Finding a weights-restricted efficient (extreme) point and using it for solving MOLP problems

Abstract

This paper deals with solving a multiple objective linear programming problem. A (preferred) weights-vector from a restricted set of weights, incorporating the case that the weighted problem may have unbounded solution, is used for generating a (preferred) efficient (extreme) point. The final solution can be found by changing (interactively) weights-restrictions or comparing weights-restricted efficient extreme points. Also, the results can be used in vector maximum and interactive methods.

Introduction

The Multiple Objective Linear Programming (MOLP) problem can be written:

$$\text{(}\text{MOLP}\text{)}\text{V}\text{Max}\text{Cx}\text{s}\text{.}\text{t}\text{.}\text{x∈X={x∈R}{}^{\mathrm{n}}\text{|Ax=b,x⩾0}.}$$

Here, C and A are k×n and m×n matrices, respectively, and b∈R^m.

Since there is usually no point which simultaneously optimizes all the objectives, the concept of efficiency is used:

Definition 1

$\text{x}\text{̄}\text{∈X}$ is efficient if and only if there does not exist another x∈X such that $\text{Cx⩾C}\text{x}\text{̄}$, $\text{Cx≠C}\text{x}\text{̄}$.

The aim of MOLP approaches is to find the most preferred solution among the efficient points. To this purpose, there are many methods which use weights. However, the ways these weights are selected, also their uses and interpretations may be different. We consider methods which use the linear problem:

$$\text{(}\text{LP}\text{(u))}\text{Max}\text{u}{}^{\mathrm{<mtext>T</mtext>}}\text{Cx}\text{s}\text{.}\text{t}\text{.}\text{x∈X,}$$

to generate one or more efficient points for presenting to the DM, where u is a vector of positive (usually normalized) weights.

The methods of MOLP can be divided into three category according to which stage of the optimization the DM expresses his/her preferences: before, during or after the optimization [1], [2].

In the first category, the DM may expresses his preferences by a positive weights-vector u. By doing one’s best to estimate u, it is hoped that (LP(u)) will produce the most preferred solution or a solution that is close enough to it. Although there are different procedures for the evaluation of the weights, obtaining their exact values is often impossible [3].

In the second category, we may use (LP(u)) to generate one or a set of efficient candidates in the beginning or during the optimization. The DM provides his preferences interactively using these candidates. By removing some poor points (usually by restricting weights) in each iteration, new candidate(s) are generated and the final solution is selected after some iterations.

In the third category, the first endeavor is to find all (or most) of the efficient solutions. Then, the DM selects the most preferred solution from this set of efficient points through some arbitrary process. In this category (LP(u)) may be used, for example, to generate an initial efficient (extreme) point.

In any case, the discussed methods are based on the optimal solution(s) of (LP(u)). Hence, it is important to select u such that (LP(u)) has (bounded) optimal solution. Note that, even in the first category, unbounded optimal solution to (LP(u)) does not result in unbounded optimal solution to the (MOLP).

In addition, each of the three pure approaches discussed above has some advantages and some disadvantages [2]. A combination of the approaches could combine the advantages of each of the individual approaches while ameliorating the disadvantages. For example, we may use some (imprecise) prior information on weights for removing unacceptable efficient solutions from further investigation and reducing the size of the efficient set (see, for example [4]).

In this paper, existing information (before or during the optimization) is assumed to be in the form of some arbitrary linear restrictions on weights. First, we will produce two methods to generate an efficient (extreme) point under existing restrictions. These methods can be used in many MOLP approaches. Also, we will propose an approach for solving an MOLP. It is assumed that some prior information (linear restrictions on weights) exists which may be refined interactively with additional information from the DM. Then, corresponding to existing information, an efficient extreme point is generated and by moving to a better adjacent efficient point, iteratively, an efficient point which is better than all its adjacent points will be found. Under some assumptions this point is the most preferred (extreme) solution.