New Method for Solving Fuzzy LR Interval Linear Systems Using Least Squares Models

Abstract

Recently, Ghanbari and Mahdavi-Amiri [2015, Soft Computing] have proposed a model to characterize exact solutions and presented an algorithm to compute approximate solutions. Here, inspired by Ghanbari and Mahdavi-Amiri, we show that solving fuzzy LR interval linear systems is equivalent to solving fuzzy LR linear systems (FLRLSs). Then we develop some necessary and sufficient conditions for the solvability of FLRLSs. We then provide a new general concept for an approximate solution when an FLRLS lacks a solution. To compute an approximate solution, we propose an algorithm using a least squares method. Finally, we show the appropriateness of our proposed approximate solution recently introduced numerically.

Keywords:

• Fuzzy LR interval number
• fuzzy LR interval linear system
• quadratic programming
• least squares method

1. Introduction

Fuzzy linear systems (FLSs) are one of the most important problems in many branches of science and technology, such as economics, engineering, and so on (see [1–4]). There are many numerical methods to solve FLSs. Zheng and Wang [5] solved FLSs by using the generalized inverses of the coefficient matrix. Ezzati [6] developed a method for solving FLSs by using the embedding approach. Otadi and Mosleh [7] used a parametric form of fuzzy number and converted an FLS into two linear systems in crisp cases.

Also, there are many non-numerical methods to solve FLSs. Allahviranloo and Salahshour [8] proposed a simple and practical method to compute maximal, minimal, and symmetric solutions of FLSs. Allahviranloo et al. [9] proposed a metric based on a modified Euclidean metric for solving fuzzy LR linear systems (FLRLSs) such that an FLRLS is transformed into a minimization problem. Allahviranloo and Ghanbari [10] proposed a new approach based on interval theory and the new concept of interval inclusion linear system.

Stanimirović and Micić [11] described the set of fuzzy relations that solve weakly linear systems to a certain degree and provided ways to compute them. They paid special attention to developing the algorithms for computing fuzzy preorders and fuzzy equivalences that are solutions to some extent to weakly linear systems. Stanimirović and Micić [11] established additional properties for the set of such approximate solutions over some particular types of complete residuated lattices. They demonstrated the advantage of this approach via many examples that arise from the problem of aggregation of fuzzy networks. Zarei et al. [12] studied first-order linear fuzzy systems under generalized differentiability and presented the general form of their solutions. Then the fuzzy optimal control problem of these systems was considered to optimize the expected values of the appropriate objective fuzzy functions. The Pontryagin maximum principle was used to obtain a necessary optimality condition in the form of a fuzzy boundary value problem. Using the necessary optimality condition, the constant formulas for the fuzzy optimal control function and the corresponding fuzzy state function were proposed. Abbasi and Allahviranloo [13] presented a method for solving the fully fuzzy linear systems by using the transmission-average-based fuzzy operations, which were introduced by Abbasi et al. [14]. Also, they presented the necessary conditions for the existence and uniqueness of the fuzzy solution. Ghanbari et al. [15] solved a dual fuzzy linear system algebraically. In considered systems, the coefficient matrices are crisp-valued matrices, and the left- and right-hand sides vectors are fuzzy number-valued vectors. Two types of solutions were defined, and the relationship between them was investigated. Based on the obtained results, a simple method was presented to obtain a unique algebraic solution for a dual fuzzy linear system. The main advantage of their proposed method over existing methods is that it does not need to convert a dual fuzzy linear system to two crisp linear systems. Phu and Hung [16] gave a new concept that was the minimum stability control problems for fuzzy linear control systems, and our observations revolve around this object. At the same time, they studied the existence of solutions for some kinds of this problem by generalized Hukuhara derivative. In addition, they presented the time-optimal fuzzy control problem for these problems. Also, they proposed an algorithm to find the time-optimal. Gasilov et al. [17] proposed a new solution method for a non-homogeneous fuzzy linear system of differential equations. The coefficients of the considered system were crisp, while forcing functions and initial values were fuzzy. They assumed each forcing function be in a special form, which they called a triangular fuzzy function and which represented a fuzzy bunch (set) of real functions. They constructed a solution as a fuzzy set of real vector-functions, not as a vector of fuzzy-valued functions, as usual. See other works for solving FLSs in [18–22].

All of these methods were based on the crisp or fuzzy linear algebraic methods. Indeed, we propose an approximate solution based on a least squares model and the distance function proposed by Ming [23]. To compute an approximate solution, we propose a quadratic programming model with linear constraints. Ghanbari et al. [24] solved FLRLSs by offering the new algorithm, using a least squares method and the ABS approach. Ghanbari and Mahdavi-Amiri [25] proposed an approach to compute the solution of FLRLSs by using a ranking function and ABS algorithm. So, Ghanbari and Mahdavi-Amiri [26] first established some necessary and sufficient conditions for the solvability of FLRLSs. Next, they proposed an algorithm based on a least squares method to compute an exact or approximate solution and showed that solving a least squares problem is equivalent to solving a quadratic 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 linear systems (FLRILSs). In Section 4, we show that solving an FLRILS is equivalent to solving an FLRLS, and we provide the necessary and sufficient conditions for the solvability of FLRLSs. In Section 5, we propose a new concept for an approximate solution of a fuzzy linear system and present a quadratic programming model to compute such solutions. In Section 6, by the numerical experiments, we show the appropriateness of our approximate solution in comparison with other approximate solutions. We conclude in Section 7.

