Fuzzy parametric iterative method for multi-objective linear fractional optimization problems

Abstract

This paper presents a fuzzy parameter based iterative method for solving multi-objective linear fractional optimization problems. Using the proposed method, number of ε-fuzzy efficient and fuzzy efficient solutions can be generated. The obtained fuzzy efficient solutions or ε-fuzzy efficient solutions have some addition features in terms of percentage contribution of each objective. The relative importance of each objective can be found in terms of satisfaction degree, value of membership function at the obtained solution, for each objective. Some theoretical results are also established for the validation of the proposed method and the method is coded in matlab (version R2014b). The efficiency of the method is verified by applying on numerical problems.

Keywords

Multi-objective optimization linear fractional programming multi-criteria decision making fuzzy optimization iterative method

1 Introduction

Optimization of ratio of linear (affine) functions under the linear constrains called the linear fractional optimization problem. It is very difficult to solve such problems due the non-convexity of linear fractional objectives. The linear fractional optimization problem mathematically can be defined as $\begin{matrix} Optimize \frac{α^{T} x + β}{γ^{T} x + θ} \\ subject to \end{matrix}$ (1) $\begin{matrix} x \in S = {x \in ℝ : C_{j} (x) \leq 0, x \geq 0, γ^{T} x + θ > 0, \\ j = 1, 2, 3, . . ., m, α, γ \in ℝ^{n} β, θ \in ℝ} \end{matrix}$

The optimization of more than one ratios together under the linear constraints is known as the multi-objective linear fractional optimization problem. The research on MOLF optimization problem is increasing continuously from past four decades because it has wide variety of applications, such as financial sector, inventory management, production planning, banking sector and others. The MOLF optimization problem is used to measure the efficiency of any particular system such as optimization of debt/equity, profit/cost, inventory/sales, actual cost/standard cost, output/employees, nurses/patients ratios etc. Mathematically MOLF optimization problem can be written as: $\begin{matrix} Optimize [\frac{α_{1}^{T} x + β_{1}}{γ_{1}^{T} x + θ_{1}}, \frac{α_{2}^{T} x + β_{2}}{γ_{2}^{T} x + θ_{2}}, \dots, \frac{α_{k}^{T} x + β_{k}}{γ_{k}^{T} x + θ_{k}}] \\ subject to \end{matrix}$ (2) $\begin{matrix} x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \\ \forall i = 1, 2, \dots k} \end{matrix}$ Also, $α_{i}^{T} x + β_{i}$ , $γ_{i}^{T} x + θ_{i}$ are real valued linear (affine) functions which are defined on convex feasible space S. There are lots of computation difficulties in using the classical approaches for the solution of MOLF optimization problem. To overcome the computational difficulties of using conventional approaches to MOLF optimization problems, the theory of fuzzy sets has been introduced in the field of linear fractional optimization. Many authors proposed the techniques for the solution of MOLFP (see, [2 , 26–28]). Costa [14] proposed a computational technique to find a solution using the weighted sum technique and the feasible region is divided into two parts and one of them is discarded on the basis of maximum weighted sum. This process is repeated with the remaining region. The drawback of the this algorithm is that it used several iteration to get one efficient solution and the uncertainty parameters are not considered. Also, Costa and Alves [15] presented a reference point technique to compute non-dominated solution for multi-objective linear fractional programming problem. The feasible space is partitioned into two parts by taking the middle point and analyzed for the deletion of one of them. The algorithm is terminated when difference between the iterative non-dominated solution becomes so small or less than the predefined error. The fuzzy iterative method is attempted by Lee and Tcha [4] who proposed a procedure by taking single fuzzy parameter and obtained compromise solution by maximizing the aggregate of membership functions. This method can find only some selective solutions in the feasible region. Valipaur, Yaghoobi and Mashinchi [1] suggested an iterative parametric approach for solving multi-objective linear fractional programming problems which only uses linear programming to obtain efficient solution and always converges to an efficient solution. Weakly efficient conditions for multiobjective fractional optimization problem was considered by Singh [25]. Chen and Chen [18] discussed a fuzzy approach for multi-level multi-objective decision- making problem. They used required minimum tolerence for higher - level decision maker’s decision variable to secure feasibility in the solution space. Dubey and Mehra [8] presented a Pareto optimal solution for multi-objective fuzzy linear programming. Yang, Li and Lai [16] developed a parameterized bilinear programming methodology for solving bimatix games with pay offs of triangular intuitionistic fuzzy numbers. Very recently, Bhati and Singh [9] proposed a novel approach for MOLF programming problem using branch and bound method. In this paper, we presented a fuzzy parametric extension of the method which was developed by Valipaur, Yaghoobi and Mashinchi [1].

