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{\textquoteright}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\%.\

}, doi = {https://doi.org/10.1016/j.artint.2019.103173}, url = {https://authors.elsevier.com/c/1ZoNa-c5JpQT}, author = {Horn, Matthias and Raidl, G{\"u}nther R and Blum, Christian} } @proceedings {56012, title = {Job Sequencing with One Common and Multiple Secondary Resources: A Problem Motivated from Particle Therapy for Cancer Treatment}, journal = {MOD 2017 -- The Third International Conference on Machine Learning, Optimization and Big Data}, year = {2017}, publisher = {Springer Verlag}, abstract = {We consider in this work the problem of scheduling a set of jobs without preemption, where each job requires two resources: (1) a common resource, shared by all jobs, is required during a part of the job{\textquoteright}s processing period, while (2) a secondary resource, which is shared with only a subset of the other jobs, is required during the job{\textquoteright}s whole processing period. This problem models, for example, the scheduling of patients during one day in a particle therapy facility for cancer treatment. First, we show that the tackled problem is NP-hard. We then present a construction heuristic and a novel A* algorithm, both on the basis of an effective lower bound calculation. For comparison, we also model the problem as a mixed-integer linear program (MILP). An extensive experimental evaluation on three types of problem instances shows that A* typically works extremely well, even in the context of large instances with up to 1000 jobs. When our A* does not terminate with proven optimality, which might happen due to excessive memory requirements, it still returns an approximate solution with a usually small optimality gap. In contrast, solving the MILP model with CPLEX is not competitive except for very small problem instances.

}, author = {Horn, Matthias and Raidl, G{\"u}nther R and Blum, Christian} }