2. Preliminaries

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

Definition 2.1

[27]

$\tilde{A}=(a,\alpha ,\beta {)}_{LR}$ with the following function $\begin{array}{r}{\mu }_{\tilde{A}}(x)=\{\begin{array}{lll}L({\displaystyle \frac{x-a}{\alpha }}),& x\le a,R({\displaystyle \frac{a-x}{\beta }}),& x\ge b,\end{array}\end{array}$ is called a fuzzy LR number, where $\alpha \ge 0$ and $\beta \ge 0$.

Definition 2.2

[27]

$\tilde{A}=(a,b,\alpha ,\beta {)}_{LR}$ with the following function $\begin{array}{r}{\mu }_{\tilde{A}}(x)=\{\begin{array}{llll}L({\displaystyle \frac{a-x}{\alpha }}),& x\le a,1,& a\le x\le b,R({\displaystyle \frac{x-b}{\beta }}),& x\ge b,\end{array}\end{array}$ is called a fuzzy LR interval number, where $\alpha \ge 0$, $\beta \ge 0$, and $b\ge a$.

Remark 2.1

[27]

For two fuzzy LR interval numbers $\tilde{a}=(a^l,a^r,a^{\alpha },a^{\beta }{)}_{LR}$ and $\tilde{b}=(b^l,b^r,b^{\theta },b^{\gamma }{)}_{LR}$ and $\delta \in \mathbb{R}$, we have the following properties:

1. $\delta \ge 0⟹\delta \tilde{a}=(\delta a^l,\delta a^r,\delta a^{\alpha },\delta a^{\beta }{)}_{LR}$.

2. $\delta \le 0⟹\delta \tilde{a}=(\delta a^r,\delta a^l,-\delta a^{\beta },-\delta a^{\alpha }{)}_{LR}$.

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

Remark 2.2

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

3. Solvability of Fuzzy LR Interval Linear System

Here, we introduce a type of FLRILSs with a crisp coefficient matrix. Next, we give a theorem for the solvability of FLRILSs, where the proof of the theorem is done inspired by the proof of the fundamental theorem given in [26].

Definition 3.1

The following system (1) $A\tilde{x}=\tilde{b},$(1) where $A\in {\mathbb{R}}^{m\times n}$, $\tilde{b}\in \mathbb{I}({\mathbb{R}}^m{)}_{LR}$, and $\tilde{x}\in \mathbb{I}({\mathbb{R}}^n{)}_{LR}$, is called an FLRILS.

Definition 3.2

We define the following matrices: (2) $\begin{array}{r}(B^{+}{)}_{ij}=\{\begin{array}{ll}{a}_{ij},& a_{ij}\ge 0,\\ 0,& a_{ij}<0,\end{array}(B^{-}{)}_{ij}=\{\begin{array}{ll}0,& a_{ij}\ge 0,\\ a_{ij},& a_{ij}<0,\end{array}\end{array}$(2) for every $i=1,\dots ,m$ and $j=1,\dots ,n$.

Theorem 3.1

Fundamental Theorem for FLRILSs

Let $A\in {\mathbb{R}}^{m\times n}$ and $\tilde{b}\in \mathbb{I}({\mathbb{R}}^m{)}_{LR}$. Then $\tilde{x}\in \mathbb{I}({\mathbb{R}}^n{)}_{LR}$ is a solution to (1(1) $A\tilde{x}=\tilde{b},$(1) ) if and only if $(x^{l^T},x^{r^T}{)}^T$ and $(x^{{\alpha }^T},x^{{\beta }^T}{)}^T$ are, respectively, solutions to the following systems: (3) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{-}{x}^l+B^{+}{x}^r=b^r,\\ x^l\le x^r,\end{array}\end{array}$(3) and (4) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^{\alpha }-B^{-}{x}^{\beta }=b^{\alpha },\\ -B^{-}{x}^{\alpha }+B^{+}{x}^{\beta }=b^{\beta },\\ x^{\alpha },x^{\beta }\ge 0.\end{array}\end{array}$(4)

Proof.

The proof is similar to the proof of the fundamental theorem given in [26].

4. Solvability of Fuzzy LR Interval Converted Linear System

In this section, we first define fuzzy LR interval converted linear systems (FLRICLSs). Next, we show that solving an FLRILS is equivalent to solving the corresponding FLRCLS (FLRCLSs are a type of FLRLSs). Finally, we compute an exact or approximate solution for FLRICLSs by using Ghanbari and Mahdavi-Amiri's method.

