A continuous time dynamic optimal control manufacturing problem

Economic production quantities address an important problem in operations management. The objective is to determine the quantity that minimises total cost required to manufacture goods and hold inventories when production is performed incrementally during the manufacturing process. The dynamic impact of costs and demand is often neglected though parameters may vary over time. In this context, optimal control theory goes beyond the suggestion of a numerical approach and allows for an analytical interpretation of optimal solutions. This paper presents a deterministic continuous time approach minimising the net present value of production and inventory holding cost with dynamic parameters. Manufacturing cost per item, holding cost, and demand rate vary over time. Applying Pontryagin’s maximum principle, the optimal policy involves intervals of production at the capacity limit with inventory build up, destocking periods, and periods of just-in-time production. A solution algorithm is presented to find the optimal manufacturing quantities and an economic interpretation of an optimal solution is provided.