On fuzzy solutions of the nonsquare fully fuzzy linear equation system with arbitrary triangular fuzzy numbers

Abstract

In this paper, Fully Fuzzy Linear Equation System (FFLS) that all parameters and variables are represented by triangular fuzzy numbers is discussed. FFLS has many important applications to branches of science, engineering and other disciplines. The objective of this paper is to find the feasible (strong) and approximate solution with a proposed method based on a mixed integer modeling of the nonsquare FFLS by removing all restrictions on the parameters and variables. The method is illustrated with numerical examples. Results of the numerical examples show that this method has the ability to generate a feasible (strong) and an approximate fuzzy solution and also to indicate no solution case of a nonsquare or square FFLS.

Keywords

Nonsquare fully fuzzy linear equation system triangular fuzzy number mixed integer programming

1 Introduction

Using the linear equation systems with fuzzy parameters and variables will be rather appropriate for developing the mathematical models to their respective problems.Due to the the lack of exact information, the parameters and variables of these models are not always exactly known. In order to express and overcome the vagueness of these parameters and variables, this imprecision are represented with fuzzy numbers.Since a fuzzy linear system can be used to apply various problems ranging from engineering to social sciences, it plays a major role and there is a vast literature on the investigation of solutions for fuzzy linear systems. Recent works in the literature are on to linear equation system called as Fuzzy Linear System (FLS) which has at least one fuzzy parameter or variable.The crispness of the components of FLS makes the modeling of real life problems restricted. Linear systems, whose components i.e. both coefficient matrix,variables and right hand vector are fuzzy, are named Fully Fuzzy Linear Equation Systems (FFLS). The main intend of FFLS is to remove restrictions and widen the scope of FLS in scientific applications.

In general, both FLS and FFLS are handled under two main headings: square (n × n) and non-square(m × n) forms. Most of the works in the literature deal with square form. Fuzzy elements of these systems can be taken as triangular, trapezoidal or generalized fuzzy numbers in general or parametric form. Another classification for FFLS can be made also depending on whether FFLS has sign restrictions on its components. These components are mostly assumed as positive (non-negative). Nowadays, studies related to FFLS with arbitrary (no restriction on sign) fuzzy numbers are increasing.

In Friedman et al. [17], a general model for solving a n × n fuzzy linear equation system Ax = b where A is a crisp matrix and b is an arbitrary fuzzy number vector is introduced. Using embedding approach, the original n × n fuzzy linear equation system is replaced by a 2n × 2n crisp linear equation system and then they gave the conditions for the existence of a unique fuzzy solution to n × n fuzzy linear equation system. Following [17], many works have been done to find the solution of the 2n × 2n crisp linear equation system [1–5 , 15]. Also some numerical iterative methods (such as Jacobi, Gauss-Seidel, etc.), steepest descent method and conjugate gradient method have been presented for solving fuzzy linear equation system [1 , 7] which is based on [17] applies Jacobi and Gauss-Seidel iterative methods for approximate of the unique solution of FLS, since solving FLS as analytically is difficult. Also he discusses convergence theorems and states that when the unique solution of system is a strong or weak fuzzy number, the approximate solution of iterative method also would be a strong or weak fuzzy number, respectively. [31] transforms the Gauss-Seidel iterative method in [7] to the successive over relaxation (SOR) method. The proposed method is followed by its convergence theorem and the algorithm is compared with the Gauss-Seidel method by solving a numerical example. Also Allahviranloo [32] applies the Adomian decomposition method and shows that this method is equivalent to the Jacobi iterative method.

Some of the main papers on the square FFLS are as follows. Dehghan and Hashemi [13] have represented fuzzy numbers in LR form and applied approximate operators between fuzzy numbers to find positive solutions of n × n FFLS, so calculating the solutions of FFLS is transformed to calculation the solutions of three crisp systems by using Adomian decomposition method. [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. For finding a non-zero solution for the FFLS, [8] introduced an algorithm. In this algorithm; at first, n × n fully fuzzy linear equation system $\tilde{A} \tilde{X} = \tilde{b}$ is transformed into a 2n × 4n fuzzy linear equation system; then again it is transformed into a 2n × 2n parametric system. In [33], Uzawa approach is considered for solving for a FFLS with trapezoidal fuzzy numbers using various relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR as well as Krylov subspace methods such as GMRES, QMR and BICGSTAB.

