A parametric approach to fuzzy multi-objective linear fractional program: An alpha cut based method

Abstract

In the multi-objective programming problem (MOPP), finding an efficient solution is challenging and partially encompasses some difficulties in practice. This paper presents an approach to address the multi-objective linear fractional programing problem with fuzzy coefficients (FMOLFPP). In the method, at first, the concept of α - cuts is used to change the fuzzy numbers into intervals. Therefore, the fuzzy problem is further changed into an interval-valued linear fractional programming problem (IVLFPP). Afterward, this problem is transformed into a linear programming problem (LPP) using a parametric approach and the weighted sum method. It is proven that the solution resulted from the LPP is at least a weakly ɛ - efficient solution. Two examples are given to illustrate the method.

Keywords

Efficient solution fuzzy numbers fuzzy programming interval arithmetic weighted sum approach

1 Introduction

Linear fractional programming problem (LFPP) has played a significant role in optimization. In [32], applications of the LFPP in economy, business, engineering, and management were demonstrated. Apart from that the LFPP was more recently utilized as an appropriate model in transportation, water consumption, medicine, and industry [1 , 37]. Charnes and Cooper [12] proposed the best ever method dealing with the LFPP. In the method, the fractional problem is transformed into the LPP using variable transformation technique. [19] showed that the fractional programming problem (FPP) can be replaced by a series of non-fractional problems. Based on this principle, many approaches have been developed [8 , 24]. Borza and Rambely [4] designed a non-iterative method to obtain the global optimal solution of the sum of linear fractional programming problem (S-LFPP) by the use of variable transformation. [22] constructed an iterative algorithm for the large-scale S-LFPP using a branch and bound technique.

In different disciplines of optimization such as engineering, business, and management, the notion of fuzzy sets has been used to design approaches [5 , 40]. Specifically, one can use fuzzy numbers when there exists an ambiguity to specify coefficients. In the LFPP, we deal with the fuzzy linear fractional programming problem (FLFPP) if the coefficients are fuzzy numbers. One way of addressing the FLFPP is to use fuzzy ranking approaches. In this manner, a fuzzy number is changed into the fixed number(s). Therefore, multiple LFPPs may be considered instead of the main fuzzy problem [3]. Although these kinds of approaches are easy and straightforward, representing a fuzzy number with fixed numbers may not be as comprehensive as we expected generally. On the other hand, using the concept of α - cut has been considered by many researchers as an efficient and comprehensive approach dealing with fuzzy numbers [16 , 38].

In general, when the concept of α – cut is used, minimizing of the FLFPP is changed into a bi-objective linear linear fractional programming problem (BOLFPP) of the form $\underset{X \in S}{Minimize} {F^{L} (X), F^{U} (X)}$ . Mehra et al. [23] developed a method treating this bi-objective problem in which only F^L (X) is used. Ignoring F^U (X) can be considered as a drawback of their approach. In order to overcome this shortcoming, convex combinations of the solutions of problems $\underset{X \in S}{Minimize} F^{L} (X)$ and $\underset{X \in S}{Minimize} F^{U} (X)$ were suggested by Stanojević and Stanojević [33]. However, their method increases the computational expenses since there is no rule to recognize which combination gives the best result. The methodology of [23] was developed by Chinnadurai and Muthukumar [14] to address the LFPP with positive fuzzy coefficients and positive fuzzy decision variables [7]. Presented an approach to deal with the LFPP with interval coefficients. In their method, the original problem is transformed into a LPP using suitable variable transformations. Recently, Borza and Rambely [6] introduced a method to address LFPP with fuzzy coefficients based on the concept of α- cut, membership functions of the objectives, suitable variable transformation, and max-min technique. In their method, the fuzzy problem is finally replaced by a LPP.

In the literature, there are several methods to deal with the multi objective linear fractional programming problem (MOLFPP). These approaches can be also employed to tackle $\underset{X \in S}{Minimize} {F^{L} (X), F^{U} (X)}$ , where F^L (X) and F^U (X) are linear fractional functions and S is a polyhedral set. Chakraborty and Gupta [10] developed a method to address MOLFPP. In their method, the multi objective problem is transformed into a multi objective linear programming problem (MOLPP). Subsequently, the membership functions are specified after identifying the fuzzy aspiration levels of the linear objectives. Finally, the MOLPP is changed into a LPP using the max-min technique. Their method was designed such that it has not been possible to prove that the outcome is efficient, which is a drawback. Motivated by Chakraborty and Gupta’s methodology, Veeramani and Sumathi [35], and De and Deb [18] introduced approaches to deal with LFPP with fuzzy coefficients and MOLFPP, respectively. Pal et al. [26] transformed the MOLFPP into a LPP using a fuzzy goal programming approach in addition to suitable variable transformations. Toksari [34] introduced an approach to tackle the MOLFPP where the membership functions of the objectives are defined and then linearized by using the first-order Taylor series about the individual optimal solutions. For some examples, Borza et al. [9] reported that the results of using the first-order Taylor series proposed by Toksari are to some extent more accurate than the results of the fuzzy goal programming used by Pal et al. Nayak and Ojha [25] introduced a method dealing with the MOLFPP with fuzzy coefficients where the fuzzy problem is altered into an interval valued LFPP using the concept of α - cut. In their method, the fuzzy problem is reduced into the MOLFPP. Afterward, they reach a MOLPP employing the first-order Taylor series. Finally, the weighted sum technique is utilized to achieve a LPP. In general, there exists a drawback to the methods which use first-order Taylor expansion since this expansion reduces the accuracy.

In the MOLFPP, if the coefficients are fuzzy numbers, then we deal with fuzzy multi-objective linear fractional programming problem (FMOLFPP). To the best of our knowledge, the most efficient method to deal with FMOLFPP was proposed by Nayak and Ojha [25]. In their method, the α - cuts of the fuzzy numbers are used. Accordingly, the problem is changed into: $\begin{matrix} \underset{X \in S}{Minimize} {[F_{i}^{L} (X), F_{i}^{U} (X)] \\ = [\frac{C_{i}^{T} X + d_{i}}{P_{i}^{T} X + q_{i}}, \frac{E_{i}^{T} X + f_{i}}{G_{i}^{T} X + h_{i}}] i = 1, \dots, k} . \end{matrix}$

Let $X_{i}^{*}$ be the optimal solution of $\underset{X \in S}{Minimize} F_{i}^{L} (X)$ , and ${\overset{˘}{F}}_{i}^{L} (X)$ be the first-order Taylor expansion of $F_{i}^{L} (X)$ around $X_{i}^{*}, i = 1, \dots, k$ . In their method, the following problem is finally solved. $\begin{matrix} \underset{X \in S}{Minimize} \sum_{i = 1}^{k} w_{i} {\overset{˘}{F}}_{i}^{L} (X) . \end{matrix}$

In general, there are two drawbacks to the method: the first one is to utilize the first-order Taylor series, which decreases the accuracy automatically. The other one is to use only $F_{i}^{L} (X), i = 1, \dots, k$ . In other word, $F_{i}^{U} (X), i = 1, \dots, k$ does not play any role in finding the solution. This paper aims to provide a method to solve the MOLFPP with fuzzy coefficients in which the parametric approach of [19] is utilized instead of the first-order Taylor series. In addition, the weighted sum technique, the method of [12], and the approach of [23] are used to design our approach. In this method, the FMOLFPP is finally changed into the LPP. It is proven that the solution resulted from the LPP is at least a weakly ɛ - efficient solution for the main fuzzy problem.

The remainder of this paper is organized as follows. In Section 2, some basic notions and definitions are given. In section 3, the main results are given. In fact, in this section, we show how the FMOLFPP is changed into the LPP. In section 4, numerical examples are solved to illustrate the method, and comparisons are made to show the efficiency. Finally, section 5 concludes the paper.

