A New Approach for Solving Fully Fuzzy Linear Systems

Abstract

Several authors have proposed different methods to find the solution of fully fuzzy linear systems (FFLSs) that is, fuzzy linear system with fuzzy coefficients involving fuzzy variables. But all the existing methods are based on the assumption that all the fuzzy coefficients and the fuzzy variables are nonnegative fuzzy numbers. In this paper a new method is proposed to solve an FFLS with arbitrary coefficients and arbitrary solution vector, that is, there is no restriction on the elements that have been used in the FFLS. The primary objective of this paper is thus to introduce the concept and a computational method for solving FFLS with no non negative constraint on the parameters. The method incorporates the principles of linear programming in solving an FFLS with arbitrary coefficients and is not only easier to understand but also widens the scope of fuzzy linear equations in scientific applications. To show the advantages of the proposed method over existing methods we solve three FFLSs.

1. Introduction

One field of applied mathematics that has many applications in various areas of science is solving a system of linear equations. Systems of simultaneous linear equations play a major role in various areas such as operational research, physics, statistics, engineering, and social sciences. When the estimation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it may be convenient to represent some or all of them with fuzzy numbers [1]. Fuzzy number arithmetic is widely applied and useful in computation of linear system whose parameters are represented by fuzzy numbers, which are called fuzzy linear systems (FLSs). Buckley and Qu [2] defined the concept of solving fuzzy equations and their work has been influential in the study of fuzzy linear systems. A general model for solving an 𝑛×𝑛 fuzzy linear system whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy vector was first proposed by Friedman et al. [3, 4]. Later some numerical methods to solve similar systems were proposed [5] and extended methods like successive overrelaxation [6] adomian decomposition [7] were also presented. Abbasbandy et al. also described LU decomposition method [8], Conjugate Gradient method [9], and Steepest descent method [10] for solving such system of fuzzy equations. Some iterative methods to solve an FLS were also extended in [11]. The condition of crispness of the coefficient matrix makes all these methods restricted with negligible applications.

In addition, another important kind of fuzzy linear systems are the fully fuzzy linear systems (FFLS) in which all the parameters are fuzzy numbers. Dehghan and Hashemi [12, 13] proposed the Adomian decomposition method, and other iterative methods to find the positive fuzzy vector solution of 𝑛×𝑛 fully fuzzy linear system. Dehghan et al. [14] proposed some computational methods such as Cramer’s rule, Gauss elimination method, LU decomposition method and linear programming approach for finding the approximated solution of FFLS. Muzzioli and Reynaerts [15] introduced an algorithm to find vector solution by transforming the system 𝐴₁𝑥+𝑏₁=𝐴₂𝑥+𝑏₂ into the FFLS 𝐴𝑥=𝑏 where 𝐴=𝐴₁−𝐴₂ and 𝑏=𝑏₂−𝑏₁.

Mosleh et al. [16] proposed a method to find the solution of fully fuzzy linear system of the form 𝐴𝑥+𝑏=𝐶𝑥+𝑑 with 𝐴, 𝐶 being square matrices of fuzzy coefficients and 𝑏, 𝑑 being fuzzy number vectors, and the unknown vector 𝑥 is vector consisting of 𝑛fuzzy numbers. Nasseri et al. [17] used certain decomposition methods of the coefficient matrix for solving fully fuzzy linear system of equations. Allahviranloo et al. [18] proposed a numerical method for solving FFLS 𝐴𝑥=𝑏,when coefficient matrix is positive. Gao [19] proposed a unified iterative scheme for solving nonsquare FFLS with nonnegative constraints.

Nasseri et al. [20] proposed Greville’s method to find the positive solution of FFLS. Mosleh et al. [21] discussed symmetric times triangular (ST) decomposition to find the positive solution of FFLS. Nasseri and Zahmatkesh [22] proposed a new method for computing the nonnegative solution of FFLS. Liu [23] developed a block homotopy perturbation method (HPM) for finding the approximation of the solution of FFLS. Nasseri and Sohrabi [24] used a certain decomposition of the coefficient matrix of the FFLS to construct a new algorithm for solving FFLS. Kumar et al. [25] proposed a new method to find the approximate solution of FFLS with trapezoidal fuzzy numbers as parameters. Recently some other methods like homotopy perturbation, splitting methods and other numerical methods have been proposed in [26–29] but possess similar restrictions for solving fuzzy systems. It may be seen that the methods assume the nonnegativity of the coefficients.

In this paper, a new method based upon the decomposition of an FFLS into a nonlinear system and subsequently a linear programming problem, is proposed to find the solution of FFLS without any restriction on coefficient and fuzzy variables. To illustrate the proposed method numerical examples are solved. The rest of paper is organized as follows. In Section 2, some basic definitions and arithmetic operations of triangular fuzzy numbers are reviewed. In Section 3, limitations of the existing methods to solve FFLS are described. In Section 4, a new computational method for solving FFLS with arbitrary coefficient matrix and arbitrary fuzzy variable is introduced. In Section 5, to exemplify the method three FFLSs are solved. The paper is concluded in Section 6.

2. Preliminaries

2.1. Basic Definitions

In this section some basic definitions of fuzzy set theory are reviewed [30, 31].

Definition 1. A fuzzy number 𝐴 is said to be nonnegative fuzzy number if the domain of its membership function is a set of nonnegative real numbers (𝑅⁺), that is, 𝜇_𝐴∶𝑅⁺→[0,1]. The set of nonnegative fuzzy numbers may be represented by 𝐹(𝑅⁺).

Definition 2. A fuzzy number 𝐴 is said to be unrestricted fuzzy number if the domain of its membership function is a set of real numbers (𝑅), that is,𝜇_𝐴∶𝑅→[0,1]. The set of unrestricted fuzzy numbers may be represented by 𝐹(𝑅).

