Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation

Abstract

In this paper, a new order relation for ranking fuzzy numbers and the fully fuzzy linear programming (FFLP) problems with flexible constraints are proposed. Based on the new order relation, three algorithms are developed to deal with FFLP problems by converting them into some equivalent crisp linear programming problems. Algorithm 1 is employed for solving FFLP problem without flexible constraint, while Algorithms 2 and 3 are developed to deal with FFLP problem with flexible constraints. A numerical example is given to illustrate the feasibility and efficiency of our proposed algorithms.

Keywords

Fully fuzzy linear programming flexible constraint order relation expected value

1 Introduction

Based on the idea of fuzzy decision proposed by Bellman and Zadeh [1], Tanaka et al. and Zimmermann introduced linear programming problems in a fuzzy environment [8 , 18]. After then, many researchers turned their attention to the fuzzy linear programming (FLP) problems. All kinds of method for solving FLP problems had been proposed [10 , 15-30]. There were two important solution approaches to the FLP problems. One solution approach was to convert the fuzzy model, with fuzzy parameters or fuzzy variables, into some normal linear programming(s) which could be solved by the classical simplex method. The other one was solving the FLP directly by the modified simplex method based on some special ranking functions [24-28 , 32]. In both of these two solution approaches, ranking method of fuzzy numbers [35 -38] always play an important role when dealing with the FLP problem. With respect to the practical application, the readers can refer to [39 -43]. In fact, FLP was widely applied in operational research, optimization management and fuzzy control.

The FLP is said to be fully fuzzy linear programming (FFLP), if all the parameters and variables are considered as fuzzy numbers. The FFLP problems have drew many researchers’ attention, after proposed by J.J. Buckley et al. [11] in 2000. To our knowledge, there are several existing methods for solving the FFLP problems [3-7 , 31-33]. We summarize them as follows.

(i) Multi-objective linear programming (MOLP) method [11 , 33-34]

J.J. Buckley and T. Feuring [11] studied the FFLP problems for the first time. The authors proposed three methods to verify the feasibility of the constraints and the fuzzy-valued objective function was equivalently transformed into three sub-objective functions. So the FFLP problem might be converted into its equivalent crisp MOLP problem and solved.

In [33], based on a new lexicographic ordering on triangular fuzzy numbers, a novel algorithm was proposed to deal with the FFLP problem by converting it to its equivalent MOLP problem and then it was solved by the lexicographic method. In [34], similarly, according to the defined order relation, a FFLP problem with L-R fuzzy numbers was converted into a classical MOLP problem and then solved.

(ii) Particle swarm optimization (POS) method [14]

A. Baykasoglu and T. Gocken [14] proposed the POS algorithm to solve the FFLP problems which the parameters and variables were represented by symmetric triangular fuzzy numbers. The ranking method based on left and right dominance (Chen et al. [2]) was used to rank the objective function values and determine the feasibility of the constraints in the PSO algorithm [14].

(iii) Lexicographic order method [4,12, 4,12]

S.M. Hashemi et al. [12] introduced the concepts of possibilistic mean value and standard deviation. The lexicographic order was introduced for comparison of fuzzy numbers. Then, a two-phase method was used to solve the FFLP problem by converting it into the equivalent linear programming problems.

In [4], all general fuzzy numbers were transformed into its nearest symmetric triangular fuzzy numbers and then the lexicographic order between symmetric triangular fuzzy numbers was constructed. Based on this lexicographic order, every FFLP model was converted into two crisp complex linear programming problems. Applying this method, the authors were able to find the fuzzy approximate solution of the FFLP problems.

(iv) Ranking function method [3 , 31]

A. Kumar et al. and T. Allahviranloo et al. [3 , 7, 31] used ranking function to solve FFLP problems with all parameters represented by triangular, trapezoidal or L-R fuzzy numbers, respectively. Usually, an order relation for comparing two arbitrary fuzzy numbers was introduced, according to the given ranking function. After then, a fully fuzzy linear programming could be equivalently changed into a crisp one, whose optimal solution could be easily found. Consequently the exact fuzzy optimal solution was obtained.

(v) Fuzzy simplex method [32]

I.U. Khan et al. [32] developed the fuzzy simplex method to FFLP problem. The basic arithmetic operations, including addition, substraction, multiplication and division, between triangular fuzzy numbers, were defined. The authors employed a ranking function to compare two arbitrary triangular fuzzy numbers. Since arithmetic operations and ranking method were given, the fuzzy simplex method could be carried out for obtaining the fuzzy optimal solutions, if they existed. In [32], the division was not the inverse operation of the multiplication.

In fact, order relation, which is introduced for comparing fuzzy numbers, paly an important role in the solution method to FFLP problem. In this paper, we define the concepts of expected value, expected interval and the degree in which $\tilde{a}$ is bigger than $\tilde{b}$ , where $\tilde{a}, \tilde{b}$ are two arbitrary fuzzy numbers. According to these concepts, we may rank two fuzzy numbers reasonable. Furthermore, a new order relation on the set of all fuzzy numbers is given. Based on the new order relation, new algorithms for solving FFLP problems are proposed.

In this paper, we consider two types of fully fuzzy linear programming problems, named FFLP1 and FFLP2. The constraints of FFLP1 are ordinary inequalities while those of FFLP2 are flexible inequalities. Both of the solution methods to these two types of FFLP problems are developed.

The remaining of the paper unfolds as follows. In Section 2, some basic definitions and the new order relation, together with its properties, are given. Two types of FFLP problems are introduced in this section too. In Sections 3 and 4, the algorithms for solving FFLP1 and FFLP2 are proposed respectively. In Section 5, a numerical example is given to illustrate the feasibility and efficiency of the proposed algorithms. Simple discussion on the proposed method is made in Section 6 while conclusion drawn in Section 7.

2 Fully fuzzy linear programming (FFLP)

2.1 Basic definitions

In this subsection, some definitions related to the fuzzy set theory, which will be used in the rest of the paper, are recalled.

Definition 2.1. [33] A fuzzy number is a fuzzy set like $u : ℝ \to I$ , which satisfies:

u is upper semi-continuous,

u (x) =0 outside some interval [c, d],

There are real numbers a, b such that c ≤ a ≤ b ≤ d and

u (x) is monotonic increasing on [c, a],

u (x) is monotonic decreasing on [b, d],

u (x) =1, a ≤ x ≤ b.

The set of all fuzzy numbers is denoted by $F (ℝ)$ . An equivalent parametric form of that is presented as follows.