2 Preliminaries

2.1 Fuzzy Numbers and intervals

Definition 1 ( α- cut). Suppose $\tilde{A}$ is a fuzzy set in X and α ∈ [0, 1]. The α- cut of the fuzzy set $\tilde{A}$ is the crisp set ${\tilde{A}}_{α}$ given by: ${\tilde{A}}_{α} = {x \in X : μ_{A} (x) ⩾ α}$ .

Definition 2 (Triangular fuzzy numbers). Assume $\tilde{A}$ is a fuzzy set. A triangular fuzzy number $\tilde{A}$ is defined as: $\begin{matrix} μ_{\tilde{A}} (x) = μ_{A} (x; a, b, c) \\ = {\begin{matrix} (x - a) / (b - a), & x \in [a, b) \\ (c - x) / (c - b), & x \in [b, c] \\ 0, & x > c or x < a . \end{matrix} \end{matrix}$

For the triangular fuzzy number $\tilde{A}$ , ${\tilde{A}}_{α} = [A_{α}^{L}, A_{α}^{U}] = [a + α (b - a), c - α (c - b)]$ .

In the literature, there exist several methods to rank fuzzy numbers. Chen and Hwang [13] reviewed 2o ranking methods. Refer to [30] for a more recent method to rank fuzzy numbers.

A fuzzy number can be represented uniquely by its α - cut. Therefore, in this paper, arithmetic operations and the ranking of fuzzy numbers are defined in terms of α - cut s of the fuzzy numbers.

According to Wu [39], the following proposition is true for given two arbitrary fuzzy numbers $\tilde{A} = [A_{α}^{L}, A_{α}^{U}]$ and $\tilde{B} = [B_{α}^{L}, B_{α}^{U}]$ .

Proposition 1. $\tilde{A} ⩾_{α} \tilde{B}$ if only if $A_{α}^{L} ⩾ B_{α}^{L}$ and $A_{α}^{U} ⩾ B_{α}^{U}, \forall α \in [0, 0.5]$ .

In the above, α ∈ [0, 0.5] means the α - cuts include the element with small membership degrees. To compensate for this shortcoming, referential fuzzy numbers are defined.

Definition 3 (Referential fuzzy number). Let $\tilde{A}$ be a triangular fuzzy number with the membership function $μ_{\tilde{A}} (x) = μ_{A} (x; a, b, c)$ , then its referential fuzzy number is a triangular fuzzy number denoted by ${\tilde{A}}^{*}$ and defined as follows: $\begin{matrix} μ_{{\tilde{A}}^{*}} (x) = μ_{{\tilde{A}}^{*}} (x; \frac{a + 2 b}{3}, b, \frac{2 b + c}{3}) \\ = {\begin{matrix} . 05 (x - a) / (b - a), & x \in [\frac{a + 2 b}{3}, b) \\ 0.5 (c - x) / (c - b), & x \in [b, \frac{2 b + c}{3}] \\ 0, & otherwise \end{matrix}, \end{matrix}$

In addition, ${\tilde{A}}_{α}^{*} = [a + 2 α (b - a), c + 2 α (b - c)], \forall α \in [\frac{1}{3}, \frac{1}{2}]$ .

It is not difficult to show ${\tilde{A}}_{α}^{*} \subset {\tilde{A}}_{α}, \forall α > 0$ . Therefore, ∃ intervals V₁ and V₂ such that:

${\tilde{A}}_{α}^{*} \cup V_{1} \cup V_{2} = {\tilde{A}}_{α}$ where $μ_{\tilde{A}} (x) < μ_{\tilde{A}} (y)$ , ∀x ∈ V₁ ∪ V₂, $\forall y \in {\tilde{A}}_{α}^{*}$ .

Definition 4 (Ranking of fuzzy numbers). Let ${\tilde{A}}^{*}$ and ${\tilde{B}}^{*}$ be referential fuzzy numbers related to triangular fuzzy numbers $\tilde{A}$ and $\tilde{B}$ , respectively. Consider ${\tilde{A}}_{α}^{*} = [A_{α}^{* L}, A_{α}^{* U}]$ and ${\tilde{B}}_{α}^{*} = [B_{α}^{* L}, B_{α}^{* U}], \forall α \in [\frac{1}{3}, \frac{1}{2}]$ . Fuzzy number $\tilde{B}$ is said to be smaller than $\tilde{A}$ and denoted by $\tilde{A} ⩾_{α} \tilde{B}$ if only if $A_{α}^{* L} ⩾ B_{α}^{* L}$ and $A_{α}^{* U} ⩾ B_{α}^{* U}, \forall α \in [\frac{1}{3}, \frac{1}{2}]$ .

Definition 5 (Arithmetic of intervals). Assume A = [A^L, A^U], B = [B^L, B^U] are intervals, and k ⩾ 0 is a scalar. Therefore, addition, multiplication, and division on the intervals are defined as follows: $\begin{matrix} \begin{matrix} A + B = [A^{L} + B^{L}, A^{U} + B^{U}], \\ - A = [- A^{U}, - A^{L}], kA = [{kA}^{L}, {kA}^{U}], \\ AB = [min {A^{L} B^{L}, A^{L} B^{U}, A^{U} B^{L}, A^{U} B^{U}}, \\ \max {A^{L} B^{L}, A^{L} B^{U}, A^{U} B^{L}, A^{U} B^{U}}], \\ \frac{A}{B} = [min {\frac{A^{L}}{B^{L}}, \frac{A^{L}}{B^{U}}, \frac{A^{U}}{B^{L}}, \frac{A^{U}}{B^{U}}}, \\ \max {\frac{A^{L}}{B^{L}}, \frac{A^{L}}{B^{U}}, \frac{A^{U}}{B^{L}}, \frac{A^{U}}{B^{U}}}] . \end{matrix} \end{matrix}$

In this paper, symbols ^′ ∼ ′ and ^′ - ′ denote the fuzziness and interval, respectively. In fact, $\tilde{a}$ means that a is a fuzzy number, and $\bar{A}$ indicates that A is an interval.

2.2 Linear fractional programming

Consider the general form of the linear fractional programming problem (LFPP) as follows:

$\begin{matrix} Minimize \frac{C^{T} X + μ}{D^{T} X + β} \\ subject to Ω = {AX ⩽ b, X ⩾ 0}, \end{matrix}$ (1) where $A \in ℝ^{m \times n}, b \in ℝ^{m \times 1}, C, D \in ℝ^{n \times 1}, α, β \in ℝ$ . In addition, D^TX + β > 0, ∀ X = (X₁, …, X_n) ∈ Ω. It is additionally assumed that feasible region Ω is a regular set i.e. a non-empty and compact set. Notify X ⩾ 0 means X_i ⩾ 0, for i = 1, …, n.

Using variable transformations $t = \frac{1}{D^{T} X + β}, Y = tX$ transforms (1) into:

$\begin{matrix} Minimize C^{T} Y + μ t \\ subject to φ = {AY - bt ⩽ 0, D^{T} Y + β t = 1, Y, t ⩾ 0}, \end{matrix}$ (2)

where φ is regular feasible set.

Lemma 1. In (2) , variable t cannot be zero.

Lemma 2. If $(\bar{Y}, \bar{t}) \in φ$ , then $\frac{\bar{Y}}{\bar{t}} \in Ω$ .

Theorem 1. If (Y^*, t^*) is optimum for (2), then $X^{*} = \frac{Y^{*}}{t^{*}}$ is optimum for (1) .

Charnes and Cooper [12] proved the above theorem and lemmas.

2.3 Multi-objective programming

Consider the general form of MOPP as follows:

$\underset{X \in S}{Minimize} {F_{1} (X), \dots, F_{k} (X)}$ (3) where S is a regular feasible region.

Definition 6. In (3), X^* ∈ S is an efficient solution if ∀X ∈ S, then ∃j∈ { 1, …, k } such that F_j (X^*) < F_j (X).

Definition 7. In (3), X^* ∈ S is a weakly efficient solution if ∀X∈ S, then ∃ j ∈ { 1, …, k } such that F_j (X^*) ⩽ F_j (X).

