Mixed backlogging and outsourcing models with inventory capacity

In this paper we consider a dynamic lot sizing model with mixed backlogging and outsourcing and bounded inventory, in which outsourcing may occur in a period even if the inventory level at that period is positive. The production cost may include setup cost and the production level is unlimited. The holding, backlogging and outsourcing cost functions are linear. Furthermore, backlogging level at each period is limited, and outsourcing level at each period cannot exceed the demand of that period. The goal is to minimize the total cost of production, inventory holding/backlogging and outsourcing. We show that this problem can be solved in O(T⁴ log T) time where T is the length of the planning horizon. Finally, the proposed algorithm is implemented in C++ and evaluated on a large variety of instances generated randomly.