{-}}_n{)}^T$, and $A_{LS}=\left[\begin{array}{cc}{A}^l& -A^r\\ A^r& -A^l\\ A^{\alpha }& A^{\beta }\\ A^{\beta }& A^{\alpha }\end{array}\right],b_{LS}=\left[\begin{array}{c}{b}^l\\ b^r\\ b^{\alpha }\\ b^{\beta }\end{array}\right].$

We outline the steps of our algorithm to solve (1(1) $\tilde{A}x=\tilde{b},$(1) ) as follows.

4. Examples and Numerical Results

In the following examples, we show the superiority of our method compared to the proposed method of [13, 14].

Example 4.1

Consider $\{\begin{array}{l}(1,2,3,1{)}_{LR}x=(-4,-2,2,6{)}_{LR},\\ (-2,1,3,2{)}_{LR}x=(-2,4,4,6{)}_{LR},\end{array}$ where $L(x)=R(x)=1-x$. After solving the corresponding quadratic programming problem, we obtain $x=-2$ with $r(x^{+},x^{-})=0$. Then x is an exact solution to the system. Also, we solve the system by the method of Amirfakharian [14], and we obtain $x=-2$ with $r(x^{+},x^{-})=0$. Indeed, according to the proposed method of [13], the system is not solvable. This example shows that two methods (Algorithm 1 and the proposed method of [14]) obtain exact solution, but the proposed method of [13] is not efficiency.

Example 4.2

Consider $\{\begin{array}{l}(1,2,2,1{)}_{LR}{x}_1+(2,4,1,3{)}_{LR}{x}_2=(5,10,4,7{)}_{LR},\\ (-1,1,1,2{)}_{LR}{x}_1+(2,3,1,2{)}_{LR}{x}_3=(1,4,2,4{)}_{LR},\\ (1,2,2,4{)}_{LR}{x}_2+(2,2,2,1{)}_{LR}{x}_3=(4,6,6,9{)}_{LR}.\end{array}$ After solving the corresponding quadratic programming problem, we obtain $x=(x_1,x_2,x_3)=(1,2,1)$ with $r(x^{+},x^{-})=0$. Then x is an exact solution of the system. Indeed, by using the method of [14], we obtain $x=(x_1,x_2,x_3)\simeq (1,2,1)$ with $r(x^{+},x^{-})=2.27E-13$. This example shows that the solution obtained by Algorithm 1 is more accurate than the solution obtained by using the method of [14].

Example 4.3

Consider $\{\begin{array}{l}(-5,10,1,2{)}_{LR}{x}_1+(2,14,1,3{)}_{LR}{x}_2+(-5,-3,1,1{)}_{LR}{x}_3=(2,6,2,4{)}_{LR},\\ (0,1,2,1{)}_{LR}{x}_2+(3,5,2,4{)}_{LR}{x}_3=(-5,-2,6,1{)}_{LR}.\end{array}$ After resolving the corresponding quadratic programming problem, we obtain $x=(x_1,x_2,x_3)\simeq (0,0.03,-0.64)$ with $r(x^{+},x^{-})=4.85$. Hence x is an approximate solution of the system. We solve the system by using the method of [14]. Then $x=(x_1,x_2,x_3)\simeq (-0.04,0.04,-1.22)$ with $r(x^{+},x^{-})=11.23$ is obtained. This example shows that the solution obtained by Algorithm 1 is more accurate than the solution obtained by using the method of [14].

Here, some large-scale numerical studies are reported. Inspired by [18], we generate the matrix $\tilde{A}=(A^l,A^r,A^{\alpha },A^{\beta }{)}_{LR}$ in (1(1) $\tilde{A}x=\tilde{b},$(1) ) in different cases. All cases are similar to cases given in [18] but in the trapezoidal form. Also, we here consider only the first two categories in [18]. Ghanbari and Mahdavi-Amiri proposed a formula to compute the relative error (see [18, Equation (41)]). We also use this formula to compare the numerical results.

We give all results in the following tables. All implementations were done in MATLAB environment version 7.14.0 and on a notebook Intel Core i5-4200M 2.50GHZ with 6 GB of RAM.

Table 1 shows the mean relative errors of Algorithm 1 on Category 1 with three initial points SIP, KKTIP, LSIP, and Amirfakharian's method [14]. Our results show that the SIP method has a better performance compared to the others in all different cases and that Amirfakharian's method [14] is not efficient numerically.

Table 1. The mean relative error for Category 1.

Sizes	Rank	Scale	SIP	KKTIP	LSIP	Amirfakharian [14]
		small	1.30E+03	$\mathbf{5.49}\mathbf{E}-\mathbf{15}$	7.11E−15	2.66E+00
$m=n$	Full rank	medium	1.38E+04	$\mathbf{5.86}\mathbf{E}-\mathbf{03}$	6.24E−03	2.93E−01
		large	2.79E+04	1.10E−02	6.39E−03	1.25E−01
		small	1.30E0.3	8.92E−15	6.47E−15	2.73E+00
m = n	Deficient rank	medium	1.38E+04	1.78E−03	5.57E−03	2.88E−01
		large	2.79E+04	1.36E−02	7.11E−03	1.29E−01
		small	6.65E−07	1.59E−05	5.18E−03	3.48E+00
m<n	Full row rank	medium	7.70E−03	1.51E−02	1.22E−01	5.38E−01
		large	1.73E−02	9.90E−02	6.77E−02	1.29E−01
		small	8.86E−07	2.93E−05	6.91E−03	3.60E+00
m<n	Deficient rank	medium	2.80E−03	5.60E−02	4.83E−02	5.07E−01
		large	1.63E−02	1.04E−01	6.37E−02	1.15E−01
		small	2.74E−06	4.23E−06	2.36E−06	3.05E+00
m>n	Full column rank	medium	1.43E−05	2.94E−05	1.16E−04	3.81E−01
		large	1.45E−02	1.96E−02	6.35E−03	1.16E−01
		small	6.11E−07	6.74E−06	1.41E−06	3.02E+00
m>n	Deficient rank	medium	7.52E−07	1.24E−04	1.33E−04	3.79E−01
		large	2.36E−03	3.15E−02	6.54E−03	1.15E−01