Definition 8. In (3), X^* ∈ S is an ɛ- efficient solution if ∀X∈ S, ∃ j ∈ { 1, …, k } such that F_j (X^*) - ɛ < F_j (X), where ɛ = inf {δ > 0 : F_j (X^*) - δ < F_j (X)}.

Definition 9. In (3), X^* ∈ S is an ɛ- weakly efficient solution if ∀X ∈ S, then ∃j∈ { 1, …, k } such that F_j (X^*) - ɛ ⩽ F_j (X), where ɛ = inf {δ > 0 : F_j (X^*) - δ ⩽ F_j (X)}.

Weighted sum approach is a classical method, which is used to change the MOPP into a single objective programming problem as follows:

$\underset{X \in S}{Minimize} \sum_{i = 1}^{k} w_{i} F_{i} (X),$ (4) where $w_{i} \in [0, 1], \sum_{i = 1}^{k} w_{i} = 1$ .

Theorem 2. The optimal solution X^* of (4) is an efficient solution for (3).

In (3), if k = 1, then the problem becomes a single objective programming problem.

Definition 10. Let X^* ∈ S is an ɛ- optimal solution for the single objective program $\underset{X \in S}{Minimize} F_{1} (X)$ if F₁ (X^*) - ɛ ⩽ F₁ (X) , ∀ X ∈ S, where ɛ = inf {δ > 0 : F₁ (X^*) - δ ⩽ F₁ (X) , ∀ X ∈ S}.

2.4 Parametric approach for solving MOFPP

In this part, the aim was to show how a MOFPP can be changed into a non-fractional single objective programming problem. For this reason, consider the following two problems.

$\underset{X \in S}{Minimize} \frac{f (X)}{g (X)}$ (5)

$\underset{X \in S}{Minimize} f (X) - θ g (X)$ (6) where f (X) and g (X) are continuous functions, and S is a non-empty compact feasible region, and θ is a parameter.

Theorem 3 ([19]). Let X^* be optimum for (5), then (5) and (6) are equivalent iff $θ = \frac{f (X^{*})}{g (X^{*})}$ .

Now, consider the MOFPP as follows:

$\underset{X \in S}{Minimize} {\frac{f_{i} (X)}{g_{i} (X)}, for i = 1, \dots, k} .$ (7)

Let us assume $\underset{X \in S}{Minimize}$ $\frac{f_{i} (X)}{g_{i} (X)} = θ_{i}^{*}$ , for i = 1, …, k.

According to theorem 3, the equivalent of (7) is:

$\underset{X \in S}{Minimize} {f_{i} (X) - θ_{i}^{*} g_{i} (X), for i = 1, \dots, k} .$ (8)

The weighted sum approach transforms (8) into:

$\underset{X \in S}{Minimize} \sum_{i = 1}^{k} w_{i} (f_{i} (X) - θ_{i}^{*} g_{i} (X)),$ (9) where $w_{i} \in [0, 1], \sum_{i = 1}^{k} w_{i} = 1$ .

Theorem 4. The optimal solution X^* of (9) is an efficient solution for (7).

Proof. Since the weighted sum approach is used to transform (8) into (9), then theorem 2 implies X^* is an efficient solution for (8). Since (8) and (7) are equivalent, then X^* is an efficient solution for (7). ■

2.5 A Method to solve LFPP with fuzzy coefficients

In this section, we briefly describe a method for solving LFPP with fuzzy parameters. Apart from that, the efficiency of the solution resulted is proved.

Consider the general form of LFPP with fuzzy parameters as follows:

$\begin{matrix} Minimize \tilde{F} (X) = \frac{\sum_{i = 1}^{n} {\tilde{c}}_{i} X_{i} + \tilde{γ}}{\sum_{i = 1}^{n} {\tilde{d}}_{i} X_{i} + \tilde{β}} \\ subject to \tilde{S} = {\begin{matrix} \sum_{j = 1}^{n} {\tilde{a}}_{ij} X_{j} ⩽ {\tilde{b}}_{i}, for i = 1, \dots, m \\ X_{j} ⩾ 0, for j = 1, \dots, n, \end{matrix} \end{matrix}$ (10) where $\sum_{i = 1}^{n} {\tilde{d}}_{i} X_{i} + \tilde{β} > 0$ , $\forall X \in \tilde{S .}$

After specifying an α level of satisfaction for the fuzzy coefficients, the fuzzy problem is changed into the following problem.

$\begin{matrix} Minimize \bar{F} (X) = \frac{\sum_{i = 1}^{n} [c_{i}^{L}, c_{i}^{U}] X_{i} + [γ^{L}, γ^{U}]}{\sum_{i = 1}^{n} [d_{i}^{L}, d_{i}^{U}] X_{i} + [β^{L}, β^{U}]} \\ subject to \bar{S} \\ = {\begin{matrix} \sum_{j = 1}^{n} [a_{ij}^{L}, a_{ij}^{U}] X_{j} ⩽ [b_{i}^{L}, b_{i}^{U}], for i = 1, \dots, m \\ X_{j} ⩾ 0, for j = 1, \dots, n . \end{matrix} \end{matrix}$ (11) where $\sum_{i = 1}^{n} [d_{i}^{L}, d_{i}^{U}] X_{i} + [β^{L}, β^{U}] > 0, \forall X \in \bar{S}$ .

Applying the interval arithmetic and ranking method of fuzzy numbers given by def. 4 changes (11) into:

$\begin{matrix} Minimize \bar{F} (X) = [F^{L} (X), F^{U} (X)] \\ subject to S = {\begin{matrix} \sum_{j = 1}^{n} a_{ij}^{L} X_{j} ⩽ b_{i}^{L}, for i = 1, \dots, m \\ \sum_{j = 1}^{n} a_{ij}^{U} X_{j} ⩽ b_{i}^{U}, for i = 1, \dots, m \\ X_{j} ⩾ 0, for j = 1, \dots, n, \end{matrix} \end{matrix}$ (12) where the following cases must be considered in order to define F^L (X) and F^U (X) such that: $\begin{matrix} \forall X \in S, F^{L} (X) ⩽ \bar{F} (X) ⩽ F^{U} (X) . \end{matrix}$

Case 1.

If $\underset{X \in S}{Minimize} \sum_{i = 1}^{n} c_{i}^{L} X_{i} + γ^{L} > 0$ , then $F^{L} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{L} X_{i} + γ^{L}}{\sum_{i = 1}^{n} d_{i}^{U} X_{i} + β^{U}}$ . Otherwise, $F^{L} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{L} X_{i} + γ^{L}}{\sum_{i = 1}^{n} d_{i}^{L} X_{i} + β^{L}}$ .

Case 2.

If $\underset{X \in S}{Maximize} \sum_{i = 1}^{n} c_{i}^{U} X_{i} + γ^{U} > 0$ , then $F^{U} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{U} X_{i} + γ^{U}}{\sum_{i = 1}^{n} d_{i}^{L} X_{i} + β^{L}}$ . Otherwise, $F^{U} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{U} X_{i} + γ^{U}}{\sum_{i = 1}^{n} d_{i}^{U} X_{i} + β^{U}}$ .

Without loss of generality, we assume $F^{L} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{L} X_{i} + γ^{L}}{\sum_{i = 1}^{n} d_{i}^{U} X_{i} + β^{U}}$ , and $F^{U} (X) = \frac{\sum_{i = 1}^{n} c_{i}^{U} X_{i} + γ^{U}}{\sum_{i = 1}^{n} d_{i}^{L} X_{i} + β^{L}}$ . Therefore, (12) turns into:

$\begin{matrix} \underset{X \in S}{Minimize} \bar{F} (X) = [F^{L} (X), F^{U} (X)] \\ = [\frac{\sum_{i = 1}^{n} c_{i}^{L} X_{i} + γ^{L}}{\sum_{i = 1}^{n} d_{i}^{U} X_{i} + β^{U}}, \frac{\sum_{i = 1}^{n} c_{i}^{U} X_{i} + γ^{U}}{\sum_{i = 1}^{n} d_{i}^{L} X_{i} + β^{L}}] . \end{matrix}$ (13)

According to [11], the minimization of interval [F^L (X) , F^U (X)] leads to the minimization of both objectives F^L (X) and F^U (X). Therefore, the above problem is considered as a bi-objective programming problem as:

$\underset{X \in S}{Minimize} {F^{L} (X), F^{U} (X)} .$ (14)

Since solving the above problem encompasses some difficulties in practice, to overcome such hardships [23], considered the following problem instead.

