Finding the fuzzy solutions of a general fully fuzzy linear equation system

Abstract

In this paper, a new algorithm is proposed to find the solutions of a general (square or nonsquare) Fully Fuzzy Linear Equation System (FFLS) with arbitrary trapezoidal fuzzy numbers, i.e. there are no sign restrictions on the variables or the parameters of the system. We introduce the “feasible (strong) fuzzy solution” and “approximate fuzzy solution” concepts, then accordingly “no solution” case of a general FFLS is defined. And a model is proposed by means of a mixed integer programming modelling of “min” and “max” concepts in the multiplication of two arbitrary trapezoidal fuzzy numbers. With the logic of goal programming, the objective function of this model is constructed within the deviation variables. Based on the proposed model, also an algorithm is presented to determine the nature of solutions of a general FFLS. The method is illustrated with some numerical examples. Our numerical results for the examples from the literature are analyzed within some distance functions.

Keywords

General fully fuzzy linear equation system trapezoidal fuzzy number fuzzy decision making mixed integer programming problem feasible and approximate fuzzy solution

1 Introduction

Wide range of real world applications in many areas including financial engineering, scientific management and engineering technology are using linear equation systems for modeling and solving their respective problems. In many situations, the estimation of the system parameters is imprecise because of the lack of exact information, environmental and changeable economic conditions, etc. The uncertainty of parameters involved in the process of actual mathematical modeling is often represented by fuzzy numbers. It enables us to consider tolerances for parameters of linear equation systems (the entries of the coefficient matrix and right hand side vector) in a more natural and direct way. Therefore, fuzzy linear equation systems seem to be more realistic and reliable than the crisp case.

There is a vast literature on the investigation of solutions for fuzzy linear equation systems. Early works in the literature are on to linear equation systems whose coefficient matrix is crisp and the right hand vector is fuzzy, that is known as Fuzzy Linear Equation System (FLS). Since the crispness of the coefficient matrix makes the modeling of real life problems restricted, FFLS whose all the parameters i.e. both coefficient matrix and right hand side vector are fuzzy is taken into account. Generally, both FLS and FFLS are handled under two main headings: square (n×n) and nonsquare (m×n) forms. In this paper, we aim to find the fuzzy solution of a general FFLS (square or nonsquare) with arbitrary coefficients and arbitrary variables.

Asady et al. [6] which is one of the pioneer studies on nonsquare FLS developed a method using the same approach in Friedman et al. [12]. They proposed a general model for by replacing the original system with a crisp linear equation system. Zheng and Wang [23] 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 [21]. 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 [14] which is an extension of [21] presented a model to solve a class of inconsistent nonsquare FLS by the embedding method. Allahviranloo and Afshar Kermani [2] investigated m×n FLS using a numerical method. They first replaced the original m×n fuzzy linear system by a 2 m×2n crisp linear equation system, and found the least-square solution by using an orthogonal matrix. Dehghan et al. [7] 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 [2], Abbasbandy et al. [1] 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 [13] 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 [19] 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 2 m×2n crisp linear equation system as the authors of Ezzati [10] and Wang, Chen and Wei [20]. Ezzati et al. [11] 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 either nonnegative or nonpositive. Moloudzadeh et al. [18] gave a solution method of an arbitrary general (square or nonsquare) FFLS with trapezoidal fuzzy numbers by converting the original system to two crisp linear equation systems based on pseudo-inverse method. Allahviranloo et al. [3] presented a new method to obtain a fuzzy trapezoidal solution namely “suitable solution”, for a general FFLS using a nonlinear programming problem. Also, they defined two new solutions as “fuzzy surrounding solution” and “fuzzy peripheral solution” based on solving two fully interval linear systems that are 1-cut and 0-cut of the related system.

In this paper, we focus on a general FFLS with arbitrary trapezoidal fuzzy numbers to find its solutions. Firstly, we introduce the feasible (strong) and approximate fuzzy solution concepts, then accordingly no solution case of a general FFLS is defined. Starting from analyzing the sign of coefficients matrix entries, a model is proposed to convert the original FFLS to a mixed integer programming problem. Based on this model, an algorithm is presented to analyze the nature of solutions. In the context of goal programming, the algorithm enables to find a feasible fuzzy solution or an approximate fuzzy solution if any, by adding deviation variables. The method is illustrated with some numerical examples. Our numerical results for the examples from the literature are analyzed by using three distance functions that are based on some distance and similarity measures of fuzzy numbers.

This paper is organized as follows: Section 2 presents brief information about trapezoidal fuzzy numbers and arithmetic operations. In Section 3, new solution concepts are introduced and some measures of fuzzy numbers are given to determine the closeness between the left and right hand side fuzzy numbers for each equation. In the same section, we also present our proposed model and algorithm. Section 4 and Section 5 consist of our numerical examples and conclusions, respectively.

2 Preliminaries

In this section, brief information about the trapezoidal fuzzy numbers is presented [15].

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

where a, b, c, d ∈ R and a ≤ b ≤ c ≤ d.

Definition 2. A trapezoidal fuzzy number $\tilde{M} = (a, b, c, d)$ 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 trapezoidal fuzzy number $\tilde{M} = (a, b, c, d)$ is called near-zero trapezoidal fuzzy number which is neither positive nor negative.

Definition 4. Two trapezoidal fuzzy numbers $\tilde{M} = (a, b, c, d)$ and $\tilde{N} = (e, f, g, h)$ are said to be equal iff a = e, b = f, c = g and d = h.

Some algebraic operations on trapezoidal fuzzy numbers that will be used in this paper are defined as follows:

Let $\tilde{M} = (a, b, c, d)$ and $\tilde{N} = (e, f, g, h)$ be arbitrary two trapezoidal fuzzy numbers:

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

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

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

The fuzzy multiplication based on the extension principle is performed via the following equation: $\tilde{M} * \tilde{N} = (l, m^{1}, m^{2}, u)$ where l = min{ ae, ah, de, dh }, m¹ = min {bf, bg, cf, cg} m² = max{ bf, bg, cf, cg } and u = max {ae, ah, de, dh}.

3 General fully fuzzy linear equation system

A system such as: $\begin{matrix} {\tilde{a}}_{11} {\tilde{x}}_{1} + {\tilde{a}}_{12} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{1 n} {\tilde{x}}_{n} = {\tilde{b}}_{1} \\ {\tilde{a}}_{21} {\tilde{x}}_{1} + {\tilde{a}}_{22} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{2 n} {\tilde{x}}_{n} = {\tilde{b}}_{2} \\ ⋮ ⋱ ⋮ \\ {\tilde{a}}_{m 1} {\tilde{x}}_{1} + {\tilde{a}}_{m 2} {\tilde{x}}_{2} + \dots + {\tilde{a}}_{mn} {\tilde{x}}_{n} = {\tilde{b}}_{m} \end{matrix}$ (1)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 general FFLS. (1) can be written as $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} = {\tilde{b}}_{i}, i = 1, 2, \dots, m$ (2)

Let us assume that ${\tilde{a}}_{ij}$ and ${\tilde{b}}_{i}$ are given by trapezoidal fuzzy numbers as (a_ij, b_ij, c_ij, d_ij) and $(l_{i}, m_{i}^{1}, m_{i}^{2}, u_{i})$ , respectively, where a_ij, b_ij, c_ij, d_ij, l_i, $m_{i}^{1}$ , $m_{i}^{2}$ and u_i are real numbers. Thus, we 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. Let the fuzzy number $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ denoted by $(t_{i}^{1}, t_{i}^{2}, t_{i}^{3}, t_{i}^{4})$ and ${\tilde{b}}_{i} = (l_{i}, m_{i}^{1}, m_{i}^{2}, u_{i})$ 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,

$l_{i} \leq t_{i}^{4}$ 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.