Considering a linear programming problem is based on a linear equation system, a new method is proposed to find the fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints in [22]. After this paper, Saberi and Edalatpanah show that this model is not correct, generally and a new version is provided in [27]. In [25], Otadi and Mosleh employ linear programming with equality constraints to find a non-negative fuzzy number matrix $\tilde{X}$ which satisfies $\tilde{A} \tilde{X} = \tilde{b}$ , where A and b are two fuzzy number matrices. If the solution of this linear programming problem has a positive artificial variable, then the original system has no non-negative solution $\tilde{X}$ . In this method, they reduce artificial variables to value zero or conclude that the original system has no non-negative solution. In [9], non-zero fuzzy number definition is given as fuzzy number which 0 is not inner point of its support. Based on this definition, an algorithm is proposed for FFLS with arbitrary coefficients. In this algorithm, n × n system $\tilde{A} \tilde{X} = \tilde{b}$ is transformed to 2n × 4n parametric system and then to 2n × 2n parametric system. Using this algorithm, one can not solve the systems that have zero solutions, i.e., it is not applicable to all FFLS. In [23], 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. In [21], a new method is proposed to find the exact fuzzy optimal solution of fully fuzzy linear programming problems with equality constraints having non-negative fuzzy coefficients and unrestricted fuzzy variables. Since this method converts the original system to a crisp non-linear programming problem, it is computationally ineffective. In [24], a simple and practical method that works only for triangular fuzzy numbers is proposed to solve a FFLS with arbitrary fuzzy parameters and variables. By using the cross-product of fuzzy numbers, this method is constructed based on the extending 0-cut and 1-cut solution of the original FFLS. In [12], three new numerical methods are discussed to solve a FFLS with triangular fuzzy numbers. The first and second methods remove the sign restriction on the coefficient matrix and the solution vector, respectively. And the third method removes joint restrictions on the coefficients as well as the solution vector; however, confining the solutions to exclude near zero fuzzy numbers. For nonsquare systems, Asady et al. [11] developed a method to solve FLS using the same approach in Friedman et al. [17]. They proposed a general model for m ≤ n by replacing the original system with a crisp linear equation system. Zheng and Wang [30] studied on m × n consistent FLS by the embedding method and then the existence and expression of the strong fuzzy solution to the system were discussed by using the generalized inverses of the coefficient matrix. These authors studied the inconsistent form in Wang and Zheng [28]. The least squares solution and the minimum norm least squares solution to the system were obtained by the same method, and they provided a sufficient condition for the least squares solution being a fuzzy vector. Gong and Guo [20] which is an extension of [28] presented a model to solve a class of inconsistent nonsquare FLS by the embedding method. Allahviranloo and AfsharKermani [10] investigated m × n FLS using a numerical method. They first replaced the original m × n fuzzy linear system by a 2m × 2n crisp linear equation system, and found the least-square solution by using an orthogonal matrix. Dehghan et al. [14] employed some heuristics based on classical methods such as Cramer’s rule, Gaussian elimination, LU decomposition method for finding the approximated solutions of FFLS and they presented a new method using linear programming to solve square and nonsquare (over-determined) FFLS. In a similar way to [10], Abbasbandy et al. [5] proposed a numerical method for finding minimal solution of a m × n general dual fuzzy linear system based on pseudo-inverse calculation. Ghanbari and Mahdavi-Amiri [18] firstly introduced the use of a ranking function to define a compromised solution for a nonsquare FLS and then proposed a methodology based on ABS class of algorithms using certain ranking functions.Otadi and Mosleh [26] developed a method to solve an arbitrary inconsistent FLS by using the embedding approach and investigated perturbation analysis in two m × n crisp linear equation systems instead of 2m × 2n crisp linear equation system as the authors of Ezzati [15] and Wang, Chen and Wei [29]. Also in [34], the similar approach is used to investigate consistent FLS. Ezzati et al. [16] proposed an approach based on the modified Huang algorithms to compute an approximate solution of FFLS, where all the parameters of the coefficient matrix are only nonnegative or nonpositive.

As it is seen, most of works in literature deals with FFLS with non-negative parameters and/or variables. For this reason, this paper focused on finding the solution of a FFLS with triangular fuzzy arbitrary coefficients and variables, additionally both square and nonsquare FFLS (m > n, m = n, m < n) are considered. In [19], a new algorithm is proposed to find the solutions of a general (square or nonsquare) FFLS with arbitrary trapezoidal fuzzy numbers.This algorithm is based on a mixed integer modeling of the original FFLS. Also in [6], a similar approach is given for a square FFLS with arbitrary triangular fuzzy numbers. However, modeling a real life problem within a square FFLS is not always possible. An actual realistic example of a nonsquare FFLS is an underdetermined linear equation system connected with Global Positioning System (GPS) for determining geographical locations [35]. Considering the widespread application area of triangular fuzzy numbers and a nonsquare FFLS, in this paper an extension of the method given in [6] for solving a general FFLS with triangular fuzzy numbers is presented.

This paper is organized as follows: Section 2 presents brief information about triangular fuzzy numbers and arithmetic operations. After the proposed method is introduced in Section 3, in Section 4, the presented method is illustrated by examples. The conclusion is given in Section 5.

2 Preliminaries

In this section, brief information about the triangular fuzzy numbers are presented.