Definition 3. A fuzzy number 𝐴=(𝑎,𝑏,𝑐) is said to be a triangular fuzzy number if its membership function is given by 𝜇_𝐴(𝑥)=0,𝑥<𝑎,𝑥−𝑎𝑏−𝑎,𝑎≤𝑥≤𝑏,𝑐−𝑥𝑐−𝑏,𝑏<𝑥≤𝑐,0,𝑥>𝑐, (1) where 𝑎,𝑏,𝑐∈𝑅.

Definition 4. A triangular fuzzy number; 𝐴=(𝑎,𝑏,𝑐)is said to be unrestricted triangular fuzzy number if 𝑎,𝑏,𝑐∈𝑅.

Definition 5. A triangular fuzzy number 𝐴=(𝑎,𝑏,𝑐)is said to be nonnegative triangular fuzzy number; that is, 𝐴≥0, that is, if and only if 𝑎≥0.

Definition 6. A triangular fuzzy number 𝐴=(𝑎,𝑏,𝑐)is said to be zero triangular fuzzy number if and only if 𝑎=0, 𝑏=0, and 𝑐=0.

Definition 7. Two triangular fuzzy numbers 𝐴₁=(𝑎,𝑏,𝑐) and 𝐴₂=(𝑒,𝑓,𝑔) are said to be equal; that is, 𝐴₁=𝐴₂ if and only if 𝑎=𝑒, 𝑏=𝑓, and 𝑐=𝑔.

Definition 8. A matrix 𝐴=(𝑎_𝑖𝑗) is called a fuzzy matrix, if each element of 𝐴 is a fuzzy number. 𝐴will be positive (negative) and denoted by 𝐴>0  (𝐴<0) if each element of 𝐴 positive (negative). 𝐴will nonpositive (nonnegative) and denoted by 𝐴≤0  (𝐴≥0) if each element of 𝐴 be nonpositive (nonnegative). We may represent 𝑛×𝑚 fuzzy matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑚 where 𝑎_𝑖𝑗=(𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗).

Definition 9. Let 𝐴=(𝑎_𝑖𝑗) and 𝐵=(𝑏̃_𝑖𝑗) be two 𝑚×𝑛 and 𝑛×𝑝 fuzzy matrices. We define 𝐴⊗𝐵=𝐶=(𝑐̃_𝑖𝑗) which is the 𝑚×𝑝 matrix where 𝑐̃_𝑖𝑗=∑^⨁_{𝑘=1,…,𝑛}𝑎_𝑖𝑘⊗𝑏̃_𝑘𝑗.

2.2. Arithmetic Operations on Triangular Fuzzy Numbers

In this subsection addition and multiplication operations between two triangular fuzzy numbers are reviewed [30, 31].

Let 𝐴₁=(𝑎,𝑏,𝑐) and 𝐴₂=(𝑒,𝑓,𝑔) be two triangular fuzzy numbers; then one has the following.(i)𝐴₁⊕𝐴₂=(𝑎,𝑏,𝑐)⊕(𝑒,𝑓,𝑔)=(𝑎+𝑒,𝑏+𝑓,𝑐+𝑔)(ii)𝐴₁⊖𝐴₂=(𝑎,𝑏,𝑐)⊖(𝑒,𝑓,𝑔)=(𝑎−𝑔,𝑏−𝑓,𝑐−𝑒)(iii) If 𝐴₁ and 𝐴₂ be two nonnegative triangular fuzzy numbers then 𝐴₁⊗𝐴₂=𝐴₂⊗𝐴₁=𝑎𝑒,𝑏𝑓,𝑐𝑔. (2)

Remark 10. Let 𝐴=(𝑎,𝑏,𝑐) and 𝑋=(𝑥,𝑦,𝑧) be two unrestricted triangular fuzzy numbers. Then ()⊗=()()()()()()()()𝑎,𝑏,𝑐𝑥,𝑦,𝑧min𝑎𝑥,𝑐𝑥,by,max𝑎𝑧,𝑐𝑧if𝑎,𝑏,𝑐≥0,thatis,if𝑎≥0,min𝑎𝑧,𝑐𝑧,by,max𝑎𝑥,𝑐𝑥if𝑎,𝑏,𝑐≤0,thatis,if𝑐≤0,min𝑎𝑧,𝑐𝑥,by,max𝑐𝑧,𝑎𝑥otherwise,thatis,if𝑎≤0≤𝑐. (3)

Remark 11. In this paper, minimum and maximum are represented by “min” and “max”, respectively, where min𝑥,𝑦=𝑥+𝑦2−𝑥−𝑦2,max𝑥,𝑦=𝑥+𝑦2+𝑥−𝑦2. (4)

3. Limitations of the Existing Methods

In this section, the limitations of the existing methods, for example, [2, 5–14, 17–29] are pointed out.(1)The existing methods can be applied only for finding the solution of an 𝑛×𝑛FFLS in which all the elements are nonnegative fuzzy numbers; that is, the existing methods presume the nonnegativity of the coefficient matrix and the solution vector. This restriction creates difficulty in using the existing methods to solve an FFLS occurring in real life situations as the nature of the unknown quantity cannot be known a priori and there exist several instances in real scenarios where the coefficients may not be entirely positive or nonnegative. The primary aim of this paper is to eliminate this restriction and to propose a concept and a computational paradigm to solve a fuzzy-based system of simultaneous equations without any nonnegative restrictions. (2)There exist several methods like [5–11] that restrict the coefficient matrix to be crisp matrix and such systems are not fully fuzzy. Such systems will be entirely spanned by fully fuzzy linear systems. Some of the methods include 𝛼-cuts to solve fuzzy linear equations which can be computationally simplified by using triangular fuzzy numbers as it is easier to operate with constants than variables.(3)Using the existing methods it is not possible to check whether the obtained solution is unique or not. There may be several cases where the FFLS may generate trivial, unique or infinitely many solutions.