1.1 Fuzzy multi-objective linear fractional (MOLF) optimization problem

Zimmermann [11] (1978) introduced fuzzy sets into the multi-objective optimization field. He gave the way for the solution of multi-objective optimization problems that had been inaccessible to unsolvable with standard optimization techniques. If an uncertain aspiration level and uncertain tolerance level is introduced to each of the objectives $\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}}$ of MOLF optimization problem, then the fuzzy multi-objective linear fractional (FMOLF) optimization problem can be defined as $\begin{matrix} Find X (x_{1}, x_{2}, . . . x_{n}) \\ such that \\ \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} ≲ {\tilde{g}}_{i} or \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} ≳ {\tilde{g}}_{i} \forall (i = 1, 2, . . ., k) \\ subject to, \end{matrix}$ (3) $\begin{matrix} x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \\ \forall i = 1, 2, \dots k} \end{matrix}$ In the above problem (3), ${\tilde{g}}_{i}$ are assumed as fuzzy goals for each i^th objective function $\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}}$ and ${\tilde{t}}_{i}$ are the fuzzy tolerance limits. The membership function μ_i (x) can be constructed as given below:

If $\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} \leq {\tilde{g}}_{i}$ , then the membership function for minimization problem can be constructed as follows:

$\begin{matrix} μ (f_{i}) (x) \\ = {\begin{matrix} 1, & if \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} < {\tilde{t}}_{i} \\ \frac{{\tilde{t}}_{i} - \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}}}{{\tilde{t}}_{i} - {\tilde{g}}_{i}}, & if {\tilde{g}}_{i} \leq \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} \leq {\tilde{t}}_{i} \\ 0 & if \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} > {\tilde{g}}_{i} \end{matrix} \end{matrix}$ (4) If $\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} \geq {\tilde{g}}_{i}$ , then the membership function for the case of maximization

$\begin{matrix} μ (f_{i}) (x) \\ = {\begin{matrix} 1, & if \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} (x) > {\tilde{g}}_{i} \\ \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, & if {\tilde{t}}_{i} \leq \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} \leq {\tilde{g}}_{i} \\ 0 & if \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} < {\tilde{t}}_{i} \end{matrix} \end{matrix}$ (5) The fuzzy multi-objective linear fractional (FMOLF) optimization problem can be written as $\begin{matrix} max μ (x) \\ = max [(\frac{\frac{α_{1}^{T} x + β_{1}}{γ_{1}^{T} x + θ_{1}} - {\tilde{t}}_{1}}{{\tilde{g}}_{1} - {\tilde{t}}_{1}}), \dots, (\frac{\frac{α_{k}^{T} x + β_{k}}{γ_{k}^{T} x + θ_{k}} - {\tilde{t}}_{k}}{{\tilde{g}}_{k} - {\tilde{t}}_{k}})] \\ subject to \end{matrix}$ (6) $\begin{matrix} x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 . . . m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \\ \forall i = 1, 2, . . . k} \end{matrix}$

1.2 Fuzzy efficient solution

Definition 1. Fuzzy efficient solution [6]: A solution x^e ∈ S is called fuzzy efficient solution to the fuzzy multi-objective linear fractional programming (FMOP) problem, if and only if there is no other solution y ∈ S such that $\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \geq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \forall i = 1, 2, . . ., k$ and $\frac{\frac{α_{r}^{T} y + β_{r}}{γ_{r}^{T} y + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}} > \frac{\frac{α_{r}^{T} x + β_{r}}{γ_{r}^{T} x + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}}$ for at least one r.

Definition 2. ɛ-fuzzy efficient solution: for any ε_i > 0 ∈ R for i = 1,. . . , k and by assume ɛ = (ε₁, ε₂,. . . , ε_k). A point x^e ∈ S is called ɛ - fuzzy efficient solution to the fuzzy multi-objective linear fractional (FMOLF) optimization problem (6), if and only if there is no other solution y ∈ S such that $\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \geq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} + ε_{i} \forall i = 1, 2, . . ., k$ and $\frac{\frac{α_{r}^{T} y + β_{r}}{γ_{r}^{T} y + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}} > \frac{\frac{α_{r}^{T} x^{e} + β_{r}}{γ_{r}^{T} x^{e} + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}} + ε_{r}$ for at least one r.

It may be noted for the problem (6) that the fuzzy efficiency is used for the feasible space and the points lie in it are known as fuzzy efficient solution. The values of fuzzy objectives are called fuzzy non-dominated solution and ε-fuzzy non-dominated solution respectively. Suppose that the fuzzy non-dominated solution of problem (6) is defined by $X_{N} = [(\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}) \in X \subseteq [0, 1]^{i}, i = 1, . . ., k]$ such that x^e is a fuzzy efficient solution of problem (6).

Definition 3. The set X_N is externally stable if for each $(\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}) \in [0, 1]^{i} \ X_{N}, \forall i$ there exists $(\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}) \in X_{N}$ such that $\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ $\geq \frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \forall i = 1, . . ., k$ and $\frac{\frac{α_{r}^{T} x^{e} + β_{r}}{γ_{r}^{T} x^{e} + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}} > \frac{\frac{α_{r}^{T} y + β_{r}}{γ_{r}^{T} y + θ_{r}} - {\tilde{t}}_{r}}{{\tilde{g}}_{r} - {\tilde{t}}_{r}}$ for at least one r.

Theorem 1. If X is the image space of FMOLF optimization problem (6) with compact feasible region and continuous membership function objectives, then X_N is externally stable.

Proof. The proof is straightforward from result [1]. It is also true that any fuzzy efficient solution x^e to the FMOLF optimization problem (6), such that $\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \in [0, 1]$ and it is also efficient solution for MOLF optimization problem (2). This fact may be shown by the following theorem.

Theorem 2. Let x^e be a fuzzy efficient solution for FMOLF optimization problem(6) iff x^e is efficient solution for MOLF optimization problem (2).

