Optimization of two-step investment in a production process

We consider the problem of optimizing the control of a production process. The control parameters are the capacity utilization and the investment in the growth of the production capacity. We assume that the investments are divided into two parts: initial investment aimed at creating production facilities, and investment aimed at increasing the capacity during the production process. The initial and increased capacities and the moment of changing the capacity are variable parameters to be specified. The price of the product is assumed to be a random process. The problem is to optimize the production process and to construct a control strategy that maximizes the average discounted profit. We propose an approach to the construction of an optimal adaptive strategy for controlling the production. The approach is based on the dynamic programming method.