Definition 1. A fuzzy number $\tilde{M} = (a, b, c)$ is said to be an arbitrary triangular fuzzy number if its membership function is given by $\begin{matrix} μ_{\tilde{M}} (x) = {\begin{matrix} \frac{x - a}{b - a}, & a \leq x < b \\ \frac{x - c}{b - c}, & b \leq x \leq c \\ 0, & Otherwise \end{matrix} \end{matrix}$ where a, b, c ∈ R and a ≤ b ≤ c.

Definition 2. An arbitrary triangular fuzzy number $\tilde{M} = (a, b, c)$ is called positive (negative), denoted by $\tilde{M} > 0 (\tilde{M} < 0)$ if its membership function $μ_{\tilde{M}} (x) = 0, \forall x < 0 (\forall x > 0) .$

Definition 3. An arbitrary triangular fuzzy number $\tilde{M} = (a, b, c)$ is called near-zero triangular fuzzy number which is neither positive nor negative.

Definition 4. Two arbitrary triangular fuzzy numbers $\tilde{M} = (a, b, c)$ and $\tilde{N} = (d, e, f)$ are said to be equal if and only if a = d, b = e and c = f. Some algebraic operations on triangular fuzzy numbers are defined as follows: Let $\tilde{M} = (a, b, c)$ and $\tilde{N} = (d, e, f)$ be arbitrary two triangular fuzzy numbers,

$\tilde{M} + \tilde{N} = (a + d, b + e, c + f)$

$- \tilde{M} = (- c, - b, - a)$

$\tilde{M} - \tilde{N} = (a - f, b - e, c - d)$ .

The fuzzy multiplication based on the extension principle is performed via the following equation:

\tilde{M} * \tilde{N} = (l, m, u)

with

\begin{matrix} m & = & be \\ l & = & min {ad, af, cd, cf} \\ u & = & max {ad, af, cd, cf} . \end{matrix}

3 Fully fuzzy linear equation system

A system such as: $\begin{matrix} {\tilde{a}}_{11} {\tilde{x}}_{1} + {\tilde{a}}_{12} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{1 n} {\tilde{x}}_{n} & = & {\tilde{b}}_{1} \\ {\tilde{a}}_{21} {\tilde{x}}_{1} + {\tilde{a}}_{22} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{2 n} {\tilde{x}}_{n} & = & {\tilde{b}}_{2} \\ ⋮ ⋱ ⋮ \end{matrix}$ (1) ${\tilde{a}}_{m 1} {\tilde{x}}_{1} + {\tilde{a}}_{m 2} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{mn} {\tilde{x}}_{n} = {\tilde{b}}_{m}$ where ${\tilde{a}}_{ij}$ , ${\tilde{b}}_{i}$ (i = 1, 2, . . . , m, j = 1, 2, . . . , n) and the unknown variables ${\tilde{x}}_{j}$ are arbitrary fuzzy numbers, is called a nonsquare FFLS. (1) can be written as

$\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} = {\tilde{b}}_{i}, i = 1, 2, . . ., m .$ (2)

Let us assume that ${\tilde{a}}_{ij}$ and ${\tilde{b}}_{i}$ are given by triangular fuzzy numbers as (a_ij, b_ij, c_ij) and (l_i, m_i, u_i), respectively, where a_ij, b_ij, c_ij, l_i, m_i and u_i are real numbers. Thus, I will investigate a fuzzy vector $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ .

Definition 5. If $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ solves (2), then it is called a feasible (strong) fuzzy solution of (2).

Definition 6. [17] Let the fuzzy number $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ denoted by $(t_{i}^{1}, t_{i}^{2}, t_{i}^{3})$ and ${\tilde{b}}_{i} = (l_{i}, m_{i}, u_{i})$ for (2). $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ is said to be an approximate fuzzy solution of (2) if it satisfies the following conditions:

$\tilde{x}$ is not a feasible fuzzy solution,

$l_{i} \leq t_{i}^{3}$ and $u_{i} \geq t_{i}^{1}$ for ∀i∈ { 1, 2, . . . , m }.

Here, the condition (i) implies that the fuzzy numbers obtained in the left hand sides of each equation are not equal to their right hand side fuzzy number. The condition (ii) ensures that the intersection of these fuzzy numbers is not an empty set.

Definition 7. If the system (2) has no feasible or approximate fuzzy solution, then it is said that the system (2) has no solution.

3.1 Analysis of the multiplication of two arbitrary triangular fuzzy numbers

In this subsection, for the presentation of the basic principle of the presented approach, the fuzzy multiplication ${\tilde{a}}_{ij} {\tilde{x}}_{j}$ multiplication of two arbitrary fuzzy numbers ${\tilde{a}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ and ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ is constructed.