$\underset{X \in S}{Minimize} F^{L} (X) .$ (15)

Let’s assume (15) is solved and the solution obtained is X^*, then X^* is called an α- acceptable solution for (10). In addition, [F^L (X^*) , F^U (X^*)] is an α- acceptable interval for the value of the objective function.

Remark 1. In the method of [23], the bi-objective problem $\underset{X \in S}{Maximize} {F^{L} (X), F^{U} (X)}$ is reduced into: $\underset{X \in S}{Maximize} F^{U} (X)$ .

Theorem 5. Suppose X^* is optimum for (15), $\overset{˘}{X}$ is optimum for $\underset{X \in S}{Minimize} F^{U} (X)$ , $ɛ = F^{U} (X^{*}) - F^{U} (\overset{˘}{X})$ , then X^* is an ɛ- optimal solution for (10).

Proof. $\tilde{F} (X^{*}) - ɛ = \tilde{F} (X^{*}) - [ɛ, ɛ] = [F^{L} (X^{*}) - ɛ, F^{U} (X^{*}) - ɛ] ⩽ [F^{L} (X), F^{U} (X)] = \tilde{F} (X)$ , ∀X ∈ φ. This implies X^* is an ɛ - optimal solution for (10). ■

3 Proposed method to solve FMOLFPP

In this part, we introduce a method to address MOLFPP with fuzzy coefficients. In addition, the efficiency of the solution obtained is demonstrated.

Let us consider the general form of the FMOLFPP as follows:

$\begin{matrix} Minimize \\ {{\tilde{F}}_{i} (X) = \frac{\sum_{j = 1}^{n} {\tilde{c}}_{ij} X_{j} + {\tilde{γ}}_{i}}{\sum_{j = 1}^{n} {\tilde{d}}_{ij} X_{j} + {\tilde{β}}_{i}}, for i = 1, \dots, k} \\ subject to \tilde{S} = {\begin{matrix} \sum_{j = 1}^{n} {\tilde{a}}_{ij} X_{j} ⩽ {\tilde{b}}_{i}, for i = 1, \dots, m \\ X_{j} ⩾ 0, for j = 1, \dots, n, \end{matrix} \end{matrix}$ (16) where $\sum_{j = 1}^{n} {\tilde{d}}_{ij} X_{j} + {\tilde{β}}_{i} > 0, \forall X = (X_{1}, \dots, X_{n}) \in \tilde{S}, for i = 1, \dots, k$ .

Taking into account the concept of α - cuts alters (16) into:

$\begin{matrix} Minimize \\ {{\bar{F}}_{i} (X) = \frac{\sum_{j = 1}^{n} [c_{ij}^{L}, c_{ij}^{U}] X_{j} + [γ_{i}^{L}, γ_{i}^{U}]}{\sum_{j = 1}^{n} [d_{ij}^{L}, d_{ij}^{U}] X_{j} + [β_{i}^{L}, β_{i}^{U}]}, for i = 1, \dots, k} \\ subject to \bar{S} = {\begin{matrix} \sum_{j = 1}^{n} [a_{ij}^{L}, a_{ij}^{U}] X_{j} ⩽ [b_{i}^{L}, b_{i}^{U}], for i = 1, \dots, m, \\ X_{j} ⩾ 0, for j = 1, \dots, n, \end{matrix} \end{matrix}$ (17)

where $\sum_{j = 1}^{n} [d_{ij}^{L}, d_{ij}^{U}] X_{j} + [β_{i}^{L}, β_{i}^{U}] > 0, \forall X \in \bar{S}, for i = 1, \dots, k$ .

Considering interval arithmetic along with def. 4 changes (17) into:

$\begin{matrix} Minimize \\ {{\bar{F}}_{i} (X) = \frac{\sum_{j = 1}^{n} [c_{ij}^{L}, c_{ij}^{U}] X_{j} + [γ_{i}^{L}, γ_{i}^{U}]}{\sum_{j = 1}^{n} [d_{ij}^{L}, d_{ij}^{U}] X_{j} + [β_{i}^{L}, β_{i}^{U}]}, for i = 1, \dots, k} \\ subject to S = {\begin{matrix} \sum_{j = 1}^{n} a_{ij}^{L} X_{j} ⩽ b_{i}^{L}, for i = 1, \dots, m, \\ \sum_{j = 1}^{n} a_{ij}^{U} X_{j} ⩽ b_{i}^{U}, for i = 1, \dots, m \\ X_{j} ⩾ 0, for j = 1, \dots, n . \end{matrix}, \end{matrix}$ (18)

The (18) is transformed into:

$\begin{matrix} \underset{X \in S}{Minimize} \\ {{\bar{F}}_{i} (X) = [F_{i}^{L} (X), F_{i}^{U} (X)], for i = 1, \dots, k}, \end{matrix}$ (19) where,

Case 1.

If $\underset{X \in S}{Minimize} \sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L} > 0$ , then $F_{i}^{L} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L}}{\sum_{j = 1}^{n} d_{ij}^{U} X_{j} + β_{i}^{U}}$ . Otherwise, $F_{i}^{L} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L}}{\sum_{j = 1}^{n} d_{ij}^{L} X_{j} + β_{i}^{L}}$ .

Case 2.

If $\underset{X \in S}{Maximize} \sum_{j = 1}^{n} c_{ij}^{U} X_{j} + γ_{i}^{U} > 0$ , then $F_{i}^{U} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{U} X_{j} + γ_{i}^{U}}{\sum_{j = 1}^{n} d_{ij}^{L} X_{j} + β_{i}^{L}}$ . Otherwise, $F_{i}^{U} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{U} X_{j} + γ_{i}^{U}}{\sum_{j = 1}^{n} d_{ij}^{U} X_{j} + β_{i}^{U}}$ .

Keep in mind that the above cases must be followed in order to:

$F_{i}^{L} (X) ⩽ {\bar{F}}_{i} (X) ⩽ F_{i}^{U} (X)$ , ∀X ∈ S, for i = 1, …, k.

Without loss of generality, let us assume $F_{i}^{L} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L}}{\sum_{j = 1}^{n} d_{ij}^{U} X_{j} + β_{i}^{U}}$ , $F_{i}^{U} (X) = \frac{\sum_{j = 1}^{n} c_{ij}^{U} X_{j} + γ_{i}^{U}}{\sum_{j = 1}^{n} d_{ij}^{L} X_{j} + β_{i}^{L}}$ .

Thus, (19) turns into:

$\begin{matrix} \underset{X \in S}{Minimize} {{\bar{F}}_{i} (X) = [F_{i}^{L} (X), F_{i}^{U} (X)] \\ = [\frac{\sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L}}{\sum_{j = 1}^{n} d_{ij}^{U} X_{j} + β_{i}^{U}}, \frac{\sum_{j = 1}^{n} c_{ij}^{U} X_{j} + γ_{i}^{U}}{\sum_{j = 1}^{n} d_{ij}^{L} X_{j} + β_{i}^{L}}] \\ for i = 1, \dots, k} . \end{matrix}$ (20)

According to [23], the following problem is solved to find an α- acceptable optimal solution and consequently an α- acceptle interval for ${\bar{F}}_{i} (X)$ , for i = 1, …, k.

