A mixed integer programming approach to a square fully fuzzy linear equation

Abstract

In this paper, a new method is proposed to find the feasible (strong) fuzzy solution of a square (n × n) fully fuzzy linear equation system (FFLS) with triangular fuzzy numbers. The main purpose of the proposed method is to remove all the sign restrictions on the parameters and variables. Our method, which is based on the multiplication of two arbitrary triangular fuzzy numbers, converts the FFLS to a mixed integer programming problem. The method is illustrated with numerical examples.

Keywords

Linear equation system fuzzy number fuzzy decision making mixed integer programming

1 Introduction

Systems of linear equations play a major role in various areas such as operational research, physics, statistics, engineering and social sciences. In many applications, the parameters of linear equation system are not always exactly known and stable. This imprecision may follow from the lack of exact information, changeable economic conditions, etc. A frequently used way of expressing the imprecision is to use the fuzzy numbers rather than crisp numbers. It enables us to consider tolerances for parameters of linear equation system in a more natural and direct way. Therefore, fuzzy linear equation systems seem to be more realistic and reliable.

Friedman et al. [1] introduced 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. 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 [1], many works have been done to find the solution of the 2n × 2n crisp linear equation system [2 –7]. 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 [2 , 8].

Another important kind of fuzzy linear systems including fuzzy numbers, in which all parameters are fuzzy numbers, is named as Fully Fuzzy Linear Equation System (FFLS). In [9], Dehghan and Hashemi have studied some methods for solving FFLS.They 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. [10] 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, [11] 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.

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 [12]. After this paper, Saberi and Edalatpanah show that this model is not correct, generally and a new version is provided in [13].

In [14], 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 [15], 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 cannot solve the systems that have zero solutions, i.e., it is not applicable to all FFLS.

In [16], a 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 [17], a 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 [18], 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 [19], three 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.

A matrix method is proposed for solving positive solutions for a positive FFLS in [20]. All coefficients on the right hand side are collected in one block matrix, while the entries on the left hand side are collected in one vector. Their methods and results are also capable of solving Left-Right Fuzzy Linear System (LR-FLS).

In [21] Mosleh and Otadi presented a note to shown that the examples presented in a paper by Basaran [22] are incorrect. The correct exact solutions are also presented and a direction for approximating fuzzy inverse matrix is proposed.

As seen, most of works in literature deals with fully fuzzy linear equation systems with non-negative parameters and/or variables. For this reason, we focused on finding the solution of FFLS with arbitrary coefficients and arbitrary variables.

In [16], for their presented approach, the authors state that a future work still remains to evolve methods with improved computational complexity due to the nonlinearity. In this paper; to overcome this complexity, we proposed a new method based upon a mixed integer modeling of the original FFLS with arbitrary fuzzy parameters and arbitrary fuzzy variables. We assumed all the parameters and variables as triangular fuzzy numbers.

This paper is organized as follows: Section 2 presents brief information about triangular fuzzy numbers and arithmetic operations. After our proposed method is introduced in Section 3, in Section 4, the presented method is illustrated by some examples from the literature. Section 5 discussed the advantages of the proposed method and the conclusion is given in Section 6.

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

$μ_{\tilde{M}} (x) = {\begin{matrix} \frac{x - a}{b - a}, & a \leq x < b \\ \frac{x - c}{b - c}, & b \leq x \leq c \\ 0, & Otherwise \end{matrix}$

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:

\begin{matrix} \tilde{M} * \tilde{N} = (l, m, u) \\ with \\ m = be \\ l = min {ad, af, cd, cf} \\ u = max {ad, af, cd, cf} \end{matrix}