In [23], the multiplication of two arbitrary fuzzy numbers is given as follows:

$\begin{matrix} {\tilde{a}}_{ij} {\tilde{x}}_{j} & = & (a_{ij}, b_{ij}, c_{ij}) (x_{j}, y_{j}, z_{j}) \\ = & (λ_{ij}, b_{ij} y_{j}, μ_{ij}) \end{matrix}$ (3) where λ_ij = min{ a_ijx_j, a_ijz_j, c_ijx_j, c_ijz_j } and μ_ij = max{ a_ijx_j, a_ijz_j, c_ijx_j, c_ijz_j }. Also, λ_ij and μ_ij values are identified according to sign of components of ${\tilde{a}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ , i.e. ${\tilde{a}}_{ij}$ is positive, negative or near-zero (mixed) in the following way:

When (i, j) indicates the position of ${\tilde{a}}_{ij}$ in the coefficient matrix of (2), $\begin{matrix} λ_{ij} & = & min {a_{ij} x_{j}, c_{ij} x_{j}} and \\ μ_{ij} & = & max {a_{ij} z_{j}, c_{ij} z_{j}} \forall (i, j) \in S^{pos} \end{matrix}$ (4) $\begin{matrix} λ_{ij} & = & min {a_{ij} z_{j}, c_{ij} x_{j}} and \\ μ_{ij} & = & max {a_{ij} x_{j}, c_{ij} z_{j}} \forall (i, j) \in S^{mix} \end{matrix}$ (5) $\begin{matrix} λ_{ij} & = & min {a_{ij} z_{j}, c_{ij} z_{j}} and \\ μ_{ij} & = & max {a_{ij} x_{j}, c_{ij} x_{j}} \forall (i, j) \in S^{neg} \end{matrix}$ (6) where $\begin{matrix} S^{pos} & = & {(i, j) | 0 < a_{ij}}, \\ S^{mix} & = & {(i, j) | a_{ij} \leq 0 \leq c_{ij}}, \\ S^{neg} & = & {(i, j) | c_{ij} < 0} . \end{matrix}$

The proposed method of [23] is based upon the decomposition of an FFLS into a nonlinear system within the following formulas $\begin{matrix} min (x, y) & = & (\frac{x + y}{2}) - | \frac{x - y}{2} | \\ max (x, y) & = & (\frac{x + y}{2}) + | \frac{x - y}{2} | . \end{matrix}$

Using a variable transformation, this method enables to obtain an feasible (strong) fuzzy solution with a nonlinear sufficient condition and an infeasible (weak) fuzzy solution. The presented approach eliminates this nonlinearity by means of a mixed integer programming modeling of “min” and “max” concepts. All the parameters and variables are assumed as triangular fuzzy numbers.

3.2 Finding the “minimum” and “maximum” of any finite set of real numbers by a mixed integer programming problem

The subsection contains the bases of the proposed approach.

Definition 8. Let P be a nonempty finite subset of $ℝ$ . The minimum element of P has the following properties:

m ∈ P

x ≥ m, for all x ∈ P

where m = min P. A similar definition can be given for the maximum.

Theorem 1. Let P ={ p₁, p₂, …, p_n } be a nonempty finite subset of $ℝ$ . P has both a minimum element and a maximum element.

The proof of Theorem 1 can be seen in [30].

Let P ={ p₁, p₂, …, p_n } and K = {k₁, k₂, …, k_n} be a nonempty finite subset of and a binary variable set, respectively. Suppose that for ∀i∈ { 1, 2, …, n }, p_i corresponds to k_i∈ { 0, 1 }.

Proposition 1. Let $(λ^{*}, k_{1}^{*}, k_{2}^{*}, \dots, k_{n}^{*})$ be the optimal solution of the following mixed integer programming:

$\begin{matrix} \max 0 \cdot λ \\ λ \leq p_{i}, i = 1, 2, \dots, n \\ p_{i} - λ \leq M k_{i}, i = 1, 2, \dots, n \\ \sum_{i = 1}^{n} k_{i} = (n - 1), k_{i} \in {0, 1} i = 1, 2, \dots, n \end{matrix}$ (7) where M is a sufficiently large constant. Then λ^* is the minimum element of P, i.e. λ = min{ p₁, p₂, …, p_n }.

Proof. It can be seen in [19]. □

Result.p_i which corresponds to $k_{i}^{*} = 0$ is the minimum of P.

Proposition 2. Let $(μ^{*}, k_{1}^{*}, k_{2}^{*}, \dots, k_{n}^{*})$ be the optimal solution of the following mixed integer programming problem:

$\begin{matrix} min 0 \cdot μ \\ μ \geq p_{i} . i = 1, 2, \dots, n \\ μ - p_{i} \leq M k_{i}, i = 1, 2, \dots, n \\ \sum_{i = 1}^{n} k_{i} = (n - 1), k_{i} \in {0, 1}, i = 1, 2, \dots, n \end{matrix}$ (8) where M is a sufficiently large constant. Then μ^* is the maximum element of P, i.e. μ = max{ p₁, p₂, …, p_n }.

Proof. Similar to Proposition 1. □

3.3 Proposed algorithm for a FFLS with triangular fuzzy numbers

The basic principle of the presented approach is based on converting the original FFLS to a mixed integer programming problem by using the propositions given in the previous subsection. To solve the original FFLS (2), the approaches presented in (7) and (8) are adapted to the analysis of the multiplication of two arbitrary fuzzy numbers. This analysis given by (4)–(6) is to find the minimum and the maximum of the sets of size two, i.e. n = 2.