$\begin{matrix} \underset{X \in S}{Minimize} F_{i}^{L} (X) \\ = \frac{\sum_{j = 1}^{n} c_{ij}^{L} X_{j} + γ_{i}^{L}}{\sum_{j = 1}^{n} d_{ij}^{U} X_{j} + β_{i}^{U}}, for i = 1, \dots, k . \end{matrix}$ (21)

Let’s consider the above problem is solved by the method of [12] and X^*i be the optimum. Thus, $[F_{i}^{L} (X^{* i}), F_{i}^{U} (X^{* i})]$ is an α- acceptable interval for the value of ${\bar{F}}_{i} (X), i = 1, \dots, k$ .

Afterward, the parametric approach of [19] changes (18) into:

$\begin{matrix} \underset{X \in S}{Minimize} {(\sum_{j = 1}^{n} [c_{ij}^{L}, c_{ij}^{U}] X_{j} + [γ_{i}^{L}, γ_{i}^{U}]) \\ - [F_{i}^{L} (X^{* i}), F_{i}^{U} (X^{* i})] (\sum_{j = 1}^{n} [d_{ij}^{L}, d_{ij}^{U}] X_{j} + [β_{i}^{L}, β_{i}^{U}]), \\ for i = 1, \dots, k} . \end{matrix}$ (22)

Using the weighted sum method in addition to interval arithmetic transforms (22) is into:

$\begin{matrix} \underset{X \in S}{Minimize} [Z^{L} (X), Z^{U} (X)] \\ = [\sum_{i = 1}^{k} w_{i} (\sum_{j = 1}^{n} (c_{ij}^{L} X_{j} + p_{ij}^{L} X_{j}) + γ_{i}^{L} + p_{i}^{L}), \\ \sum_{i = 1}^{k} w_{i} (\sum_{j = 1}^{n} (c_{ij}^{U} X_{j} + p_{ij}^{U} X_{j}) + γ_{i}^{U} + p_{i}^{U})] . \end{matrix}$ (23) where, w_i ∈ [0, 1] i = 1, …, k is specified by the decision maker as the weight for the objective function ${\tilde{F}}_{i} (X)$ . Moreover,

$\begin{matrix} \begin{matrix} p_{ij}^{L} = \min {- F_{i}^{L} d_{ij}^{L}, - F_{i}^{L} d_{ij}^{U}, - F_{i}^{U} d_{ij}^{L}, - F_{i}^{U} d_{ij}^{U}} \\ p_{i}^{L} = \min {- F_{i}^{L} β_{i}^{L}, - F_{i}^{L} β_{i}^{U}, - F_{i}^{U} β_{i}^{L}, - F_{i}^{U} β_{i}^{U}}, \\ p_{ij}^{U} = \max {- F_{i}^{L} d_{ij}^{L}, - F_{i}^{L} d_{ij}^{U}, - F_{i}^{U} d_{ij}^{L}, - F_{i}^{U} d_{ij}^{U}}, \\ p_{i}^{U} = \max {- F_{i}^{L} β_{i}^{L}, - F_{i}^{L} β_{i}^{U}, - F_{i}^{U} β_{i}^{L}, - F_{i}^{U} β_{i}^{U}} . \end{matrix}, \end{matrix}$

cording to [11], the (23) is changed into:

$\underset{X \in S}{Minimize} {Z^{L} (X), Z^{U} (X)} .$ (24)

The weighted sum method changes (24) into:

$\underset{X \in S}{Minimize} (Z^{L} (X) + Z^{U} (X)) .$ (25)

Theorem 6. The optimal solution of (25) is a weakly ɛ - efficient solution for (16).

Proof.

Suppose $\overset{⌣}{X}$ is the optimal solution of (25). More-over, let’s suppose for i = 1, …, k, X^**i is optimum for $\underset{X \in S}{Minimize} F_{i}^{U} (X)$ , $F_{i}^{L} (\overset{˘}{X}) - F_{i}^{L} (X^{* i}) = ɛ_{i 1}$ , $F_{i}^{U} (\overset{˘}{X}) - F_{i}^{U} (X^{* * i}) = ɛ_{i 2}, ɛ_{i} = max {ɛ_{i 1}, ɛ_{i 2}}$ , and ɛ = min { ɛ_i, for i = 1, …, k } = ɛ_p. Therefore, ${\tilde{F}}_{p} (\overset{˘}{X}) - ɛ = {\tilde{F}}_{p} (\overset{˘}{X}) - [ɛ, ɛ] = [F_{p}^{L} (\overset{˘}{X}) - ɛ, F_{p}^{U} (\overset{˘}{X}) - ɛ] ⩽ [F_{p}^{L} (X), F_{p}^{U} (X)] = {\tilde{F}}_{p} (X), \forall X \in S$ . This proves $\overset{˘}{X}$ is at least a weakly ɛ - efficient solution for (16). ■

3.1 Algorithm to the proposed method

This algorithm summarizes the procedure of finding efficient solutions for the multi-objective linear fractional programming problem with fuzzy coefficients or with interval coefficients.

Initial Step. Define the referential fuzzy numbers, and then specify their α - cuts .

Step 1. Specify $α \in [\frac{1}{3}, \frac{1}{2}]$ , and w_i ∈ [0, 1], for i = 1, …, k.

Step 2. Formulate (18). Subsequently, identify $F_{i}^{L} (X)$ , $F_{i}^{U} (X)$ , for i = 1, … k.

Step 3 . Formulate (21). Next, compute $[F_{i}^{L} (X^{* i}), F_{i}^{U} (X^{* i})]$ as an α- acceptable interval of ${\bar{F}}_{i} (X)$ , for i = 1, …, k .

Step 4. Formulate (23). Afterward, formulate (25) and find $\overset{˘}{X}$ as the optimal solution. Consider $\overset{˘}{X}$ as a weakly ɛ- efficient solution for (16) and $[F_{i}^{L} (\overset{˘}{X}), F_{i}^{U} (\overset{˘}{X})]$ as an α- acceptable interval for the value of i^th objective function of (16).

Step 5. Calculate $ɛ = \min {Max {(F_{i}^{L} (\overset{˘}{X}) - F_{i}^{L} (X^{* i}), (F_{i}^{U} (\overset{˘}{X}) - F_{i}^{U} (X^{* * i})} for i = 1, \dots, k}$ as the criteria to evaluate the accuracy of the solution obtained.

Remark 2. For a maximization problem, the value of ɛ is calculated as follows: $\begin{matrix} ɛ = \min {Max {(F_{i}^{U} (X^{* i}) - F_{i}^{L} (\overset{˘}{X}), \\ (F_{i}^{U} (X^{* * i}) - F_{i}^{U} (\overset{˘}{X})} for i = 1, \dots, k} . \end{matrix}$

4 Numerical example

4.1 Example 1

In this section, a numerical example is given to illustrate the method. The example was taken from Nayak and Ojha [24], but the parameters were changed into triangular fuzzy numbers.

$\begin{matrix} Minimize \\ {{\tilde{F}}_{1} = \frac{- \tilde{1} X_{1} + \tilde{3} X_{2} + \tilde{2}}{\tilde{1} X_{1} + \tilde{2} X_{2} + \tilde{1}}, {\tilde{F}}_{2} = \frac{\tilde{5} X_{1} + \tilde{2} X_{2} + \tilde{2}}{\tilde{2} X_{1} + \tilde{3} X_{2} + \tilde{1}}} \\ subject to \tilde{S} = {\tilde{2} X_{1} + \tilde{1} X_{2} ⩽ \tilde{4}, \tilde{3} X_{1} - \tilde{2} X_{2} \\ ⩽ \tilde{5,} \tilde{1} X_{1} + \tilde{2} X_{2} ⩽ \tilde{3}, - \tilde{1} X_{1} - \tilde{3} X_{2} ⩽ - \tilde{2}, X_{1}, X_{2} ⩾ 0,} \end{matrix}$ (26)

where, $\begin{matrix} \begin{matrix} μ_{\tilde{1}} (X) = μ_{\tilde{1}} (X; 0.5, 1, 1.25) μ_{\tilde{2}} (X) = μ_{\tilde{2}} (X; 1, 2, 3) \\ μ_{\tilde{3}} (X) = μ_{\tilde{3}} (X; 2.5, 3, 4) μ_{\tilde{4}} (X) = μ_{\tilde{4}} (X; 3.5, 4, 4.5) \\ μ_{\tilde{5}} (X) = μ_{\tilde{5}} (X; 4, 5, 6) . \end{matrix} \end{matrix}$

