A Multi-objective Method for Solving Fuzzy Linear Programming Based on Semi-infinite Model

ABSTRACT In this paper, we present a new method to solve a fuzzy linear programming problem with fuzzy coefficients in the constraints and the objective function based on solving an associated multi-objective model. In particular, we present a weighted method for linear semi-infinite programming (LSIP) model to solve the original problem. Finally, a numerical example is included to illustrate the suggested solving process.


Introduction
After successful applications of fuzzy sets theory in the various fields, this theory has been applied in the optimisation area. In particular, fuzzy linear programming (FLP) has a long history as well as fuzzy sets theory. As a pioneering study, Delgado et al. [1] studied a general model for solving FLP problems which involve fuzziness both in the coefficients and in the accomplishment of the constraints. Wu et al. [2] used an analytic centre based on a cutting plane method to solve linear semi-infinite programming (LSIP) problems. It is shown that a near optimal solution can be obtained by generating a polynomial number of cuts. Goberna et al. [3] analyse the effect on the optimal value of a given LSIP problem of the kind of perturbations which more frequently arise in practical applications. Goberna and López [4] present a state-of-the-art survey on LSIP theory and its extensions (in particular, convex semi-infinite programming). Recently, Nasseri et al. [5] shown that such problems can be reduced to an LSIP problem with fuzzy cost coefficients. This paper is organised into five sections. In Section 2, we give some necessary concepts and definitions which is useful throughout the paper. In Section 3, the mentioned novel solving approach is given to solve the FLP model. Section 4 is assigned to the illustrative example and finally, we will give the conclusions in Section 5. Definition 2.1: Let R be the real line. A fuzzy setÃ in R is defined to be a set of ordered pairsÃ = {(x, μÃ(x)) : x ∈ R}, where μÃ(x) is called the membership function for the fuzzy set. The membership function maps each element of R to a membership value between 0 and 1. A general form of a membership function μÃ(x) is given by We symbolically show every fuzzy numberÃ byÃ = (m − l, m, m + u) LR based on its membership function.
We named every fuzzy set as a triangular fuzzy number, where L(x) an R(x) are linear functions. We also show the set of all triangular fuzzy number by F(R).

A Proposed Method
In this section, we will present a new approach for solving an FLP problem which is defined in (1). In the process of solving an FLP problem, we first need to solve an LSIP problem with multi-objectives. In this way, we show that this problem will reduce to one target by a weighted method for objective function. Finally, we will use a cutting plan method for its constraints.
The constraint n j=1 a ij x j ≥ α b i , i = 1, . . . , m based on Definition 2.4 is equivalent to the following constraints: We define The objective function of Problem (1) can be reduced as follows: Then, we define are the weights of the mentioned functions, which are determined by the decision maker. Hence, Problem (1) can be reduced to Problem (2), where T is a compact metric space, f ij (t), b i (t), i = 1, . . . , m, j = 1, . . . , n are real-valued continuous functions on T.
Problem (2) is a linear semi-infinite programming with weighted function (WFLSIP) problem with n variables and infinitely many constraints. In the sequel, we denote its feasible region and its optimal objective value as FP and v(LSIP), respectively. Here, for solving the following LSIP problem, we reduce it to the form of Problem (2). We will use a 'cutting plane approach' to solve LSIP problems as well as used in [9]. By using a cutting plan approach, we can design an iterative algorithm that adds m constraints at a time until an optimal solution is identified. To be more specific, at the rth iteration, given T r = {t 1 , t 2 , . . . , t r }, where t r = (t r 1 , . . . , t r m ) ∈ T m , and r ≥ 1. We consider the following model: Let F r is the feasible region of Problem (3). If x r = (x r 1 , x r 2 , . . . , x r n ) is an optimal solution for Problem (3), then consider the 'constraint violation functions' as follows: Since f ij (t) and b i (t) are continuous over T, and also T is a compact set, then the function Below, we present the main steps of the suggested approach.

Numerical Example
In this section, the solving process of an FLP problem, which is proposed in the last section, will be illustrated.
We consider a company which produces two products A 1 and A 2 . These products are processed on two different machines A 1 and A 2 . The time required for the production of one unit of each product on both machines is uncertain which is given together with the daily capacities of the machines in Table 1.
Note that the time availability can vary from day to day due to the breakdown of the machines, overtime work, etc. Finally, the profit for each product can also vary due to variations in price. At the same time, the company wants to keep the profit somehow close to 8 for P 1 and 4 for P 2 . The company wants to determine the range of each product to be produced per day to maximise its profit. It is assumed that all the amounts produced are consumed in the market.
Since the profit from each product and the availability time of each machine are uncertain, the number of units to be produced on each product will also be uncertain. So, we may formulate this problem as an FLP problem. Note that, we will use triangular fuzzy numbers for each uncertain value.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
The authors did not receive any funding for this research.