Using (3), (2) can be written as

$\begin{matrix} \sum_{j = 1}^{n} (λ_{ij}, b_{ij} y_{j}, μ_{ij}) = (l_{i}, m_{i}, u_{i}); \\ i = 1, 2, \dots, m \end{matrix}$ (9)

Using the equality of fuzzy numbers (Definition 4), (2) is equivalent to the following system: $\sum_{j = 1}^{n} λ_{ij} = l_{i}, i = 1, 2, \dots, m$ (10) $\sum_{j = 1}^{n} b_{ij} y_{j} = m_{i}, i = 1, 2, \dots, m$ (11) $\sum_{j = 1}^{n} μ_{ij} = u_{i}, i = 1, 2, \dots, m$ (12)

The proposed method that is based on the sign of entries of the coefficient matrix can be adapted to the original system (2) with the following step functions: $\begin{matrix} ɛ_{1} ({\tilde{a}}_{ij}) = {\begin{matrix} 1, & if (i, j) \in S^{pos} \\ 0, & otherwise \end{matrix} \\ ɛ_{2} ({\tilde{a}}_{ij}) = {\begin{matrix} 1, & if (i, j) \in S^{mix} \\ 0, & otherwise \end{matrix} \\ ɛ_{3} ({\tilde{a}}_{ij}) = {\begin{matrix} 1, & if (i, j) \in S^{neg} \\ 0, & otherwise . \end{matrix} \end{matrix}$

The mixed integer programming problem which is equivalent to (10)–(12) can be written as follows:

$\begin{matrix} min Z & = & \sum_{i = 1}^{m} (R_{i}^{l +} + R_{i}^{l -} + α (R_{i}^{m +} + R_{i}^{m -}) + R_{i}^{u +} + R_{i}^{u -}) \\ ɛ_{1} ({\tilde{a}}_{ij}) (λ_{ij} - a_{ij} x_{j}) + ɛ_{2} ({\tilde{a}}_{ij}) (λ_{ij} - a_{ij} z_{j}) + ɛ_{3} ({\tilde{a}}_{ij}) (λ_{ij} - a_{ij} z_{j}) \leq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (λ_{ij} - c_{ij} x_{j}) + ɛ_{2} ({\tilde{a}}_{ij}) (λ_{ij} - c_{ij} z_{j}) + ɛ_{3} ({\tilde{a}}_{ij}) (λ_{ij} - c_{ij} z_{j}) \leq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (a_{ij} x_{j} - λ_{ij} - {Mk}_{ij}^{λ}) + ɛ_{2} ({\tilde{a}}_{ij}) (a_{ij} z_{j} - λ_{ij} - {Mk}_{ij}^{λ}) + ɛ_{3} ({\tilde{a}}_{ij}) (a_{ij} z_{j} - λ_{ij} - {Mk}_{ij}^{λ}) \leq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (c_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ})) + ɛ_{2} ({\tilde{a}}_{ij}) (c_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ})) \\ + ɛ_{3} ({\tilde{a}}_{ij}) (c_{ij} z_{j} - λ_{ij} - M (1 - k_{ij}^{λ})) \leq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} z_{j}) + ɛ_{2} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j}) + ɛ_{3} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j}) \geq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} z_{j}) + ɛ_{2} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} z_{j}) + ɛ_{3} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} x_{j}) \geq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} z_{j} - {Mk}_{ij}^{μ}) + ɛ_{2} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j} - {Mk}_{ij}^{μ}) + ɛ_{3} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j} - {Mk}_{ij}^{μ}) \leq 0, \forall (i, j) \\ ɛ_{1} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} z_{j} - M (1 - k_{ij}^{μ})) + ɛ_{2} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} z_{j} - M (1 - k_{ij}^{μ})) \\ + ɛ_{3} ({\tilde{a}}_{ij}) (μ_{ij} - c_{ij} x_{j} - M (1 - k_{ij}^{μ})) \leq 0, \forall (i, j) \\ \sum_{j = 1}^{n} λ_{ij} - R_{i}^{l +} + R_{i}^{l -} = l_{i} \forall i \\ \sum_{j = 1}^{n} b_{ij} y_{j} - R_{i}^{m +} + R_{i}^{m -} = m_{i}, \forall i \\ \sum_{j = 1}^{n} μ_{ij} - R_{i}^{u +} + R_{i}^{u -} = u_{i} \forall i \\ x_{j} \leq y_{j} \leq z_{j}, \forall j \\ λ_{ij}, μ_{ij} is free, \forall (i, j) \\ x_{j}, y_{j}, z_{j} is free,, \forall j \\ k_{ij}^{λ}, k_{ij}^{μ} \in {0, 1} . \\ R_{i}^{l +}, R_{i}^{l -}, R_{i}^{m +}, R_{i}^{m -}, R_{i}^{u +}, R_{i}^{u -} \geq 0, \forall i \end{matrix}$ (13) where $R_{i}^{l +}, R_{i}^{m +}, R_{i}^{u +}$ $(R_{i}^{l -}, R_{i}^{m -}, R_{i}^{u -})$ represent the deviations of the left hand side values in the right (left) direction from l_i, $m_{i}$ and u_i for equation i∈ { 1, 2, …, m } of the original FFLS, respectively and the weightiness parameter α > 1 is a constant value that is chosen sufficiently large number to minimize the deviations of the full satisfaction values (the core).If the optimal objective function value of (13) is zero (Z^* = 0), then the original FFLS has a feasible fuzzy solution. Otherwise, i.e. Z^* > 0, it should be checked whether the optimal solution of (13) is satisfied the Definition 6 and the nature of the solution of (2) can bedetermined.

