ISSN:
1573-2878
Keywords:
Vector performance indexes
;
multicriteria optimization
;
Pareto optimality
;
cooperative games
;
vector maximization problems
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In solving many practical problems, we have to deal with conflictive multiple objectives (in performance, cost, gain, or payoff, etc). Can all such objectives be achieved simultaneously? The general answer is negative. That is, most multiple-objective problems do not have supreme solutions that can satisfy all of the objectives. Many broader definitions of optimality like Pareto optimum, efficient point, noninferior point, etc, have been introduced in various contexts, so that most multiple-objective problems can have optimal solutions. But such optimal solutions do not in general yield unique vectors of optimal indexes of the conflictive multiple objectives. In most cases, we have to make appropriate tradeoffs, compromises, or choices, among those optimal solutions. To obtain the set of all such optimal solutions (in particular, the set of all optimal index vectors), say for a comprehensive study on appropriate tradeoffs, compromises, or choices, a usual practice is to optimize linear combinations of the multiple-objective functions for various weights. The success of such approach relies heavily on a certain directional convexity condition; in other words, if such convexity is absent, this method will fail to obtain essential subsets. The method of proper equality constraints (PEC), however, relies on no convexity condition at all, and through it we can obtain the entire set. In this paper, we attempt to lay the foundation for the method of PEC. We are mainly concerned with obtaining the set of all maximal index vectors, for most of the broader-sense optimal solutions are actually expressed in terms of maximal index vectors (Ref. 1). First, we introduce the notion of quasisupremal vector as a substantially equivalent substitute for, but a rather practical and useful extension of, the notion of maximal vector. Then, we propose and develop the method of PEC for computing the set of all quasisupremal (or maximal) index vectors. An illustrative example in the allocation of funds is given. One of the important conclusions is that optimizing the index of one objective with the indexes of all other objectives equated to some arbitrary constants may still result in inferior solutions. The sensitivity to variations in these constants are examined, and various tests for quasisupremality (maximality, or optimality) are derived in this paper.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01767453
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