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 Distance and similarity measures of fuzzy numbers

Since our approach has the ability to generate approximate fuzzy solutions, we need to measure the distance or similarity between $\sum_{j} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ and ${\tilde{b}}_{i}$ for each equation i, that is to find an approximate fuzzy solution which is close to feasible fuzzy solution as much as possible. For this purpose, some preliminaries and then the distance and similarity measures for two arbitrary trapezoidal fuzzy numbers $\tilde{M} = (a, b, c, d)$ and $\tilde{N} = (e, f, g, h)$ are given as follows:

The centroid point of a trapezoidal fuzzy number $\tilde{M}$ is obtained by Wang et al. [22] by $COG (\tilde{M}) = (α_{\tilde{M}}, β_{\tilde{M}}),$ where $\begin{matrix} α_{\tilde{M}} = \frac{1}{3} [a + b + c + d - \frac{dc - ab}{(d + c) - (a + b)}] \\ β_{\tilde{M}} = \frac{1}{3} [1 + \frac{c - b}{(d + c) - (a + b)}] . \end{matrix}$

And the left and right spreads of $\tilde{M}$ are $\begin{matrix} L_{\tilde{M}} = b - a \\ R_{\tilde{M}} = d - c . \end{matrix}$

The area of $\tilde{M}$ is $s = \frac{1}{2} ((d + c) - (a + b))$ .

Different authors have constructed different measures between two arbitrary trapezoidal fuzzy numbers $\tilde{M} = (a, b, c, d)$ and $\tilde{N} = (e, f, g, h)$ . The followings are used to analyze our numerical results:

In [9], Ebadi et al. proposed a distance measure using their centroids and left/right spreads for two trapezoidal fuzzy numbers $\tilde{M}$ and $\tilde{N}$ given below: $\begin{matrix} d (\tilde{M}, \tilde{N}) = max {| α_{\tilde{M}} - α_{\tilde{N}} |, \\ | β_{\tilde{M}} - β_{\tilde{N}} |, | L_{\tilde{M}} - L_{\tilde{N}} |, | R_{\tilde{M}} - R_{\tilde{N}} |} \end{matrix}$

In 2013, Allahviranloo et al. [4] proposed another distance measure. Firstly, they define the convex combination of a fuzzy number’s centroid point on the x-axis and area as follows: $C_{λ} (\tilde{M}) = λ α_{\tilde{M}} + (1 - λ) s$ where the index λ ∈ [0, 1] reflects the decision maker’s preferences. And the distance between $\tilde{M}$ and $\tilde{N}$ is defined as $d (\tilde{M}, \tilde{N}) = C_{λ} (\tilde{M}) - C_{λ} (\tilde{N}) .$

In [17], Lee define the similarity between $\tilde{M}$ and $\tilde{N}$ as follows: $d (\tilde{M}, \tilde{N}) = 1 - \frac{{∥ \tilde{M} - \tilde{N} ∥}_{l_{p}}}{∥ U ∥} . 4^{\frac{- 1}{p}}$ where l_p metric is defined to be $\begin{matrix} {∥ \tilde{M} - \tilde{N} ∥}_{l_{p}} = ({| a - e |}^{p} \\ + {| b - f |}^{p} + {| c - g |}^{p} + {| d - h |}^{p})^{\frac{1}{p}} \end{matrix}$ and U is the universe of discourse as $∥ U ∥ = max (U) - min (U) .$

We note that $d (\tilde{M}, \tilde{N})$ values in [9] and [4] go to zero, one can conclude that the mentioned fuzzy numbers are identical, that is, the closer $d (\tilde{M}, \tilde{N})$ value to zero, these fuzzy numbers are so close to each other. Also, $d (\tilde{M}, \tilde{N})$ value in [17] goes to 1, one can conclude that the mentioned fuzzy numbers are identical.

These distance and similarity measures can be used to determine the closeness between $\sum_{j} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ and ${\tilde{b}}_{i}$ for each equation i, individually. To generalize this approach to FFLS, we use the following function which is similar to the distance function between two fuzzy vectors given in [5]: $D (\tilde{A} \tilde{x}, \tilde{b}) = \sum_{i = 1}^{m} | d (\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}, {\tilde{b}}_{i}) |$ where $\tilde{A} \tilde{x} = {[\begin{matrix} \sum_{j = 1}^{n} {\tilde{a}}_{1 j} {\tilde{x}}_{j} & \sum_{j = 1}^{n} {\tilde{a}}_{2 j} {\tilde{x}}_{j} & \dots & \sum_{j = 1}^{n} {\tilde{a}}_{mj} {\tilde{x}}_{j} \end{matrix}]}^{T}$ and $\tilde{b} = {[\begin{matrix} {\tilde{b}}_{1} & {\tilde{b}}_{2} & \dots & {\tilde{b}}_{m} \end{matrix}]}^{T}$ .

If $D (\tilde{A} \tilde{x}, \tilde{b}) = 0$ for the solution $\tilde{x}$ , this means that $\tilde{x}$ is a feasible fuzzy solution. For an approximate solution $\tilde{x}$ , the value $D (\tilde{A} \tilde{x}, \tilde{b})$ provides an insight into the qualification of $\tilde{x}$ according to the related distance or similarity measure.

3.2 Analysis of the multiplication of two arbitrary trapezoidal 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 ${\tilde{a}}_{ij} = (a_{ij}, b_{ij}, c_{ij}, d_{ij})$ and ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j}, w_{j})$ .

