Optimal inventory control of continuous product in regeneration theory with determinate delay of the delivery and the period of real replenishment

The problem of inventory control in the stochastic model of regeneration is investigated. The purpose is to find a theoretical solution of the optimal control problem. To solve the problem, the authors have found the distribution function of a random time from the replenishment moment till the next order which determines the optimal replenishing product plan. It is proved that the initial problem solution exists and it is attained on the degenerate probability distribution. In addition, the optimal deterministic value of the control parameter is the global extreme point of a function which was expressed in an analytical representation. The existence of deterministic optimal control assertion was proved. The authors have specified the analytical conditions on the initial model characteristics under which the optimal control exists.