Definition 2.2. [9, 33] A fuzzy number $\tilde{u}$ in parametric form is a pair $(\underline{u}, \bar{u})$ of functions $\underline{u} (r), \bar{u} (r)$ , 0 ≤ r ≤ 1, with these functions satisfy the following requirements:

$\underline{u} (r)$ is bounded monotonic increasing right continuous function,

$\bar{u} (r)$ is bounded monotonic decreasing left continuous function,

$\underline{u} (r) \leq \bar{u} (r), 0 \leq r \leq 1$ .

A popular fuzzy number is the triangular fuzzy number $\tilde{u} = (a, b, c)$ , with membership function given as follows: $μ_{\tilde{u}} (x) = {\begin{matrix} \frac{x - a}{b - a}, a \leq x \leq b, \\ \frac{x - c}{b - c}, b < x \leq c, \\ 0, otherwise . \end{matrix}$ Its parametric form is $(\underline{u} (r), \bar{u} (r))$ with $\underline{u} (r) = a + (b - a) r, \bar{u} (r) = c + (b - c) r, 0 \leq r \leq 1$ . The set of all triangular fuzzy numbers is denoted by $TF (ℝ)$ .

Definition 2.3. [9,33, 9,33] A triangular fuzzy number $\tilde{u} = (x_{1}, y_{1}, z_{1})$ is said to be a non-negative triangular fuzzy number if and only if x ₁ ≥ 0. The set of all non-negative triangular fuzzy numbers is denoted by $TF (ℝ)^{+}$ .

Definition 2.4. [9, 33] Two triangular fuzzy numbers $\tilde{u} = (x_{1}, y_{1}, z_{1})$ and $\tilde{v} = (x_{2}, y_{2}, z_{2})$ are said to be equal, $\tilde{u} = \tilde{v}$ , if and only if x ₁ = x ₂, y ₁ = y ₂ and z ₁ = z ₂.

Definition 2.5. [9, 33] The arithmetic operations between triangular fuzzy numbers are defined by the extension principle and can be equivalent represented as follows:

Let $\tilde{u} = (x_{1}, y_{1}, z_{1})$ and $\tilde{v} = (x_{2}, y_{2}, z_{2})$ be two triangular fuzzy numbers and $k \in ℝ$ . Then

k ≥ 0, $k \tilde{u} = ({kx}_{1}, {ky}_{1}, {kz}_{1})$ ,

k ≤ 0, $k \tilde{u} = ({kz}_{1}, {ky}_{1}, {kx}_{1})$ ,

$\tilde{u} \oplus \tilde{v} = (x_{1} + x_{2}, y_{1} + y_{2}, z_{1} + z_{2})$ ,

$\tilde{u} ⊖ \tilde{v} = (x_{1} - z_{2}, y_{1} - y_{2}, z_{1} - x_{2})$ ,

$\tilde{v} = (x_{2}, y_{2}, z_{2})$ is non-negative, $\tilde{u} \otimes \tilde{v} = {\begin{matrix} (x_{1} x_{2}, y_{1} y_{2}, z_{1} z_{2}), x_{1} \geq 0, \\ (x_{1} z_{2}, y_{1} y_{2}, z_{1} z_{2}), x_{1} < 0, z_{1} \geq 0, \\ (x_{1} z_{2}, y_{1} y_{2}, z_{1} x_{2}), z_{1} < 0 . \end{matrix}$

2.2 Expected value and the new order relation

In this subsection, we shall introduce the concepts of expected interval and expected value. Based on the expected interval, we define the degree in which $\tilde{a}$ is bigger than $\tilde{b}$ , with $\tilde{a}$ and $\tilde{b}$ being two arbitrary fuzzy numbers.

In fact, the order relation, which is used to compare two fuzzy numbers, play an important role in solving the FFLP problems. Based on the order relation, we may check the constraints and then get a feasible solution. Not only that, we may use the order relation to compare the corresponding objective values of the feasible solutions and verify the optimality.

Definition 2.6. Let $\tilde{a}$ be a fuzzy number with parametric form $\tilde{a} = (\underline{a} (r), \bar{a} (r)), r \in [0, 1]$ . The expected interval and expected value of $\tilde{a}$ , written as $EI (\tilde{a})$ and $EV (\tilde{a})$ respectively, are defined as: $EI (\tilde{a}) = [E_{\tilde{a}}^{-}, E_{\tilde{a}}^{+}] = [\frac{\int_{0}^{1} r \underline{a} (r) dr}{\int_{0}^{1} rdr}, \frac{\int_{0}^{1} r \bar{a} (r) dr}{\int_{0}^{1} rdr}],$ $EV (\tilde{a}) = \frac{E_{\tilde{a}}^{-} + E_{\tilde{a}}^{+}}{2}$

Definition 2.7. For any pair of fuzzy number $\tilde{a}$ and $\tilde{b}$ , the degree in which $\tilde{a}$ is bigger than $\tilde{b}$ is defined as: $P (\tilde{a}, \tilde{b}) = {\begin{matrix} (i) 0, if E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} < 0, \\ (ii) \frac{E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-}}{E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} - (E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+})}, \\ if E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} \geq 0 and E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+} \leq 0, \\ (iii) 1 if E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+} > 0 . \end{matrix}$

Definition 2.8. Let $\tilde{a}$ and $\tilde{b}$ be two arbitrary fuzzy numbers. Then the order relation between fuzzy numbers is defined as:

$\tilde{a} ⪰ \tilde{b}$ if and only if $P (\tilde{a}, \tilde{b}) \geq \frac{1}{2}$ .

$\tilde{a} ⪯ \tilde{b}$ if and only if $P (\tilde{b}, \tilde{a}) \geq \frac{1}{2}$ .

$\tilde{a} ≃ \tilde{b}$ if and only if $P (\tilde{a}, \tilde{b}) = \frac{1}{2}$ .

The notation ^″ ≾ ″ will be employed to formulate the fully fuzzy linear programming with flexible constraints. Here we explain the meaning of the flexible inequality $\tilde{a} ≾ \tilde{b}$ first. $\tilde{a}$ is said to be less than $\tilde{b}$ in a degree of α, written as $\tilde{a} ⪯_{α} \tilde{b}$ , if and only if $P (\tilde{b}, \tilde{a}) \geq α$ . In this paper $\tilde{a} ⪯_{α} \tilde{b}$ and $\tilde{b} ⪰_{α} \tilde{a}$ mean the same thing. The flexible inequality $\tilde{a} ≾ \tilde{b}$ represents that $\tilde{a} ⪯_{α} \tilde{b}$ , where α is a flexible parameter and $\frac{1}{2} \leq α \leq 1$ .