Proof. Let x^e not be fuzzy efficient solution of (6), then there exists a feasible point x ∈ S such that for i = 1, 2, …, k . $\begin{matrix} \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} > \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \\ \Rightarrow \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i} > \frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i} \\ \Rightarrow \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} > \frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} \end{matrix}$ which contradicts the efficiency of x^e for MOLF optimization problem (2).

Converse: Suppose x^e is not efficient solution for MOLF optimization problem, then there exists another x ∈ S with aspiration levels $\tilde{g_{i}}$ and tolerance limit $\tilde{t_{i}}$ such that $\begin{matrix} \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} > \frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} = \tilde{g_{i}} (say) \\ \Rightarrow \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i} > \frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i} \\ \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} > \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \end{matrix}$ which is a contradiction that x^e is fuzzy efficient solution.

2 Fuzzy parametric approach

The solution of multi-objective optimization problem can be obtained by many methods and there is no unique solution for any particular MOP. Most of the methods give the compromise solution to the MOP and the solution may be ranked according to the value of objective function obtained in the given feasible region. But fuzzy approaches also give the satisfaction degree for each objective in terms of membership function of each objective. It gives additional information about the solution by saying that how much contribution of each objective is presented in the compromise solution.

For the solution of fuzzy multi-objective linear fractional optimization problem (6), auxiliary variables (parameters) λ_i ∀ i = 1, 2, …, k are introduced for each objective function in problem (6) and the well known Zimmerman’s [11] model can be as follows $\begin{matrix} max {λ_{1}, λ_{2}, \dots, λ_{k}} \\ subject to \\ λ_{i} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \forall i = 1, 2, \dots, k . \end{matrix}$ (7) $\begin{matrix} x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \\ i = 1, 2, \dots k} \\ and λ_{i} \in [0, 1] \forall i = 1, 2, \dots, k . \end{matrix}$

If we assume the auxiliary variable constraints in problem (7) as a function of x ∈ Rⁿ and λ ∈ [0, 1] ^k such that $G (x, λ) = \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i} \geq 0 .$ (8) Then the problem (7) is transformed into the following form $\begin{matrix} max G (x, λ) \\ = Max [(\frac{\frac{α_{1}^{T} x + β_{1}}{γ_{1}^{T} x + θ_{1}} - {\tilde{t}}_{1}}{{\tilde{g}}_{1} - {\tilde{t}}_{1}} - λ_{1}), \dots, \\ (\frac{\frac{α_{k}^{T} x + β_{k}}{γ_{k}^{T} x + θ_{k}} - {\tilde{t}}_{k}}{{\tilde{g}}_{k} - {\tilde{t}}_{k}} - λ_{k})] \\ subject to \end{matrix}$ (9) $\begin{matrix} λ_{i} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \\ x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \\ \forall i = 1, 2, \dots k} \\ and λ_{i} \in [0, 1] \forall i = 1, 2, \dots, k . \end{matrix}$ After simplification and using equal weights w_i = 1 each membership function of objectives in problem (9) $\begin{matrix} max η (x) \\ = max \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i}) - (\tilde{g_{i}} λ_{i} + (1 - λ_{i}) {\tilde{t}}_{i}) \\ (γ_{i}^{T} x + θ_{i})] \\ = max \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i}) - (\tilde{g_{i}} λ_{i} + (1 - λ_{i}) {\tilde{t}}_{i}) \\ (γ_{i}^{T} x + θ_{i})] \\ subject to \end{matrix}$ (10) $\begin{matrix} λ_{i} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \\ x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n} β_{i}, θ_{i} \in ℝ \forall i = 1, 2, \\ \dots, k} and λ_{i} \in [0, 1] \forall i = 1, 2, \dots, k . \end{matrix}$

Remark 1. In problem (10), λ ∈ [0, 1] ^k is a fixed vector, the constraints $λ_{i} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ for i = 1, 2,. . . k are linear because ${\tilde{t}}_{i}$ and $\tilde{g_{i}}$ are the numerical values.

Theorem 3. If the feasible space S of Problem (10) is nonempty, then it has a non-negative fuzzy efficient value.

Proof. Let x be feasible solution of problem (10), then $\begin{matrix} λ_{i} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, for i = 1, . . . k \\ \Rightarrow \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i} \geq 0, for i = 1, . . ., k \\ \Rightarrow \frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i} - ({\tilde{g}}_{i} - {\tilde{t}}_{i}) λ_{i} \geq 0, \\ for i = 1, . . ., k \\ \Rightarrow (α_{i}^{T} x + β_{i}) - [(\tilde{g_{i}} λ_{i} + (1 - λ_{i}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})] \\ \geq 0 for i = 1, . . ., k \end{matrix}$ For equal weights w_i = 1 for all i = 1,. . . , k, we have $\begin{matrix} [(α_{i}^{T} x + β_{i}) - (\tilde{g_{i}} λ_{i} + (1 - λ_{i}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})] \\ \geq 0 for i = 1, . . ., k \\ \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i}) - (\tilde{g_{i}} λ_{i} + (1 - λ_{i}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})] \\ \geq 0 for i = 1, . . ., k \end{matrix}$ Thus the optimal value is non-negative.