Referential fuzzy numbers related to the above fuzzy numbers are defined as follows: $\begin{matrix} \begin{matrix} μ_{{\tilde{1}}^{*}} (X) = μ_{{\tilde{1}}^{*}} (X; \frac{2.5}{3}, 1, \frac{3.25}{3}) μ_{{\tilde{2}}^{*}} (X) = μ_{{\tilde{2}}^{*}} (X; \frac{5}{3}, 2, \frac{7}{3}) \\ μ_{{\tilde{3}}^{*}} (X) = μ_{{\tilde{3}}^{*}} (X; \frac{8.5}{3}, 3, \frac{10}{3}) μ_{{\tilde{4}}^{*}} (X) = μ_{{\tilde{4}}^{*}} (X; \frac{11.5}{3}, 4, \frac{12.5}{3}) \\ μ_{{\tilde{5}}^{*}} (X) = μ_{{\tilde{5}}^{*}} (X; \frac{14}{3}, 5, \frac{16}{3}) . \end{matrix} \end{matrix}$

The α–cuts of the corresponding referential fuzzy coefficients are given as below, $\forall α \in [\frac{1}{3}, \frac{1}{2}]$ .

$\begin{matrix} \begin{matrix} {[{\tilde{1}}^{*}]}_{α} = [α + 0.5, 1.25 - 0.5 α] {[{\tilde{2}}^{*}]}_{α} = [2 α + 1, 3 - 2 α] \\ {[{\tilde{3}}^{*}]}_{α} = [α + 2.5, 4 - 2 α] {[{\tilde{4}}^{*}]}_{α} = [α + 3.5, 4.5 - α] \\ {[{\tilde{5}}^{*}]}_{α} = [2 α + 4, 6 - 2 α] . \end{matrix} \end{matrix}$

Step 1. Set α = 0.4, w₁ = w₂ = 0.5.

Step 2 . Formulate (18) for this example as follows:

$\begin{matrix} Minimize {{\bar{F}}_{1} (X) = \frac{{\bar{F}}_{1}^{N} (X)}{{\bar{F}}_{1}^{D} (X)} = \frac{[- 1.05, - 0.9] X_{1} + [2.9, 3.2] X_{2} + [1.8, 2.2]}{[0.9, 1.05] X_{1} + [1.8, 2.2] X_{2} + [0.9, 1.05]}, \\ {\bar{F}}_{2} (X) = \frac{{\bar{F}}_{2}^{N} (X)}{{\bar{F}}_{2}^{D} (X)} = \frac{[4.8, 5.2] X_{1} + [1.8, 2.2] X_{2} + [1.8, 2.2]}{[1.8, 2.2] X_{1} + [2.9, 3.2] X_{2} + [0.9, 1.05]}} \\ subject to S = {1.8 X_{1} + 0.9 X_{2} ⩽ 3.9, 2.2 X_{1} + 1.05 X_{2} ⩽ 4.1, \\ 2.9 X_{1} - 2.2 X_{2} ⩽ 4.8, 3.2 X_{1} - 1.8 X_{2} ⩽ 5.2, \\ 0.9 X_{1} + 1.8 X_{2} ⩽ 2.9, 1.05 X_{1} + 2.2 X_{2} ⩽ 3.2, \\ - 1.05 X_{1} - 3.2 X_{2} ⩽ - 2.2, - 0.9 X_{1} - 2.9 X_{2} ⩽ - 1.8, X_{1}, X_{2} ⩾ 0 .} \end{matrix}$ (27)

According to (27), $\begin{matrix} \begin{matrix} F_{1}^{L} (X) = \frac{- 1.05 X_{1} + 2.9 X_{2} + 1.8}{1.05 X_{1} + 2.2 X_{2} + 1.05}, F_{1}^{U} (X) = \frac{- 0.9 X_{1} + 3.2 X_{2} + 2.2}{0.9 X_{1} + 1.8 X_{2} + 0.9} \\ F_{2}^{L} (X) = \frac{4.8 X_{1} + 1.8 X_{2} + 1.8}{2.2 X_{1} + 3.2 X_{2} + 1.05}, F_{2}^{U} (X) = \frac{5.2 X_{1} + 2.2 X_{2} + 2.2}{1.8 X_{1} + 2.9 X_{2} + 0.9} . \end{matrix} \end{matrix}$

Step 3. The (21) is formulated as follows:

$\underset{X \in S}{Minimize} F_{1}^{L} (X) = \frac{- 1.05 X_{1} + 2.9 X_{2} + 1.8}{1.05 X_{1} + 2.2 X_{2} + 1.05} .$ (28)

$\underset{X \in S}{Minimize} F_{2}^{L} (X) = \frac{4.8 X_{1} + 1.8 X_{2} + 1.8}{2.2 X_{1} + 3.2 X_{2} + 1.05} .$ (29)

The (28) is solved and the solution obtained is: X^*1 = (1.6983, 0.1303).

Thus, $[F_{1}^{L} (X^{* 1}), F_{1}^{U} (X^{* 1})] = [0.1265, 0.4087]$ .

The (29) is solved and the solution resulted is: X^*2 = (0, 1.4545).

Thus, $[F_{2}^{L} (X^{* 2}), F_{2}^{U} (X^{* 2})] = [0.7745, 1.0551]$ .

Step 4. The (23) is formulated for (27) as follows:

$\begin{matrix} \underset{X \in S}{Minimize} [Z^{L} (X), Z^{U} (X)] \\ = w_{1} ({\bar{F}}_{1}^{N} - [F_{1}^{L} (X^{* 1}), F_{1}^{U} (X^{* 1})] {\bar{F}}_{1}^{D}) \\ + w_{2} ({\bar{F}}_{2}^{N} - [F_{2}^{L} (X^{* 2}), F_{2}^{U} (X^{* 2})] {\bar{F}}_{2}^{D}) \\ = [0.5 X_{1} + 0.2123 X_{2} + 1.0315, 1.396 X_{1} \\ + 1.4631 X_{2} + 1.7946] . \end{matrix}$ (30)

One can normalize Z^L (X) and Z^U (X) if necessary and then proceed to the next step.

Form (25) for the above problem as follows:

$\begin{matrix} \underset{X \in S}{Minimize} (Z^{L} (X) + Z^{U} (X)) \\ = 1.8960 X_{1} + 1.6754 X_{2} + 2.8261 . \end{matrix}$ (31)

The above problem is solved and the solution obtained is: $\overset{˘}{X} = (0, 0.6875)$ .

Furthermore, the acceptable intervals for the objective functions ${\tilde{F}}_{1}, {\tilde{F}}_{2}$ of the fuzzy multi-objective problem are $[F_{1}^{L} (\overset{˘}{X}), F_{1}^{U} (\overset{˘}{X})] =$ $[1.4805 2.0585], [F_{2}^{L} (\overset{˘}{X}), F_{2}^{U} (\overset{˘}{X})] = [0.9346 1.2829]$ , respectively.