Definition 4.1

Fuzzy LR interval converted number

For each fuzzy LR interval number $\tilde{a}=(a^l,a^r,a^{\alpha },a^{\beta }{)}_{LR}$, we define ${\tilde{a}}_{left}=(a^l,a^{\alpha },a^r-a^l+a^{\beta }{)}_{LR},{\tilde{a}}_{right}=(a^r,a^r-a^l+a^{\alpha },a^{\beta }{)}_{LR}.$

Note 1

By the definition of matrices $B^{+}$ and $B^{-}$ in (2(2) $\begin{array}{r}(B^{+}{)}_{ij}=\{\begin{array}{ll}{a}_{ij},& a_{ij}\ge 0,\\ 0,& a_{ij}<0,\end{array}(B^{-}{)}_{ij}=\{\begin{array}{ll}0,& a_{ij}\ge 0,\\ a_{ij},& a_{ij}<0,\end{array}\end{array}$(2) ), we define the matrix $A_{new}$ as follows: (5) $\begin{array}{r}{A}_{new}={\left[\begin{array}{cc}{B}^{+}& B^{-}\\ B^{-}& B^{+}\end{array}\right]}_{2m\times 2n}.\end{array}$(5) Then (6) $\begin{array}{r}{B}_{new}^{+}={\left[\begin{array}{cc}{B}^{+}& 0\\ 0& B^{+}\end{array}\right]}_{2m\times 2n},B_{new}^{-}={\left[\begin{array}{cc}0& B^{-}\\ B^{-}& 0\end{array}\right]}_{2m\times 2n}.\end{array}$(6)

Note 2

From Definition 4.1, for two fuzzy LR interval vectors $\tilde{x}$ and $\tilde{b}$ in (1(1) $A\tilde{x}=\tilde{b},$(1) ), we can write $\begin{array}{rl}{\tilde{x}}_{left}& =(x^l,x^{\alpha },x^r-x^l+x^{\beta }{)}_{LR},{\tilde{x}}_{right}=(x^r,x^r-x^l+x^{\alpha },x^{\beta }{)}_{LR},\\ {\tilde{b}}_{left}& =(b^l,b^{\alpha },b^r-b^l+b^{\beta }{)}_{LR},{\tilde{b}}_{right}=(b^r,b^r-b^l+b^{\alpha },b^{\beta }{)}_{LR}.\end{array}$ Now, let (7) $\begin{array}{r}{\tilde{x}}_{new}=\left[\begin{array}{c}{\tilde{x}}_{left}\\ {\tilde{x}}_{right}\end{array}\right],{\tilde{b}}_{new}=\left[\begin{array}{c}{\tilde{b}}_{left}\\ {\tilde{b}}_{right}\end{array}\right],\end{array}$(7) where ${\tilde{x}}_{new}=(x_{new},x_{new}^{\alpha },x_{new}^{\beta }{)}_{LR}$ and ${\tilde{b}}_{new}=(b_{new},b_{new}^{\alpha },b_{new}^{\beta }{)}_{LR}$. Then $\begin{array}{rl}{x}_{new}& =\left[\begin{array}{c}{x}^l\\ x^r\end{array}\right],x_{new}^{\alpha }=\left[\begin{array}{c}{x}^{\alpha }\\ x^r-x^l+x^{\alpha }\end{array}\right],x_{new}^{\beta }=\left[\begin{array}{c}{x}^r-x^l+x^{\beta }\\ x^{\beta }\end{array}\right],\\ b_{new}& =\left[\begin{array}{c}{b}^l\\ b^r\end{array}\right],b_{new}^{\alpha }=\left[\begin{array}{c}{b}^{\alpha }\\ b^r-b^l+b^{\alpha }\end{array}\right],b_{new}^{\beta }=\left[\begin{array}{c}{b}^r-b^l+b^{\beta }\\ b^{\beta }\end{array}\right].\end{array}$

Definition 4.2

The new system (8) $A_{new}{\tilde{x}}_{new}={\tilde{b}}_{new}$(8) is called an FLRICLS.

Now, we show that solving an FLRILS is equivalent to solving an FLRICLS by using the following theorems.

Theorem 4.1

Extended Fundamental Theorem for FLRLSs

Consider $A_{new}\in {\mathbb{R}}^{2m\times 2n}$ and ${\tilde{b}}_{new}\in \mathbb{F}({\mathbb{R}}^{2m}{)}_{LR}$. Then ${\tilde{x}}_{new}\in \mathbb{F}({\mathbb{R}}^{2n}{)}_{LR}$ is a solution to (8(8) $A_{new}{\tilde{x}}_{new}={\tilde{b}}_{new}$(8) ) if and only if $(x^{l^T},x^{r^T}{)}^T$ and $(x_{new}^{{\alpha }^T},x_{new}^{{\beta }^T}{)}^T$ are, respectively, solutions for the following two systems: (9) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{+}{x}^r+B^{-}{x}^l=b^r,\\ x^l\le x^r.\end{array}\end{array}$(9) and (10) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta },\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0.\end{array}\end{array}$(10)

Proof.