Theorem 4. Let x^e ∈ S and $λ^{e} = [(\frac{\frac{α_{1}^{T} x^{e} + β_{1}}{γ_{1}^{T} x^{e} + θ_{1}} - {\tilde{t}}_{1}}{{\tilde{g}}_{1} - {\tilde{t}}_{1}})$ , $(\frac{\frac{α_{2}^{T} x^{e} + β_{2}}{γ_{2}^{T} x^{e} + θ_{2}} - {\tilde{t}}_{2}}{{\tilde{g}}_{2} - {\tilde{t}}_{2}}), \dots, (\frac{\frac{α_{k}^{T} x^{e} + β_{k}}{γ_{k}^{T} x^{e} + θ_{k}} - {\tilde{t}}_{k}}{{\tilde{g}}_{k} - {\tilde{t}}_{k}})]$ Then x^e is a fuzzy efficient solution of FMOLF optimization problem (6) if and only if the optimal value of problem (10) is zero.

Proof. Let x^e be a fuzzy efficient solution of FMOLF optimization problem (6) and assume that the fuzzy efficient value of problem (10) is nonzero. Therefore, by theorem (3) there exists a feasible point y ∈ S such that $λ_{i}^{e} \leq \frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ , for i = 1, 2,. . . , k $\Rightarrow (\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i}^{e}) \geq 0$ , for i = 1, 2,. . . , k $\Rightarrow (\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i}^{e}) \geq 0$ , then there exists index j such that $(\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} - λ_{j}^{e}) > 0 \Rightarrow (\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} - λ_{j}^{e}) > 0 \Rightarrow \frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > λ_{j}^{e} = \frac{\frac{α_{j}^{T} x^{e} + β_{j}}{γ_{j}^{T} x^{e} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}}$ . Also, $λ_{i}^{e} = \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ for i = 1, 2,. . . , k . Hence, $\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \geq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ for i = 1, 2,. . . , k and there exists j such that $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > \frac{\frac{α_{j}^{T} x^{e} + β_{j}}{γ_{j}^{T} x^{e} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} .$ Which is contradiction with the fuzzy efficiency of x^e.

Conversely. Let a fuzzy efficient value of problem (10) is zero and suppose that x^e is not a fuzzy efficient solution of FMOLF optimization problem (6). Then, there exists another feasible point y ∈ S such that $\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \geq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} for i = 1, 2, . . ., k$ and $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > \frac{\frac{α_{j}^{T} x^{e} + β_{j}}{γ_{j}^{T} x^{e} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} for some j .$

Thus y is a feasible solution of problem (6). Further $\begin{matrix} \frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i}^{e} \geq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i}^{e} \\ for i = 1, 2, . . ., k . \\ \Rightarrow \sum_{i = 1}^{k} [(α_{i}^{T} y + β_{i}) - (\tilde{g_{i}} λ_{i}^{e} + (1 - λ_{i}^{e}) {\tilde{t}}_{i}) (γ_{i}^{T} y + θ_{i})] \\ \geq \sum_{i = 1}^{k} [(α_{i}^{T} x^{e} + β_{i}) - (\tilde{g_{i}} λ_{i}^{e} + (1 - λ_{i}^{e}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{e} + θ_{i})] \end{matrix}$ which contradicts the fuzzy efficiency of x^e.

3 Proposed method

In this section, the proposed method is outlined in a systematic way

Input: For the solution of FMOLF optimization Problem (6) ∀ i = 1,. . . , k and assume that acceptable error in the obtained solution is ε > 0, and make partition feasible space using the same technique as defined by Valipour, Yaghoobi and Mashinchi [1].

Step 1: Let x⁰ ∈ S be any feasible starting point and assume that $λ^{0} = (\frac{\frac{α_{1}^{T} x^{0} + β_{1}}{γ_{1}^{T} x^{0} + θ_{1}} - {\tilde{t}}_{1}}{{\tilde{g}}_{1} - {\tilde{t}}_{1}}, . . ., \frac{\frac{α_{k}^{T} x^{0} + β_{k}}{γ_{k}^{T} x^{0} + θ_{k}} - {\tilde{t}}_{k}}{{\tilde{g}}_{k} - {\tilde{t}}_{k}})$ and set p = 0 for x^p = x⁰. The feasibleregion can be defined as $S^{p} = S \cap (x \in R^{n} : λ_{i}^{p} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, i = 1, . . ., k)$ where pand S^p are the iteration stage and the feasible region in the p^th iteration, respectively.

If, any λ_i’s equal to 0 for the initial point, then discard that i^th constraints and solve the programming to find new λ_i’s.

Step 2: Find the fuzzy efficient solution x^p+1 for the problem (6) and η^p+1 as fuzzy efficient value for the problem (6). $\begin{matrix} max η^{p + 1} \\ = max \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i}) \\ - (\tilde{g_{i}} λ_{i}^{p} + (1 - λ_{i}^{p}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})] \\ subject to \end{matrix}$ (11) $\begin{matrix} λ_{i}^{p} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \\ x \in S = {x \in ℝ^{n} : C_{j} (x) \leq 0, x \geq 0, γ_{i}^{T} x + θ_{i} > 0, \\ j = 1, 2, 3 \dots m, α_{i}, γ_{i} \in ℝ^{n}, β_{i}, θ_{i} \in ℝ \\ \forall i = 1, 2, \dots k} \\ and λ_{i} \in [0, 1] \forall i = 1, 2, \dots, k . \end{matrix}$ By setting ξ^p+1 = η^p+1.

