A Dynamic Programming Solution to the Dynamic, Multi-Item, Single-Machine Scheduling Problem

Abstract

This article presents a dynamic programming algorithm for scheduling, on a single machine, production of multiple items with time-varying deterministic demands. We formulate the scheduling problem with the objective of minimizing the sum of changeover and inventory holding costs. The formulation is appealing in that it represents changeover costs directly instead of by the familiar approximate technique of including setup costs in the objective. Our algorithm, which we developed using an approach similar to C. R. Glassey's that minimizes the total number of changeovers, casts the optimal schedule as a shortest path through a network embedded in a state space. It generates optimal schedules under two assumptions. First, we assume that in each time period within the planning horizon, the machine must either be shut down or be producing some one item for the entire time period. Second, we assume that inventory holding costs are representable as a nondecreasing function of aggregate inventory. We provide a number of numerical examples that we solved using the algorithm.