We consider a sequencing problem that arises, for example, in the context of scheduling patients in particle therapy facilities for cancer treatment. A set of non-preemptive jobs needs to be scheduled, where each job requires two resources: (1) a common resource that is shared by all jobs and (2) a secondary resource, which is shared with only a subset of the other jobs. While the common resource is only required for a part of the job's processing time, the secondary resource is required for the whole duration. The objective is to minimize the makespan. First we show that the tackled problem is NP-hard and provide three different lower bounds for the makespan. These lower bounds are then exploited in a greedy construction heuristic and a novel exact anytime A algorithm, which uses an advanced diving mechanism based on Beam Search and Local Search to find good heuristic solutions early. For comparison we also provide a basic Constraint Programming model solved with the ILOG CP optimizer. An extensive experimental evaluation on two types of problem instances shows that the approach works even for large instances with up to 2000 jobs extremely well. It typically yields either optimal solutions or solutions with an optimality gap of less than 1%.

We gratefully acknowledge the financial support of this project by the Doctoral Program “Vienna Graduate School on Computational Optimization” funded by the Austrian Science Foundation

Peer reviewed