The proof is very similar to the proof of fundamental theorem given in [26].

Theorem 4.2

It holds that $\tilde{x}=(x^l,x^r,x^{\alpha },x^{\beta })$ is a solution to (1(1) $A\tilde{x}=\tilde{b},$(1) ) if and only if $(x^{l^T},x^{r^T}{)}^T$ and $(x_{new}^{{\alpha }^T},x_{new}^{{\beta }^T}{)}^T$ are, respectively, solution to (9(9) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{+}{x}^r+B^{-}{x}^l=b^r,\\ x^l\le x^r.\end{array}\end{array}$(9) ) and (10(10) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta },\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0.\end{array}\end{array}$(10) ).

Proof.

We must show that $\tilde{x}$ is a solution to (1(1) $A\tilde{x}=\tilde{b},$(1) ) if and only if $(x^{l^T},x^{r^T}{)}^T$ and $(x_{new}^{{\alpha }^T},x_{new}^{{\beta }^T}{)}^T$ are, respectively, solutions to (9(9) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{+}{x}^r+B^{-}{x}^l=b^r,\\ x^l\le x^r.\end{array}\end{array}$(9) ) and (10(10) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta },\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0.\end{array}\end{array}$(10) ).

Based on the fundamental theorem for FLRILSs, $\tilde{x}$ is a solution to (1(1) $A\tilde{x}=\tilde{b},$(1) ) if and only if $(x^{l^T},x^{r^T}{)}^T$ and $(x^{{\alpha }^T},x^{{\beta }^T}{)}^T$ are, respectively, solutions to (3(3) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{-}{x}^l+B^{+}{x}^r=b^r,\\ x^l\le x^r,\end{array}\end{array}$(3) ) and (4(4) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^{\alpha }-B^{-}{x}^{\beta }=b^{\alpha },\\ -B^{-}{x}^{\alpha }+B^{+}{x}^{\beta }=b^{\beta },\\ x^{\alpha },x^{\beta }\ge 0.\end{array}\end{array}$(4) ). Then $x_{new}^{\alpha },x_{new}^{\beta }\ge 0$. In the other hand, system (3(3) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{-}{x}^l+B^{+}{x}^r=b^r,\\ x^l\le x^r,\end{array}\end{array}$(3) ) is solvable if and only if the system (9(9) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{+}{x}^r+B^{-}{x}^l=b^r,\\ x^l\le x^r.\end{array}\end{array}$(9) ) is solvable. Also, if system (3(3) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{-}{x}^l+B^{+}{x}^r=b^r,\\ x^l\le x^r,\end{array}\end{array}$(3) ) is solvable, then it can be written (11) $B^{+}{x}^r+B^{-}{x}^l-B^{+}{x}^l-B^{-}{x}^r=b^r-b^l.$(11) Now, by using the solvability of (4(4) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^{\alpha }-B^{-}{x}^{\beta }=b^{\alpha },\\ -B^{-}{x}^{\alpha }+B^{+}{x}^{\beta }=b^{\beta },\\ x^{\alpha },x^{\beta }\ge 0.\end{array}\end{array}$(4) ) and (11(11) $B^{+}{x}^r+B^{-}{x}^l-B^{+}{x}^l-B^{-}{x}^r=b^r-b^l.$(11) ), we have (12) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta }.\end{array}\end{array}$(12) Clearly, system (10(10) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta },\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0.\end{array}\end{array}$(10) ) is solvable if and only if system (12(12) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta }.\end{array}\end{array}$(12) ) is solvable. Therefore, we show that $\tilde{x}$ is a solution to (1(1) $A\tilde{x}=\tilde{b},$(1) ) if and only if $x_{new}=(x^{l^T},x^{r^T}{)}^T$ and $(x_{new}^{{\alpha }^T},x_{new}^{{\beta }^T}{)}^T$ are, respectively, solutions for (9(9) $\begin{array}{r}\{\begin{array}{l}{B}^{+}{x}^l+B^{-}{x}^r=b^l,\\ B^{+}{x}^r+B^{-}{x}^l=b^r,\\ x^l\le x^r.\end{array}\end{array}$(9) ) and (10(10) $\begin{array}{r}\{\begin{array}{l}{B}_{new}^{+}{x}_{new}^{\alpha }-B_{new}^{-}{x}_{new}^{\beta }=b_{new}^{\alpha },\\ B_{new}^{+}{x}_{new}^{\beta }-B_{new}^{-}{x}_{new}^{\alpha }=b_{new}^{\beta },\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0.\end{array}\end{array}$(10) ), and the proof is complete.

Remark 4.1

Based on the mentioned theorems, FLRILS (1(1) $A\tilde{x}=\tilde{b},$(1) ) with n variables is equal to FLRICLS (8(8) $A_{new}{\tilde{x}}_{new}={\tilde{b}}_{new}$(8) ) with $2n$ variables.