Step 3: If ξ^p+1 < ε, then go to Step 4, else, $p = p + 1, λ^{p} = (\frac{\frac{α_{1}^{T} x^{p} + β_{1}}{γ_{1}^{T} x^{p} + θ_{1}} - {\tilde{t}}_{1}}{{\tilde{g}}_{1} - {\tilde{t}}_{1}}, . . ., \frac{\frac{α_{k}^{T} x^{p} + β_{k}}{γ_{k}^{T} x^{p} + θ_{k}} - {\tilde{t}}_{k}}{{\tilde{g}}_{k} - {\tilde{t}}_{k}}), S^{p} = S \cap [x \in R^{p} : λ_{i}^{p} = \frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, i = 1, . . ., k]$ and return to step 2, end of it.

Step 4: If ξ^p+1 = 0, then stop the method and the solution obtained x^p+1 in a fuzzy efficient solution and stop the iteration.

else

if ξ^p+1 < ε, then x^p+1 is an ε-fuzzy efficient solution, where, ε = (ε₁,. . . , ε_k). Also, every x ∈ S^p is an ε- fuzzy efficient solution which approximates the fuzzy efficient solution. end of it.

Output: A fuzzy efficient or an approximate fuzzy efficient solution is obtained.

Theorem 5. The following statements hold for the proposed method

∅ ≠ S^p+1 ⊂ S^p for p = 1, 2, 3 . . .

S^p contains at least one fuzzy efficient solution ${\bar{x}}^{p},$ for every p = 1, 2, 3,. . .

$(λ_{i}^{p + 1} \leq \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})$ for i = 1,. . . , k, are nondecreasing convergent sequence of membership degrees.

{ξ^p} is a non-increasing sequence of nonnegative numbers.

Proof.

x^p+1 is the fuzzy efficient solution of problem (11) in step 2 of the method. Thus, $x^{p + 1} \in S^{p} \Rightarrow λ_{i}^{p + 1} \leq \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ for i = 1,. . . , k⇒ $(x \in S : λ_{i}^{p + 1} \leq \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, for i = 1, . . ., k)$ $\subset (x \in S : λ_{i}^{p} \leq \frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}, for i = 1, . . ., k)$ ⇒S^p+1 ⊂ S^p .

If x^p is a fuzzy efficient solution, then the statement is correct. Otherwise, $[\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}] \in X \ X_{N} .$ By Theorem 1, the non-dominated set of FMOLFP Problem (6) is externally stable. Therefore, there exists $[\frac{\frac{α_{i}^{T} {\bar{x}}^{p} + β_{i}}{γ_{i}^{T} {\bar{x}}^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}] \in X_{N}$ such that $[\frac{\frac{α_{i}^{T} {\bar{x}}^{p} + β_{i}}{γ_{i}^{T} {\bar{x}}^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}] \geq [\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}]$ for i = 1,. . . , k and $[\frac{\frac{α_{j}^{T} {\bar{x}}^{p} + β_{j}}{γ_{j}^{T} {\bar{x}}^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}}] > [\frac{\frac{α_{j}^{T} x^{p} + β_{j}}{γ_{j}^{T} x^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}}]$ for at least one j. Hence, ${\bar{x}}^{p} \in S^{p}$ . Since ${\bar{x}}^{p}$ is a fuzzy efficient solution, the assertion follows.

$\forall p, x^{p + 1} \in S^{p} \Rightarrow [\frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}] \geq [\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}]$ for i = 1,. . . , k. It implies that $[λ_{i}^{p} = (\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})]$ are non-decreasing sequence for i = 1,. . . , k.

Since the membership function $\frac{\frac{α_{j}^{T} x^{p} + β_{j}}{γ_{j}^{T} x^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}}$ is a bounded set, $(λ_{i}^{p})$ are bounded sequences for i = 1,. . . , k. Hence, sequences ${λ_{i}^{p}}$ are convergent for i = 1,. . . , k.

∀ p, η^p is a fuzzy efficient value of problem (11). Since problems (11) and (10) are the same, then by theorem 2, η^p and ξ^p are nonnegative. For the completion of proof, we will show that $ξ^{p + 1} - ξ^{p} \leq 0 .$

$\begin{matrix} ξ^{p + 1} - ξ^{p} = \sum_{i = 1}^{k} [(α_{i}^{T} x^{p + 1} + β_{i}) - (\tilde{g_{i}} λ_{i}^{p} + \\ (1 - λ_{i}^{p}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p + 1} + θ_{i})] - \sum_{i = 1}^{k} [(α_{i}^{T} x^{p} + β_{i}) \\ - (\tilde{g_{i}} λ_{i}^{p - 1} + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p} + θ_{i})] \leq \sum_{i = 1}^{k} \\ [(α_{i}^{T} x^{p + 1} + β_{i}) - (α_{i}^{T} x^{p} + β_{i})) - (\tilde{g_{i}} λ_{i}^{p - 1} + \\ (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p + 1} + θ_{i}) + (\tilde{g_{i}} λ_{i}^{p - 1} + \\ (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p} + θ_{i})] \end{matrix}$ (12) In addition, since x^p+1 ∈ S^p ⊂ S^p-1 and x^p is the fuzzy efficient solution of problem (11) in iteration p - 1,

