Optimal Control of Continuous Inventory in the Regeneration Model with Random Delivery Delay and Random Cost Characteristics

In this work, a stochastic model of continuous inventory control in the form of a regenerative process is situated. The key feature of the investigated model is the existence of random cost characteristics. Specifically, the price of delivery of the product to the consumer is a random function. The carrying and deficiency costs are also random functions. Under specific conditions for these functions the regenerative process describing inventory level was developed. The control parameter in this model is a random time from the moment of resupply till the moment of the next reorder point. The performance indicator has a meaning of an average profit margin. It was proved that the indicator takes the form of a linear-fractional functional of a random distribution of the control parameter. In addition, the explicit representation for the numerator and denominator integrands of the mentioned functional were obtained. In terms of the general linear-fractional integral functional extreme theorem it was found that the optimal control is deterministic and is defined by the global maximum point of the prescribed function. Accessory analytical conditions at the original cost characteristics of the system were obtained so that the mentioned function, subject to this proviso reaches it’s global maximum. Accordingly, a solution to the given optimal control problem exists.