A polynomial algorithm for a lot-sizing problem with backlogging, outsourcing and limited inventory

This paper addresses a real-life production planning problem arising in a manufacturer of luxury goods. This problem can be modeled as a single item dynamic lot-sizing model with backlogging, outsourcing and inventory capacity. Setup cost is included in the production cost function, and the production level at each period is unbounded. The holding, backlogging and outsourcing cost functions are assumed to be linear. The backlogging level at each period is also limited. The goal is to satisfy all demands in the planning horizon at minimal total cost. We show that this problem can be solved in O(T4 log T) time where T is the number of periods in the planning horizon.

► A real-life production planning is modeled as a single item lot-sizing problem.
► The main features include outsourcing, backlogging and limited inventory.
► The problem is proved to be solvable in polynomial time with dynamic programming.
► The algorithm can be implemented with Excel tables.
► The computation time is negligible even for real-life instances.