REGULATION OF THE RELATIONSHIP BETWEEN ADDITIVE REDUCTION AND METRICS METHODS
https://doi.org/10.21822/2073-6185-2017-44-2-107-117
Abstract
Objectives. The aim of the work is to determine the relationship between generalised criterion and target programming methods.
Methods. The paper considers the aggregation operation that underlies many decision-making procedures used in input-output models, in neural network technologies and in the study of multi-purpose systems. The use of certain metrics within the framework of target programming can lead to solutions that are not Pareto-optimal. Therefore, in targeted programming, a significant place is given to finding the conditions under which the use of one or another metric obviously leads to Pareto-optimal solutions. The necessary (Carlin's theorem) and sufficient conditions of Pareto-optimality are known to perform the additive reduction. For a generalised criterion on the basis of order operators of weighted aggregation, two theorems proven by the author (the theorem on the inclusion of the set of Pareto-optimal solutions into a set of effective solutions and the Pareto optimality theorem for the solution obtained) are presented.
Results. The proof of the Pareto optimality theorem of the solution is given, maximising the generalised criterion obtained on the basis of the order operations of weighted aggregation, which justifies the use of operations of this type for solving the problems of vector optimisation or multicriteria choice. The theorem on the existence of an additive reduction for a metric is true only in the particular case and is based on Carlin's theorem, according to which a subset of Pareto-set points maximises some additive reduction.
Conclusion. In the paper a relationship between the additive reduction and metrics methods is established. An assertion concerning the relationship between the parameters of the distance function in the target programming method and the weighting coefficients of the additive reduction is formulated and proved, which ensures the equivalence of the optimal Pareto solutions.
About the Author
E. M. Aristova
Voronezh State University
Russian Federation
Russian Federation
Cand. Sci.(Physics and Mathematical), Assoc.Prof.,
1 Universitetskaya square, Voronezh 394018
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Review
For citations:
Aristova E.M. REGULATION OF THE RELATIONSHIP BETWEEN ADDITIVE REDUCTION AND METRICS METHODS. Herald of Dagestan State Technical University. Technical Sciences. 2017;44(2):107-117. (In Russ.) https://doi.org/10.21822/2073-6185-2017-44-2-107-117
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