Solving Multiobjective Linear Programming Problems with Interval Parameters

In the present paper, a multiobjective linear programming problem under uncertainty, particularly when parameters are given in interval forms, is investigated. In this case, it is assumed that objective coefficients and constraints parameters have arrived in interval numbers. Considering a suitable order relation for interval numbers, a solution procedure for dealing with such a problem is developed. A numerical example is provided to illustrate the efficiency of the solution procedure.


Introduction
Multiobjective programming (MOP) problems involving several, often, conflicting and incommensurate objective functions have gained the attention of many researchers in earlier decades [1]. When handling real-world MOP problems, it is often necessary to treat inexact or uncertain input data due to various measurement errors or estimations. Throughout the years, several approaches for dealing with MOP problems with imprecise data based on different sources of uncertainty have emerged.
Uncertainty may be interpreted as randomness or fuzziness [2]. To deal with randomness in MOP problems, stochastic programming approaches are applied [3,4]. On the other hand, fuzzy programming techniques are used for dealing with fuzziness in MOP problems [5][6][7]. In stochastic programming and fuzzy programming, the uncertain coefficients are assumed as random variables with known distributions and fuzzy numbers with known membership functions, respectively. However, determining distributions of random variables and determining membership functions of fuzzy numbers are not easy. It is due to the fact that sometimes they do not perfectly match the real situations.
Interval programming as an alternative choice could be considered to deal with uncertainty in MOP problems. In interval programming, it is assumed that the coefficients perturb independently within the given lower and upper bounds [8]. Actually, uncertain coefficients are modelled by closed intervals in this approach. Since interval programming does not require stringent applicability conditions, therefore, it has been considered a suitable tool for modelling uncertainty in many practical applications [9][10][11][12][13][14][15][16][17][18][19][20].
CONTACT S. Rivaz srivaz@nit.ac.ir This paper focuses on interval programming for dealing with uncertainty in multiobjective linear programming (MOLP) problems. MOLP problems with interval coefficients have been investigated by some authors. Bitran [21] discussed MOLP problems with interval objective function coefficients and introducing two types of solutions. Urli and Nadeau [22] used an interactive method for solving MOLP problems with interval coefficients. Oliveira and Antunes [23] provided an overview of MOLP problems with interval coefficients. Also, Oliveira and Antunes [24] presented an interactive method to solve such problems. Wu [25] proposed Karush-Kuhn-Tucker optimality conditions for a multiobjective programming problem with interval objective function coefficients. Some new solution concepts and algorithms were suggested by Rivaz and Yaghoobi to MOLP problems with interval objective function coefficients in [2,19].
The current research tries to propose a solution procedure to interval MOLPs. In this sense, an order relation for interval numbers is used and a solution concept according to interval MOLPs is defined. Further, in order to solve an interval MOLP problem, a bi-objective linear programming problem is presented.
The remainder of the paper is organised as follows. In Section 2, some preliminaries are discussed. Section 3 is devoted to introduce an interval MOLP problem. In addition, a solution procedure for dealing with interval MOLP problems is discussed in the same section. In Section 4, with the aid of a numerical example, the solution procedure is illustrated. Finally, Section 5 states conclusions and proposes directions for future research.

Preliminaries
In this section, we recall some concepts of interval arithmetic and multiobjective optimisation which are used later [1,26]. Throughout the paper, capital letters indicate closed intervals. There are two different representations of an interval. One may represent the interval A by its left bound a L and right bound a R as Another representation of interval A is by its centre point a C and half-width length (or radius) a W as Definition 2.1: Let * ∈ {+, −, ·, ÷} be a binary operation on R. If A and B are two arbitrary closed intervals, then In the case of division, it is supposed that 0 ∈ B.
From Definition 2.1, it could be shown that In what follows, an order relation which represents the decision maker's preference between intervals are defined for minimisation problems.
A MOLP problem can be formulated as follows: To present the following definitions, two order relations in R p are needed. Consider two Consider (w 1 , . . . , w p ) such that w k ≥ 0, k = 1, . . . , p. The weighted sum linear programming problem (for short, weighted sum problem) associated with the MOLP Problem (1) is as follows:  (w 1 , . . . , w p ) with w i > 0, i = 1, . . . , p, such that x 0 is an optimal solution to Problem (2).

Problem Statement and Main Results
Let the interval MOLP problem be given as It should be noted that if each of the intervals is a real value, then Problem (3) z 1 (x), . . . , z p (x)) RC (z 1 (x 0 ), . . . , z p (x 0 )), i.e. z i (x) ≤ RC z i (x 0 ) for i = 1, . . . , p and there is at least one 1 ≤ q ≤ p with z q (x) < RC z q (x 0 ).

Definition 3.2: A feasible solution
In order to solve Problem (3), an attempt is being made to obtain an equivalent crisp problem. To do that, firstly by using the weighted sum approach [1], the following linear programming problem with interval parameters is achieved.
where w k ≥ 0, k = 1, . . . , p denotes the weight of the kth interval objective function. Finally, considering the order relation ≤ RC , the following deterministic bi-objective problem is obtained.
which means z i (x) ≤ RC z i (x) for i = 1, . . . , p and z q (x) < RC z q (x) for some 1 ≤ q ≤ p. This is a contradiction to the optimality ofx for Problem (4) and the proof is completed.

Numerical Example
Suppose that a factory can produce three products, namely, P 1 , P 2 and P 3 . According to the past experience, for selling each unit of P 1 , P 2 and P 3 , the factory can earn income that are intervals [7,8], [2,3] and [4,6] (in $), respectively. With respect to the workers' experience, producing each unit of P 1 , P 2 and P 3 , consume the quantity of a rare resource that are [6,9], [2,4] and [4,5] (in kg), respectively. The production cost of each unit of P where x i , i = 1, 2, 3, denotes the amount of product P i , i = 1, 2, 3, that should be produced.By applying the proposed method with w 1 = w 2 = 0.5, the bi-objective linear program (7) is yielded.
It should be noted that the smallest feasible region is used in Problem (7) which could be changed according to the decision maker's idea. An efficient solution to Problem (7) by using the weighted sum method with the weight vector (0.5, 0.5) is

Conclusion
In this paper, a multiobjective linear programming problem with interval parameters was considered. To deal with such a problem, a new solution procedure based on a suitable order relation of intervals, was presented. A numerical example was provided to illustrate the efficiency of the proposed method. Try to propose new solution methods with suitable properties for dealing with MOLP problems could be considered as a general topic for further research.