Based on [16], the multiplication of two arbitrary trapezoidal fuzzy numbers can be given as follows:

$\begin{matrix} {\tilde{a}}_{ij} {\tilde{x}}_{j} = (a_{ij}, b_{ij}, c_{ij}, d_{ij}) (x_{j}, y_{j}, z_{j}, w_{j}) \\ = (λ_{ij}, λ_{ij}^{'}, μ_{ij}^{'}, μ_{ij}) \end{matrix}$ (3)where $\begin{matrix} λ_{ij} = min {a_{ij} x_{j}, a_{ij} w_{j}, d_{ij} x_{j}, d_{ij} w_{j}}, \\ λ_{ij}^{'} = min {b_{ij} y_{j}, b_{ij} z_{j}, c_{ij} y_{j}, c_{ij} z_{j}}, \\ μ_{ij}^{'} = max {b_{ij} y_{j}, b_{ij} z_{j}, c_{ij} y_{j}, c_{ij} z_{j}}, \\ μ_{ij} = max {a_{ij} x_{j}, a_{ij} w_{j}, d_{ij} x_{j}, d_{ij} w_{j}} . \end{matrix}$

Additionally, the multiplication of ${\tilde{a}}_{ij} {\tilde{x}}_{j}$ can be identified according to sign of components of ${\tilde{a}}_{ij}$ , i.e. it is positive, negative or near-zero (mixed) fuzzy number in the following way:

When (i, j) indicates the position of ${\tilde{a}}_{ij}$ in the coefficient matrix of (2), $\begin{matrix} (min {a_{ij} x_{j}, d_{ij} x_{j}}, min {b_{ij} y_{j}, c_{ij} y_{j}}, \\ max {b_{ij} z_{j}, c_{ij} z_{j}}, max {a_{ij} w_{j}, d_{ij} w_{j}}), \forall (i, j) \in S^{pos} \end{matrix}$ (4) $\begin{matrix} (min {a_{ij} w_{j}, d_{ij} x_{j}}, min {b_{ij} y_{j}, c_{ij} y_{j}}, \\ max {b_{ij} z_{j}, c_{ij} z_{j}}, max {a_{ij} x_{j}, d_{ij} w_{j}}), \forall (i, j) \in S_{1}^{mix} \end{matrix}$ (5) $\begin{matrix} (min {a_{ij} w_{j}, d_{ij} x_{j}}, min {b_{ij} z_{j}, c_{ij} y_{j}}, \\ max {b_{ij} y_{j}, c_{ij} z_{j}}, max {a_{ij} x_{j}, d_{ij} w_{j}}), \forall (i, j) \in S_{2}^{mix} \end{matrix}$ (6) $\begin{matrix} (min {a_{ij} w_{j}, d_{ij} x_{j}}, min {b_{ij} z_{j}, c_{ij} z_{j}}, \\ max {b_{ij} y_{j}, c_{ij} y_{j}}, max {a_{ij} x_{j}, d_{ij} w_{j}}), \forall (i, j) \in S_{3}^{mix} \end{matrix}$ (7) $\begin{matrix} (min {a_{ij} w_{j}, d_{ij} w_{j}}, min {b_{ij} z_{j}, c_{ij} z_{j}}, \\ max {b_{ij} y_{j}, c_{ij} y_{j}}, max {a_{ij} x_{j}, d_{ij} x_{j}}), \forall (i, j) \in S^{neg} \end{matrix}$ (8)

where S^pos ={ (i, j) | 0 < a_ij }, $S_{1}^{mix} = {(i, j) | a_{ij} \leq 0 < b_{ij}}$ , $S_{2}^{mix} = {(i, j) | b_{ij} \leq 0 < c_{ij}}$ , $S_{3}^{mix} = {(i, j) | c_{ij} \leq 0 < d_{ij}}$ , S^neg ={ (i, j) | d_ij ≤ 0 }.

In this paper, we propose an approach by means of a mixed integer programming modeling of “min” and “max” concepts and fuzzy programming approach [24].

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

This subsection contains the bases of our proposed approach.

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

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 [8].

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

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

$\begin{matrix} max 0 \cdot λ \\ λ \leq p_{i}, & i = 1, 2, \dots, n \\ p_{i} - λ \leq M k_{i}, & i = 1, 2, \dots, n \\ \sum_{i = 1}^{n} k_{i} = (n - 1), \\ k_{i} \in {0, 1}, & i = 1, 2, \dots, n \end{matrix}$ (9)where M is a sufficiently large constant. Then λ^* is the minimum element of P, i.e. λ = min{ p₁, p₂, …, p_n }.

Proof. For the optimal solution of (9), the constraint $\sum_{i = 1}^{n} k_{i}^{*} = (n - 1)$ requires that only one binary variable $k_{i}^{*}$ (i = 1, 2, …, n) is zero and the rest of $k_{i}^{*}$ are 1. Suppose that $k_{1}^{*} = 0$ , i.e. i = 1. In this case, the first and second type constraints will be λ^* ≤ p₁ and p₁ - λ^* ≤ 0, respectively and imply that λ^* = p₁. For i = 2, …, n, the first and second type constraints will be λ^* ≤ p_i and p_i - λ^* ≤ M, respectively. According to the Definition 6, the constraints λ^* = p₁ and λ^* ≤ p_i (i = 2, …, n) proves that p₁, namely λ^*, is the minimum element of P. □.

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:

$\begin{matrix} min 0 \cdot μ \\ μ \geq p_{i}, & i = 1, 2, \dots, n \\ μ - p_{i} \leq M k_{i}, & i = 1, 2, \dots, n \\ \sum_{i = 1}^{n} k_{i} = (n - 1), \\ k_{i} \in {0, 1}, & i = 1, 2, \dots, n \end{matrix}$ (10)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. □.

3.4 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 previous subsection. To solve the original FFLS (2), the approaches presented in (9) and (10) are adapted to the analysis of the multiplication of two arbitrary fuzzy numbers. This analysis given by (4–8) 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

$\begin{matrix} \sum_{j = 1}^{n} (λ_{ij}, λ_{ij}^{'}, μ_{ij}^{'}, μ_{ij}) & = & (l_{i}, m_{i}^{1}, m_{i}^{2}, u_{i}); \\ i = 1, 2, \dots, m \end{matrix}$ (11)

Using the equality of fuzzy numbers (Definition 4), (2) is equivalent to the following system: $\sum_{j = 1}^{n} λ_{ij} = l_{i}, i = 1, 2, \dots, m$ (12) $\sum_{j = 1}^{n} λ_{ij}^{'} = m_{i}^{1}, i = 1, 2, \dots, m$ (13) $\sum_{j = 1}^{n} μ_{ij}^{'} = m_{i}^{2}, i = 1, 2, \dots, m$ (14) $\sum_{j = 1}^{n} μ_{ij} = u_{i}, i = 1, 2, \dots, m .$ (15)

Our proposed method that is based on the sign of entries of the coefficient matrix can be adapted

to the original system (2) with the followingfunction:

$ɛ ({\tilde{a}}_{ij}) = {\begin{matrix} [\begin{matrix} 1 & 0 & 0 & 0 & 0 \end{matrix}], if (i, j) \in S^{pos} \\ [\begin{matrix} 0 & 1 & 0 & 0 & 0 \end{matrix}], if (i, j) \in S_{1}^{mix} \\ [\begin{matrix} 0 & 0 & 1 & 0 & 0 \end{matrix}], if (i, j) \in S_{2}^{mix} \\ [\begin{matrix} 0 & 0 & 0 & 1 & 0 \end{matrix}], if (i, j) \in S_{3}^{mix} \\ [\begin{matrix} 0 & 0 & 0 & 0 & 1 \end{matrix}], if (i, j) \in S^{neg} . \end{matrix}$

