About One Task of Managing a Hierarchical System

The metrological support system for special equipment and special objects considered in the paper is modeled by a three-level hierarchical system. The metrological support system is considered as park of measuring equipment: high-precision instruments installed, working standards and working measuring instruments in a hierarchical sequence. The statement and solution of the problem of program - target planning for the development of the measuring equipment park is given. The goal of the development of the park is to provide the required number of verifications with working measuring instruments. The verified devices are measuring devices installed at special techniques and special objects. As the control actions on the park, the procurement of high-precision installations, working standards and working measuring instruments is strictly used in a certain proportion, determined in accordance with the hierarchy of the system

The mathematical model of the relationship between hierarchies is proposed. The model is based on the equation of the temporal balance of the park's potential power and the required amount of time for verification, taking into account the frequency of verifications. In the general case, in a hierarchical model, correspondences between levels can be of the type: one to one, one to several and several to one. This circumstance leads to the necessity of solving NP – complete problem of discrete optimization. The article gives an algorithm for setting this NP – complete problem for different cases of violation of the uniqueness of correspondence between elements of different levels. The effective algorithm for estimating the dimension of this problem has been developed. Assessment of the dimension of the problem allows you to choose the appropriate algorithm for solving the discrete optimization problem (branch and bound method, genetic algorithm, ant algorithm, exhaustive search method). The results of mathematical modeling are presented.