$\begin{matrix} \sum_{i = 1}^{k} [(α_{i}^{T} x^{p + 1} + β_{i}) - (\tilde{g_{i}} λ_{i}^{p - 1} + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) \\ (γ_{i}^{T} x^{p + 1} + θ_{i})] \leq \sum_{i = 1}^{k} [(α_{i}^{T} x^{p} + β_{i}) - (\tilde{g_{i}} λ_{i}^{p - 1} \\ + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p} + θ_{i})] \\ \Rightarrow \sum_{i = 1}^{k} [((α_{i}^{T} x^{p + 1} + β_{i}) - (α_{i}^{T} x^{p} + β_{i})) \\ - (\tilde{g_{i}} λ_{i}^{p - 1} + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p + 1} + θ_{i}) \\ + (\tilde{g_{i}} λ_{i}^{p - 1} + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p} + θ_{i})] \leq 0 \end{matrix}$ (13) Now, (12) and (13) imply that ξ^p+1 ≤ ξ^p . Therefor, {ξ^p} is a non-increasing sequences of nonnegative numbers.

Theorem 6. The sequence {ξ^p} converge to zero.

Proof. $\begin{matrix} lim_{p \to \infty} ξ^{p} = lim_{p \to \infty} \sum_{i = 1}^{k} [(α_{i}^{T} x^{p} + β_{i}) \\ - (\tilde{g_{i}} λ_{i}^{p - 1} + (1 - λ_{i}^{p - 1}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{p} + θ_{i})] \\ lim_{p \to \infty} \sum_{i = 1}^{k} [(\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - λ_{i}^{p - 1}) (\frac{γ_{i}^{T} x^{p} + θ_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})] \\ = lim_{p \to \infty} \sum_{i = 1}^{k} [(λ_{i}^{p} - λ_{i}^{p - 1}) (\frac{γ_{i}^{T} x^{p} + θ_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})] \end{matrix}$

By part of Theorem 5, the sequence $(λ_{i}^{p}$ is convergent. Also $(\frac{γ_{i}^{T} x^{p} + θ_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})$ is a bounded sequence, since $γ_{i}^{T} x + θ_{i}$ is bounded on S. Therefore, $lim_{p \to \infty} [(λ_{i}^{p} - λ_{i}^{p - 1})$ $(\frac{γ_{i}^{T} x^{p} + θ_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})] = 0$ for i = 1,. . . , k. Hence, $lim_{p \to \infty} ξ^{p} = lim_{p \to \infty} \sum_{i = 1}^{p} [(λ_{i}^{p} - λ_{i}^{p - 1}) (\frac{γ_{i}^{T} x^{p} + θ_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}})] = 0$ .

Corollary 1. The proposed method is stopped within the finite steps.

Proof. At the stage of step 3 of proposed method, the stoping criteria is ξ^p+1 = η^p+1 < ε. By Theorem 6, the sequence {ξ^p+1} is convergent to zero and the proof is complete.

Theorem 7. In the proposed method, suppose that ξ^p+1 < ε. Then, every x ∈ S^p is an ɛ- fuzzy efficient solution of FMOLF optimization problem (6) where ɛ = (ε,. . . , ε).

Proof. Let ξ^p+1 < ɛ and assume that x ∈ S^p is not a ɛ- fuzzy efficient solution of FMOLF optimization problem (6). So, there exists y ∈ S such that $\frac{\frac{α_{i}^{T} y + β_{i}}{γ_{i}^{T} y + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \geq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} + ε$ for i = 1,. . . , k and $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > \frac{\frac{α_{j}^{T} x + β_{j}}{γ_{j}^{T} x + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} + ɛ$ for at least one j so that y ∈ S^p.

Therefore, $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > \frac{\frac{α_{j}^{T} x + β_{j}}{γ_{j}^{T} x + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} + ɛ \geq \frac{\frac{α_{j}^{T} x^{p} + β_{j}}{γ_{j}^{T} x^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} + ɛ$ for at least one j which implies that $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} > \frac{\frac{α_{j}^{T} x^{p} + β_{j}}{γ_{j}^{T} x^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} + ɛ = λ_{j}^{p} + ɛ$ .

Thus, $\frac{\frac{α_{j}^{T} y + β_{j}}{γ_{j}^{T} y + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} - λ_{j}^{p} > ɛ$ $\Rightarrow [(α_{j}^{T} y + β_{j}) - (\tilde{g_{j}} λ_{j}^{p} + (1 - λ_{j}^{p}) {\tilde{t}}_{j}) (γ_{j}^{T} y + θ_{j})]$ $> ɛ ({\tilde{g}}_{j} - {\tilde{t}}_{j}) (γ_{j}^{T} y + θ_{j})$ Since, $(γ_{j}^{T} y + θ_{j}) \geq 1$ and $({\tilde{g}}_{j} - {\tilde{t}}_{j}) \geq 1$ ,therefore, $[(α_{j}^{T} y + β_{j}) - (\tilde{g_{j}} λ_{j}^{p} + (1 - λ_{j}^{p}) {\tilde{t}}_{j}) (γ_{j}^{T} y + θ_{j})] > ɛ$ Since, by step 1-3 of the proposed method,then $ξ^{p + 1} = \sum_{i = 1}^{k} [(α_{i}^{T} y + β_{i}) - (\tilde{g_{i}} λ_{i}^{p} + (1 - λ_{i}^{p}) {\tilde{t}}_{i})$ $(γ_{i}^{T} y + θ_{i})] \geq [(α_{j}^{T} y + β_{j}) - (\tilde{g_{j}} λ_{j}^{p} + (1 - λ_{j}^{p}) {\tilde{t}}_{j})$ $(γ_{j}^{T} y + θ_{j})] > ε$

which is a contradiction that ξ^p+1 < ε . Hence, the proof is complete.