Next, some properties of the new order relation will be given in the form of some propositions and remarks. The detailed analysis will be shown in another paper which is being constructed.

Proposition 2.1. Let $\tilde{a}, \tilde{b} \in F (ℝ)$ , then $P (\tilde{a}, \tilde{b}) + P (\tilde{b}, \tilde{a}) = 1$ .

Proof. Suppose the expected intervals of $\tilde{a}$ and $\tilde{b}$ are $EI (\tilde{a}) = [E_{\tilde{a}}^{-}, E_{\tilde{a}}^{+}]$ and $EI (\tilde{b}) = [E_{\tilde{b}}^{-}, E_{\tilde{b}}^{+}]$ . This proposition may be proved though the following cases.

case 1. If $E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} < 0$ , then $E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+} > 0$ . According to Definition 2.7 we get $P (\tilde{a}, \tilde{b}) + P (\tilde{b}, \tilde{a}) = 0 + 1 = 1$ .

case 2. If $E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+} > 0$ , then $E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} < 0$ . According to Definition 2.7 we get $P (\tilde{a}, \tilde{b}) + P (\tilde{b}, \tilde{a}) = 1 + 0 = 1$ .

case 3. If $E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} \geq 0 and E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+} \leq 0$ , then

$P (\tilde{a}, \tilde{b}) + P (\tilde{b}, \tilde{a})$

$= \frac{E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-}}{E_{\tilde{a}}^{+} - E_{\tilde{b}}^{-} - (E_{\tilde{a}}^{-} - E_{\tilde{b}}^{+})} + \frac{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+})}$

= 1 . □

Proposition 2.2. Let $\tilde{a}, \tilde{b}, \tilde{c}$ are three arbitrary fuzzy numbers. Then

$\tilde{a} ⪯ \tilde{a}$ ,

$\tilde{a} ⪯ \tilde{b}$ and $\tilde{b} ⪯ \tilde{a} \Rightarrow$ $\tilde{a} ≃ \tilde{b},$

$\tilde{a} ⪯ \tilde{b}$ and $\tilde{b} ⪯ \tilde{c}$ ⇒ $\tilde{a} ⪯ \tilde{c} .$

Either $\tilde{a} ⪯ \tilde{b}$ or $\tilde{a} ⪰ \tilde{b}$ holds.

Proof. i) Since $P (\tilde{a}, \tilde{a}) = \frac{E_{\tilde{a}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{a}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{a}}^{-} - E_{\tilde{a}}^{+})} = \frac{1}{2} \geq \frac{1}{2},$ we have $\tilde{a} ⪯ \tilde{a}$ .

ii) From $\tilde{a} ⪯ \tilde{b}$ and $\tilde{b} ⪯ \tilde{a}$ we get that $P (\tilde{b}, \tilde{a}) \geq \frac{1}{2}$ and $P (\tilde{a}, \tilde{b}) \geq \frac{1}{2}$ . Considering $P (\tilde{a}, \tilde{b}) + P (\tilde{b}, \tilde{a}) = 1$ , it follows that $P (\tilde{a}, \tilde{b}) = P (\tilde{b}, \tilde{a}) = \frac{1}{2}$ . So we have $\tilde{a} ≃ \tilde{b}$ according to Definition 2.8.

iii) $\tilde{a} ⪯ \tilde{b}$ and $\tilde{b} ⪯ \tilde{c}$ imply that $P (\tilde{b}, \tilde{a}) \geq \frac{1}{2}, P (\tilde{c}, \tilde{b}) \geq \frac{1}{2} .$

That is $\frac{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+})} \geq \frac{1}{2},$ $\frac{E_{\tilde{c}}^{+} - E_{\tilde{b}}^{-}}{E_{\tilde{c}}^{+} - E_{\tilde{b}}^{-} - (E_{\tilde{c}}^{-} - E_{\tilde{b}}^{+})} \geq \frac{1}{2} .$

These formulae may be changed into $E_{\tilde{a}}^{-} + E_{\tilde{a}}^{+} \leq E_{\tilde{b}}^{-} + E_{\tilde{b}}^{+}, E_{\tilde{b}}^{-} + E_{\tilde{b}}^{+} \leq E_{\tilde{c}}^{-} + E_{\tilde{c}}^{+} .$

Hence $E_{\tilde{a}}^{-} + E_{\tilde{a}}^{+} \leq E_{\tilde{c}}^{-} + E_{\tilde{c}}^{+},$ and then $\frac{E_{\tilde{c}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{c}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{c}}^{-} - E_{\tilde{a}}^{+})} \geq \frac{1}{2},$ i.e. $P (\tilde{c}, \tilde{a}) \geq \frac{1}{2} .$

By Definition 2.8 we get $\tilde{a} ⪯ \tilde{c} .$

iv) Since $P (\tilde{a}, \tilde{b}) \in [0, 1]$ , it is obvious that either $P (\tilde{a}, \tilde{b}) \geq \frac{1}{2}$ or $P (\tilde{a}, \tilde{b}) \leq \frac{1}{2}$ holds, and the proof is complete. □

From Proposition 2.1 and 2.2, it is easy to check the following remarks.

Remark 2.1. The order relation ⪯ is a total order on the set $F (ℝ)$ .

Remark 2.2. Let $\tilde{a}, \tilde{b} \in F (ℝ)$ , then the following statements are equivalent.

$\tilde{a} ≃ \tilde{b}$ ;

$\tilde{a} ⪯ \tilde{b}$ and $\tilde{b} ⪯ \tilde{a}$ ;

$P (\tilde{a}, \tilde{b}) = \frac{1}{2}$ ;

$P (\tilde{b}, \tilde{a}) = \frac{1}{2}$ .

2.3 Two types of fully fuzzy linear programming problems

In this subsection, we introduce two types of fully fuzzy linear programming problems.

Before constructing the optimization models, we give some necessary assumptions first.

Assumption I: The optimization model can be reduced into fuzzy linear programming. That is to say, all of the objective function and constraint can be expressed by linear formulae.

Assumption II: All of the parameters and variables can be valued by triangular fuzzy numbers.

In the following we establish these two types of optimization models.

Type I: Fully fuzzy linear programming without flexible constraint

The fully fuzzy linear programming without flexible constraint may be described as: $\begin{matrix} (FFLP 1) \\ \max z (\tilde{x}) = {\tilde{c}}_{1} \otimes {\tilde{x}}_{1} \oplus {\tilde{c}}_{2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{c}}_{n} \otimes {\tilde{x}}_{n} \\ s . t . {\begin{matrix} {\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n} ⪯ {\tilde{b}}_{i}, \\ {\tilde{x}}_{j} \in TF (ℝ)^{+}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (1) where ${\tilde{c}}_{j}, {\tilde{b}}_{i}, {\tilde{a}}_{ij}, {\tilde{x}}_{j},$ (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n)are triangular fuzzy numbers. The order relations for comparison between triangular fuzzy numbers are as shown in Definition 2.8.

