Multiperiod production planning with demand and cost fluctuation

We consider the production planning of a single product over a finite planning horizon. The latter is made up of time periods that may not necessarily be of the same length. In each time period, the product is subject to a constant demand and a finite production capacity. If demand is not satisfied immediately, it is backlogged. The cost rates of carrying inventory or backlogging a unit of the product may also vary from one time period to the next. The objective is to determine an optimal production plan for the product so as to minimize the total inventory and backlog costs incurred over the planning horizon. The problem is formulated as an optimal control problem. First, the optimal solution of the one-time-period problem is determined. Then, based on the one-time-period optimal solution, the optimal solution for the whole planning horizon is obtained by solving a nonlinear programming problem.