J. Software Engineering & Applications, 2009, 2: 237-247 doi:10.4236/jsea.2009.24031 Published Online November 2009 (http://www.SciRP.org/journal/jsea) Copyright © 2009 SciRes JSEA 237 A New Interactive Method to Solve Multiobjective Linear Programming Problems Mahmood REZAEI SADRABADI 1 , Seyed Jafar SADJADI 2 1 Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands; 2 Department of Industrial Engineering, Iran University of Science & Technology, Tehran, Iran. Email: m.rezaei.sadrabadi@student.tue.nl Received May 20 th , 2009; revised June 19 th , 2009; accepted June 29 th , 2009. ABSTRACT Multiobjective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. A lot of methods have been proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP by starting from a utopian point, which is usually infeasible, and moving towards the feasible region via stepwise movements and a simple continuous interaction with decision maker. We consider the case where all objective functions and constraints are linear. The implementation of the pro- posed algorithm is demonstrated by two numerical examples. Keywords: Multiobjective Linear Programming, Multiobjective Decision Making, Interactive Methods 1. Introduction During the past four decades, many methods and algo- rithms have been developed to solve Multiobjective Pro- gramming (MOP), in which some objectives are con- flicting and the utility function of the Decision Maker (DM) is imprecise in nature. MOP is believed to be one of the fastest growing areas in management science and operations research, in that many decision making prob- lems can be formulated in this domain. For some engi- neering applications of MOP problems the interested reader is referred to [1,2]. Decision making problems with several conflicting objectives are common in prac- tice. Hence, for such problems, a single objective func- tion is not sufficient to seek the real desired solution. Because of this limitation, an MOP method is needed to solve many real world optimization problems [3]. Although different solution procedures have been in- troduced, the interactive approaches are generally be- lieved to be the most promising ones, in which the pre- ferred information of the DM is progressively articulated during the solution process and is incorporated into it [4]. The purpose of MOP in the mathematical programming framework is to optimize r different objective functions, subject to a set of systematic constraints. A mathematical formulation of an MOP is also known as the vector maximization (or minimization) problem. Generally, MOP can be divided into four different categories. The first and the oldest group of MOP need not to get any information from DM during the process of finding an efficient solution. These types of algorithms rely solely on the pre-assumptions about DM's preferences. In this category, L-P Metric methods are noticeable, algo- rithms whose objectives are minimization of deviations of the objective functions from the ideal solution. Since different objectives have different scales, they must be normalized before the process of minimization of devia- tions starts. Therefore, a new problem is minimized which has no scale [5]. The second group of MOP includes gathering cardinal or ordinal preferred information before the solving proc- ess initiates. In the method of utility function [6], for example, we determine DM's utility as a function of ob- jective functions and then we maximize the overall func- tion under the initial constraints. The other method in this group, which is extensively used by many researchers, is Goal Programming (GP) [7] in which DM determines the least (the most) acceptable level of Max (Min) functions. Since attaining these values might lead to an infeasible point, the constraints are allowed to exceed, but we try to minimize these weighted deviations. The third group of MOP provides a set of efficient so- lutions in which DM has the opportunity to choose his preferred solution among the efficient ones. Although finding an efficient solution in MOP is not difficult, but finding all efficient solutions to render DM is not a trivial task. Many papers have discussed this important issue [8–11]. The set of all efficient feasible solutions in a Multiobjective Linear Programming (MOLP) can be represented by convex combination of efficient extreme