Integrated production and delivery scheduling with disjoint windows

Consider a company that manufactures perishable goods. The company relies on a third party to deliver goods, which picks up delivery products at regular or irregular times. At each delivery time, there is a time window that products can be produced to be delivered at that delivery time. The time windows are disjoint. Suppose we have a set of jobs with each job specifying its delivery time, processing time and profit. The company can earn the profit if the job is produced and delivered at its specified delivery time. From the company point of view, we are interested in picking a subset of jobs to produce and deliver so as to maximize the total profit. The unpicked jobs will be discarded without penalty. We consider both the single machine case and the parallel and identical machine case. In this article we consider three kinds of profits: (1) arbitrary profit, (2) equal profit, and (3) profit proportional to its processing time. In the first case, we give a fully polynomial time approximation scheme (FPTAS) for a single machine with running time O (frac(n³, ε{lunate})). Using the bound improvement technique of Kovalyov, the running time can be further reduced to O (frac(n², ε{lunate}) + n² log n). In the second case, we give an O (n log n)-time optimal algorithm for a single machine. In the third case, we give an FPTAS for a single machine with running time O (frac(n², ε{lunate})). All of our algorithms can be extended to parallel and identical machines with a degradation of performance ratios.