3 Fully fuzzy linear equation system

A matrix system such as: $\begin{matrix} {\tilde{a}}_{11} {\tilde{x}}_{1} + {\tilde{a}}_{12} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{1 n} {\tilde{x}}_{n} = {\tilde{b}}_{1} \\ {\tilde{a}}_{21} {\tilde{x}}_{1} + {\tilde{a}}_{22} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{2 n} {\tilde{x}}_{n} = {\tilde{b}}_{2} \\ ⋮ ⋱ ⋮ \\ {\tilde{a}}_{n 1} {\tilde{x}}_{1} + {\tilde{a}}_{n 2} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{nn} {\tilde{x}}_{n} = {\tilde{b}}_{n} \end{matrix}$ (1) where ${\tilde{a}}_{ij}$ , ${\tilde{b}}_{i}$ (i = 1, 2, …, n, j = 1, 2, …, n) and the unknown variables ${\tilde{x}}_{j}$ are arbitrary fuzzy numbers, is called a Fully Fuzzy Linear Equation System (FFLS). (1) can be written as $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} = {\tilde{b}}_{i}, i = 1, 2, \dots, n$ (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, we will investigate arbitrary triangular fuzzy number ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ .

Definition 5. If ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ solves (2), then it is called a feasible (strong) fuzzy solution of (2).

The structure of the system (2) changes according to the sign and value of the components of ${\tilde{a}}_{ij}$ , ${\tilde{x}}_{j}$ (i = 1, 2, …, n, j = 1, 2, …, n). Assuming parameters and variables as arbitrary, (2) cannot be decomposed and be solved easily. Therefore, there are not lots of papers in the literature for this case. In this paper, we assume all parameters and variables in system (2) as arbitrary triangular fuzzy numbers and present a new approach for (2).

3.1 Analysis of the multiplication of two arbitrary triangular fuzzy numbers

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

In [16], the multiplication of two arbitrary fuzzy numbers is given as follows: ${\tilde{a}}_{ij} {\tilde{x}}_{j} = (a_{ij}, b_{ij}, c_{ij}) (x_{j}, y_{j}, z_{j}) = (λ_{ij}, b_{ij} y_{j}, μ_{ij})$ (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 S^pos ={ (i, j) | 0 < a_ij }, S^mix = {(i, j) | a_ij ≤ 0 ≤ c_ij}, S^neg ={ (i, j) | c_ij < 0 }. The proposed method of [16] 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 a feasible (strong) fuzzy solution with a nonlinear sufficient condition and an infeasible (weak) fuzzy solution. In our approach, we eliminate this nonlinearity by means of a mixed integer programming modeling of “min” and “max” concepts and fuzzy programming approach [23]. And we note that we only aim to find a feasible fuzzy solution.

3.2 Our proposed approach

The basic principle of our approach is based on converting the original FFLS to a mixed integer programming problem by using the propositions given in the subsection 3.3. of our paper [24]. To solve the original FFLS (2), the approaches presented in (9) and (10) in [24] 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 $\sum_{j = 1}^{n} (λ_{ij}, b_{ij} y_{j}, μ_{ij}) = (l_{i}, m_{i}, u_{i}), i = 1, 2, \dots, n$ (7)

Using the equality of fuzzy numbers (Definition 4), (7) implies the followings: $\sum_{j = 1}^{n} λ_{ij} = l_{i}, i = 1, 2, \dots, n$ (8) $\sum_{j = 1}^{n} b_{ij} y_{j} = m_{i}, i = 1, 2, \dots, n$ (9) $\sum_{j = 1}^{n} μ_{ij} = u_{i}, i = 1, 2, \dots, n .$ (10)

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

Now, we state the essential theorem of our approach.

