Выбор решений при проектировании сложных систем на основе анализа вариантов со случайными весами

Abstract

A new one-parameter approach to the selection of optimal solutions for the design of complex systems is proposed. The approach is based on the analysis of a tree of variants with random weights (here weight is a certain non-negative quantity: for example, cost, mass, energy consumption, etc.). At the root of the method suggested in the work lies the fact that the tree of variants forms a matroid, on which the optimal solution can be found using the greedy algorithm. The basis of the method proposed in the paper is the fact that in case of non-negative values of the mathematical expectation and variance of the elements of the variants tree they can be considered as components of vectors belonging to a semiring. It is shown that the appropriate definition of the operations of addition and multiplication makes it possible to define a function on the semiring. This function satisfies the norm axioms of vectors and coincides in structure with the expression for the upper bound of the confidence interval. After determining the weight of the tree elements through the introduced norm function the upper bound of the confidence interval of the variant tree with the minimum weight was found. The approach suggested in the work can be used at various stages of designing complex systems, including, among other things, the development of system profiles, and makes it possible to increase the validity of the decisions made.