To overcome the above shortcomings, in Section 4, a new computational method is proposed for solving an FFLS with triangular fuzzy numbers.

4. Proposed Method

In this section, a new method is proposed to solve a fully fuzzy linear system with no restrictions on the parameters.

The steps of the proposed method are as follows.

Step 1. Substituting 𝐴=(𝑎_𝑖𝑗)_𝑚×𝑛, 𝑥=(𝑥_𝑗)_𝑛×1, and 𝐵=(𝐵_𝑖)_𝑚×1 the 𝑚×𝑛 FFLS 𝐴⊗𝑥=𝐵, may be written as ^⊕_{𝑗=1,…,𝑛}𝑎_𝑖𝑗⊗𝑥_𝑗=𝐵_𝑖∀𝑖=1,2,…,𝑚. (5)

Step 2. If all the parameters 𝑎_𝑖𝑗, 𝑥_𝑗, and𝑏̃_𝑖 are represented by triangular fuzzy numbers (𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗), (𝑥_𝑗,𝑦_𝑗,𝑧_𝑗), and (𝐵_𝑖,𝐺_𝑖,𝐻_𝑖), respectively, then the FFLS, obtained in Step 1, may be written as ^⊕_{𝑗=1,…,𝑛}𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗⊗𝑥_𝑗,𝑦_𝑗,𝑧_𝑗=𝐵_𝑖,𝐺_𝑖,𝐻_𝑖,∀𝑖=1,2,…,𝑚. (6)

Step 3. Assuming (𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗)⊗(𝑥_𝑗,𝑦_𝑗,𝑧_𝑗)=(𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗), the FFLS, obtained in Step 2, may be written as ^⊕_{𝑗=1,…,𝑛}𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗=𝐵_𝑖,𝐺_𝑖,𝐻_𝑖∀𝑖=1,2,…,𝑚, (7) where 𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗⊗𝑥_𝑗,𝑦_𝑗,𝑧_𝑗. (8) As 𝑎_𝑖𝑗 and 𝑥_𝑗 are both arbitrary fuzzy numbers, the above product can be defined in three subcases.

Case 1. If (𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗)≥0, that is, 𝑎_𝑖𝑗≥0, 𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑥_𝑗−𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑥_𝑗,𝑏_𝑖𝑗𝑦_𝑗,𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑧_𝑗+𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑧_𝑗. (9) Using Definition 7, the above equation may be written as, 𝑓_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑥_𝑗−𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑥_𝑗,𝑝_𝑖𝑗=𝑏_𝑖𝑗𝑦_𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑧_𝑗+𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑧_𝑗. (10) To solve this system we define variables 𝑥^′_𝑗 and 𝑥_𝑗^′′ as 𝑥^′_𝑗=𝑥_𝑗if𝑥_𝑗𝑥>0,0otherwise,_𝑗′′=−𝑥_𝑗if𝑥_𝑗<0,0otherwise. (11) Hence substitute 𝑥_𝑗=𝑥^′_𝑗−𝑥_𝑗^′′ and |𝑥_𝑗|=𝑥^′_𝑗+𝑥_𝑗^′′; similarly define variables 𝑧^′_𝑗 and 𝑧_𝑗^′′; substitute 𝑧_𝑗=𝑧^′_𝑗−𝑧_𝑗^′′ and |𝑧_𝑗|=𝑧^′_𝑗+𝑧_𝑗^′′ where 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′≥0.Therefore we obtain the following linear system of equation: 𝑓_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑥^′_𝑗−𝑥_𝑗^′′−𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑥^′_𝑗+𝑥_𝑗^′′,𝑝_𝑖𝑗=𝑏_𝑖𝑗𝑦_𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑧^′_𝑗−𝑧_𝑗^′′+𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑧^′_𝑗+𝑧_𝑗^′′. (12)

Case 2. If (𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗)≤0, that is, 𝑐_𝑖𝑗≤0, 𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑧_𝑗−𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑧_𝑗,𝑏_𝑖𝑗𝑦_𝑗,𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑥_𝑗+𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑥_𝑗. (13) This can be similarly reduced as follows.
Putting 𝑥_𝑗=𝑥^′_𝑗−𝑥_𝑗^′′ and |𝑥_𝑗|=𝑥^′_𝑗+𝑥_𝑗^′′, 𝑧_𝑗=𝑧^′_𝑗−𝑧_𝑗^′′, and |𝑧_𝑗|=𝑧^′_𝑗+𝑧_𝑗^′′, where 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′≥0. Thus𝑓_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑧^′_𝑗−𝑧_𝑗^′′−𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑧^′_𝑗+𝑧_𝑗^′′,𝑝_𝑖𝑗=𝑏_𝑖𝑗𝑦_𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗+𝑐_𝑖𝑗2𝑥^′_𝑗−𝑥_𝑗^′′+𝑐_𝑖𝑗−𝑎_𝑖𝑗2𝑥^′_𝑗+𝑥_𝑗^′′. (14)