5. Least Squares Solution

Ghanbari and Mahdavi-Amiri [26] proposed a method based on the least squares method to solve an FLRLS and showed that solving a least squares problem is equivalent to solving a quadratic programming problem. Then to compute an exact or approximate solution for system (8(8) $A_{new}{\tilde{x}}_{new}={\tilde{b}}_{new}$(8) ), we can solve the following quadratic programming problem by using Ghanbari and Mahdavi-Amiri's method [26]: (13) $\begin{array}{r}\{\begin{array}{l}min\frac{1}{2}[x_{new}^T{x}_{new}^{{\alpha }^T}{x}_{new}^{{\beta }^T}]Q\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+f^T\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]\\ s.t.\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0,\end{array}\end{array}$(13) where (14) $\begin{array}{r}r({\tilde{x}}_{new})=\frac{1}{2}[x_{new}^T{x}_{new}^{{\alpha }^T}{x}_{new}^{{\beta }^T}]Q\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+f^T\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+c,\end{array}$(14) in which from [26], (15) $\begin{array}{rl}Q& =\left[\begin{array}{ccc}8A^{T}A& -2A^{T}A& 2A^{T}A\\ -2A^{T}A& 2(B^{{+}^T}{B}^{+}+B^{{-}^T}{B}^{-})& -2(B^{{+}^T}{B}^{-}+B^{{-}^T}{B}^{+})\\ 2A^{T}A& -2(B^{{+}^T}{B}^{-}+B^{{-}^T}{B}^{+})& 2(B^{{+}^T}{B}^{+}+B^{{-}^T}{B}^{-})\end{array}\right],\end{array}$(15) (16) $\begin{array}{rl}f& =\left[\begin{array}{c}-8A^{T}b+2A^T{b}^l-2A^T{b}^r\\ 2A^{T}b-2B^{{+}^T}{b}^l+2B^{{-}^T{b}^r}\\ -2A^{T}b+2B^{{-}^T{b}^l}-2B^{{+}^T{b}^r}\end{array}\right],\end{array}$(16) (17) $\begin{array}{rl}c& =4b^{T}b+b^{l^T}{b}^l+2b^T(b^r-b^l).\end{array}$(17) Now, we suppose that ${\tilde{x}}_{new}$ is the solution obtained from solving quadratic programming (13(13) $\begin{array}{r}\{\begin{array}{l}min\frac{1}{2}[x_{new}^T{x}_{new}^{{\alpha }^T}{x}_{new}^{{\beta }^T}]Q\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+f^T\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]\\ s.t.\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0,\end{array}\end{array}$(13) ) by using Ghanbari and Mahdavi-Amiri's method [26]. Then we compute an exact or approximate solution for system (1(1) $A\tilde{x}=\tilde{b},$(1) ) by using the following definition.

Definition 5.1

Let ${\tilde{x}}_{new}$ be a solution to (13(13) $\begin{array}{r}\{\begin{array}{l}min\frac{1}{2}[x_{new}^T{x}_{new}^{{\alpha }^T}{x}_{new}^{{\beta }^T}]Q\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+f^T\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]\\ s.t.\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0,\end{array}\end{array}$(13) ). Then we define ${\alpha }_1=x^{\alpha },{\alpha }_2=x^r-x^l+x^{\alpha },{\alpha }_3=x^r-x^l+x^{\beta },{\alpha }_4=x^{\beta },$ and $\begin{array}{r}{\tilde{x}}_{left}=(x_i^l,min\{{\alpha }_i^1,{\alpha }_i^2\},max\{{\alpha }_i^3,{\alpha }_i^4\}{)}_{LR},\\ {\tilde{x}}_{right}=(x_i^r,max\{{\alpha }_i^1,{\alpha }_i^2\},min\{{\alpha }_i^3,{\alpha }_i^4\}{)}_{LR}.\end{array}$ Now, for every $i=1,\dots ,n$, we have the following properties:

1. If $x_i^l\le x_i^r$, then ${\tilde{x}}_i=(x_i^l,x_i^r,x_i^{\alpha },x_i^{\beta }{)}_{LR}$ is the i-component of vector solution of system (1(1) $A\tilde{x}=\tilde{b},$(1) ).

2. Otherwise, ${\tilde{x}}_i=(x_i^r,x_i^l,x_i^r-x_i^l+x_i^{\alpha },x_i^r-x_i^l+x_i^{\beta }{)}_{LR}$ is the i-component of vector solution of system (1(1) $A\tilde{x}=\tilde{b},$(1) ). Therefore, $\tilde{x}=({\tilde{x}}_1,\dots ,{\tilde{x}}_n)$ is a solution of system (1(1) $A\tilde{x}=\tilde{b},$(1) )

Algorithm 1 shows the proposed method.

For convenience, our proposed algorithm is shown in flowchart form in Figure 1.

Figure 1. Diagram flow for computing an exact or approximate solution.

6. Examples and Numerical Results

In the following examples, we show superiority of our method compared to proposed method in [28].