Type II: Fully fuzzy linear programming with flexible constraints

The fully fuzzy linear programming with flexible constraints may be described as: $\begin{matrix} (FFLP 2) \\ \max z (\tilde{x}) = {\tilde{c}}_{1} \otimes {\tilde{x}}_{1} \oplus {\tilde{c}}_{2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{c}}_{n} \otimes {\tilde{x}}_{n} \\ s . t . {\begin{matrix} {\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n} ≾ {\tilde{b}}_{i}, \\ {\tilde{x}}_{j} \in TF (ℝ)^{+}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (2)

Here ${\tilde{c}}_{j}, {\tilde{b}}_{i}, {\tilde{a}}_{ij}, {\tilde{x}}_{j}$ have the same meanings as in FFLP1 and ≾ means ⪯_α, where α is flexible and ranges from $\frac{1}{2}$ to 1, i.e. $\frac{1}{2} \leq α \leq 1 .$

3 Algorithm for solving FFLP1

The concepts of feasible region, feasible solution, optimal solution and optimal value of the FFLP are similar to that of the crisp linear programming. We introduce these concepts by the following definitions.

Definition 3.1. In the above FFLP1 problem, $D = {\tilde{x} \in (TF (R)^{+})^{n} | \tilde{A} \otimes \tilde{x} ⪯ \tilde{b},}$ is said to be the feasible region, and any $\tilde{x} \in D$ is said to be the feasible solution.

Definition 3.2. ${\tilde{x}}^{*} \in D$ is said to be the optimal solution of FFLP1, if $z (\tilde{x}) ⪯ z ({\tilde{x}}^{*})$ holds for all $\tilde{x} \in D$ . The corresponding objective function value $z ({\tilde{x}}^{*})$ is said to be the optimal value.

In order to solve the FFLP1, we try to change it into a crisp linear programming equivalently.

Lemma 3.1. $\tilde{a} ⪯ \tilde{b} \Leftrightarrow EV (\tilde{a}) \leq EV (\tilde{b})$ .

Proof. $\tilde{a} ⪯ \tilde{b}$ if and only if $\frac{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+})} \geq \frac{1}{2},$ if and only if $E_{\tilde{a}}^{-} + E_{\tilde{a}}^{+} \leq E_{\tilde{b}}^{-} + E_{\tilde{b}}^{+},$ if and only if $EV (\tilde{a}) \leq EV (\tilde{a}) . □$

Theorem 3.1. The model FFLP1 is equivalent to the following model LP1. $\begin{matrix} (LP 1) \\ \max EV (z (\tilde{x})) = EV (\tilde{c} \otimes \tilde{x}) \\ s . t . {\begin{matrix} EV ({\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n}) \\ \leq EV ({\tilde{b}}_{i}), \\ {\tilde{x}}_{j} \in TF (ℝ)^{+}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (3)

Proof. Suppose the feasible regions of FFLP1 and LP1 are D _FFLP1 and D _LP1 respectively. By the Lemma 3.1, we get ${\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n} ⪯ {\tilde{b}}_{i}$ if and only if $EV ({\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n}) \leq EV ({\tilde{b}}_{i}),$ i = 1, 2, ⋯ , m. So FFLP1 and LP1 have the same feasible region, i.e. D _FFLP1 = D _LP1.

If ${\tilde{x}}^{*}$ is the optimal solution of FFLP1, then $z (\tilde{x}) \leq z ({\tilde{x}}^{*})$ for all $\tilde{x}$ in D _FFLP1. According to the Lemma 3.1, we may obtain that $EV (z (\tilde{x})) \leq EV (z ({\tilde{x}}^{*}))$ for all $\tilde{x}$ in D _LP1 = D _FFLP1. This shows that ${\tilde{x}}^{*}$ is the optimal solution of LP1.

In the similar way we may prove that any optimal solution of LP1 is also that of FFLP1. Hence the model FFLP1 is equivalent to LP1.□

Lemma 3.2. Let $\tilde{a} = (a_{1}, a_{2}, a_{3}) \in TF (ℝ)$ . Then $E_{\tilde{a}}^{-} = \frac{1}{3} (a_{1} + 2 a_{2})$ , $E_{\tilde{a}}^{+} = \frac{1}{3} (2 a_{2} + a_{3})$ , and $EV (\tilde{a}) = \frac{1}{6} (a_{1} + 4 a_{2} + a_{3})$ .

Proof. Suppose the parametric form of $\tilde{a}$ is $(\underline{a} (r), \bar{a} (r))$ , r ∈ [0, 1]. Then $\underline{a} (r) = a_{1} + r (a_{2} - a_{1}), \bar{a} (r) = a_{3} - r (a_{3} - a_{2}) .$

So we obtain that $\begin{matrix} E_{\tilde{a}}^{-} & = \frac{\int_{0}^{1} r \underline{a} (r) dr}{\int_{0}^{1} rdr} \\ = 2 \int_{0}^{1} r (a_{1} + r (a_{2} - a_{1})) dr = \frac{1}{3} (a_{1} + 2 a_{2}), \end{matrix}$ $\begin{matrix} E_{\tilde{a}}^{+} & = \frac{\int_{0}^{1} r \bar{a} (r) dr}{\int_{0}^{1} rdr} \\ = 2 \int_{0}^{1} r (a_{3} - r (a_{3} - a_{2})) dr = \frac{1}{3} (2 a_{2} + a_{3}) \end{matrix}$ and then $EV (\tilde{a}) = \frac{1}{2} (E_{\tilde{a}}^{-} + E_{\tilde{a}}^{+}) = \frac{1}{6} (a_{1} + 4 a_{2} + a_{3}) .$ □

Theorem 3.2. LP1 is a crisp linear programming.