Case 3. If (𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗)≈0, that is, 𝑎_𝑖𝑗<0<𝑐_𝑖𝑗, 𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗𝑧_𝑗+𝑐_𝑖𝑗𝑥_𝑗2−𝑐_𝑖𝑗𝑥_𝑗−𝑎_𝑖𝑗𝑧_𝑗2,𝑏_𝑖𝑗𝑦_𝑗,𝑎_𝑖𝑗𝑥_𝑗+𝑐_𝑖𝑗𝑧_𝑗2+𝑐_𝑖𝑗𝑧_𝑗−𝑎_𝑖𝑗𝑥_𝑗2. (15) Using Definition 5, the above equation may be written as 𝑓_𝑖𝑗=𝑎_𝑖𝑗𝑧_𝑗+𝑐_𝑖𝑗𝑥_𝑗2−𝑐_𝑖𝑗𝑥_𝑗−𝑎_𝑖𝑗𝑧_𝑗2,𝑝_𝑖𝑗=𝑏_𝑖𝑗𝑦_𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗𝑥_𝑗+𝑐_𝑖𝑗𝑧_𝑗2+𝑐_𝑖𝑗𝑧_𝑗−𝑎_𝑖𝑗𝑥_𝑗2. (16) This can be further reduced as follows.
Putting 𝑥_𝑗=𝑥^′_𝑗−𝑥_𝑗^′′ and |𝑥_𝑗|=𝑥^′_𝑗+𝑥_𝑗^′′, 𝑧_𝑗=𝑧^′_𝑗−𝑧_𝑗^′′ and |𝑧_𝑗|=𝑧^′_𝑗+𝑧_𝑗^′′, 𝑐_𝑖𝑗𝑥_𝑗−𝑎_𝑖𝑗𝑧_𝑗=𝑢^′_𝑖𝑗−𝑢^′′_𝑖𝑗 and |𝑐_𝑖𝑗𝑥_𝑗−𝑎_𝑖𝑗𝑧_𝑗|=𝑢^′_𝑖𝑗+𝑢^′′_𝑖𝑗, 𝑐_𝑖𝑗𝑧_𝑗−𝑎_𝑖𝑗𝑥_𝑗=𝑣^′_𝑖𝑗−𝑣^′′_𝑖𝑗 and |𝑐_𝑖𝑗𝑧_𝑗−𝑎_𝑖𝑗𝑥_𝑗|=𝑣^′_𝑖𝑗+𝑣^′′_𝑖𝑗, where 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′, 𝑢^′_𝑖𝑗, 𝑢^′′_𝑖𝑗, 𝑣^′_𝑖𝑗, 𝑣^′′_𝑖𝑗≥0.
On simplifying we obtain𝑓_𝑖𝑗=𝑎_𝑖𝑗𝑧^′_𝑗−𝑧_𝑗^′′+𝑐_𝑖𝑗𝑥^′_𝑗−𝑥_𝑗^′′2−𝑢^′_𝑖𝑗+𝑢^′′_𝑖𝑗2,𝑝_𝑖𝑗=𝑏_𝑖𝑗𝑦_𝑗,𝑞_𝑖𝑗=𝑎_𝑖𝑗𝑥^′_𝑗−𝑥_𝑗^′′+𝑐_𝑖𝑗𝑧^′_𝑗−𝑧_𝑗^′′2+𝑣^′_𝑖𝑗+𝑣^′′_𝑖𝑗2. (17)

Step 4. Having broken down the FFLS from Step 3 into a set of linear equations, rewrite the given system as: ^𝑛_𝑗=1𝑓_𝑖𝑗,^𝑛_𝑗=1𝑝_𝑖𝑗,^𝑛_𝑗=1𝑞_𝑖𝑗=𝐵_𝑖,𝐺_𝑖,𝐻_𝑖∀𝑖=1,2,…,𝑚. (18) Or equivalently ^𝑛_𝑗=1𝑓_𝑖𝑗=𝐵_𝑖;𝑖=1,2,…,𝑚,_𝑛𝑗=1𝑝_𝑖𝑗=𝐺_𝑖;𝑖=1,2,…,𝑚,_{𝑛𝑗=𝑖}𝑞_𝑖𝑗=𝐻_𝑖;𝑖=1,2,…,𝑚, (19) with appropriate constraints obtained from Step 3.

