Finding a weights-restricted efficient (extreme) point and using it for solving MOLP problems
Introduction
The Multiple Objective Linear Programming (MOLP) problem can be written:
Here, C and A are k×n and m×n matrices, respectively, and b∈Rm.
Since there is usually no point which simultaneously optimizes all the objectives, the concept of efficiency is used: Definition 1 is efficient if and only if there does not exist another x∈X such that , . 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: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.
Section snippets
Basic theorems and definitions
First, we state the following Theorem to relate efficient points with a family of solutions to the weighted-sums problem (LP(u)) and give a simple proof: Theorem 1 Let be a given vector of positive weights. Then, y∈X is efficient if and only if there exists such that y solves (LP(u)). Proof Let y be efficient then there does not exist x∈X such that Cx⩾Cy, Cx≠Cy. Hence, the optimal objective values of (P) Max s.t. s−Cx=−Cy, Ax=b, x⩾0, s⩾0, and its dual (D) Min-uTCy+vTb s.t. −uTC+vTA⩾0, ,
are
Generating an efficient point using (LP(u))
To generate a candidate from EWR (or E), using (LP(u)), we need to find weights-vector u from UWR (or U). For this purpose, we use the following problem:
Which f(u) may define the DM’s preferences for selecting a good representative of UWR; or simply f(u)=0 when we need only a candidate (not necessarily the preferred one) of UWR. Without loss of generality, for example by deleting inconsistent restrictions on weights and normalizing the weights, we can
Solving MOLP
The previous methods can be used in many MOLP approaches. For example, we can use them to generate an initial efficient (extreme) point for use in vector maximum or interactive algorithms [3], [6], [7], [8], by f(u)=0 and replacing Fu⩽f with , in which ū is a (preferred) positive vector (we may select ū arbitrarily for example that is a vector of ones). These methods have some advantages over other existing methods. for example in the first method we need only to solve one linear
Example
Consider the multiple objective linear problem defined by (MOLP) with:
It can be seen that (LP(u)) has unbounded optimal value, and hence fails to generate an efficient point, for every positive vector u=[u1,u2]T with u1>6/7u2.
Now, for instance, let the restrictions on weights be u1+u2=1, 2u1−u2>1, u1>0, and u2>0. Also, assume that we prefer to use the middlemost representative of these weights (as is used, for example, in Zionts–Wallenius
Conclusions
The weighted-sums problem (LP(u)) has been used in many MOLP approaches for generating efficient points, using one or a set of weights-vectors u selected from a (restricted) set of weights. The problem may be arisen is when (LP(u)) has no optimal solution (is unbounded) for a selected weights-vector u. To overcome this problem, we have produced two methods which prevent selecting such weights using the restrictions uTCdj⩽0, j=1,2,…,l, or CTu−ATv⩽0, respectively. A convenient function may also
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