Proof. Let ${\tilde{c}}_{j} = (c_{j}^{l}, c_{j}^{c}, c_{j}^{r})$ , ${\tilde{x}}_{j} = (x_{j}^{l}, x_{j}^{c}, x_{j}^{r})$ , ${\tilde{a}}_{ij} = (a_{ij}^{l}, a_{ij}^{c}, a_{ij}^{r})$ , ${\tilde{b}}_{i} = (b_{i}^{l}, b_{i}^{c}, b_{i}^{r})$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Since ${\tilde{x}}_{j} \in TF (R)^{+}, j = 1, 2, \dots, n$ , the results of $z (\tilde{x}) = \tilde{c} \otimes \tilde{x}$ and $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} = {\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n}$ , i = 1, 2, ⋯ , m may be calculated by the arithmetic operations on TF (R). Let $X = (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}, x_{2}^{l}, x_{2}^{c}, x_{2}^{r}, \dots, x_{n}^{l}, x_{n}^{c}, x_{n}^{r}) .$

The results of $\tilde{c} \otimes \tilde{x}$ and $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ are still triangular fuzzy numbers, and written as $\tilde{c} \otimes \tilde{x} = (f^{l} (X), f^{c} (X), f^{r} (X)),$ and $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j} = (g_{i}^{l} (X), g_{i}^{c} (X), g_{i}^{r} (X)),$ where all the functions $f^{l} (X), f^{c} (X), f^{r} (X), g_{i}^{l} (X)$ , $g_{i}^{c} (X) and g_{i}^{r} (X)$ are linear functions with variable vector X, i = 1, 2, ⋯ , m .

By the Lemma 3.2, we get $EV (\tilde{c} \otimes \tilde{x}) = \frac{1}{6} (f^{l} (X) + 4 f^{c} (X) + f^{r} (X)) .$ $EV (\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}) = \frac{1}{6} (g_{i}^{l} (X) + 4 g_{i}^{c} (X) + g_{i}^{r} (X)),$ $EV ({\tilde{b}}_{i}) = \frac{1}{6} (b_{i}^{l} + 4 b_{i}^{c} + b_{i}^{r}) .$

So the model LP1 may be written as $\begin{matrix} (LP 1^{'}) \\ \max \frac{1}{6} (f^{l} (X) + 4 f^{c} (X) + f^{r} (X)) \\ s . t . {\begin{matrix} \frac{1}{6} (g_{i}^{l} (X) + 4 g_{i}^{c} (X) + g_{i}^{r} (X)) \\ \leq \frac{1}{6} (b_{i}^{l} + 4 b_{i}^{c} + b_{i}^{r}), \\ x_{j}^{l} \geq 0, x_{j}^{l} \leq x_{j}^{c} \leq x_{j}^{r}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (4) which is a crisp linear programming with decision variable vector $X = (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}, \dots, x_{n}^{l}, x_{n}^{c}, x_{n}^{r}) . □$

From Theorem 3.1 and 3.2, it is easy to get the following remark.

Remark 3.1. FFLP1 has optimal solution if and only if the linear programming LP1’ has optimal solution. Moreover, ${\tilde{x}}^{*} = ((x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}), (x_{2}^{l *}, x_{2}^{c *}, x_{2}^{r *}), \dots,$ $(x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *}))$ is the optimal solution of FFLP1 if and only if $X^{*} = (x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}, \dots, x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *})$ is the optimal solution of LP1’.

Based on the above-mentioned theorems, we develop Algorithm 1 for solving FFLP1 below.

Algorithm 1.

Step 2. Solving the linear programming LP1’ obtained in step 1.

Step 3. If LP1’ has no optimal solution, then FFLP1 has no optimal solution and stop. Otherwise, suppose $X^{*} = (x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}, \dots, x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *})$ is one of the optimal solutions of LP1’ and go to step 4.

Step 4. Generate the optimal solutions ${\tilde{x}}^{*}$ of FFLP1 by X ^* as follows and stop. $\begin{matrix} {\tilde{x}}^{*} = ({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*}, \dots, {\tilde{x}}_{n}^{*}) \\ = ((x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}), (x_{2}^{l *}, x_{2}^{c *}, x_{2}^{r *}), \dots, (x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *})) . \end{matrix}$

4 Algorithms for solving FFLP2

Let α ∈ [0, 1], we denote the following programming as FFLP_α. $\begin{matrix} ({FFLP}_{α}) \\ \max z (\tilde{x}) = {\tilde{c}}_{1} \otimes {\tilde{x}}_{1} \oplus {\tilde{c}}_{2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{c}}_{n} \otimes {\tilde{x}}_{n} \\ s . t . {\begin{matrix} {\tilde{a}}_{i 1} \otimes {\tilde{x}}_{1} \oplus {\tilde{a}}_{i 2} \otimes {\tilde{x}}_{2} \oplus \dots \oplus {\tilde{a}}_{in} \otimes {\tilde{x}}_{n} ⪯_{α} {\tilde{b}}_{i}, \\ {\tilde{x}}_{j} \in TF (ℝ)^{+}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (5)

Next, we will introduce a theorem which helps us to develop the method for solving FFLP_α.

Theorem 4.1. Let $\tilde{a} = (a_{1}, a_{2}, a_{3})$ and $\tilde{b} = (b_{1}, b_{2}, b_{3})$ are two triangular fuzzy numbers, α ∈ [0, 1]. Then $\tilde{a} ⪯_{α} \tilde{b}$ if and only if $\frac{2 b_{2} + b_{3} - a_{1} - 2 a_{2}}{b_{3} - b_{1} + a_{3} - a_{1}} \geq α .$

Proof. $\tilde{a} ⪯_{α} \tilde{b}$ if and only if $P (\tilde{b}, \tilde{a}) \geq α,$ if and only if $E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} \geq 0 and \frac{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+})} \geq α .$

Since $E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} = \frac{1}{3} (2 b_{2} + b_{3}) - \frac{1}{3} (a_{1} + 2 a_{2}),$ $\frac{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-}}{E_{\tilde{b}}^{+} - E_{\tilde{a}}^{-} - (E_{\tilde{b}}^{-} - E_{\tilde{a}}^{+})} = \frac{2 b_{2} + b_{3} - a_{1} - 2 a_{2}}{b_{3} - b_{1} + a_{3} - a_{1}},$ and $b_{3} - b_{1} + a_{3} - a_{1} \geq 0, 0 \leq α \leq 1,$ so we get $\tilde{a} ⪯_{α} \tilde{b}$ if and only if $\frac{2 b_{2} + b_{3} - a_{1} - 2 a_{2}}{b_{3} - b_{1} + a_{3} - a_{1}} \geq α . □$

We can solve FFLP_α by the following Algorithm 2, when the parameter α is fixed in the interval [0, 1].

Algorithm 2.