Remark 1. If the problem (13) has a unique optimal solution with Z^* = 0, then the original FFLS has a unique feasible fuzzy solution.

Remark 2. If the problem (13) has alternative optimal solutions with Z^* = 0, then the original FFLS has many feasible fuzzy solutions.

Remark 3. If Z^* ≠ 0, the original system (2) has no feasible fuzzy solution, i.e. it has an approximate fuzzy solution or no solution.

Remark 4. The problem (13) always has an optimal solution.

Remark 5. With the proposed approach, a m × n FFLS results into a mixed integer programming problem with a number of 8mn + 3m + 2nconstraints.

4 Numerical experiments

The mixed integer programming problems are solved by using the software package GAMS for each numerical example.

Example 4.1. Solve the following 2 × 2 fuzzy system taken from [23]: $\begin{matrix} (- 2, 3, 4) {\tilde{x}}_{1} + (- 2, 2, 3) {\tilde{x}}_{2} & = & (- 13, 8, 14) \\ (1, 2, 2) {\tilde{x}}_{1} + (4, 4, 5) {\tilde{x}}_{2} & = & (- 14, 8, 14) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ and ${\tilde{x}}_{2} = (x_{2}, y_{2}, z_{2})$ are arbitrary triangular fuzzy numbers.

Firstly, the following sets are determined: S^pos ={ (2, 1) , (2, 2) }, S^mix ={ (1, 1) , (1, 2) }, S^neg ={ }. Since n = 2 and the mixed integer programming problem corresponding to (13) has 8n² + 5n = 42 constraints and can be given as follows:

$\begin{matrix} min Z & = & \sum_{i = 1}^{m} (R_{i}^{l +} + R_{i}^{l -} + α (R_{i}^{m +} + R_{i}^{m -}) + R_{i}^{u +} + R_{i}^{u -}) \\ λ_{11} + 2 z_{1} \leq 0, λ_{11} - 4 x_{1} \leq 0, - 2 z_{1} - λ_{11} - M k_{11}^{λ} \leq 0, 4 x_{1} - λ_{11} - M (1 - k_{11}^{λ}) \leq 0 \\ λ_{12} + 2 z_{2} \leq 0, λ_{12} - 3 x_{2} \leq 0, - 2 z_{2} - λ_{12} - M k_{12}^{λ} \leq 0, 3 x_{2} - λ_{12} - M (1 - k_{12}^{λ}) \leq 0 \\ λ_{21} - x_{1} \leq 0, λ_{21} - 2 x_{1} \leq 0, x_{1} - λ_{21} - M k_{21}^{λ} \leq 0, 2 x_{1} - λ_{21} - M (1 - k_{21}^{λ}) \leq 0 \\ λ_{22} - 4 x_{2} \leq 0, λ_{22} - 5 x_{2} \leq 0, 4 x_{2} - λ_{22} - M k_{22}^{λ} \leq 0, 5 x_{2} - λ_{22} - M (1 - k_{22}^{λ}) \leq 0 \\ μ_{11} + 2 x_{1} \geq 0, μ_{11} - 4 z_{1} \geq 0, μ_{11} + 2 x_{1} - M k_{11}^{μ} \leq 0, μ_{11} - 4 z_{1} - M (1 - k_{11}^{μ}) \leq 0 \\ μ_{12} + 2 x_{2} \geq 0, μ_{12} - 3 z_{2} \geq 0, μ_{12} + 2 x_{2} - M k_{12}^{μ} \leq 0, μ_{12} - 3 z_{2} - M (1 - k_{12}^{μ}) \leq 0 \\ μ_{21} - z_{1} \geq 0, μ_{21} - 2 z_{1} \geq 0, μ_{21} - z_{1} - M k_{21}^{μ} \leq 0, μ_{21} - 2 z_{1} - M (1 - k_{21}^{μ}) \leq 0 \\ μ_{22} - 4 z_{2} \geq 0, μ_{22} - 5 z_{2} \geq 0, μ_{22} - 4 z_{2} - M k_{22}^{μ} \leq 0, μ_{22} - 5 z_{2} - M (1 - k_{22}^{μ}) \leq 0 \\ λ_{11} + λ_{12} - R_{1}^{l +} + R_{1}^{l -} = - 13, λ_{21} + λ_{22} - R_{2}^{l +} + R_{2}^{l -} = - 14 \\ 3 y_{1} + 2 y_{2} - R_{1}^{m +} + R_{1}^{m -} = 8, 2 y_{1} + 4 y_{2} - R_{2}^{m +} + R_{2}^{m -} = 8 \\ μ_{11} + μ_{12} - R_{1}^{u +} + R_{1}^{u -} = 14, μ_{11} + μ_{12} - R_{2}^{u +} + R_{2}^{u -} = 14 \\ x_{1} \leq y_{1} \leq z_{1}, x_{2} \leq y_{2} \leq z_{2} \\ λ_{ij}, μ_{ij} is free, \forall (i, j) \\ x_{j}, y_{j}, z_{j} is free, \forall j \\ k_{ij}^{λ}, k_{ij}^{μ} \in {0, 1}, \forall (i, j) \\ R_{i}^{l +}, R_{i}^{l -}, R_{i}^{m +}, R_{i}^{m -}, R_{i}^{u +}, R_{i}^{u -} \geq 0, \forall i \end{matrix}$ (14)

