Preemptive scheduling in a two-stage supply chain to minimize the makespan

This paper deals with the problem of preemptive scheduling in a two-stage supply chain framework. The supply chain environment contains two stages: production and transportation. In the production stage jobs are processed on a manufacturer's bounded serial batching machine, preemptions are allowed, and set-up time is required before a new batch is processed. In the transportation stage each batch is delivered to a customer by a single vehicle. The objective is to minimize the makespan by making decisions for both mutually coordinated stages. Specifically, two versions are studied. The first one is that all jobs are available to be processed at time zero, and the second one is that jobs have different release times. An time algorithm is developed for the first version, and we show that it can produce the optimal schedule for the entire problem. For the second version, based on some useful properties we have designed an time heuristic and a novel lower bound. The worst-case performance ratio of our algorithm is bounded by 2. Our computational study with random instances of different scales shows high-quality solutions for either small-scale or large-scale instances returned in a reasonable PC times.