Theorem. The feasible (strong) solution of the system (2) is the optimal solution of the following mixed integer programming problem: $max \sum_{i = 1}^{n} \sum_{j = 1}^{n} 0 \cdot λ_{ij} - \sum_{i = 1}^{n} \sum_{j = 1}^{n} 0 \cdot μ_{ij}$ (11) $\begin{matrix} 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} x_{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} - M k_{ij}^{λ}) \\ + ɛ_{2} ({\tilde{a}}_{ij}) (a_{ij} z_{j} - λ_{ij} - M k_{ij}^{λ}) \\ + ɛ_{3} ({\tilde{a}}_{ij}) (a_{ij} z_{j} - λ_{ij} - M k_{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} \leq 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} - M k_{ij}^{μ}) \\ + ɛ_{2} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j} - M k_{ij}^{μ}) \\ + ɛ_{3} ({\tilde{a}}_{ij}) (μ_{ij} - a_{ij} x_{j} - M k_{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} = l_{i}, \forall i \\ \sum_{j = 1}^{n} b_{ij} y_{j} = m_{i}, \forall i \\ \sum_{j = 1}^{n} μ_{ij} = 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} . \end{matrix}$

Proof. Considering the subsection 3.3. of our paper [24] and (8–10), the proof of the theorem is straightforward.

Remark 1. The problem (11) has the same solution structure with the system (2) which may have unique feasible (strong) fuzzy solution, infinitely many feasible (strong) fuzzy solutions or no feasible (strong) fuzzy solution. Corresponding to the solution structure of the system (2), the problem (11) may have unique optimal solution, alternative solutions or the problem (11) may be infeasible.

Remark 2. If the problem (11) is infeasible, then the system (2) has no feasible (strong) fuzzy solution.

Remark 3. With our proposed approach, a n × n FFLS results into a mixed integer programming problem with a number of 8n² + 5n constraints.

4 Numerical experiments

Example 4.1. Consider the following FFLS $(3, 8, 12) {\tilde{x}}_{1} + (- 4, - 3, 0) {\tilde{x}}_{2} + (- 2, 3, 7) {\tilde{x}}_{3} = (- 43, 12, 104) (- 6, 0, 1) {\tilde{x}}_{1} + (- 20, - 10, 2) {\tilde{x}}_{2} + (- 13, - 6, - 1) {\tilde{x}}_{3} = (- 61, 24, 125) (- 30, - 27, - 21) {\tilde{x}}_{1} + (5, 6, 7) {\tilde{x}}_{2} + (12, 19, 22) {\tilde{x}}_{3} = (- 356, - 157, - 26)$ 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.

Firstly, we should determine the following sets: $\begin{matrix} S^{pos} = {(1, 1), (3, 2), (3, 3)}, \\ S^{mix} = {(1, 2), (1, 3), (2, 1), (2, 2)}, \\ S^{neg} = {(2, 3), (3, 1)} . \end{matrix}$

Since n = 3 and the mixed integer programming problem corresponding to (9) has 8n² + 5n = 87 constraints, we will give the constraints for any element of S^pos, instead of giving all constraints of the problem, explicitly. If we consider (3, 2) ∈ S^pos, the step function values will be ɛ₁ = 1 and ɛ₂ = ɛ₃ = 0. Thus, the corresponding constraints can be constructed for (3, 2) ∈ S^pos as follows: $\begin{matrix} λ_{32} \leq 5 x_{2}, & μ_{32} \geq 5 z_{2}, \\ λ_{32} \leq 7 x_{2}, & μ_{32} \geq 7 z_{2}, \\ 5 x_{2} - λ_{32} \leq M k_{32}^{λ}, & μ_{32} - 5 z_{2} \leq M k_{32}^{μ}, \\ 7 x_{2} - λ_{32} \leq M (1 - k_{32}^{λ}) & μ_{32} - 7 z_{2} \leq M (1 - k_{32}^{μ}) \\ k_{32}^{λ} \in {0, 1} & k_{32}^{μ} \in {0, 1} \end{matrix}$ where M is a sufficiently large constant. For all elements of S^pos, S^mix and S^neg, the constraints can be constructed similarly. So, the optimal solution of the mixed integer programming problem corresponding to (9) is obtained as ${\tilde{x}}_{1} = (1, 3, 7)$ , ${\tilde{x}}_{2} = (- 2, 0, 1)$ and ${\tilde{x}}_{3} = (- 6, - 4, - 1)$ .