The mixed integer programming problem which is equivalent to (12–15) can be written as follows: $\begin{matrix} min Z = \sum_{i = 1}^{m} (R_{i}^{λ +} + R_{i}^{λ -} + R_{i}^{λ^{'} +} + R_{i}^{λ^{'} -} + R_{i}^{μ^{'} +} + R_{i}^{μ^{'} -} + R_{i}^{μ +} + R_{i}^{μ -}) \\ ɛ ({\tilde{a}}_{ij}) {[λ_{ij} - a_{ij} x_{j} λ_{ij} - a_{ij} w_{j} λ_{ij} - a_{ij} w_{j} λ_{ij} - a_{ij} w_{j} λ_{ij} - a_{ij} w_{j}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[λ_{ij} - a_{ij} x_{j} λ_{ij} - d_{ij} x_{j} λ_{ij} - d_{ij} x_{j} λ_{ij} - d_{ij} x_{j} λ_{ij} - d_{ij} w_{j}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [d_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ}) d_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ}) d_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ}) \\ {d_{ij} x_{j} - λ_{ij} - M (1 - k_{ij}^{λ}) d_{ij} w_{j} - λ_{ij} - M (1 - k_{ij}^{λ})]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} λ_{ij}^{'} - b_{ij} y_{j} & λ_{ij}^{'} - b_{ij} y_{j} & λ_{ij}^{'} - b_{ij} z_{j} & λ_{ij}^{'} - b_{ij} z_{j} & λ_{ij}^{'} - b_{ij} z_{j} \end{matrix}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} λ_{ij}^{'} - c_{ij} y_{j} & λ_{ij}^{'} - c_{ij} y_{j} & λ_{ij}^{'} - c_{ij} y_{j} & λ_{ij}^{'} - c_{ij} z_{j} & λ_{ij}^{'} - c_{ij} z_{j} \end{matrix}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [b_{ij} y_{j} - λ_{ij}^{'} - M k_{ij}^{λ^{'}} b_{ij} y_{j} - λ_{ij}^{'} - M k_{ij}^{λ^{'}} b_{ij} z_{j} - λ_{ij}^{'} - M k_{ij}^{λ^{'}} b_{ij} z_{j} - λ_{ij}^{'} - M k_{ij}^{λ^{'}} \\ {b_{ij} z_{j} - λ_{ij}^{'} - M k_{ij}^{λ^{'}}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [c_{ij} y_{j} - λ_{ij}^{'} - M (1 - k_{ij}^{λ^{'}}) c_{ij} y_{j} - λ_{ij}^{'} - M (1 - k_{ij}^{λ^{'}}) c_{ij} y_{j} - λ_{ij}^{'} - M (1 - k_{ij}^{λ^{'}}) \\ {c_{ij} z_{j} - λ_{ij}^{'} - M (1 - k_{ij}^{λ^{'}}) c_{ij} z_{j} - λ_{ij}^{'} - M (1 - k_{ij}^{λ^{'}})]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} μ_{ij}^{'} - b_{ij} z_{j} & μ_{ij}^{'} - b_{ij} z_{j} & μ_{ij}^{'} - b_{ij} y_{j} & μ_{ij}^{'} - b_{ij} y_{j} & μ_{ij}^{'} - b_{ij} y_{j} \end{matrix}]}^{T} \geq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} μ_{ij}^{'} - c_{ij} z_{j} & μ_{ij}^{'} - c_{ij} z_{j} & μ_{ij}^{'} - c_{ij} z_{j} & μ_{ij}^{'} - c_{ij} y_{j} & μ_{ij}^{'} - c_{ij} y_{j} \end{matrix}]}^{T} \geq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [μ_{ij}^{'} - b_{ij} z_{j} - M k_{ij}^{μ^{'}} μ_{ij}^{'} - b_{ij} z_{j} - M k_{ij}^{μ^{'}} μ_{ij}^{'} - b_{ij} y_{j} - M k_{ij}^{μ^{'}} μ_{ij}^{'} - b_{ij} y_{j} - M k_{ij}^{μ^{'}} \\ {μ_{ij}^{'} - b_{ij} y_{j} - M k_{ij}^{μ^{'}}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [μ_{ij}^{'} - c_{ij} z_{j} - M (1 - k_{ij}^{μ^{'}}) μ_{ij}^{'} - c_{ij} z_{j} - M (1 - k_{ij}^{μ^{'}}) μ_{ij}^{'} - c_{ij} z_{j} - M (1 - k_{ij}^{μ^{'}}) \\ {μ_{ij}^{'} - c_{ij} y_{j} - M (1 - k_{ij}^{μ^{'}}) μ_{ij}^{'} - c_{ij} y_{j} - M (1 - k_{ij}^{μ^{'}})]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} μ_{ij} - a_{ij} w_{j} & μ_{ij} - a_{ij} x_{j} & μ_{ij} - a_{ij} x_{j} & μ_{ij} - a_{ij} x_{j} & μ_{ij} - a_{ij} x_{j} \end{matrix}]}^{T} \geq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) {[\begin{matrix} μ_{ij} - d_{ij} w_{j} & μ_{ij} - d_{ij} w_{j} & μ_{ij} - d_{ij} w_{j} & μ_{ij} - d_{ij} w_{j} & μ_{ij} - d_{ij} x_{j} \end{matrix}]}^{T} \geq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [μ_{ij} - a_{ij} w_{j} - M k_{ij}^{μ} μ_{ij} - a_{ij} x_{j} - M k_{ij}^{μ} μ_{ij} - a_{ij} x_{j} - M k_{ij}^{μ} μ_{ij} - a_{ij} x_{j} - M k_{ij}^{μ} \\ {μ_{ij} - a_{ij} x_{j} - M k_{ij}^{μ}]}^{T} \leq 0, \forall (i, j) \\ ɛ ({\tilde{a}}_{ij}) [μ_{ij} - d_{ij} w_{j} - M (1 - k_{ij}^{μ}) μ_{ij} - d_{ij} w_{j} - M (1 - k_{ij}^{μ}) μ_{ij} - d_{ij} w_{j} - M (1 - k_{ij}^{μ}) \\ {μ_{ij} - d_{ij} w_{j} - M (1 - k_{ij}^{μ}) μ_{ij} - d_{ij} x_{j} - M (1 - k_{ij}^{μ})]}^{T} \leq 0, \forall (i, j) \\ \sum_{j = 1}^{n} λ_{ij} - R_{i}^{λ +} + R_{i}^{λ -} = l_{i}, \forall i \\ \sum_{j = 1}^{n} λ_{ij}^{'} - R_{i}^{λ^{'} +} + R_{i}^{λ^{'} -} = m_{i}^{1}, \forall i \\ \sum_{j = 1}^{n} μ_{ij}^{'} - R_{i}^{μ^{'} +} + R_{i}^{μ^{'} -} = m_{i}^{2}, \forall i \\ \sum_{j = 1}^{n} μ_{ij} - R_{i}^{μ +} + R_{i}^{μ -} = u_{i}, \forall i \\ x_{j} \leq y_{j} \leq z_{j} \leq w_{j}, \forall j \\ λ_{ij}, λ_{ij}^{'}, μ_{ij}^{'}, μ_{ij} is free, \forall (i, j) \\ x_{j}, y_{j}, z_{j}, w_{j} is free, \forall j \\ k_{ij}^{λ}, k_{ij}^{λ^{'}}, k_{ij}^{μ^{'}}, k_{ij}^{μ} \in {0, 1}, \forall (i, j) \\ R_{i}^{λ +}, R_{i}^{λ -}, R_{i}^{λ^{'} +}, R_{i}^{λ^{'} -}, R_{i}^{μ^{'} +}, R_{i}^{μ^{'} -}, R_{i}^{μ +}, R_{i}^{μ -} \geq 0, \forall i \end{matrix}$ (16)

