Studying the optimization problem in discrete semi-markov model of continuous inventory control

The stochastic model of inventory control of a certain product, whose volume may take the values within an interval bounded above that belongs to the set of real numbers, is considered and investigated. This model consists of two components. One of them is named a basic process and describes the inventory level in the system under study; another is named a concomitant process and presents a controlled semi-Markov process with a finite set of states. The use of the concomitant process allows the theory of semi-Markov process control to be applied for solving the problem. The probabilistic characteristics of the concomitant semi-Markov process, as well as the characteristics of time-independent cost functionals connected with this process are determined. It is proved that a deterministic strategy is the optimal strategy for control. The explicit representation for the time-independent functional describing the quality ofprocess control is obtained. It is found that the optimal control strategy in the semi-Markov model is determined by a point ofglobal extremum of the function of several real variables. The explicit analytical expression of this function is found.

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