Step 1. Suppose ${\tilde{c}}_{j} = (c_{j}^{l}, c_{j}^{c}, c_{j}^{r})$ , ${\tilde{x}}_{j} = (x_{j}^{l}, x_{j}^{c}, x_{j}^{r})$ , ${\tilde{a}}_{ij} = (a_{ij}^{l}, a_{ij}^{c}, a_{ij}^{r})$ , ${\tilde{b}}_{i} = (b_{i}^{l}, b_{i}^{c}, b_{i}^{r})$ , i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n. Then the FFLP_α may be written as: $\begin{matrix} \max (f^{l} (X), f^{c} (X), f^{r} (X)) \\ s . t . {\begin{matrix} (g_{i}^{l} (X), g_{i}^{c} (X), g_{i}^{r} (X)) ⪯_{α} (b_{i}^{l}, b_{i}^{c}, b_{i}^{r}), \\ {\tilde{x}}_{j} \in TF (ℝ)^{+}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (6) where $f^{l} (X), f^{c} (X), f^{r} (X), g_{i}^{l} (X), g_{i}^{c} (X), g_{i}^{r} (X)$ are linear functions with variable vector $X = (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}, x_{2}^{l}, x_{2}^{c}, x_{2}^{r}, \dots, x_{n}^{l}, x_{n}^{c}, x_{n}^{r}),$ i = 1, 2, ⋯ , m .

Step 2. By Theorem 3.1, 3.2 and 4.1, the model obtained in step 1 may be transformed into its equivalent crisp linear programming as follows. $\begin{matrix} ({LP}_{α}) \\ \max \frac{1}{6} (f^{l} (X) + 4 f^{c} (X) + f^{r} (X)) \\ s . t . {\begin{matrix} \frac{2 b_{i}^{c} + b_{i}^{r} - g_{i}^{l} (X) - 2 g_{i}^{c} (X)}{b_{i}^{r} - b_{i}^{l} + g_{i}^{r} (X) - g_{i}^{l} (X)} \geq α, \\ x_{j}^{l} \geq 0, x_{j}^{l} \leq x_{j}^{c} \leq x_{j}^{r}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (7)

Step 3. Solving the linear programming LP_α obtained in step 2.

Step 4. If LP_α has no optimal solution, then FFLP_α has no optimal solution and stop. Otherwise, suppose $X_{α}^{*} = (x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}, \dots, x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *})$ is one of the optimal solutions of LP_α and go to step 5.

Step 5. Generate the optimal solution ${\tilde{x}}_{α}^{*}$ of FFLP_α by $X_{α}^{*}$ as follows and stop. $\begin{matrix} {\tilde{x}}_{α}^{*} = ({\tilde{x}}_{1}^{*}, {\tilde{x}}_{2}^{*}, \dots, {\tilde{x}}_{n}^{*}) \\ = ((x_{1}^{l *}, x_{1}^{c *}, x_{1}^{r *}), (x_{2}^{l *}, x_{2}^{c *}, x_{2}^{r *}), \dots, (x_{n}^{l *}, x_{n}^{c *}, x_{n}^{r *})) . \end{matrix}$

Theorem 4.2. Let α₁, α₂ ∈ [0, 1] and α₁ ≤ α₂. ${\tilde{x}}_{α_{1}}^{*}$ and ${\tilde{x}}_{α_{2}}^{*}$ are the optimal solutions of FFLP_{α
₁} and FFLP_{α
₂}, respectively. Then $z ({\tilde{x}}_{α_{1}}^{*}) ⪰ z ({\tilde{x}}_{α_{2}}^{*}) .$

Proof. Suppose LP_{α
₁} and LP_{α
₂} are the equivalent linear programmings of FFLP_{α
₁} and FFLP_{α
₂}, respectively. The feasible regions of LP_{α
₁} and LP_{α
₂} are D _{α
₁} and D _{α
₂}, respectively.

In fact, $z ({\tilde{x}}_{α_{1}}^{*}) ⪰ z ({\tilde{x}}_{α_{2}}^{*})$ if and only if $EV (z ({\tilde{x}}_{α_{1}}^{*})) \geq EV (z ({\tilde{x}}_{α_{2}}^{*})) .$

Here, $EV (z ({\tilde{x}}_{α_{1}}^{*}))$ and $EV (z ({\tilde{x}}_{α_{2}}^{*}))$ are the optimal objective values of LP_{α
₁} and LP_{α
₂}, respectively. In addition, LP_{α
₁} and LP_{α
₂} have the same objective function. Hence, we only need to prove that D _{α
₂} ⊆ D _{α
₁} below.

If $X_{α} = (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}, \dots, x_{n}^{l}, x_{n}^{c}, x_{n}^{r}) \in D_{α_{2}}$ , then $\frac{2 b_{i}^{c} + b_{i}^{r} - g_{i}^{l} (X) - 2 g_{i}^{c} (X)}{b_{i}^{r} - b_{i}^{l} + g_{i}^{r} (X) - g_{i}^{l} (X)} \geq α_{2},$ and $x_{j}^{l} \geq 0, x_{j}^{l} \leq x_{j}^{c} \leq x_{j}^{r},$ i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . Since α ₂ ≥ α ₁, we get $\frac{2 b_{i}^{c} + b_{i}^{r} - g_{i}^{l} (X) - 2 g_{i}^{c} (X)}{b_{i}^{r} - b_{i}^{l} + g_{i}^{r} (X) - g_{i}^{l} (X)} \geq α_{1},$ and $x_{j}^{l} \geq 0, x_{j}^{l} \leq x_{j}^{c} \leq x_{j}^{r},$ i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n . This indicates X _α ∈ D _{α
₁}. Hence, D _{α
₂} ⊆ D _{α
₁}. □

We divide $[\frac{1}{2}, 1]$ into n subintervals equally, where n is a positive integer. Take $α_{k} = \frac{1}{2} + \frac{k}{2 n}$ and suppose ${\tilde{x}}_{α_{k}}^{*}$ is the optimal solution of FFLP_{α
_k}, k = 0, 1, ⋯ , n. It is clear that $\frac{1}{2} = α_{0} \leq α_{1} \leq \dots \leq α_{n} = 1 .$ From Theorem 4.2, we get $z ({\tilde{x}}_{α_{0}}^{*}) ⪰ z ({\tilde{x}}_{α_{1}}^{*}) ⪰ \dots ⪰ z ({\tilde{x}}_{α_{n}}^{*}),$ and $EV (z ({\tilde{x}}_{α_{0}}^{*})) \geq EV (z ({\tilde{x}}_{α_{1}}^{*})) \geq \dots \geq EV (z ({\tilde{x}}_{α_{n}}^{*})) .$