where $R_{i}^{λ +}, R_{i}^{λ^{'} +}, R_{i}^{μ^{'} +}, R_{i}^{μ +} (R_{i}^{λ -}, R_{i}^{λ^{'} -}, R_{i}^{μ^{'} -}, R_{i}^{μ -})$ represent the deviations of the left hand side values in the right (left) direction from l_i, $m_{i}^{1}$ , $m_{i}^{2}$ and u_i for equation i∈ { 1, 2, …, m } of the original FFLS, respectively. If the optimal objective function value of (16) is zero (Z^* = 0), then the original FFLS has a feasible (strong) fuzzy solution. Otherwise, i.e. Z^* > 0, the system has an approximate fuzzy solution or no solution.

Remark 1. If the problem (16) has a unique optimal solution with Z^* = 0, then the original FFLS has a unique feasible fuzzy solution.

Remark 2. If the problem (16) 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 (16) always has an optimal solution.

Remark 5. With our proposed approach, a m × n FFLS results into a mixed integer programming problem with a number of 16mn + 4m + 3n constraints.

Based on the proposed model (16), our algorithm to determine the nature of the solutions of (2) can be given as follows:

Step 0. Determine the sets S^pos, $S_{1}^{mix}$ , $S_{2}^{mix}$ , $S_{3}^{mix}$ , S^neg and then accordingly $ɛ ({\tilde{a}}_{ij})$ for ∀ (i, j).

Step 1. Construct the mixed integer programming problem corresponding to (16) and obtain its optimal solution ${\tilde{x}}_{j}^{*} = (x_{j}^{*}, y_{j}^{*}, z_{j}^{*}, w_{j}^{*}),$ $(λ_{ij}^{*}, λ_{ij}^{' *}, μ_{ij}^{' *}, μ_{ij}^{*}),$ $(k_{ij}^{λ *}, k_{ij}^{λ^{'} *}, k_{ij}^{μ^{'} *}, k_{ij}^{μ *}),$ $(R_{i}^{λ * +}, R_{i}^{λ^{'} * +}$ , $R_{i}^{μ^{'} * +}, R_{i}^{μ * +}),$ $(R_{i}^{λ * -}, R_{i}^{λ^{'} * -}, R_{i}^{μ^{'} * -}, R_{i}^{μ * -})$ for ∀ i, j.

If Z^* = 0, then ${\tilde{x}}_{j}^{*}$ (j = 1, 2, …, n) is a feasible fuzzy solution of (2), stop. Otherwise, i.e. Z^* > 0, determine whether ${\tilde{x}}_{j}^{*}$ (j = 1, 2, …, n) is an approximate fuzzy solution by using Definition 6. If it is an approximate fuzzy solution, then go to Step 2, otherwise go to Step 3.

Step 2. (Approximate fuzzy solution) If all of $R_{i}^{λ^{'} * +}, R_{i}^{λ^{'} * -}, R_{i}^{μ^{'} * +}, R_{i}^{μ^{'} * -}$ (i = 1, 2, …, m) are equal to zero, then stop. Otherwise, i.e. at least one of $R_{i}^{λ^{'} * +}, R_{i}^{λ^{'} * -}, R_{i}^{μ^{'} * +}, R_{i}^{μ^{'} * -}$ (i = 1, 2, …, m) is not equal to zero and if the decision maker is not satisfied with the approximate fuzzy solution ${\tilde{x}}_{j}^{*}$ , then update the objective function of (16) with the following

$\begin{matrix} Z & = & \sum_{i = 1}^{m} (R_{i}^{λ +} + R_{i}^{λ -} + α (R_{i}^{λ^{'} +} + R_{i}^{λ^{'} -} \\ + R_{i}^{μ^{'} +} + R_{i}^{μ^{'} -}) + R_{i}^{μ +} + R_{i}^{μ -}) \end{matrix}$ (17)where the weightiness parameter α > 1. And obtain another approximate fuzzy solution(s) depending upon the decision maker’s preferences for α, then stop.

We note that choosing α as greater than 1 provides to minimize the deviations of the full satisfaction values (the core) of the fuzzy numbers $(λ_{ij}^{*}, λ_{ij}^{' *}, μ_{ij}^{' *}, μ_{ij}^{*})$ for each i∈ { 1, 2, …, m } and j∈ { 1, 2, …, n } as much as possible.

Step 3. (No solution) Considering the constraints in condition (ii) of Definition 6, restrict the feasible region of (16) to find an approximate fuzzy solution, i.e. by adding the following constraints $\sum_{j = 1}^{n} λ_{ij} \leq u_{i}, \forall i$ (18) $\sum_{j = 1}^{n} μ_{ij} \geq l_{i}, \forall i$ (19) and solve the obtained mixed integer programming problem. If this new problem has an optimal solution, then obtained ${\tilde{x}}_{j}^{*}$ (j = 1, 2, …, n) is an approximate fuzzy solution of (2), go to Step 2. Otherwise, i.e. if this new problem is infeasible, then the system (2) has no solution, stop.

It is obvious that, even if the optimal solution which is obtained in Step 2 does not satisfy the inequalities (18) and (19) for at least one i, the mixed integer programming problem of Step 3 will be infeasible; that is, the corresponding FFLS has no solution. We also note that the distance function $D (\tilde{A} \tilde{x}, \tilde{b})$ can be used to provide an insight for the decision maker according to the related distance or similarity measure.

4 Numerical experiments

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

Example 4.1. Consider the following FFLS (Example 1 in [18]): $\begin{matrix} (0.9, 1, 1.5, 1.7) {\tilde{x}}_{1} = (0.8, 2, 4.5, 6.6) \\ (- 2.7, - 2, - 1, - 0.7) {\tilde{x}}_{1} = (- 8.3, - 6, - 2, - 0.4) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1}, w_{1})$ .