The optimal solution of (14) is: $\begin{matrix} {\tilde{x}}_{1}^{*} = (x_{1}^{*}, y_{1}^{*}, z_{1}^{*}) & = & (1, 2, 2) {\tilde{x}}_{2}^{*} = (x_{2}^{*}, y_{2}^{*}, z_{2}^{*}) = (- 3, 1, 2) \\ (λ_{11}^{*}, μ_{11}^{*}) & = & (- 4, 8) (λ_{12}^{*}, μ_{12}^{*}) = (- 9, 6); \end{matrix}$ $\begin{matrix} (λ_{21}^{*}, μ_{21}^{*}) & = & (1, 4) (λ_{22}^{*}, μ_{22}^{*}) = (- 15, 10) \\ (k_{11}^{λ *}, k_{11}^{μ *}) & = & (0, 1) (k_{12}^{λ *}, k_{12}^{μ *}) = (1, 1) \\ (k_{21}^{λ *}, k_{21}^{μ *}) & = & (0, 1) (k_{22}^{λ *}, k_{22}^{μ *}) = (1, 1) \\ (R_{1}^{l * +}, R_{1}^{m * +}, R_{1}^{u * +}) & = & (0, 0, 0) (R_{1}^{l -}, R_{1}^{m -}, R_{1}^{u -}) = (0, 0, 0) \\ (R_{2}^{l * +}, R_{2}^{m * +}, R_{2}^{u * +}) & = & (0, 0, 0); (R_{2}^{l -}, R_{2}^{m -}, R_{2}^{u -}) = (0, 0, 0); \\ Z^{*} & = & 0 \end{matrix}$

Since Z^* = 0, using Definition 6, $({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*})$ is a feasible (strong) fuzzy solution which is exactly the same as in [23].

Example 4.2. Solve the following 2 × 3 FFLS taken from [23]: $\begin{matrix} (1, 2, 3) {\tilde{x}}_{1} + (- 2, - 1, - 1) {\tilde{x}}_{2} + (2, 3, 4) {\tilde{x}}_{3} \\ = (- 7, 0, 8) \\ (- 3, - 2, - 1) {\tilde{x}}_{1} + (2, 4, 5) {\tilde{x}}_{2} + (3, 4, 5) {\tilde{x}}_{3} \\ = (- 26, - 10, - 4) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ , ${\tilde{x}}_{2} = (x_{2}, y_{2}, z_{2})$ and ${\tilde{x}}_{3} = (x_{3}, y_{3}, z_{3})$ are arbitrary triangular fuzzy numbers.

The sets S^pos ={ (1, 1) , (1, 3) , (2, 2) , (2, 3) }, S^neg ={ (1, 2) , (2, 1) }, $S^{mix} = {}$ . Since m = 2 and n = 3, the mixed integer programming problem corresponding to (13) which has 8mn + 3m + 2n = 60 constraints can be constructed and the following optimal ${\tilde{x}}_{j}^{*}$ (j = 1, 2, 3) values are obtained as ${\tilde{x}}_{1}^{*} = (0, 1.889, 2.093)$ , ${\tilde{x}}_{2}^{*} = (- 2.194, - 0.222, 0)$ , ${\tilde{x}}_{3}^{*} = (- 1.75, - 1.333, - 1.333)$ with the optimal objective value Z^* = 0. Since Z^* = 0, $({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*}, {\tilde{x}}_{3}^{*})$ is a feasible fuzzy solution. In [23] the feasible fuzzy solution is obtained as ${\tilde{x}}_{1}^{*} = (1, 1, 2)$ , ${\tilde{x}}_{2}^{*} = (- 2, - 1, 0)$ , ${\tilde{x}}_{3}^{*} = (- 2, - 1, - 1)$ . It is clear that an alternative feasible fuzzy solution is found by the proposed method.