Define the membership function of $z (\tilde{x})$ as follow: $\begin{matrix} μ (z (\tilde{x})) \\ = {\begin{matrix} (i) 0, if EV (z (\tilde{x})) < EV (z ({\tilde{x}}_{α_{n}}^{*})), \\ (ii) \frac{EV (z (\tilde{x})) - EV (z ({\tilde{x}}_{α_{n}}^{*}))}{EV (z ({\tilde{x}}_{α_{0}}^{*})) - EV (z ({\tilde{x}}_{α_{n}}^{*}))}, \\ if EV (z ({\tilde{x}}_{α_{n}}^{*})) \leq EV (z (\tilde{x})) \leq EV (z ({\tilde{x}}_{α_{0}}^{*})), \\ (iii) 1, if EV (z (\tilde{x})) > EV (z ({\tilde{x}}_{α_{0}}^{*})) . \end{matrix} \end{matrix}$

Then we may get $1 = μ (z ({\tilde{x}}_{α_{0}}^{*})) \geq μ (z ({\tilde{x}}_{α_{1}}^{*})) \geq \dots \geq μ (z ({\tilde{x}}_{α_{n}}^{*})) = 0 .$

Let β _k = 2α _k - 1, k = 0, 1, ⋯ , n. then $0 = β_{0} \leq β_{1} \leq \dots \leq β_{n} = 1 .$

If $min_{k} {| β_{k} - μ (z ({\tilde{x}}_{α_{k}}^{*})) |} = | β_{k_{0}} - μ (z ({\tilde{x}}_{α_{k_{0}}}^{*})) |$ , we take ${\tilde{x}}_{α_{k_{0}}}^{*}$ as the fuzzy optimal solution of fully fuzzy linear programming FFLP2.

Below, we propose Algorithm 3 for solving FFLP2.

Algorithm 3.

Step 1. Fixed a positive integer n and calculate the value of α _k by $α_{k} = \frac{1}{2} + \frac{k}{2 n}$ , k = 0, 1, ⋯ , n.

Step 2. Solving FFLP_{α
₀}, FFLP_{α
₂}, ⋯, FFLP_{α
_n} by Algorithm 2 respectively.

Step 3. Suppose ${\tilde{x}}_{α_{k}}^{*}$ is one of the optimal solutions of FFLP_{α
_k}, k = 0, 1, ⋯ , n. Calculate the optimal objective values $z ({\tilde{x}}_{α_{0}}^{*})$ , $z ({\tilde{x}}_{α_{1}}^{*}), \dots$ , $z ({\tilde{x}}_{α_{n}}^{*})$ , and then give the membership functions $μ (z (\tilde{x})) .$

Step 4. Calculate β _k = 2α _k - 1 and $μ (z ({\tilde{x}}_{α_{k}}^{*}))$ , k = 0, 1, ⋯ , n.

Step 5. Calculate $min_{k} {| β_{k} - μ (z ({\tilde{x}}_{α_{k}}^{*})) |}$ .

Step 6. Suppose $min_{k} {| β_{k} - μ (z ({\tilde{x}}_{α_{k}}^{*})) |} = | β_{k_{0}} - μ (z ({\tilde{x}}_{α_{k_{0}}}^{*})) | .$

Then the fuzzy optimal solution of FFLP2 is ${\tilde{x}}^{*} = {\tilde{x}}_{α_{k_{0}}}^{*}$ .

Remark 4.1. Here, we only consider the situation that every fully fuzzy linear programming FFLP_{α
_k}(k ∈ {0, 1, ⋯ , n}) has one optimal solution at least, in Step 3.

5 Numerical example

In this section we illustrate the proposed method by a numerical example.

Example 5.1. Consider the following fully fuzzy linear programming with flexible constraints: $\begin{matrix} (FFLP 21) \\ \max z (\tilde{x}) = (45, 50, 52) \otimes {\tilde{x}}_{1} \oplus (56, 60, 63) \otimes {\tilde{x}}_{2} \\ s . t . \\ {\begin{matrix} (9, 10, 12) \otimes {\tilde{x}}_{1} \oplus (3, 5, 6) \otimes {\tilde{x}}_{2} ≾ (180, 200, 210), \\ (0, 1, 1) \otimes {\tilde{x}}_{1} \oplus (2, 3, 4) \otimes {\tilde{x}}_{2} ≾ (45, 50, 55), \\ (3, 4, 6) \otimes {\tilde{x}}_{1} \oplus (3, 4, 5) \otimes {\tilde{x}}_{2} ≾ (90, 100, 120), \\ {\tilde{x}}_{1}, {\tilde{x}}_{2} \in TF (ℝ)^{+} . \end{matrix} \end{matrix}$ (8)

Solution:

Step 1. Take n = 10, and then $α_{k} = \frac{1}{2} + \frac{k}{20}$ , k = 0, 1, ⋯ , n.

Step 2. Suppose ${\tilde{x}}_{j} = (x_{j}^{l}, x_{j}^{c}, x_{j}^{r}), j = 1, 2$ . The fully fuzzy linear programming FFLP_{α
_k} may be written as: $\begin{matrix} ({FFLP}_{α_{k}}) \\ \max z (\tilde{x}) = (45, 50, 52) \otimes (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}) \oplus (56, 60, 63) \\ \otimes (x_{2}^{l}, x_{2}^{c}, x_{2}^{r}) \\ s . t . \\ {\begin{matrix} (9, 10, 12) \otimes (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}) \oplus (3, 5, 6) \otimes (x_{2}^{l}, x_{2}^{c}, x_{2}^{r}) \\ ⪯_{α_{k}} (180, 200, 210), \\ (0, 1, 1) \otimes (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}) \oplus (2, 3, 4) \otimes (x_{2}^{l}, x_{2}^{c}, x_{2}^{r}) \\ ⪯_{α_{k}} (45, 50, 55), \\ (3, 4, 6) \otimes (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}) \oplus (3, 4, 5) \otimes (x_{2}^{l}, x_{2}^{c}, x_{2}^{r}) \\ ⪯_{α_{k}} (90, 100, 120), \\ (x_{1}^{l}, x_{1}^{c}, x_{1}^{r}), (x_{2}^{l}, x_{2}^{c}, x_{2}^{r}) \in TF (ℝ)^{+} . \end{matrix} \end{matrix}$ (9) k = 0, 1, ⋯ , n . We solve these fully fuzzy linear programming models by Algorithm 2.