Step 5. In order to solve the system obtained in Step 4 we decompose the FFLS into an equivalent linear programming problem LPP. For notational convenience we construct the following step functions: 𝜀₁(=𝜀𝐴=𝑎,𝑏,𝑐)1if𝐴≥0,thatis,𝑎≥0,0otherwise₂()=𝜀𝐴=𝑎,𝑏,𝑐1if𝐴≤0,thatis,𝑐<0,0otherwise₃𝐴=(𝑎,𝑏,𝑐)=1if𝐴≈0,thatis,𝑎<0<𝑐,0otherwise. (20) The equivalent LPP can thus be directly written as:
Minimize𝑅=^𝑚_𝑖=1𝑅_𝑖+^𝑚_𝑖=1𝑅^′_𝑖+^𝑚_𝑖=1𝑅_𝑖^′′ (21) subject to the following constraints.
First Set of Constraints:^𝑛_𝑗=1𝑏_𝑖𝑗𝑦_𝑗+𝑅_𝑖=𝐺_𝑖,𝑖=1,2,…,𝑚. (22) Second Set of Constraints: ^𝑛_𝑗=1𝜀₁𝑎_𝑖𝑗𝑎_𝑖𝑗𝑥^′_𝑗−𝑐_𝑖𝑗𝑥_𝑗^′′+^𝑛_𝑗=1𝜀₂𝑎_𝑖𝑗𝑎_𝑖𝑗𝑧^′_𝑗−𝑐_𝑖𝑗𝑧_𝑗^′′+^𝑛_𝑗=1𝜀₃𝑎_𝑖𝑗0.5𝑎_𝑖𝑗𝑧^′_𝑗−𝑧_𝑗^′′+𝑐_𝑖𝑗𝑥^′_𝑗−𝑥_𝑗^′′−0.5𝑢^′_𝑖𝑗+𝑢^′′_𝑖𝑗+𝑅^′_𝑖=𝐵_𝑖∀𝑖=1,2,…,𝑚. (23) Third Set of Constraints: ^𝑛_𝑗=1𝜀₁𝑎_𝑖𝑗𝑐_𝑖𝑗𝑧^′_𝑗−𝑎_𝑖𝑗𝑧_𝑗^′′+^𝑛_𝑗=1𝜀₂𝑎_𝑖𝑗𝑐_𝑖𝑗𝑥^′_𝑗−𝑎_𝑖𝑗𝑥_𝑗^′′+^𝑛_𝑗=1𝜀₃𝑎_𝑖𝑗0.5𝑐_𝑖𝑗𝑧^′_𝑗−𝑧_𝑗^′′+𝑎_𝑖𝑗𝑥^′_𝑗−𝑥_𝑗^′′+0.5𝑣^′_𝑖𝑗+𝑣^′′_𝑖𝑗+𝑅_𝑖^′′=𝐻_𝑖.∀𝑖=1,2,…,𝑚. (24) Fourth Set of Constraints: 𝜀₃𝑎_𝑖𝑗𝑐_𝑖𝑗𝑥_𝑗−𝑎_𝑖𝑗𝑧_𝑗−𝑢^′_𝑖𝑗+𝑢^′′_𝑖𝑗=0,𝑖=1,2,…𝑚,𝑗=1,2,…,𝑛. (25) Fifth Set of Constraint: 𝜀₃𝑎_𝑖𝑗𝑐_𝑖𝑗𝑧_𝑗−𝑎_𝑖𝑗𝑥_𝑗−𝑣^′_𝑖𝑗+𝑣^′′_𝑖𝑗=0,𝑖=1,2,…𝑚,𝑗=1,2,…,𝑛. (26) Nonnegative Constraint: 𝑥^′_𝑗,𝑥_𝑗^′′,𝜀₃𝑎_𝑖𝑗𝑢^′_𝑖𝑗,𝜀₃𝑎_𝑖𝑗𝑢^′′_𝑖𝑗,𝜀₃𝑎_𝑖𝑗𝑣^′_𝑖𝑗,𝜀₃𝑎_𝑖𝑗𝑣^′′_𝑖𝑗,𝑧^′_𝑗,𝑧_𝑗^′′,𝑅_𝑖,𝑅^′_𝑖,𝑅_𝑖^′′≥0,𝑖=1,2,…𝑚,𝑗=1,2,…,𝑛. (27)

Step 6. Solve the above LPP and put the above values in 𝑥_𝑗=𝑥^′_𝑗−𝑥_𝑗^′′,𝑦_𝑗,𝑧^′_𝑗−𝑧_𝑗^′′∀𝑗=1,2,…,𝑛, (28) to find the solution of FFLS. The solution would be termed as (feasible) strong fuzzy solution if 𝑦_𝑗−𝑥^′_𝑗+𝑥_𝑗^′′≥0 and 𝑧^′_𝑗−𝑧_𝑗^′′−𝑦_𝑗≥0. Otherwise the solution would be termed as (infeasible) weak fuzzy solution.

Remark 12. The necessary condition for the above LPP to generate a feasible solution is that the value of the objective function 𝑅 must be zero, that is, ∑^𝑚_𝑖=1𝑅_𝑖+∑^𝑚_𝑖=1𝑅^′_𝑖+∑^𝑚_𝑖=1𝑅_𝑖^′′=0.

Remark 13. The sufficient conditions for the solution to be termed as a strong (feasible) solution for 𝑖=1,2,…𝑚, 𝑗=1,2,…,𝑛, are as follows:(1)𝑦_𝑗−𝑥^′_𝑗+𝑥_𝑗^′′≥0 and 𝑧^′_𝑗−𝑧_𝑗^′′−𝑦_𝑗≥0,(2)𝑥^′_𝑗⋅𝑥_𝑗^′′=𝑢^′_𝑖𝑗⋅𝑢^′′_𝑖𝑗=𝑣^′_𝑖𝑗⋅𝑣^′′_𝑖𝑗=𝑧^′_𝑗⋅𝑧_𝑗^′′=𝑅=0.

Remark 14. The nature of the solutions of the FFLS depends upon the nature of the solutions of the LPP which may be unique, trivial, or infinitely many, that is, the FFLS may yield no solution, unique solution, or infinitely many solutions.

Remark 15. The decomposition of the 𝑚×𝑛 FFLS results into an LPP with a minimum number of 3𝑚 constraints (when all the coefficients are either nonnegative or nonpositive) and a maximum number of 3𝑚+2𝑚𝑛 constraints (when all the coefficients are neither negative nor positive).

5. Numerical Example

In this section, to illustrate the proposed method and also to show the advantages of the proposed method we solve three FFLSs. We take two FFLSs which cannot be solved by any of the available methods in the literature due to the restrictions imposed by the methods. Also, we solve one FFLSs randomly chosen from literature and apply our method to show that the proposed method can equivalently solve the examples from literature. Hence the FFLS that we solve belong to the following categories:(1)a square FFLS with unrestricted fuzzy variables, (2)a nonsquare FFLS with unrestricted fuzzy variables that generates infinite solutions,(3)a square FFLS with restriction of crispness of the coefficient matrix randomly chosen from the literature [5].

Example 1. Solve the following FFLS by the proposed method: ()()−2,3,4)(−2,2,3𝑥1,2,2)(4,4,5𝑥¹₂=()()−13,8,14−14,8,14, (29) where 𝑥₁ and 𝑥₂ are arbitrary triangular fuzzy numbers.