Example 6.1

Consider $\begin{array}{r}\{\begin{array}{l}3{\tilde{x}}_1+2{\tilde{x}}_2=(4,13,5,8{)}_{LR},\\ -5{\tilde{x}}_1+3{\tilde{x}}_2=(-18,-4,13,8{)}_{LR}.\end{array}\end{array}$ Then $\begin{array}{r}{A}_{new}=\left[\begin{array}{cccc}3& 2& 0& 0\\ 0& 3& -5& 0\\ 0& 0& 3& 2\\ -5& 0& 0& 3\end{array}\right],{\tilde{b}}_{new}=\left[\begin{array}{c}(4,5,17{)}_{LR}\\ (-18,13,22{)}_{LR}\\ (13,14,8{)}_{LR}\\ (-4,27,8{)}_{LR}\end{array}\right].\end{array}$ Now, we must solve the following fuzzy LR linear system: $\begin{array}{r}\{\begin{array}{l}3{\tilde{y}}_1+2{\tilde{y}}_2=(4,5,17{)}_{LR},\\ 3{\tilde{y}}_2-5{\tilde{y}}_3=(-18,13,22{)}_{LR},\\ 3{\tilde{y}}_3+2{\tilde{y}}_4=(13,14,8{)}_{LR},\\ -5{\tilde{y}}_1+3{\tilde{y}}_4=(-4,27,8{)}_{LR}.\end{array}\end{array}$ We compute the solution of the system by using Ghanbari and Mahdavi-Amiri's method [26]. Now, we have $\begin{array}{r}{\tilde{x}}_{left}=\left[\begin{array}{c}{\tilde{y}}_1\\ {\tilde{y}}_2\end{array}\right]=\left[\begin{array}{c}(2,1,3{)}_{LR}\\ (-1,1,4{)}_{LR}\end{array}\right],{\tilde{x}}_{right}=\left[\begin{array}{c}{\tilde{y}}_3\\ {\tilde{y}}_4\end{array}\right]=\left[\begin{array}{c}(3,2,2{)}_{LR}\\ (2,4,1{)}_{LR}\end{array}\right].\end{array}$ By Definition 5.1, $x_i^l\le x_i^r$ for $i=1,2$. Therefore ${\tilde{x}}_1=(2,3,1,2{)}_{LR},{\tilde{x}}_2=(-1,2,1,1{)}_{LR}.$ Then $\tilde{x}=({\tilde{x}}_1,{\tilde{x}}_2{)}^T$ with $r(\tilde{x})=0$ is an exact solution of the FLRILS. We solve the system by using method in [28] and obtain the same solution.

Example 6.2

Consider Example 6.1 with vector right side $\begin{array}{r}\tilde{b}=\left[\begin{array}{c}(4,13,5,18{)}_{LR}\\ (7,9,8,11{)}_{LR}\end{array}\right].\end{array}$ Since $x_i^l\le x_i^r$ for $i=1,2$, from Definition 5.1, we can write ${\tilde{x}}_1=(0.57,0.57,0,1.42{)}_{LR},{\tilde{x}}_2=(2.63,4.47,0.96,4.64{)}_{LR}.$ Then $\tilde{x}=({\tilde{x}}_1,{\tilde{x}}_2{)}^T$ with $r(\tilde{x})=139.17$ is an approximate solution of the FLRILS. Indeed, by using the method in [28], we obtain ${\tilde{x}}_1=(7.42,8.57,1.42,7.42{)}_{LR},{\tilde{x}}_2=(0.57,3.25,3.57,6.57{)}_{LR}.$ Then $\tilde{x}=({\tilde{x}}_1,{\tilde{x}}_2{)}^T$ with $r(\tilde{x})=1.03\times {10}^{+04}$. This example shows that our solution is better than that obtained by using a method in [28].

We report some numerical results by inspired [26, 29]. For our test problems, the coefficient matrix A must be generated such that it is singular, nonsingular, full rank, or rank deficient, and m and n are selected from the following three sets: $\begin{array}{rl}small& =\{10,20,\dots ,90\},\\ medium& =\{100,200,\dots ,500\},\\ large& =\{600,\dots ,1000\}.\end{array}$ We investigated the given results in two categories:

1. Category 1: It is generated completely random.

2. Category 2: It is generated such that the corresponding crisp system of 13(13) $\begin{array}{r}\{\begin{array}{l}min\frac{1}{2}[x_{new}^T{x}_{new}^{{\alpha }^T}{x}_{new}^{{\beta }^T}]Q\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]+f^T\left[\begin{array}{c}{x}_{new}\\ x_{new}^{\alpha }\\ x_{new}^{\beta }\end{array}\right]\\ s.t.\\ x_{new}^{\alpha },x_{new}^{\beta }\ge 0,\end{array}\end{array}$(13) is solvable.

Now, by inspired [26, 29], we compute the relative error for each solution. Next, we compare mean relative errors for an approximate solution in Tables 1 and 2.

Table 1. The mean relative error for Category1.

