Minimizing Makespan in Flowshops with Pallet Requirements

We treat the problem of scheduling a owshop with pallet requirements. Each job in the owshop needs a pallet the entire time, from the start of its rst operation until the completion of the last operation, and the number of pallets in the shop at any given time is limited by an integer K. We prove that the problem is NP-hard in the strong sense for m 2 and K 3, and for m 3 and K 2, where m is the number of machines. We focus on the impact of the number of pallets on makespan. While it is an NP-hard problem to nd the minimum number of pallets subject to an upper bound on makespan, we prove a worst-case bound on the minimum K that guarantees the least possible makespan in a two-machine owshop. We propose several heuristic algorithms to solve the scheduling problem, and using simulation we study the impact of K and the performance of the proposed algorithms. We nd that the Gilmore-Gomory algorithm performs very well in a two-machine owshop if K 4, and the well-known Nawaz-Enscore-Ham sequencing heuristic is superior to other proposed methods in general. We further conclude that an appropriate number for K is between 1:5m and 2m in the sense that the marginal reduction in makespan is negligible when K increases beyond 2m.