Solution 1. Step 1. The chosen FFLS is as follows: (−2,3,4)⊗𝑥₁⊕(−2,2,3)⊗𝑥₂=(),()⊗𝑥−13,8,141,2,2₁⊕(4,4,5)⊗𝑥₂=(−14,8,14), (30) where 𝑥₁,𝑥₂ are unrestricted triangular fuzzy numbers.Step 2. Assuming 𝑥₁=(𝑥₁,𝑦₁,𝑧₁) and 𝑥₂=(𝑥₂,𝑦₂,𝑧₂) the FFLS, chosen in Step 1, may be written as (−2,3,4)⊗𝑥₁,𝑦₁,𝑧₁⊕(−2,2,3)⊗𝑥₂,𝑦₂,𝑧₂=(),()⊗𝑥−13,8,141,2,2₁,𝑦₁,𝑧₁⊕(4,4,5)⊗𝑥₂,𝑦₂,𝑧₂=(−14,8,14), (31) where (𝑥₁,𝑦₁,𝑧₁) and (𝑥₂,𝑦₂,𝑧₂) are unrestricted triangular fuzzy numbers.Step 3. Using Step 3 we may obtain (𝑓_𝑖𝑗,𝑝_𝑖𝑗,𝑞_𝑖𝑗) as follows: 𝑓₁₁,𝑝₁₁,𝑞₁₁=(−2,3,4)⊗𝑥₁,𝑦₁,𝑧₁,𝑓₁₁=𝑎₁₁𝑧^′₁−𝑧₁^′′+𝑐₁₁𝑥^′₁−𝑥₁^′′2−𝑢^′₁₁+𝑢^′′₁₁2,𝑝₁₁=𝑏₁₁𝑦₁,𝑞₁₁=𝑎₁₁𝑥^′₁−𝑥₁^′′+𝑐₁₁𝑧^′₁−𝑧₁^′′2+𝑣^′₁₁+𝑣^′′₁₁2. (32) Or we may have 𝑝₁₁=3𝑦₁,𝑓₁₁=−2𝑧^′₁−𝑧₁^′′+4𝑥^′₁−𝑥₁^′′2−𝑢^′₁₁+𝑢^′′₁₁2,𝑞₁₁=4𝑧^′₁−𝑧₁^′′−2𝑥^′₁−𝑥₁^′′2+𝑣^′₁₁+𝑣^′′₁₁2, (33) with two additional constraints: 2𝑧^′₁−𝑧₁^′′+4𝑥^′₁−𝑥₁^′′−𝑢^′₁₁+𝑢^′′₁₁2𝑥=0,_′1−𝑥₁^′′+4𝑧^′₁−𝑧₁^′′−𝑣^′₁₁+𝑣^′′₁₁=0. (34) Similarly we obtain the other coefficients as 𝑝₁₂=2𝑦₂,𝑓₁₂=−2𝑧^′₂−𝑧₂^′′+3𝑥^′₂−𝑥₂^′′2−𝑢^′₁₂+𝑢^′′₁₂2,𝑞₁₂=3𝑧^′₂−𝑧₂^′′−2𝑥^′₂−𝑥₂^′′2+𝑣^′₁₂+𝑣^′′₁₂2, (35) with two additional equations: 2𝑧^′₂−𝑧₂^′′+3𝑥^′₂−𝑥₂^′′−𝑢^′₁₂+𝑢^′′₁₂2𝑥=0,_′2−𝑥₂^′′+3𝑧^′₂−𝑧₂^′′−𝑣^′₁₂+𝑣^′′₁₂𝑝=0,₂₁=2𝑥₁,𝑓₂₁=𝑥^′₁−2𝑥₁^′′,𝑞₂₁=2𝑧^′₁−𝑧₁^′′,𝑝₂₂=4𝑥₂,𝑓₂₂=4𝑥^′₂−5𝑥₂^′′,𝑞₂₂=5𝑧^′₂−4𝑧₂^′′. (36)Step 4. Using Step 4 of the proposed method the FFLS obtained in Step 3 may be written as 3𝑦₁+2𝑦₂=8,2𝑦₁+4𝑦₂𝑓=8,₁₁+𝑓₁₂𝑓=−13,₂₁+𝑓₂₂𝑞=−14,₁₁+𝑞₁₂𝑞=14,₂₁+𝑞₂₂=14. (37)Step 5. Convert the system of equation as a linear programming problem as follows.
Minimize 𝑅=𝑅₁+𝑅₂+𝑅^′₁+𝑅^′₂+𝑅₁^′′+𝑅₂^′′subject to the following 10 constraints:3𝑦₁+2𝑦₂+𝑅₁=8,2𝑦₁+4𝑦₂+𝑅₂=8,−𝑧^′₁+𝑧₁^′′+2𝑥^′₁−2𝑥₁^′′−0.5𝑢^′₁₁−0.5𝑢^′′₁₁−𝑧^′₂+𝑧₂^′′+1.5𝑥^′₂−1.5𝑥₂^′′−0.5𝑢^′₁₂−0.5𝑢^′′₁₂+𝑅^′₁𝑥=−13,_′1−2𝑥₁^′′+4𝑥^′₂−5𝑥₂^′′+𝑅^′₂=−14,2𝑧^′₁−2𝑧₁^′′−𝑥^′₁+𝑥₁^′′+0.5𝑣^′₁₁+0.5𝑣^′′₁₁+1.5𝑧^′₂−1.5𝑧₂^′′−𝑥^′₂+𝑥₂^′′+0.5𝑣^′₁₂+0.5𝑣^′′₁₂+𝑅₁^′′=14,2𝑧^′₁−𝑧₁^′′+5𝑧^′₂−4𝑧₂^′′+𝑅₂^′′2𝑧=14,_′1−𝑧₁^′′+4𝑥^′₁−𝑥₁^′′−𝑢^′₁₁+𝑢^′′₁₁2𝑥=0,_′1−𝑥₁^′′+4𝑧^′₁−𝑧₁^′′−𝑣^′₁₁+𝑣^′′₁₁2𝑧=0,_′2−𝑧₂^′′+3𝑥^′₂−𝑥₂^′′−𝑢^′₁₂+𝑢^′′₁₂2𝑥=0,_′2−𝑥₂^′′+3𝑧^′₂−𝑧₂^′′−𝑣^′₁₂+𝑣^′′₁₂𝑥=0,_′1,𝑥₁^′′,𝑥^′₂,𝑥₂^′′,𝑢^′₁₁,𝑢^′′₁₁,𝑣^′₁₁,𝑣^′′₁₁,𝑧^′₁,𝑧₁^′′,𝑧^′₂,𝑧₂^′′,𝑢^′₁₂,𝑢^′′₁₂,𝑣^′₁₂,𝑣^′′₁₂,𝑅₁,𝑅₂,𝑅^′₁,𝑅^′₂,𝑅₁^′′,𝑅₂^′′≥0. (38) On solving the above LPP, we get𝑦₁=2, 𝑦₂=1, 𝑥^′₁=1, 𝑥₁^′′=0, 𝑢^′₁₁=8, 𝑢^′′₁₁=0, 𝑣^′₁₁=10, 𝑣^′′₁₁=0, 𝑥^′₂=0, 𝑥₂^′′=3, 𝑧^′₁=2, 𝑧₁^′′=0, 𝑧^′₂=2, 𝑧₂^′′=0, 𝑢^′₁₂=0, 𝑢^′′₁₂=5, 𝑣^′₁₂=0, 𝑣^′′₁₂=0, 𝑅₁=0, 𝑅₂=0, 𝑅^′₁=0, 𝑅^′₂=0, 𝑅₁^′′=0, and 𝑅₂^′′=0.Step 6. Put the values of 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′, 𝑦_𝑗, obtained from Step 5, in 𝑥_𝑗=(𝑥^′_𝑗−𝑥_𝑗^′′,𝑦_𝑗,𝑧^′_𝑗−𝑧_𝑗^′′) for all 𝑗=1,2,…,𝑛. We find the solution of FFLS as 𝑥₁=(1,2,2),𝑥₂=(−3,1,2). (39) Clearly, the FFLS has a feasible and a strong fuzzy solution.Example 2. Solve the following 2×3 FFLS: ()()𝑥1,2,3)(−2,−1,−1)(2,3,4−3,−2,−1)(2,4,5)(3,4,5¹𝑥₂𝑥₃=()().−7,0,8−26,−10,−4 (40)