Theorem 8. In the proposed method, suppose that ξ^p+1 < ε . Then, there exist a fuzzy efficient solution x^e such that ${∥ \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} ∥}_{\infty} < ɛ .$ Also, all points of S^p are approximations to efficient solutions.

Proof. ξ^p+1 = η^p+1, for i = 1,. 2,. . , k .

Further, $η^{p + 1} = max_{x \in S^{p}} \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i}) - (\tilde{g_{i}} λ_{i}^{p}$

$+ (1 - λ_{i}^{p}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})] = \sum_{i = 1}^{k} [(α_{i}^{T} x + β_{i})$

$- (\tilde{g_{i}} λ_{i}^{p} + (1 - λ_{i}^{p}) {\tilde{t}}_{i}) (γ_{i}^{T} x + θ_{i})]$ where, $S^{p} = S \cap [x \in R^{p} : λ_{i}^{p} = \frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \leq \frac{\frac{α_{i}^{T} x + β_{i}}{γ_{i}^{T} x + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}},$

i = 1,. . . , k]. If x^p+1 is a fuzzy efficient solution, let x^e = x^p+1 then ${∥ \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} ∥}_{\infty}$

<ɛ . Otherwise, μ (x^p+1) ∈ X ∖ X_N . Now, Theorem 1 implies that the non-dominated set of FMOLF optimization problem (6) is externally stable. Therefore, there exists $\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \in X_{N}$ such that $\frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \leq \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} \leq \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}}$ for i = 1,. . . k and $\frac{\frac{α_{j}^{T} x^{p} + β_{j}}{γ_{j}^{T} x^{p} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} < \frac{\frac{α_{j}^{T} x^{p + 1} + β_{j}}{γ_{j}^{T} x^{p + 1} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}} < \frac{\frac{α_{j}^{T} x^{e} + β_{j}}{γ_{j}^{T} x^{e} + θ_{j}} - {\tilde{t}}_{j}}{{\tilde{g}}_{j} - {\tilde{t}}_{j}}$ for some j. So, x^e is a fuzzy efficient and it belongs to S^p. Therefore, $[(α_{i}^{T} x^{e} + β_{i}) - (\tilde{g_{i}} λ_{i}^{p} + (1 - λ_{i}^{p}) {\tilde{t}}_{i}) (γ_{i}^{T} x^{e} + θ_{i})] < η^{p + 1} < ɛ,$ for i = 1,. . . , k.

$\Rightarrow \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} < \frac{ɛ}{({\tilde{g}}_{i} - {\tilde{t}}_{i}) (γ_{i}^{T} \hat{x} + θ_{i})}$ for i = 1,. . . , k

As we know that $γ_{i}^{T} x^{e} + θ_{i} \geq 1, ({\tilde{g}}_{i} - {\tilde{t}}_{i}) \geq 1$ for i = 1, 2, 3,. . . , k, therefore

$\frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p} + β_{i}}{γ_{i}^{T} x^{p} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} < ɛ$

$\Rightarrow \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} < ɛ$

$\Rightarrow {∥ \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} ∥}_{\infty} < ɛ .$ for i = 1,. . . , k . This shows that for each feasible point x ∈ S^p, there always exist a fuzzy efficient solution x^e such that ${∥ \frac{\frac{α_{i}^{T} x^{e} + β_{i}}{γ_{i}^{T} x^{e} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} - \frac{\frac{α_{i}^{T} x^{p + 1} + β_{i}}{γ_{i}^{T} x^{p + 1} + θ_{i}} - {\tilde{t}}_{i}}{{\tilde{g}}_{i} - {\tilde{t}}_{i}} ∥}_{\infty}$ <ɛ .

Thus, we can say that every feasible point in S^p is an approximation to a fuzzy efficient solution.

4 Computational experiment

For the efficiency measurement of the developed method and for the applicability of the fuzziefied method, a numerical problem from Valipour et al. [1] is taken for the analysis.

Experiment 1. Consider the following multi-objective linear fractional(MOLF) optimization problem [Valipour et al. [1]] $\begin{matrix} max {Z_{1} = \frac{x_{1} - 4}{x_{2} + 3}, Z_{2} = \frac{- x_{1} + 4}{x_{2} + 1}, \\ Z_{3} = - x_{1} + x_{2}} \\ subject to \end{matrix}$ (14) $\begin{matrix} - x_{1} + 4 x_{2} \leq 0, \\ x_{1} - \frac{1}{2} x_{2} \leq 4, \\ x_{1}, x_{2} \geq 0 . \end{matrix}$ The approximate fuzzy efficient solutions for this problem are summarized in Table 1. Figure 1 described the possible fuzzy efficient space and Fig. 2 described the satisfaction degrees laid on the common surfaces of all three surfaces are the compromise fuzzy efficient.