Step 5. The solutions X^**1 = (1.6983, 0.1303) and X^**2 = (0, 1.4554) are optimum for $\underset{X \in S}{Minimize} F_{1}^{U} (X)$ and $\underset{X \in S}{Minimize} F_{2}^{U} (X)$ , respectively. Thus, $ɛ = \min {max {F_{1}^{L} (\overset{˘}{X}) - F_{1}^{L} (X^{* 1}), F_{1}^{U} (\overset{˘}{X}) - F_{1}^{U} (X^{* * 1})}, \max {F_{2}^{L} (\overset{˘}{X}) - F_{2}^{L} (X^{* 2}), F_{2}^{U} (\overset{˘}{X}) - F_{2}^{U} (X^{* * 2})}} = 0.2341 .$

4.1.1 Numerical analysis

In Table 1, the extreme points of feasible region S and values of ${\tilde{F}}_{1} (X)$ and ${\tilde{F}}_{2} (X)$ at extreme points are listed. As we observe numerically, ∀ extreme point $\hat{X}$ :

Table 1
Extreme points and their objective function values for the fuzzy multi-objective problem for α = 0.4

Extreme point $\hat{X}$ ${\tilde{F}}_{1} (\hat{X}) = [F_{1}^{L} (\hat{X}), F_{1}^{U} (\hat{X})]$ ${\tilde{F}}_{2} (\hat{X}) = [F_{2}^{L} (\hat{X}), F_{2}^{U} (\hat{X})]$

(0, 1.4545) [1.4160, 1.9483] [0.7745, 1.0551]

(1.5144, 0.7318) [0.5487, 0.8879] [1.5448, 2.0328]

(1.7831, 0.1687) [0.1266, 0.4042] [1.9342, 2.5753]

(1.7541, 0.2295) [0.1836, 0.4688] [1.8841, 2.5040]

(1.6983, 0.1303) [0.1265, 0.4087] [1.9577, 2.6109]

$(0, 0.6875) = \bar{X}$ [1.4805, 2.0585] [0.9348, 1.2829]

ɛ = 0.2341 $[F_{1}^{L} (\overset{⌣}{X}) - ɛ, F_{1}^{U} (\overset{⌣}{X}) - ɛ] =$ $[F_{2}^{L} (\overset{⌣}{X}) - ɛ, F_{2}^{U} (\overset{⌣}{X}) - ɛ] =$

[1.2464, 1.8244] [0.7007, 1.0488]

Extreme point $\hat{X}$	${\tilde{F}}_{1} (\hat{X}) = [F_{1}^{L} (\hat{X}), F_{1}^{U} (\hat{X})]$	${\tilde{F}}_{2} (\hat{X}) = [F_{2}^{L} (\hat{X}), F_{2}^{U} (\hat{X})]$
(0, 1.4545)	[1.4160, 1.9483]	[0.7745, 1.0551]
(1.5144, 0.7318)	[0.5487, 0.8879]	[1.5448, 2.0328]
(1.7831, 0.1687)	[0.1266, 0.4042]	[1.9342, 2.5753]
(1.7541, 0.2295)	[0.1836, 0.4688]	[1.8841, 2.5040]
(1.6983, 0.1303)	[0.1265, 0.4087]	[1.9577, 2.6109]
$(0, 0.6875) = \bar{X}$	[1.4805, 2.0585]	[0.9348, 1.2829]
ɛ = 0.2341	$[F_{1}^{L} (\overset{⌣}{X}) - ɛ, F_{1}^{U} (\overset{⌣}{X}) - ɛ] =$	$[F_{2}^{L} (\overset{⌣}{X}) - ɛ, F_{2}^{U} (\overset{⌣}{X}) - ɛ] =$
[1.2464, 1.8244]	[0.7007, 1.0488]

$F_{2}^{L} (\overset{⌣}{X}) - ɛ < F_{2}^{L} (\overset{⌣}{X})$ and $F_{2}^{U} (\overset{⌣}{X}) - ɛ < F_{2}^{U} (\hat{X})$ . Therefore, the convexity of S in addition to pseudoconvexity of $F_{2}^{L} (X)$ , $F_{2}^{U} (X)$ results that:

$F_{2}^{L} (\overset{⌣}{X}) - ɛ < F_{2}^{L} (X)$ and $F_{2}^{U} (\overset{⌣}{X}) - ɛ < F_{2}^{U} (X), \forall X \in S .$ As a consequence, ${\tilde{F}}_{2} (\overset{⌣}{X}) - ɛ = [F_{2}^{L} (\overset{⌣}{X}) - ɛ, F_{2}^{U} (\overset{⌣}{X}) - ɛ] < [F_{2}^{L} (X), F_{2}^{U} (X)] = {\tilde{F}}_{2} (X), \forall X \in S$ . This indicates $\overset{˘}{X}$ is an ɛ- efficient solution for the fuzzy multi-objective problem.

Apart from the above analysis, according to the Figs. 1–4 and considering the average of the lower bounds and the average of the upper bounds of the intervals related to the objectives, it can be partially deduced that:

$\begin{matrix} 0.2712 ⩽ {\tilde{F}}_{1} (X) ⩽ 1.6758, 0.7838 ⩽ {\tilde{F}}_{2} (X) ⩽ 4.2090 . \end{matrix}$

Fig. 1

Objective $1 = F_{1}^{L}$ , Objective $2 = F_{2}^{L}$ .

Fig. 2

Objective $1 = F_{1}^{u}$ , Objective $2 = F_{2}^{U}$ .

Fig. 3

Objective $1 = F_{1}^{L}$ , Objective $2 = F_{2}^{U}$ .

Fig. 4

Objective $1 = F_{1}^{u}$ , Objective $2 = F_{2}^{L}$ .

This demonstrates that the proposed solution $\overset{˘}{X}$ is ɛ- efficient due to the fact that: $[F_{2}^{L} (\overset{˘}{X}) - ɛ, F_{2}^{U} (\overset{˘}{X}) - ɛ] = [0.7007 1.0488]$ .

4.2 Example 2

In this section, a production problem taken partially from [27] is considered, where the coefficients of the objective functions are set to be triangular fuzzy numbers.

$\begin{matrix} Maximize {{\tilde{F}}_{1} (X), {\tilde{F}}_{2} (X)} = \\ {\frac{\begin{matrix} (55000, 60000, 65000) X_{1} + (20000, 23000, 25000) X_{2} + (25000, 30000, 33000) X_{3} + \\ (56000, 59000, 61000) X_{4} + (38000, 40000, 42000) X_{5} + (57000, 60000, 64000) X_{6} \end{matrix}}{\begin{matrix} (32000, 35000, 37000) X_{1} + (10000, 13000, 15000) X_{2} + (16000, 18000, 20000) X_{3} + (36000, 39000, 42000) X_{4} + \\ (20000, 23000, 27000) X_{5} + (21000, 24000, 26000) X_{6} + (450000, 500000, 600000) \end{matrix}}, \\ \frac{\begin{matrix} (55000, 60000, 65000) X_{1} + (20000, 23000, 25000) X_{2} + (25000, 30000, 33000) X_{3} + \\ (56000, 59000, 61000) X_{4} + (38000, 40000, 42000) X_{5} + (57000, 60000, 64000) X_{6} \end{matrix}}{\begin{matrix} (90, 96, 100) X_{1} + (110, 120, 130) X_{2} + (140, 144, 150) X_{3} + (140, 144, 155) X_{4} + \\ (80, 84, 90) X_{5} + (110, 120, 130) X_{6} + (450, 480, 500) \end{matrix}}} \\ S . t S = {0.3 X_{1} + 0.4 X_{2} + 0.4 X_{3} + 0.98 X_{4} + 0.97 X_{5} + 0.98 X_{6} ⩽ 600, 2280000 X_{1} + 9200 X_{2} + 16000 X_{3} + 22500 X_{4} \\ + 20000 X_{5} + 20000 X_{6} ⩽ 20000000, 650 X_{1} + 630 X_{2} + 320 X_{3} + 660 X_{4} + 360 X_{5} + 640 X_{6} ⩽ 500000, 20 X_{1} + 22 X_{2} \\ + 20 X_{3} + 18 X_{4} + 20 X_{5} + 17 X_{6} ⩽ 15000, 11400 X_{1} + 3220 X_{2} + 1800 X_{3} + 12750 X_{4} + 3250 X_{5} + 3000 X_{6} ⩽ 6000000, \\ 148 X_{1} + 238 X_{4} + 135 X_{6} ⩽ 50000, 180 X_{1} + 220 X_{2} + 200 X_{3} + 150 X_{4} + 100 X_{5} + 160 X_{6} ⩽ 120000, 60 X_{1} + 40 X_{2} \\ + 35 X_{3} + 50 X_{4} + 30 X_{5} + 45 X_{6} ⩽ 30000, 30 X_{1} + 32 X_{2} + 28 X_{3} + 35 X_{4} + 26 X_{5} + 20 X_{6} ⩽ 200000, 15 X_{1} + 18 X_{2} \\ + 16 X_{3} + 14 X_{4} + 17 X_{5} + 18 X_{6} ⩽ 10000, 42 X_{1} + 38 X_{2} + 36 X_{3} + 40 X_{4} + 37 X_{5} + 35 X_{6} ⩽ 25000, X_{i} ⩾ 0, i = 1, \dots, 6} . \end{matrix}$ (32)