Solution 2. Clearly this is a nonsquare FFLS with arbitrary triangular fuzzy numbers as coefficients and hence cannot be solved by any of the existing methods. We apply our method to solve this system.
Rewrite the FFLS as(1,2,3)⊗𝑥₁,𝑦₁,𝑧₁⊕(−2,−1,−1)⊗𝑥₂,𝑦₂,𝑧₂⊕(2,3,4)⊗𝑥₃,𝑦₃,𝑧₃=(),()⊗𝑥−7,0,8−3,−2,−1₁,𝑦₁,𝑧₁⊕(2,4,5)⊗𝑥₂,𝑦₂,𝑧₂⊕(3,4,5)⊗𝑥₃,𝑦₃,𝑧₃=(−26,−10,−4). (41) Using the proposed method we obtain the following LPP:
Minimize 𝑅=𝑅₁+𝑅₂+𝑅^′₁+𝑅^′₂+𝑅₁^′′+𝑅₂^′′ subject to the following 6 constraints:2𝑦₁−𝑦₂+3𝑦₃+𝑅₁=0,−2𝑦₁+4𝑦₂+4𝑦₃+𝑅₂𝑥=−10,_′1−3𝑥₁^′′−2𝑧^′₂+𝑧₂^′′+2𝑥^′₃−4𝑥₃^′′+𝑅^′₁=−7,3𝑧^′₁−𝑧₁^′′−𝑥^′₂+2𝑥₂^′′+4𝑧^′₃−2𝑧₃^′′+𝑅₁^′′=8,−3𝑧^′₁+𝑧₁^′′+2𝑥^′₂−5𝑥₂^′′+3𝑥^′₃−5𝑥₃^′′+𝑅^′₂=−26,−𝑥^′₁+3𝑥₁^′′+5𝑧^′₂−2𝑧₂^′′+5𝑧^′₃−3𝑧₃^′′+𝑅₂^′′𝑥=−4,_′1,𝑥₁^′′,𝑥^′₂,𝑥₂^′′,𝑥^′₃,𝑥₃^′′,𝑦^′₁,𝑦₁^′′,𝑦^′₂,𝑦₂^′′,𝑦^′₃,𝑦₃^′′,𝑅₁,𝑅₂,𝑅^′₁,𝑅^′₂,𝑅₁^′′,𝑅₂^′′≥0. (42) The above LPP on solving by traditional algebra yields many solutions and one of the feasible solutions is𝑦₁=1, 𝑦₂=−1, 𝑦₃=−1, 𝑥^′₁=1, 𝑥₁^′′=0, 𝑥^′₂=0, 𝑥₂^′′=2, 𝑥^′₃=0, 𝑥₃^′′=2, 𝑧^′₁=2, 𝑧₁^′′=0, 𝑧^′₂=0, 𝑧₂^′′=0, 𝑧^′₃=0, 𝑧₃^′′=1, 𝑅₁=0, 𝑅₂=0, 𝑅^′₁=0, 𝑅^′₂=0, 𝑅₁^′′=0, and 𝑅₂^′′=0.
Put the values of 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′, 𝑦_𝑗obtained from Step 5, in 𝑥_𝑗=(𝑥^′_𝑗−𝑥_𝑗^′′,𝑦_𝑗,𝑧^′_𝑗−𝑧_𝑗^′′) for all 𝑗=1,2,…,𝑛. we find the solution of FFLS as𝑥₁=(1,1,2),𝑥₂=(−2,−1,0),𝑥₃=(−2,−1,−1). (43) Clearly, the obtained solution is a feasible (strong) fuzzy solution.

