Development of method for optimal inventory control under continuouse supply of product and random demand

The strategy of work of the supply firm under the conditions of a casual fluctuation of demand at a continuous replenishment of production stock in periods of availability of demand is studied. It assumes the use of adaptive management, which consists in the fact that during periods of lack of demand, replenishment of the stock level does not occur. For a formalized description of the firm's work, it is proposed to use the apparatus of semi-Markov processes with drift, in which the discrete component describes the state of the market environment, and the continuous component - the random fluctuations in the level of the stock in the warehouse. To find the limiting (for) probability distribution of states of a semi-Markov process with drift, a system of integral equations of convolution type on the half-axis is derived, the solution of which is found in closed form for the special case. With the help of the found solution, the problem of stochastic optimization is formulated to find the intensity of supply of homogeneous products to the warehouse of the supply firm in periods of availability of demand for it, which minimizes the average total costs of the firm per unit time. The generalization of the stochastic optimization model to the case of several types of products is considered.