DISCRETE-CONTINUOUS MODEL OF SALES MANAGEMENT IN REAL TIME

Abstract

The subject of the research is the development of a strategy for dynamic price management when selling products over a fixed time interval. We consider the case when the demand for the seller's products can be represented as a combination of two random processes: 1) Poisson flow of potential consumers; 2) the purchase of goods by an individual consumer, the probability of which is inversely related to the price of the product. Consumers need at most one unit of the good and have independent equally distributed estimates of its consumer value. Such demand structure allows to formalize the choice of the optimal pricing strategy as an optimal control problem. Employing dynamic programming methods to solving this problem yields a system of Riccati differential equations. The optimal solution is obtained in the closedloop form as a function of the time to expiration of the product value and unsold inventory levels. Examples of a practical solution to the optimal pricing problem are given for special cases when it is possible to find an analytical solution. For the general case, it is shown how to find the optimal prices using numerical methods. Calculations show that optimal prices are decreasing functions of time and inventory levels. The combination of these factors, together with the random nature of the product sales, leads to rather complex observed price trajectories, examples of which were obtained using computer simulations. In particular, in many cases, the implementation of the proposed strategy results in cyclical price behavior, the prevalence of which in retail is well documented. The problem of optimizing the expected income of the seller when using constant prices was also solved. Comparison of the expected income of the seller under static and dynamic prices indicates a significant advantage of the latter. The economic effect of using dynamic pricing is most significant near the expiration of the product value.