Step 3. After solving FFLP_{α
_k}, k = 0, 1, ⋯ , n, we get the optimal objective values as follows:

$z ({\tilde{x}}_{α_{0}}^{*}) = (1270.60, 1381.81, 1445.18),$

$z ({\tilde{x}}_{α_{1}}^{*}) = (1251.46, 1361.04, 1423.44),$

$z ({\tilde{x}}_{α_{2}}^{*}) = (1232.67, 1340.65, 1402.10),$

$z ({\tilde{x}}_{α_{3}}^{*}) = (1214.28, 1320.70, 1381.22),$

$z ({\tilde{x}}_{α_{4}}^{*}) = (1196.26, 1301.14, 1360.76),$

$z ({\tilde{x}}_{α_{5}}^{*}) = (1178.60, 1281.97, 1340.69),$

$z ({\tilde{x}}_{α_{6}}^{*}) = (1161.30, 1263.18, 1321.03),$

$z ({\tilde{x}}_{α_{7}}^{*}) = (1144.33, 1244.75, 1301.75),$

$z ({\tilde{x}}_{α_{8}}^{*}) = (1127.70, 1226.68, 1282.86),$

$z ({\tilde{x}}_{α_{9}}^{*}) = (1111.39, 1208.96, 1264.32),$

$z ({\tilde{x}}_{α_{10}}^{*}) = (1095.39, 1191.58, 1246.14) .$

Their expected values are

$EV (z ({\tilde{x}}_{α_{0}}^{*})) = 1373.85, EV (z ({\tilde{x}}_{α_{1}}^{*})) = 1353.17,$

$EV (z ({\tilde{x}}_{α_{2}}^{*})) = 1332.90, EV (z ({\tilde{x}}_{α_{3}}^{*})) = 1313.05,$

$EV (z ({\tilde{x}}_{α_{4}}^{*})) = 1293.60, EV (z ({\tilde{x}}_{α_{5}}^{*})) = 1274.53,$

$EV (z ({\tilde{x}}_{α_{6}}^{*})) = 1255.83, EV (z ({\tilde{x}}_{α_{7}}^{*})) = 1237.52,$

$EV (z ({\tilde{x}}_{α_{8}}^{*})) = 1219.55, EV (z ({\tilde{x}}_{α_{9}}^{*})) = 1201.93,$

$EV (z ({\tilde{x}}_{α_{10}}^{*})) = 1184.65 .$

The membership function of $z (\tilde{x})$ is $μ (z (\tilde{x})) = {\begin{matrix} (i) 0, if z (\tilde{x}) < 1184.65, \\ (ii) \frac{EV (z (\tilde{x})) - 1184.65}{1373.85 - 1184.65}, \\ if 1184.65 \leq EV (z (\tilde{x})) \leq 1373.85, \\ (iii) 1, if z (\tilde{x}) > 1373.85 . \end{matrix}$

Step 4. Calculate β _k = 2α _k - 1 and $μ (z ({\tilde{x}}_{α_{k}}^{*}))$ , k = 0, 1, ⋯ , n. The results are shown in Table 1.

Step 5.

$min_{k} {| β_{k} - μ (z ({\tilde{x}}_{α_{k}}^{*})) |}$

= {1, 0.7907, 0.5836, 0.3786, 0.1758, 0.0249, 0.2238,

0.4206, 0.6155, 0.8087, 1}

= 0.0249

$= | β_{5} - μ (z ({\tilde{x}}_{α_{5}}^{*})) |$ .

Step 6. The fuzzy optimal solution of FFLP21 is ${\tilde{x}}^{*} = {\tilde{x}}_{α_{5}}^{*}$ , ${\tilde{x}}_{1}^{*} = (10.7377, 10.7377, 10.7377)$ , ${\tilde{x}}_{2}^{*} = (12.4180, 12.4180, 12.4180)$ . and the optimal objective value is $z ({\tilde{x}}_{α_{5}}^{*}) = (1178.60, 1281.97, 1340.69) .$

6 Discussion on the proposed method

(i) In most existing works [2 , 44-45], fuzzy numbers were ranked by a specific function and a absolute comparison result was obtained. Moreover, using different ranking methods, the comparison results of two fuzzy numbers might be different. To avoid this confusion, a more flexible ranking method is proposed in this paper. As shown in Definition 2.7, we introduce the degree in which $\tilde{a}$ is bigger than $\tilde{b}$ for ranking tow triangular fuzzy numbers. We give the following example to illustrate this concept with comparison with the existing methods.

Example 6.1. Let $\tilde{a} = (0.2, 0.5, 0.8)$ and $\tilde{b} = (0.3,$ 0.4, 0.9). Please rank these two triangular fuzzy numbers.

We show some different comparison results according to the existing methods in the following Table 2.

However, when using our proposed method, the degree in which $\tilde{a}$ is bigger than $\tilde{b}$ is $\frac{7}{12}$ . This comparison result is reasonable and flexible.

(ii) In most of the existing works which investigated fully fuzzy linear programming problem, such as [3-7 , 31-33], the authors just only considered the FFLP problems with normal constraint, excluding FFLP problems with flexible constraint. However, in this paper, our proposed method is applicable to both of these two types of FFLP problems.

(iii) As shown in Theorem 3.1, FFLP 2 will be reduced to some linear programming problems. Hence, the optimal solution(s) of FFLP 2 may be not unique.

(iv) Due to the soft constraint of FFLP 2, the optimal solution of FFLP 2 is exactly the approximate optimal solution. Different values of n will lead to different solutions. Furthermore, when n is bigger, the approximate optimal solution is more closed to the accurate optimal solution.

7 Conclusion

Many of the existing literatures have employed the ranking function to solve FFLP problems with ordinary inequalities constraints. However, in this paper, we define a new order relation for comparing fuzzy numbers and two types of FFLP problems named FFLP1 and FFLP2. The constraints of FFLP1 are ordinary inequalities while those of FFLP2 are flexible inequalities. Three algorithms are developed to solve these problems. Algorithm 1 is developed for solving FFLP1 and Algorithm 2 and 3 for solving FFLP2. In Algorithm 1 and 2, FFLP1 and FFLP_α are converted into their equivalent crisp linear programming problems and then solved. Algorithm 2 is the foundation of Algorithm 3. In Algorithm 3, FFLP2 is transformed into FFLP_α, which may be solved by Algorithm 2. For the limited length of the paper, we just give a example to illustrate the algorithms for solving fully fuzzy linear programming with flexible constraints.

Acknowledgments

We would like to express our appreciation to the editor and the anonymous reviewers for their valuable comments, which have been very helpful in improving the paper. This work is supported by the PhD Start-up Fund of Natural Science Foundation of Guangdong Province, China (S2013040012506) and the China Postdoctoral Science Foundation funded project (2014M562152).

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