Experiment 2. Consider an inventory problem model where we are simultaneously interested in maximizing the profit ratio to the holding cost and minimizing the back orders ratio to the total ordered quantities. For simplicity, we assume that there is only one period in the cycle time. Finally, we consider two constraints in our modeling formulation: warehouse space and budget. Therefore, the following multi-objective fractional problem will be formulated $\begin{matrix} max F_{1} = \frac{\sum_{i = 1}^{n} (S_{i} - P_{i}) Q_{i}}{\sum_{i = 1}^{n} \frac{h_{i} Q_{i}}{2}} \\ min F_{2} = \frac{\sum_{i = 1}^{n} (D_{i} - Q_{i})}{\sum_{i = 1}^{n} Q_{i}} \\ subject to, \end{matrix}$ (15) $\begin{matrix} \sum_{i = 1}^{n} P_{i} Q_{i} \leq B, \\ \sum_{i = 1}^{n} f_{i} Q_{i} \leq F, \end{matrix}$ where:

Q_i: Ordering quantity of product i.

P_i: Purchasing price of product i.

S_i: Selling price of product i.

h_i: Inventory holding cost of product i.

D_i: Demand for product i per unit time.

B: Maximum available budget.

f_i: Space required for product i.

F: Maximum available space.

Now the following numerical values are considered for the implementation of proposed application.

Data for inventory Problem:

Item h_i P_i S_i D_i f_i B F

i= 1 20 600 650 8000 1 400000 240

i=2 24 720 750 4000 2 400000 240

Based on the above numerical values, the MOLFP inventory problem with two objective functions is summarized as follows $\begin{matrix} Max F_{1} = \frac{50 Q_{1} + 30 Q_{2}}{10 Q_{1} + 12 Q_{2}} \\ Min F_{2} = \frac{8000 - Q_{1} + 4000 - Q_{2}}{Q_{1} + Q_{2}} \\ subject to, \end{matrix}$ (16) $\begin{matrix} 600 Q_{1} + 720 Q_{2} \leq 400000, \\ Q_{1} + 2 Q_{2} \leq 240, \\ Q_{1}, Q_{2} \geq 0 . \end{matrix}$ For simplification, we may assume that Q_i = x_i, for all i, then the problem becomes $\begin{matrix} Max F_{1} (x) = \frac{50 x_{1} + 30 x_{2}}{10 x_{1} + 12 x_{2}} \\ Max F_{2} (x) = \frac{x_{1} + x_{2} - 12000}{x_{1} + x_{2}} \\ subject to, \end{matrix}$ (17) $\begin{matrix} 600 x_{1} + 720 x_{2} \leq 400000, \\ x_{1} + 2 x_{2} \leq 240, \\ x_{1}, x_{2} \geq 0 . \end{matrix}$ The fuzzy efficient solution for experiment 2 are described in Table 2 and Figs. 3 and 4 described the possible fuzzy efficient space.

5 Comparison

In this section, we are giving some comparisons with the already reported work of Costa [14], Costa and Alves [15], Lee and Tacha [4] and Valipaur, Yaghoobi and Mashinchi [1]. Following are the strength of the proposed method

The methods of costa [14] and costa and Alves [15] are in used several programs to find one efficient solution and take the weight initially for showing the relative importance of the objective functions but in our proposed method the importance of the objective function is obtained after getting the solution point in the feasible region.

The computational cost of the method of Costa [14] and Costa and Alves [15] is having more computational cost because, the number of constraints are increased in each step but in our proposed method the number of constraints are fixed initially i.e. (m+k).

In the method of Costa [14] and Costa and Alves [15], at each iteration, feasible space is partitioned into two sub feasible space and stooping criterion is the minimum distance between the best value and the worst value of the objective function in the non-partitioned feasible space or less than the admissible tolerance. In this respect, the fuzzy efficient solution x^p obtained in the p^th iteration, all the fuzzy efficient solutions having the satisfaction degree less than x^p are ignored. Hence, the number of iterations are reduced.

Valipaur, Yaghoobi and Mashinchi [1] presented the algorithm by taking the weights initially for giving the relative importance of the objective functions and also the concept of fuzzy is not considered. But we are finding the relative importance in terms of the satisfaction degree of each objective at the obtained points and we find the fuzzy efficient and ε-fuzzy efficient solution.

Lee and Tacha [4] proposed and fuzzy iterative technique using fuzzy parameter and find the solution in the feasible space. Sometimes the solution does not exist in this case. In our algorithm if the solution does not exist we can discard the infeasible constraint and obtain the fuzzy efficient solution.

The proposed method is presented with the fuzzy concept and all theoretical results are established. Hence the proposed method is more informative and innovative as application of fuzzy set theory.

We compared the solution of proposed method with classical approaches (Valipaur, Yaghoobi and Mashinchi [1] (2014) and Costa [14] (2007)) for optimal value at (0.000, 0.000) in Table 3.

6 Conclusion

This paper presents the fuzzified version of the algorithm proposed by Valipaur, Yaghoobi and Mashinchi [1] with some additional features. This method can provide an additional information about the obtained fuzzy efficient solution. Using this method, solver can give the information to the decision maker that how much an objective function is satisfied with the obtained fuzzy efficient solution. Also, the proposed method does not need any kind of relative importance in terms of weight to the objective functions. Thus, we can take equal weights to make single objective optimization problem. The relative importance of the objective can be obtained in terms of membership satisfaction degree.

Footnotes

Acknowledgments

Authors would like to thank anonymous reviewer for their valuable suggestions and comments to improve the manuscript.

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