Sizes	Rank	Scale	Friedman [30]	Nasseri and Gholami [28]	SIP	KKTIP	LSIP
$m=n$	Full rank	Small	$2.60\times {10}^{+01}$	$2.95\times {10}^{+01}$	$2.12\times {10}^{-06}$	$3.56\times {10}^{-05}$	$\mathbf{1.01}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
		Medium	$1.38\times {10}^{+02}$	$3.58\times {10}^{+01}$	$2.14\times {10}^{-07}$	$3.17\times {10}^{-07}$	$\mathbf{4.58}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
		Large	$2.45\times {10}^{+02}$	$3.25\times {10}^{+03}$	$3.32\times {10}^{-08}$	$2.18\times {10}^{-08}$	$\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
$m=n$	Deficient rank	Small	$3.25\times {10}^{+02}$	$4.16\times {10}^{+01}$	$5.23\times {10}^{-08}$	$\mathbf{2.03}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$2.38\times {10}^{-08}$
		Medium	$1.03\times {10}^{+02}$	$2.32\times {10}^{+01}$	$6.14\times {10}^{-07}$	$3.24\times {10}^{-08}$	$\mathbf{2.01}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
		Large	$4.32\times {10}^{+02}$	$4.96\times {10}^{+01}$	$\mathbf{8.88}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$2.59\times {10}^{-07}$	$3.25\times {10}^{-06}$
$m=n$	Full rank	Small	$1.56\times {10}^{+02}$	$2.46\times {10}^{+01}$	$\mathbf{1.06}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$	$5.36\times {10}^{-07}$	$3.33\times {10}^{-06}$
		Medium	$4.31\times {10}^{+02}$	$5.16\times {10}^{+01}$	$6.32\times {10}^{-08}$	$\mathbf{5.24}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$2.87\times {10}^{-07}$
		Large	$3.16\times {10}^{+02}$	$3.99\times {10}^{+01}$	$\mathbf{6.66}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$3.41\times {10}^{-07}$	$1.89\times {10}^{-07}$
$m=n$	Deficient rank	Small	$2.74\times {10}^{+02}$	$3.3\times {10}^{+01}$	$2.02\times {10}^{-07}$	$4.44\times {10}^{-07}$	$\mathbf{3.22}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
		Medium	$2.98\times {10}^{+02}$	$3.40\times {10}^{+01}$	$7.33\times {10}^{-08}$	$\mathbf{5.21}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$6.12\times {10}^{-07}$
		Large	$3.24\times {10}^{+02}$	$3.58\times {10}^{+01}$	$2.31\times {10}^{-07}$	$2.52\times {10}^{-07}$	$\mathbf{1.35}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$
$m=n$	Full rank	Small	$2.83\times {10}^{+02}$	$3.16\times {10}^{+01}$	$2.31\times {10}^{-09}$	$\mathbf{1.92}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$4.68\times {10}^{-09}$
	Full rank	Medium	$1.47\times {10}^{+02}$	$1.87\times {10}^{+01}$	$\mathbf{2.33}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$6.15\times {10}^{-09}$	$4.31\times {10}^{-09}$
		Large	$3.85\times {10}^{+02}$	$4.01\times {10}^{+01}$	$\mathbf{6.39}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$8.75\times {10}^{-09}$	$2.50\times {10}^{-08}$
$m=n$	Deficient rank	Small	$2.52\times {10}^{+02}$	$3.15\times {10}^{+01}$	$2.14\times {10}^{-09}$	$2.01\times {10}^{-09}$	$\mathbf{1.17}\times {\mathbf{10}}^{\mathbf{-}\mathbf{10}}$
		Medium	$4.36\times {10}^{+02}$	$5.1\times {10}^{+01}$	$\mathbf{8.74}\times {\mathbf{10}}^{\mathbf{-}\mathbf{10}}$	$8.21\times {10}^{-09}$	$3.01\times {10}^{-09}$
		Large	$2.03\times {10}^{+02}$	$3.51\times {10}^{+01}$	$\mathbf{5.07}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$3.24\times {10}^{-08}$	$1.02\times {10}^{-08}$

Table 2. The mean relative error for Category2.