Example 4.3. Solve $\tilde{A} \tilde{x} = \tilde{b}$ taken from [16], where $\begin{matrix} \tilde{A} & = & [\begin{matrix} (- 1 . 6 008, - 1 . 484, - 1 . 3596) & (- 5 . 3721, - 5 . 2718, - 4.0683) & (7 . 25 02, 8 . 0951, 8 . 255) \\ (- 6.3172, - 6.2145, - 4.8308) & (2.82, 3 . 1244, 3 . 31 05) & (- 7 . 3871, - 7 . 2331, - 6 . 7218) \\ (6 . 1715, 7 . 5437, 7 . 7266) & (1 . 7692, 1 . 8174, 1 . 9952) & (0.3857, 0.4531, 0.4555) \\ (- 2.8307, - 2.806, - 2.6388) & (6.0766, 7 . 7925, 7 . 9551) & (4.9626, 6 . 133, 6 . 2951) \end{matrix}] \\ \tilde{b} & = & [\begin{matrix} (- 23 . 5351, - 13 . 2664, 0) \\ (- 55.8273, 49 . 5008, - 29.9415) \\ (50.3095, 64 . 4898, 67 . 4468) \\ (26.9409, 52 . 118, 56 . 8597) \end{matrix}] and \tilde{x} = [\begin{matrix} {\tilde{x}}_{1} \\ {\tilde{x}}_{2} \\ {\tilde{x}}_{3} \end{matrix}] \end{matrix}$

where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ , ${\tilde{x}}_{2} = (x_{2}, y_{2}, z_{2})$ and ${\tilde{x}}_{3} = (x_{3}, y_{3}, z_{3})$ are arbitrary triangular fuzzy numbers.

After the sets S^pos, $S^{mix}$ , S^neg are determined, the mixed integer programming problem corresponding to (13) with 8mn + 3m + 2n = 114 constraints is constructed and the following solution is obtained:

${\tilde{x}}_{1}^{*} = (6 . 470, 6 . 823, 6 . 864)$ , ${\tilde{x}}_{2}^{*} = (5 . 224, 6 . 248, 6 . 381)$ and ${\tilde{x}}_{3}^{*} = (2 . 947, 3 . 681, 3 . 681)$ with the optimal objective value Z^* = 1 . 232.

Since Z^* > 0, by checking whether $({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*}, {\tilde{x}}_{3}^{*})$ is satisfied the Definition 6, it is seen that this solution is an approximate fuzzy solution with the total deviation 1 . 232. In [16] the approximate solution of the system is obtained as ${\tilde{x}}_{1}^{*} = (6.4756, 6.8225, 6.901)$ , ${\tilde{x}}_{2}^{*} = (5.5054, 6.248, 6.353)$ and ${\tilde{x}}_{3}^{*} = (3.0467, 3.6808, 3.6981)$ .

After this solution is substituted in each equation, the total deviation can be calculated as 6.859. Thus, the proposed method generates more accurate solution for this example.

Example 4.4. Consider the following FFLS: $\begin{matrix} (1, 2, 3) {\tilde{x}}_{1} & = & (2, 6, 12) \\ (30, 40, 50) {\tilde{x}}_{1} & = & (4, 5, 6) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ is an arbitrary triangular fuzzy number.

After the sets S^pos, $S^{mix}$ , S^neg are determined, the mixed integer programming problem corresponding to (13) with 8mn + 3m + 2n = 24 constraints is constructed and the optimal solution is obtained as ${\tilde{x}}_{1}^{*} = (0.125, 0.125, 0.125)$ with a nonzero Z^* value.

Since Z^* > 0, it should be checked whether ${\tilde{x}}_{1}^{*}$ is an approximate fuzzy solution by Definition 6. Hence the inequality $μ_{11}^{1 *} \geq 2$ is not satisfied with this optimal solution, ${\tilde{x}}_{1}^{*}$ is not an approximate fuzzy solution, and the corresponding system has no solution.

5 Conclusion

Expressing the parameters as crisp numbers, modeling of real world applications is the major problem in many scientific areas. Thus in this paper, the nonsquare FFLS with triangular fuzzy numbers having no restrictions on all parameters and variables is examined. For this system, the feasible (strong) and approximate fuzzy solution concepts and no solution case are defined. And a new method is presented based on converting the original FFLS to a mixed integer programming problem. It is seen that, the proposed method has the ability to determine the solution types of a FFLS. And also the results of the examples taken from the literature shows that this method can generate more accurate approximate fuzzy feasible solution. As a future work, the proposed method can be extended to solve fully fuzzy Sylvester matrix equations or complex fuzzy systems which are more general.

Footnotes

Acknowledgments

This research has been supported by Yıldız Technical University Scientific Research Projects Coordination Department. Project Number: 2013-07-03-GEP01.

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