Example 4.2. Consider the following FFLS (Example 2 in [18]) $\begin{matrix} (- 3, - 1, 2) {\tilde{x}}_{1} + (1, 2, 4) {\tilde{x}}_{2} = (- 15, - 2, 15) \\ (- 4, - 3, - 2) {\tilde{x}}_{1} + (1, 3, 6) {\tilde{x}}_{2} = (- 15, 4, 20) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ and ${\tilde{x}}_{2} = (x_{2}, y_{2}, z_{2})$ .

For this system, the mixed integer programming problem corresponding to (9) is infeasible. Thus, the original system has no feasible (strong) fuzzy solution.

Example 4.3. Consider the following FFLS with arbitrary coefficients (test 3.2. in [10]) $\begin{matrix} (4, 5, 6) {\tilde{x}}_{1} + (5, 6, 8) {\tilde{x}}_{2} = (40, 50, 67) \\ (6, 7, 7) {\tilde{x}}_{1} + (4, 4, 5) {\tilde{x}}_{2} = (43, 48, 55) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1})$ and ${\tilde{x}}_{2} = (x_{2}, y_{2}, z_{2})$ .

While the results in [10] are given as ${\tilde{x}}_{1} = (3.9091, 4, 4)$ and ${\tilde{x}}_{2} = (4.9091, 5, 5.5)$ , our proposed method generates the following feasible (strong) fuzzy solution ${\tilde{x}}_{1} = (3.929, 4, 4.038)$ and ${\tilde{x}}_{2} = (4.857, 5, 5.346)$ .

Substituting these solutions in the above system, left hand sides of the first and the second equation are calculated as (40.1819, 50, 68) and (43.0910, 48, 55.5) in [10], on the other hand (40.0010, 50, 66.9960) and (43.0020, 48, 54.9960) in our method, respectively. These results show that our solutions are more accurate according to in [10].

5 Advantages of the proposed method over the existing methods

In this section, the shortcomings of the existing methods for solving FFLS and advantages of our proposed method are pointed out. Most of works in literature deals with FFLS which has sign restrictions on parameters and variables [1 –14]. Besides, there exists a few works that removed sign restrictions, i.e., [15 , 19].

Allahviranloo and Mikaeilvand [15] classifies the solutions as zero and nonzero. The existing method in [15] is applicable only if all the solution elements are nonzero, e.g., it is not possible to find the solution of FFLS, chosen in Example 4.1 by using the existing method in [15]. The advantage of our proposed method over the existing method in [15] is that it can be applied for solving FFLS which has zero solutions.

While [16] presents a nonlinear programming approach for solving FFLS, our proposed method overcomes the nonlinearity and decreases the complexity.

The method in [18] can generate a fuzzy solution providing that theorem 3.2 of [18] is satisfied. Thus, this method is not applicable to all FFLS. It can be seen that our Example 4.1 could not be solved with this method.

The existing method in [19] is applicable only if all the solution elements are nonzero, just as [15].

Our proposed method can be applied to all FFLS, that is, there are no sign restrictions on parameters and variables. Note that, however there exists some studies which are oriented to find the infeasible (weak) fuzzy solutions of FFLS, our method aims to find only the feasible (strong) fuzzy solution. If (9) is infeasible, this yields non-existence of the strong fuzzy solution, i.e. infeasible fuzzy solution of the original FFLS.

6 Conclusion

In this paper, a new method for finding the feasible fuzzy solution of FFLS with arbitrary fuzzy parameters and arbitrary fuzzy variables is presented. The proposed method is easy to understand and can be applied in real life situations. Our method is based on expressing the “min” and “max” concepts in fuzzy multiplication as a mixed integer programming problem by means of fuzzy programming approach of Bellman-Zadeh. The method is illustrated with some numerical examples.

Footnotes

Acknowledgments

This paper is supported by the “Office of Scientific Research Project Coordination” in Yildiz Technical University with the project number of “2013-07-03-GEP01”.

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