Firstly, we should determine the following sets $S^{pos} = {(1, 1)}, S_{1}^{mix} = S_{2}^{mix} = S_{3}^{mix} = {}, S^{neg} = {(2, 1)},$ then accordingly $ɛ ((0.9, 1, 1.5, 1.7)) = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \end{matrix}]$ for ${\tilde{a}}_{11}$ and $ɛ ((- 2.7, - 2, - 1, - 0.7)) = [\begin{matrix} 0 & 0 & 0 & 0 & 1 \end{matrix}]$ for ${\tilde{a}}_{21}$ (Step 0).

Since m = 2 and n = 1, the mixed integer programming problem corresponding to (16) which has 16mn + 4m + 3n = 43 constraints can be given as follows: $\begin{matrix} min Z & = & \sum_{i = 1}^{2} (R_{i}^{λ +} + R_{i}^{λ -} + R_{i}^{λ^{'} +} + R_{i}^{λ^{'} -} + R_{i}^{μ^{'} +} + R_{i}^{μ^{'} -} + R_{i}^{μ +} + R_{i}^{μ -}) \\ λ_{11} + 0.9 w_{1} \leq 0; λ_{11} - 1.7 x_{1} \leq 0; \\ - 0.9 w_{1} - λ_{11} - M k_{11}^{λ} \leq 0; 1.7 x_{1} - λ_{11} - M (1 - k_{11}^{λ}) \leq 0 \\ λ_{21} + 2.7 w_{1} \leq 0; λ_{21} + 0.7 w_{1} \leq 0; \\ - 2.7 w_{1} - λ_{21} - M k_{21}^{λ} \leq 0; - 0.7 w_{1} - λ_{21} - M (1 - k_{21}^{λ}) \leq 0 \\ λ_{11}^{'} - y_{1} \leq 0; λ_{11}^{'} - 1.5 y_{1} \leq 0; \\ y_{1} - λ_{11}^{'} - M k_{11}^{λ^{'}} \leq 0; 1.5 y_{1} - λ_{11}^{'} - M (1 - k_{11}^{λ^{'}}) \leq 0 \\ λ_{21}^{'} + 2 z_{1} \leq 0; λ_{21}^{'} + z_{1} \leq 0; \\ - 2 z_{1} - λ_{21}^{'} - M k_{21}^{λ^{'}} \leq 0; - z_{1} - λ_{21}^{'} - M (1 - k_{21}^{λ^{'}}) \leq 0 \\ μ_{11}^{'} - z_{1} \geq 0; μ_{11}^{'} - 1.5 z_{1} \geq 0; \\ μ_{11}^{'} - z_{1} - M k_{11}^{μ^{'}} \leq 0; μ_{11}^{'} - 1.5 z_{1} - M (1 - k_{11}^{μ^{'}}) \leq 0 \\ μ_{21}^{'} + 2 y_{1} \geq 0; μ_{21}^{'} + y_{1} \geq 0; \\ μ_{21}^{'} + 2 y_{1} - M k_{21}^{μ^{'}} \leq 0; μ_{21}^{'} + y_{1} - M (1 - k_{21}^{μ^{'}}) \leq 0 \\ μ_{11} + 0.9 x_{1} \geq 0; μ_{11} - 1.7 w_{1} \geq 0; \\ μ_{11} + 0.9 x_{1} - M k_{11}^{μ} \leq 0; μ_{11} - 1.7 w_{1} - M (1 - k_{11}^{μ}) \leq 0 \\ μ_{21} + 2.7 x_{1} \geq 0; μ_{21} + 0.7 x_{1} \geq 0; \\ μ_{21} + 2.7 x_{1} - M k_{21}^{μ} \leq 0; μ_{21} + 0.7 x_{1} - M (1 - k_{21}^{μ}) \leq 0 \\ λ_{11} - R_{1}^{λ +} + R_{1}^{λ -} = 0.8; λ_{21} - R_{2}^{λ +} + R_{2}^{λ -} = - 8.3 \\ λ_{11}^{'} - R_{1}^{λ^{'} +} + R_{1}^{λ^{'} -} = 2; λ_{21}^{'} - R_{2}^{λ^{'} +} + R_{2}^{λ^{'} -} = - 6 \\ μ_{11}^{'} - R_{1}^{μ^{'} +} + R_{1}^{μ^{'} -} = 4.5; μ_{21}^{'} - R_{2}^{μ^{'} +} + R_{2}^{μ^{'} -} = - 2 \\ μ_{11} - R_{1}^{μ +} + R_{1}^{μ -} = 6.6; μ_{21} - R_{2}^{μ +} + R_{2}^{μ -} = - 0.4 \\ x_{1} \leq y_{1} \leq z_{1} \leq w_{1} \\ λ_{ij}, λ_{ij}^{'}, μ_{ij}^{'}, μ_{ij} is free, \forall (i, j) \\ x_{j}, y_{j}, z_{j}, w_{j} is free, \forall j \\ k_{ij}^{λ}, k_{ij}^{λ^{'}}, k_{ij}^{μ^{'}}, k_{ij}^{μ} \in {0, 1}, \forall (i, j) \\ R_{i}^{λ +}, R_{i}^{λ -}, R_{i}^{λ^{'} +}, R_{i}^{λ^{'} -}, R_{i}^{μ^{'} +}, R_{i}^{μ^{'} -}, R_{i}^{μ +}, R_{i}^{μ -} \geq 0, \forall i . \end{matrix}$ (20)The optimal solution of (20) is: $\begin{matrix} {\tilde{x}}_{1}^{*} = (x_{1}^{*}, y_{1}^{*}, z_{1}^{*}, w_{1}^{*}) = (0.889, 2, 3, 3.074) \\ (λ_{11}^{*}, λ_{11}^{' *}, μ_{11}^{' *}, μ_{11}^{*}) = (0.8, 2, 4.5, 5.226) \\ (λ_{21}^{*}, λ_{21}^{' *}, μ_{21}^{' *}, μ_{21}^{*}) = (- 8.3, - 6, - 2, - 0.622) \\ (k_{11}^{λ *}, k_{11}^{λ^{'} *}, k_{11}^{μ^{'} *}, k_{11}^{μ *}) = (0, 0, 1, 1) \\ (k_{21}^{λ *}, k_{21}^{λ^{'} *}, k_{21}^{μ^{'} *}, k_{21}^{μ *}) = (0, 0, 1, 1) \\ (R_{1}^{λ * +}, R_{1}^{λ^{'} * +}, R_{1}^{μ^{'} * +}, R_{1}^{μ * +}) = (0, 0, 0, 0) \\ (R_{2}^{λ * +}, R_{2}^{λ^{'} * +}, R_{2}^{μ^{'} * +}, R_{2}^{μ * +}) = (0, 0, 0, 0) \\ (R_{1}^{λ * -}, R_{1}^{λ^{'} * -}, R_{1}^{μ^{'} * -}, R_{1}^{μ * -}) = (0, 0, 0, 1.374) \\ (R_{2}^{λ * -}, R_{2}^{λ^{'} * -}, R_{2}^{μ^{'} * -}, R_{2}^{μ * -}) = (0, 0, 0, 0.222) \\ Z^{*} = 1.596 \end{matrix}$

Since Z^* > 0, using Definition 6, ${\tilde{x}}_{1}^{*} = (0.889, 2, 3, 3.074)$ is an approximate fuzzy solution with the total deviation 1.596 (Step 1). Due to the fact that $R_{i}^{λ^{'} +}, R_{i}^{λ^{'} -}, R_{i}^{μ^{'} +}, R_{i}^{μ^{'} -}$ are equal to zero, the objective function does not need to be weighted (Step 2).

