Attainability of optimal solutions to a linear problem of multicriterion optimization with respect to the weighed sum of
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ATTAINABILITY OF OPTIMAL SOLUTIONS TO A LINEAR PROBLEM OF MULTICRITERION OPTIMIZATION WITH RESPECT TO THE WEIGHED SUM OF VARIOUSLY IMPORTANT AND TRANSITIVELY SUBORDINATED CRITERIA
UDC 519.7
A. Yu. Brila
This paper presents methods of finding criteria coefficients such that optimal solutions are obtained to a linear problem of multicriteria optimization with respect to the weighed sum of variously important and transitively subordinated criteria. The case of a partial transitive subordination is also considered, and a method is founded that finds coefficients such that optimal solutions to the problem of multicriteria optimization are attainable with respect to the weighed sum of variously important and partially transitively subordinated criteria. Keywords: attainability, subordination, chain of criteria, linear lexicographic multicriteria optimization problem. In recent years, there has been an increased interest in the investigation of multicriteria models of continuous and discrete optimization. Different aspects of investigation and construction of methods for solving multicriteria optimization problems are considered In [1–9]. Let us consider a linear multicriteria problem with linear criteria c k ( x ), k = 1, 2, K , q,
(1)
and a set of admissible solutions X Ì R n . Let X V be the set of admissible solutions that are vertices of the set X . If a choice is based on many criteria, the question arises as to the admissible alternative that can be the result of such a choice. Its determination is based on the comparison of the alternatives obtained. The result of this comparison underlies a unique ordering introduced on the set of the alternatives being considered. The criterion that underlies such an ordering is called the criteria convolution [1]. We note that criteria convolutions are usually determined by comparing criteria. As a result of their quantitative comparison, i.e., the comparison of their values (estimates), convolutions of these criteria into a unique scalar criterion are obtained. Some methods of construction of such convolutions are considered in [2–4]. In comparing criteria on the basis of their relative importance with a view to estimating alternatives, these criteria are convoluted into a unique vector criterion with a vector function of estimates. If the optimal alternative of a multicriteria optimization problem can be obtained as the optimal solution of the corresponding one-criterion optimization problem with the objective function that is a linear convolution of the criteria of the multicriteria problem, then this optimal alternative is assumed to be attainable with respect to the weighed sum of variously important criteria. A criteria convolution also can be specified in a system of interrelations between criteria that expresses the subordination between the criteria. The system of pair subordination, which is also called a subordination specified over a set of criteria, is widespread. For any two criteria, such a subordination establishes the fact that one of them is subordinated
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