For Category 2, the mean relative errors are shown in Table 2. It is obvious that the performance of SIP and LSIP is better than Amirfakharian's method [14].

Table 2. The mean relative error for Category 2.

Sizes	Rank	Scale	SIP	KKTIP	LSIP	Amirfakharian [14]
		small	1.30E+03	2.23E−15	9.64E−15	2.66E+00
m = n	Full rank	medium	1.38E+04	4.16E−03	6.31E−03	2.94E−01
		large	2.79E+04	7.11E−02	5.40E−03	1.24E−01
		small	1.30E+03	9.19E−15	3.73E−15	2.73E+00
m = n	Deficient rank	medium	1.38E+04	5.06E−05	5.55E−03	2.88E−01
		large	2.79E+04	6.36E−02	7.06E−03	1.29E−01
		small	6.74E−07	8.11E−07	5.18E−03	3.48E+00
m<n	Full row rank	medium	2.70E−03	9.76E−02	1.16E−01	5.29E−01
		large	1.73E−02	9.90E−02	6.77E−02	1.29E−01
		small	1.36E−06	1.69E−07	1.22E−07	1.10E+00
$m<n$	Deficient rank	medium	3.52E−03	9.91E−02	8.77E−03	4.33E−01
		large	7.38E−03	8.34E−02	2.51E−02	1.13E−01
		small	1.24E−05	1.71E−05	7.36E−06	2.35E+00
$m>n$	Full column rank	medium	9.80E−07	1.93E−06	6.57E−07	2.72E+00
		large	1.78E−07	9.47E−08	3.59E−03	5.98E−01
		small	3.16E−06	9.77E−06	8.47E−06	2.59E+00
$m>n$	Deficient rank	medium	2.11E−05	1.16E−05	1.02E−06	2.50E+00
		large	8.24E−08	1.17E−02	1.39E−06	6.03E−01

Therefore, from our numerical study, we can say our proposed method is very more efficient and robust than Amirfakharian's method [14].

Remark 4.1

In our numerical study, we always have ${x^{+}}^T{x}^{-}=0$ after solving (10(10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) ). In other words, in all test problems when we solved (10(10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) ), we have ${x^{+}}^T{x}^{-}=0$. So, the ‘otherwise’ part of step 5 in Algorithm 1 was not called in all our test problems from the obtained solution to (10(10) $\{\begin{array}{l}min\frac{1}{2}\left[\begin{array}{cc}{x^{+}}^T& {x^{-}}^T\end{array}\right]Q\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]+f^T\left[\begin{array}{c}{x}^{+}\\ x^{-}\end{array}\right]\\ \mathrm{s}.\mathrm{t}.\\ x^{+},x^{-}\ge 0.\end{array}$(10) ), but we could not prove it in general form. Moreover, we could not find a counterexample.

5. Conclusion

In this paper, we first investigated the solvability conditions of FLRIALS with fuzzy LR interval coefficients matrix and fuzzy LR interval hand-right vector. Next, we computed a solution to FLRIALS using a least squares approach. We showed that solving the least squares problem is equivalent to solving a quadratic programming problem. Nevertheless, instead of FLRIALS, this quadratic programming has a non-convex constraint. Thus, we relaxed this constraint to have a convex search space. Then, from the solution obtained from quadratic programming on a convex search space, we gave an approximate solution for FLRIALS.

Acknowledgment

The first and second authors thank the Research Council of Ferdowsi University of Mashhad and Optimization Laboratory of Ferdowsi University of Mashhad and the third author thanks the Mosaheb Institute of Mathematics, Kharazmi University for supporting this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Mehrnoosh Salari

Mehrnoosh Salari graduated student of Ferdowsi University of Mashhad, Iran.

Reza Ghanbari

Reza Ghanbari is an Associate Professor of Mathematical Sciences at Ferdowsi University of Mashhad, Iran. He received his B.S. degree in Applied Mathematics from Ferdowsi University of Mashhad, Iran, in 2002, and his M.S. and Ph.D. degrees in Applied Mathematics from Sharif University of Technology, Iran, in 2004 and 2009, respectively. He is president of Khorasan science and technology park. He also is the manager of the optimization laboratory of Ferdowsi University. His research interests include algorithmic operational research, optimization and soft computing.

Khatere Ghorbani-Moghadam

Khatere Ghorbani-Moghadam is Assistant professor of Mosaheb Institute of Mathematics, Kharazmi University, Tehran, Iran. She received her Ph.D. from Mathematical Sciences in Applied Mathematics at Sharif University of Technology, Iran in 2018. She received her B.S. and M.S. degrees in Applied Mathematics from Ferdowsi University of Mashhad, Iran, in 2009 and 2011, respectively.