The obtained ${\tilde{a}}_{ij} {\tilde{x}}_{j}$ and ${\tilde{b}}_{i}$ values for each equation i and the measurement results are given in Table 1 and Table 2, respectively. In Table 1, the values that deviate from the components of $\tilde{b}$ are written in bold.

Example 4.2. Consider the following FFLS (Example 2 in [18]): $\begin{matrix} (0.9, 1, 2, 2.4) {\tilde{x}}_{1} + (1.2, 1.5, 2, 2.1) {\tilde{x}}_{2} \\ + (0.8, 1, 1.5, 1.7) {\tilde{x}}_{3} = (- 2.6, - 1, 5.25, 8.4) \\ (- 1.3, - 1, 0, 0.2) {\tilde{x}}_{1} + (1.6, 2, 3, 3.3) {\tilde{x}}_{2} \\ + (- 0.1, 0, 1, 1.1) {\tilde{x}}_{3} = (- 10.15, - 7.5, 0.5, 2.25) \end{matrix}$ where ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j}, w_{j})$ for j = 1, 2, 3.

The sets S^pos = { (1, 1) , (1, 2) , (1, 3) , (2, 2) } , $S_{1}^{mix} = S^{neg} = {},$ $S_{2}^{mix} = {(2, 3)},$ $S_{3}^{mix} = {(2, 1)},$ then accordingly $ɛ ({\tilde{a}}_{ij})$ vectors can be determined easily. Since m = 2 and n = 3, the mixed integer programming problem corresponding to (16) which has 16mn + 4m + 3n = 113 constraints is constructed and the following optimal ${\tilde{x}}_{j}^{*}$ (j = 1, 2, 3) values are obtained as: ${\tilde{x}}_{1}^{1 *} = (1.47, 2.361, 2.458, 3.066), {\tilde{x}}_{2}^{1 *} = (- 1.868, - 1.681, 0.167, 0.496) {\tilde{x}}_{3}^{1 *} = (0, 0, 0, 0)$ with the optimal objective value Z^* = 0. Also, our mixed integer programming problem has alternative optimal solutions, such as: ${\tilde{x}}_{1}^{2 *} = (1.787, 2.25, 2.25, 2.354), {\tilde{x}}_{2}^{2 *} = (- 2.099$ , $- 1.750, 0, 0), {\tilde{x}}_{3}^{2 *} = (0.25, 0.25, 0.5, 1.617)$ . Since Z^* = 0, ${\tilde{x}}^{1 *}$ and ${\tilde{x}}^{2 *}$ are feasible fuzzy solutions. The obtained $\sum_{j = 1}^{3} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ and ${\tilde{b}}_{i}$ values for each equation i corresponding to these solutions are given in Table 3. It shows that our proposed method generates many alternative feasible fuzzy solutions for Example 2.

Example 4.3. Consider the FFLS (Example 6.3 in [3]) $\tilde{A} \tilde{x} = \tilde{b}$ where $\begin{matrix} \tilde{A} \\ = [\begin{matrix} (2, 4, 5, 6) & (- 11, - 4, - 3, - 2) & (3, 8, 10, 15) & (- 13, - 5, - 5, - 3) \\ (3, 7, 9, 11) & (7, 10, 12, 17) & (- 13, - 9, - 7, - 5) & (- 9, - 3, - 1, 0) \\ (1, 3, 7, 10) & (- 12, - 6, - 4, - 1) & (- 9, - 8, - 6, - 3) & (3, 5, 5, 9) \\ (0, 3, 6, 7) & (3, 8, 10, 11) & (1, 6, 10, 12) & (- 11, - 7, - 4, 0) \end{matrix}] \end{matrix}$ and $\tilde{b} = [\begin{matrix} (- 295, - 65, 9, 166) \\ (- 97, 62, 135, 386) \\ (- 249, - 51, 33, 147) \\ (- 157, - 8, 102, 261) \end{matrix}]$ .

The index sets can be written as: $\begin{matrix} S^{pos} & = & {(1, 1), (1, 3), (2, 1), (2, 2), \\ (3, 1), (3, 4), (4, 1), (4, 2), (4, 3)} \\ S^{neg} & = & {(1, 2), (1, 4), (2, 3), (2, 4), (3, 2), \\ (3, 3), (3, 4)} \\ S_{1}^{mix} & = & S_{2}^{mix} = S_{3}^{mix} = {} . \end{matrix}$

Then accordingly $ɛ ({\tilde{a}}_{ij})$ vectors can be determined easily. Since m = 4 and n = 4, the mixed integer programming problem corresponding to (16) which has 16mn + 4m + 3n = 284 constraints can be constructed and the optimal ${\tilde{x}}_{j}^{*}$ (j = 1, 2, 3, 4) values are obtained as: $\begin{matrix} {\tilde{x}}_{1}^{*} & = & (- 1.955, 1, 2, 3.032) \\ {\tilde{x}}_{2}^{*} & = & (- 1.173, 5, 6, 11.368) \\ {\tilde{x}}_{3}^{*} & = & (- 5.016, - 3, - 2, - 0.379) \\ {\tilde{x}}_{4}^{*} & = & (- 10.464, - 6, 3, 6.383) \end{matrix}$ with the optimal objective value Z^* = 0. The obtained $\sum_{j = 1}^{4} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ and ${\tilde{b}}_{i}$ values for each equation i corresponding to these solutions are given in Table 4. It shows that our proposed method generates a feasible fuzzy solution.

Example 4.4. Consider the following FFLS: $\begin{matrix} (1, 2, 3, 4) {\tilde{x}}_{1} = (2, 6, 12, 20) \\ (20, 30, 40, 50) {\tilde{x}}_{1} = (4, 5, 6, 7) \end{matrix}$ where ${\tilde{x}}_{1} = (x_{1}, y_{1}, z_{1}, w_{1})$ .

