Solution to the Problem of Optimal Control of Discrete Semi-Markov Stochastic Inventory Model of Continuous Product with Constant Consumption

This work is a continuation of the bachelor's work and is devoted to the study of a discrete stochastic semi-Markovian model that describes the functioning of some control system of continuous supply of product at constantly occurring consumption. Intensity of consumption depends on the period of evolution of the original system. The stochastic model considered here is a pair of random processes (x (t), ζ (t)), in which the main process x (t) describes the current storage volume in the system at time t, and the attendant ζ (t) describes a semi-Markov process with finite set States, which is the basis for using a number of theoretical statements. The problem of optimal control is posed in relation to the stationary indicator associated with the accompanying process. In terms of its ideological content, this indicator is the average specific profit obtained with the evolution of the initial supply control system. In this paper we perform formal analytic transformations of the integral representations obtained earlier for the probabilistic characteristics of this model, which are necessary for solving the optimization problem, namely, integral representations of transition probabilities and expressions for the mathematical expectation of the accompanying process ζ (t) in state i, as well as expressions for the mathematical expectation of profit, which will be obtained during the time of finding ζ (t) in state i. These transformations made it possible to use the theorem on the analytic representation of the stationary index of control quality in fractional linear form, proved by P.V. Shnurkov. In the sequel, we use the general theorem on the extremum of a fractional-linear integral functional. This theorem makes it possible to reduce the problem of optimal reserve management to the study of the global extremum of a given function of a finite number of real nonnegative variables.