Cyclic lot-sizing problems with sequencing costs

We study a single-machine lot-sizing problem, where n types of products need to be scheduled on the machine. Each product is associated with a constant demand rate, maximum production rate and inventory costs per time unit. Every time when the machine switches production between products, sequencing costs are incurred. These sequencing costs depend both on the product the machine just produced and on the product the machine is about to produce. The goal is to find a cyclic schedule minimizing total average costs, subject to the condition that all demands are satisfied. We establish the complexity of the problem, and we prove a number of structural properties largely characterizing optimal solutions. Moreover, we present two algorithms approximating the optimal schedules by augmenting the problem input. Due to the high-multiplicity setting, even trivial cases of the corresponding conventional counterparts become highly non-trivial with respect to the output sizes and computational complexity, even without sequencing costs. In particular, the length of an optimal solution can be exponential in the input size of the problem. Nevertheless, our approximation algorithms produce schedules of a polynomial length and with a good quality compared to the optimal schedules of exponential length.