The sets $S_{1}^{mix} = S_{2}^{mix} = S_{3}^{mix} = S^{neg} = {},$ S^pos = { (1, 1) , (2, 1) } , and then accordingly $ɛ ((1, 2, 3, 4)) = ɛ ((20, 30, 40, 50)) = [\begin{matrix} 1 & 0 & 0 & 0 & 0 \end{matrix}]$ for ${\tilde{a}}_{11}$ and ${\tilde{a}}_{21}$ (Step 0). Since m = 2 and n = 1, the mixed integer programming problem corresponding to (16) which has 16mn + 4m + 3n = 43 constraints can be constructed and the optimal solution is obtained as follows: $\begin{matrix} {\tilde{x}}_{1}^{1 *} = (0.15, 0.15, 0.15, 0.15) \\ (λ_{11}^{1 *}, λ_{11}^{' 1 *}, μ_{11}^{' 1 *}, μ_{11}^{1 *}) = (0.15, 0.3, 0.45, 0.6) \\ (λ_{21}^{1 *}, λ_{21}^{' 1 *}, μ_{21}^{' 1 *}, μ_{21}^{1 *}) = (3, 4.5, 6, 7.5) \\ (k_{11}^{λ 1 *}, k_{11}^{λ^{'} 1 *}, k_{11}^{μ^{'} 1 *}, k_{11}^{μ 1 *}) = (0, 0, 1, 1) \\ (k_{21}^{λ 1 *}, k_{21}^{λ^{'} 1 *}, k_{21}^{μ^{'} 1 *}, k_{21}^{μ 1 *}) = (0, 0, 1, 1) \\ (R_{1}^{λ 1 * +}, R_{1}^{λ^{'} 1 * +}, R_{1}^{μ^{'} 1 * +}, R_{1}^{μ 1 * +}) = (0, 0, 0, 0) \\ (R_{2}^{λ 1 * +}, R_{2}^{λ^{'} 1 * +}, R_{2}^{μ^{'} 1 * +}, R_{2}^{μ 1 * +}) = (0, 0, 0, 0.5) \\ (R_{1}^{λ 1 * -}, R_{1}^{λ^{'} 1 * -}, R_{1}^{μ^{'} 1 * -}, R_{1}^{μ 1 * -}) \\ = (1.85, 5.7, 11.55, 19.4) \\ (R_{2}^{λ 1 * -}, R_{2}^{λ^{'} 1 * -}, R_{2}^{μ^{'} 1 * -}, R_{2}^{μ 1 * -}) = (1, 0.5, 0, 0) \\ Z^{1 *} = 40.5 . \end{matrix}$

We should determine whether ${\tilde{x}}_{1}^{1 *}$ is an approximate fuzzy solution. Since the inequality $μ_{11}^{1 *} \geq 2$ is not satisfied with this optimal solution, ${\tilde{x}}_{1}^{1 *}$ is not an approximate fuzzy solution (Step1). Thus, Step 3 should be implemented, that is the following constraints are added to the mixed integer programming problem which is constructed in Step 1: $\begin{matrix} λ_{11} \leq 20 and μ_{11} \geq 2 for i = 1 \\ λ_{21} \leq 7 and μ_{21} \geq 4 for i = 2 . \end{matrix}$

And the new optimal solution is obtained as follows: $\begin{matrix} {\tilde{x}}_{1}^{2 *} = (0.167, 0.167, 0.167, 0.5) \\ (λ_{11}^{2 *}, λ_{11}^{' 2 *}, μ_{11}^{' 2 *}, μ_{11}^{2 *}) = (0.167, 0.333, 0.5, 2) \\ (λ_{21}^{2 *}, λ_{21}^{' 2 *}, μ_{21}^{' 2 *}, μ_{21}^{2 *}) = (3.333, 5, 6.667, 25) \\ (k_{11}^{λ 2 *}, k_{11}^{λ^{'} 2 *}, k_{11}^{μ^{'} 2 *}, k_{11}^{μ 2 *}) = (0, 0, 1, 1) \\ (k_{21}^{λ 2 *}, k_{21}^{λ^{'} 2 *}, k_{21}^{μ^{'} 2 *}, k_{21}^{μ 2 *}) = (0, 0, 1, 1) \\ (R_{1}^{λ 2 * +}, R_{1}^{λ^{'} 2 * +}, R_{1}^{μ^{'} 2 * +}, R_{1}^{μ 2 * +}) = (0, 0, 0, 0) \\ (R_{2}^{λ 2 * +}, R_{2}^{λ^{'} 2 * +}, R_{2}^{μ^{'} 2 * +}, R_{2}^{μ 2 * +}) = (0, 0, 0.667, 18) \\ (R_{1}^{λ 2 * -}, R_{1}^{λ^{'} 2 * -}, R_{1}^{μ^{'} 2 * -}, R_{1}^{μ 2 * -}) \\ = (1.833, 5.667, 11.5, 18) \\ (R_{2}^{λ 2 * -}, R_{2}^{λ^{'} 2 * -}, R_{2}^{μ^{'} 2 * -}, R_{2}^{μ 2 * -}) = (0.667, 0, 0, 0) \\ Z^{2 *} = 56.33 . \end{matrix}$

Since this problem has an optimal solution, then obtained ${\tilde{x}}_{1}^{2 *}$ is an approximate fuzzy solution. Indeed, the inequalities $λ_{11}^{2 *} \leq 20$ , $μ_{11}^{2 *} \geq 2$ , $λ_{21}^{2 *} \leq 7$ and $μ_{21}^{2 *} \geq 4$ are satisfied for the new optimal solution. Now, Step 2 should be implemented. Since all of $R_{i}^{λ^{'} 2 * +}, R_{i}^{λ^{'} 2 * -}, R_{i}^{μ^{'} 2 * +}, R_{i}^{μ^{'} 2 * -} (i = 1, 2)$ are not equal to zero, the objective function can be updated with (17) and a new approximate fuzzy solution(s) can be obtained depending upon the decision maker’s preferences for different values of the weightiness parameter α. In this example, the approximate fuzzy solution ${\tilde{x}}_{1}^{3 *} = (0.15, 0.15, 0.15, 0.5)$ is obtained for α = 5.

This example shows that although ${\tilde{x}}_{1}^{1 *}$ is neither a feasible nor an approximate fuzzy solution, our algorithm enabled to generate the approximate fuzzy solutions ${\tilde{x}}_{1}^{2 *}$ and ${\tilde{x}}_{1}^{3 *}$ which can be offered to the decision maker.

5 Consclusion

Representing the most general structure of a linear equation system, a general FFLS provides to model real life problems in a more flexible way. In this paper, by introducing the “feasible fuzzy solution”, “approximate fuzzy solution” and “no solution” concepts, we aim to solve a general FFLS with arbitrary trapezoidal fuzzy parameters and variables. A new model which is based on expressing the “min” and “max” concepts in fuzzy multiplication operation as a mixed integer programming problem is presented. Within this model, an algorithm is constructed to analyze the nature of solutions of a general FFLS. The prominent feature of our algorithm is the ability to convert a no solution case to an approximate fuzzy solution, if possible. The algorithm is illustrated with some numerical examples.

Generally, the papers related to FLS and FFLS give numerical examples after presenting the theoretical approach. In this sense, our algorithm which determines the nature of solutions of a general FFLS with arbitrary trapezoidal fuzzy parameters and variables can be improved by the addition of real life applications. This will give an opportunity to increase the employability of a FLS and FFLS in various areas.

Footnotes

Acknowledgments

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

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