Example 3. Solve the following 2×2 fuzzy system randomly taken from [5]: 𝑥₁−𝑥₂=(),𝑥𝑟,2−𝑟₁+3𝑥₂=(4+𝑟,7−2𝑟). (44)

Solution 3. The fuzzy linear system is based upon the concept of 𝛼-cuts and has assumed restricted nature of FFLS by considering the coefficients crisp rather than fuzzy. We first convert this system into an FFLS with triangular fuzzy numbers and solve it by the proposed method to obtain the same result. The above system can be rewritten as ()()1,1,1)(−1,−1,−1𝑥1,1,1)(3,3,3𝑥¹₂=()()0,1,24,5,7, (45) where𝑥₁ and 𝑥₂ are arbitrary triangular fuzzy numbers.
Assuming 𝑥₁=(𝑥₁,𝑦₁,𝑧₁) and 𝑥₂=(𝑥₂,𝑦₂,𝑧₂), the FFLS may be written as(1,1,1)⊗𝑥₁,𝑦₁,𝑧₁⊕(−1,−1,−1)⊗𝑥₂,𝑦₂,𝑧₂=(),()⊗𝑥0,1,21,1,1₁,𝑦₁,𝑧₁⊕(3,3,3)⊗𝑥₂,𝑦₂,𝑧₂=(4,5,7), (46) where (𝑥₁,𝑦₁,𝑧₁) and (𝑥₂,𝑦₂,𝑧₂) are unrestricted triangular fuzzy numbers.
Using the proposed method we obtain the following LPP.
Minimize 𝑅=𝑅₁+𝑅₂+𝑅^′₁+𝑅^′₂+𝑅₁^′′+𝑅₂^′′ subject to the following 6 constraints:
𝑦₁−𝑦₂+𝑅₁𝑦=1,₁+3𝑦₂+𝑅₂𝑥=5,_′1−𝑥₁^′′−𝑧^′₂+𝑧₂^′′+𝑅^′₁𝑥=0,_′1−𝑥₁^′′+3𝑥^′₂−3𝑥₃^′′+𝑅^′₂𝑧=4,_′1−𝑧₁^′′−𝑥^′₂+𝑥₂^′′+𝑅₁^′′𝑧=2,_′1−𝑧₁^′′+3𝑧^′₂−3𝑧₂^′′+𝑅₂^′′𝑥=7,_′1,𝑥₁^′′,𝑥^′₂,𝑥₂^′′,𝑦^′₁,𝑦₁^′′,𝑦^′₂,𝑦₂^′′,𝑅₁,𝑅₂,𝑅^′₁,𝑅^′₂,𝑅₁^′′,𝑅₂^′′≥0. (47) The above LPP on solving by traditional algebra yields the following solution: 𝑦₁=2, 𝑦₂=1, 𝑥^′₁=1.375, 𝑥₁^′′=0, 𝑥^′₂=0.875, 𝑥₂^′′=0, 𝑧^′₁=2.875, 𝑧₁^′′=0, 𝑧^′₂=1.375, 𝑧₂^′′=0, 𝑅₁=0, 𝑅₂=0, 𝑅^′₁=0, 𝑅^′₂=0, 𝑅₁^′′=0, and 𝑅₂^′′=0.
Substituting the values of 𝑥^′_𝑗, 𝑥_𝑗^′′, 𝑧^′_𝑗, 𝑧_𝑗^′′, 𝑦_𝑗, obtained from Step 5, in 𝑥_𝑗=(𝑥^′_𝑗−𝑥_𝑗^′′,𝑦_𝑗,𝑧^′_𝑗−𝑧_𝑗^′′) for all 𝑗=1,2,…,𝑛 we find the solution of FFLS as𝑥₁=(1.375,2,2.875),𝑥₂=(0.875,1,1.375). (48) Clearly, the obtained solution is exactly the same as in [5] and is clearly feasible.

6. Conclusion

In this paper, we propose a new method for solving FFLS with arbitrary coefficients that not only overcomes the limitations of the several existing methods available in the literature like [2, 5–14, 17–29] but also amplifies the scope of incorporating fuzzy linear equation in real scenarios. The proposed method can successfully solve the following types of FFLS: (i)FFLS having nonnegative fuzzy coefficients and nonnegative fuzzy variables,(ii)FFLS having nonnegative fuzzy coefficients and unrestricted fuzzy variables,(iii)FFLS having unrestricted fuzzy coefficients and nonnegative fuzzy variables,(iv)FFLS having unrestricted fuzzy coefficients and unrestricted fuzzy variables,(v)Fuzzy linear systems with crisp coefficient matrix and unrestricted fuzzy variables.

The numerical examples that are solved in Section 5 clearly demonstrate that the proposed method can solve FFLS with arbitrary coefficients as well as FFLS with nonnegative and other several types of restrictions. The proposed method thus fruitfully removes the constraints that are imposed by researchers and is therefore better to use in several engineering and scientific applications. However a future work still remains to evolve methods with improved computational complexity and to solve an arbitrary FFLS with nonlinear membership functions.