Considering the concept of referential fuzzy numbers and their α - cuts along with $α = \frac{5}{12}$ transforms (32) into:

$\begin{matrix} \underset{X \in S}{Maximize} {{\bar{F}}_{1} (X), {\bar{F}}_{2} (X)} = \\ {\frac{\begin{matrix} [55000, 60833] X_{1} + [22500, 23333] X_{2} + [29167, 30500] X_{3} + [58500, 59333] X_{4} \\ + [39667, 40333] X_{5} + [59500, 60667] X_{6} \end{matrix}}{\begin{matrix} [34500, 39500] X_{1} + [12500, 13333] X_{2} + [17667, 18333] X_{3} + [38500, 39500] X_{4} + \\ [22500, 23667] X_{5} + [23500, 24333] X_{6} + [491670, 516670] \end{matrix}}, \\ \frac{\begin{matrix} [55000, 60833] X_{1} + [22500, 23333] X_{2} + [29167, 30500] X_{3} + [58500, 59333] X_{4} + \\ [39667, 40333] X_{5} + [59500, 60667] X_{6} \end{matrix}}{\begin{matrix} [95, 96.7] X_{1} + [118.3, 121.7] X_{2} + [143.3, 145] X_{3} + [143.3, 145] X_{4} + \\ [83.3, 85] X_{5} + [118.3, 121.7] X_{6} + [475, 483.3] \end{matrix}}} \end{matrix}$ (33)

By using the interval arithmetic, the (33) is further changed into:

$\begin{matrix} \underset{X \in S}{Maximize} {[F_{1}^{L} (X), F_{1}^{U} (X)], [F_{2}^{L} (X), F_{2}^{U} (X)]} = \end{matrix}$ (34)

where

$\begin{matrix} \begin{matrix} F_{1}^{L} (X) = \frac{55000 X_{1} + 22500 X_{2} + 29167 X_{3} + 58500 X_{4} + 39667 X_{5} + 59500 X_{6}}{39500 X_{1} + 13333 X_{2} + 18333 X_{3} + 39500 X_{4} + 23667 X_{5} + 24333 X_{6} + 516670}, \\ F_{1}^{U} (X) = \frac{60833 X_{1} + 23333 X_{2} + 30500 X_{3} + 59333 X_{4} + 40333 X_{5} + 60667 X_{6}}{34500 X_{1} + 12500 X_{2} + 17667 X_{3} + 38500 X_{4} + 22500 X_{5} + 23500 X_{6} + 491670}, \\ F_{2}^{L} (X) = \frac{55000 X_{1} + 22500 X_{2} + 29167 X_{3} + 58500 X_{4} + 39667 X_{5} + 59500 X_{6}}{96.7 X_{1} + 121.7 X_{2} + 145 X_{3} + 145 X_{4} + 85 X_{5} + 121.7 X_{6} + 483.3}, \\ F_{2}^{U} (X) = \frac{60833 X_{1} + 23333 X_{2} + 30500 X_{3} + 59333 X_{4} + 40333 X_{5} + 60667 X_{6}}{95 X_{1} + 118.3 X_{2} + 143.3 X_{3} + 143.3 X_{4} + 83.3 X_{5} + 118.3 X_{6} + 475} . \end{matrix} \end{matrix}$

Following the technique proposed by [23], $\underset{X \in S}{Maximize} F_{1}^{U} (X)$ and $\underset{X \in S}{Maximize} F_{2}^{U} (X)$ are solved and respectively the solutions obtained are:

X^*1 = (0, 0, 0, 0, 0, 370.3704) and X^*2 = (50.5767, 0, 0, 0, 0, 364.2565). Furthermore, respectively the α- acceptable optimal intervals for ${\bar{F}}_{1} (X)$ and ${\bar{F}}_{2} (X)$ are:

$[F_{1}^{L} (X^{* 1}), F_{1}^{U} (X^{* 1})] = [2.3127, 2.4435]$ and $[F_{2}^{L} (X^{* 2}), F_{2}^{U} (X^{* 2})] = [484.6467, 508.8314]$ .

If we set w₁ = 0, w₂ = 1, then $\begin{matrix} \begin{matrix} Z^{L} (X) = 5658.6 X_{1} - 39425 X_{2} - 44614 X_{3} \\ - 15281 X_{4} - 3583.7 X_{5} - 2424.8 X_{6} - 245920, \\ Z^{U} (X) = 14792 X_{1} - 34001 X_{2} - 38950 X_{3} \\ - 10117 X_{4} - 38.0701 X_{5} + 3333.3 X_{6} - 230210 . \end{matrix} \end{matrix}$

Thus, the following problem is formulated: $\begin{matrix} \underset{X \in S}{Maximize} Z^{L} (X) + Z^{U} (X) \end{matrix}$ (35)

The above problem is solved and the obtained solution is: $\overset{˘}{X} = (5.5767, 0, 0, 0, 0, 364.2567)$ . At the solution $\overset{˘}{X}$ :

$\begin{matrix} \begin{matrix} {\tilde{F}}_{1} (X) = [F_{1}^{L} (X), F_{1}^{U} (X)] = [2.2895, 2.4272], \\ {\tilde{F}}_{2} (X) = [F_{2}^{L} (X), F_{2}^{U} (X)] = [484.6467, 508.8314] . \end{matrix}, \end{matrix}$

For the maximization problem, the accuracy of the proposed solution is calculated as follows: $\begin{matrix} ɛ = \min {\max {(F_{i}^{U} (X^{* i}) - F_{i}^{U} (\overset{⌣}{X}), \\ (F_{i}^{L} (X^{* * i}) - F_{i}^{U} (\overset{⌣}{X})}, i = 1, 2}, \end{matrix}$ where $X^{* i} = \underset{X \in S}{argmax} F_{i}^{U} (X)$ and $X^{* * i} = \underset{X \in S}{argmax} F_{i}^{L} (X)$ .

Therefore, ɛ = min { max { 2.4435 - 2.4272, 2.3127 - 2.2895 } , max { 508.8314 - 508.8314, 484.6467 - 484.6467 }} = 0 ; this means $\overset{˘}{X}$ is at least a weakly efficient solution.

5 Conclusion

In this paper, we proposed an approach to address the multi-objective linear fractional programming problem with fuzzy coefficients (FMOLFPP). In the method, the multi-objective problem is finally changed into the linear programming problem (LPP). It was proven that the solution resulted from the LPP is at least a weakly ɛ- efficient for the main problem. To design our method, the concept of α - cuts was used to transform the fuzzy numbers into the intervals in addition to rank the fuzzy numbers; a parametric approach was utilized to linearize the fractions; the weighted sum method was considered to end up with the LPP.

Two examples were solved and the results showed that this study proposed ɛ- efficient outcomes. Moreover, we applied the genetic algorithm (GA) of the Optimization Toolbox of MATLAB R2016b and found that: the GA results are efficient solutions for example 1. However, for example 2, the GA failed to reach an efficient solution. In fact, the GA tends to $\vec{0}$ which is completely dominated by the solution proposed by this study. Therefore, we don’t suggest using the GA to address the partially large-scale FMOLFPPs.

Footnotes

Acknowledgment

The authors would like to thank UKM for the financial support through the research grant ST-2019-016.

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