Sizes	Rank	Scale	Friedman [30]	Nasseri and Gholami [28]	SIP	KKTIP	LSIP
$m=n$	Full rank	Small	$3.22\times {10}^{+03}$	$4.25\times {10}^{+04}$	$7.41\times {10}^{-03}$	$\mathbf{4.24}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$2.91\times {10}^{-01}$
		Medium	$3.26\times {10}^{+03}$	$5.28\times {10}^{+04}$	$2.44\times {10}^{-07}$	$5.71\times {10}^{-06}$	$\mathbf{9.06}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$
		Large	$2.36\times {10}^{+03}$	$5.14\times {10}^{+04}$	$2.39\times {10}^{-08}$	$\mathbf{1.98}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$3.66\times {10}^{-07}$
$m=n$	Deficient rank	Small	$3.95\times {10}^{+03}$	$2.04\times {10}^{+04}$	$\mathbf{2.33}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$2.36\times {10}^{-08}$	$1.03\times {10}^{-07}$
		Medium	$2.45\times {10}^{+03}$	$3.24\times {10}^{+04}$	$2.03\times {10}^{-07}$	$5.01\times {10}^{-06}$	$\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$
		Large	$1.89\times {10}^{+03}$	$2.53\times {10}^{+04}$	$3.27\times {10}^{-08}$	$1.05\times {10}^{-07}$	$\mathbf{2.17}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$
$m=n$	Full rank	Small	$4.15\times {10}^{+03}$	$3.65\times {10}^{+04}$	$\mathbf{2.04}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$3.44\times {10}^{-08}$	$1.32\times {10}^{-06}$
		Medium	$3.45\times {10}^{+03}$	$4.98\times {10}^{+04}$	$2.16\times {10}^{-07}$	$\mathbf{1.04}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$	$1.12\times {10}^{-07}$
		Large	$4.37\times {10}^{+03}$	$4.26\times {10}^{+04}$	$3.02\times {10}^{-07}$	$5.33\times {10}^{-03}$	$\mathbf{1.02}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$
$m=n$	Deficient rank	Small	$3.59\times {10}^{+03}$	$3.66\times {10}^{+04}$	$8.55\times {10}^{-08}$	$\mathbf{3.22}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$4.91\times {10}^{-08}$
		Medium	$5.29\times {10}^{+03}$	$4.15\times {10}^{+04}$	$\mathbf{1.56}\times {\mathbf{10}}^{\mathbf{-}\mathbf{08}}$	$2.35\times {10}^{-08}$	$1.19\times {10}^{-07}$
		Large	$2.16\times {10}^{+03}$	$2.54\times {10}^{+04}$	$3.14\times {10}^{-07}$	$6.34\times {10}^{-07}$	$\mathbf{1.03}\times {\mathbf{10}}^{\mathbf{-}\mathbf{07}}$
$m=n$	Full rank	Small	$3.17\times {10}^{+03}$	$1.03\times {10}^{+04}$	$\mathbf{7.01}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$8.14\times {10}^{-09}$	$2.00\times {10}^{-08}$
		Medium	$1.23\times {10}^{+03}$	$6.13\times {10}^{+04}$	$\mathbf{4.98}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$6.35\times {10}^{-09}$	$8.21\times {10}^{-08}$
		Large	$1.65\times {10}^{+03}$	$2.13\times {10}^{+04}$	$1.33\times {10}^{-08}$	$\mathbf{7.54}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$7.11\times {10}^{-08}$
$m=n$	Deficient rank	Small	$1.11\times {10}^{+03}$	$3.21\times {10}^{+04}$	$1.12\times {10}^{-09}$	$2.73\times {10}^{-09}$	$\mathbf{1.04}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$
		Medium	$1.56\times {10}^{+03}$	$3.54\times {10}^{+04}$	$3.11\times {10}^{-08}$	$4.57\times {10}^{-09}$	$\mathbf{5.80}\times {\mathbf{10}}^{\mathbf{-}\mathbf{10}}$
		Large	$3.29\times {10}^{+03}$	$3.98\times {10}^{+04}$	$\mathbf{2.66}\times {\mathbf{10}}^{\mathbf{-}\mathbf{09}}$	$3.1\times {10}^{-08}$	$3.31\times {10}^{-08}$

In Table 1, we compare the mean relative error to compute the approximate solution by Algorithm 1 in category 1 with three initial points, simple initial point (SIP) [26, 29], Karush–Kuhn–Tucker initial point (KKTIP) [26, 29], and local search initial point (LSIP) [26, 29]. Results show that SIP and LSIP methods have better performance in all different cases.

Also, in Table 2, we compare the mean relative error to compute the approximate solution by Algorithm 1 in category 2 with three initial points SIP, KKTIP, and LSIP. Results show that LSIP, SIP, and KKTIP methods have similar performance.

We give all results in the MATLAB environment version R2022a and run the algorithm on a notebook Intel Core i5-4200M 2.50GHZ with 6 GB of RAM.

Also, all problems are solved by the methods due to Friedman [30] and Nassri and Gholami [28]. According to the obtained results, it is observed that the mean relative error to compute approximate solution by Algorithm 1 by three different initial points is less than the ones due to other methods on all test problems.

7. Conclusions and Future Work

Here, we showed that solving an FLRILS is equivalent to solving the corresponding FLRLS. Next, we developed some necessary and sufficient conditions for the solvability of FLRLSs. We used the proposed method of Ghanbari and Mahdavi-Amiri for solving FLRLSs. Therefore, we showed that the proposed method by Ghanbari and Mahdavi-Amiri can be extended for solving FLRILSs. We then provided a new general concept for an approximate solution when the FLRLS lacked solutions. To compute an approximate solution, we proposed an algorithm using a least squares method. We showed numerically the appropriateness of our proposed approximate solution for large-scale problems in comparison with other recently proposed approximate solutions. The numerical results showed that our proposed algorithm produces significantly more accurate solutions. In future work, the method used in this paper can be extended to other classes of problems, such as fuzzy complex matrix equations and their dual forms [15].

Acknowledgments

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 author(s).

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.