The supply of pharmaceuticals is one important factor in a functioning health care system. In the German health care system, the chambers of pharmacists are legally obliged to ensure that every resident can find an open pharmacy at any day and night time within an appropriate distance. To that end, the chambers of pharmacists create an out-of-hours plan for a whole year in which every pharmacy has to take over some 24 hours shifts. These shifts are important for a reliable supply of pharmaceuticals in the case of an emergency but also unprofitable and stressful for the pharmacists. Therefore, an efficient planning that meets the needs of the residents and reduces the load of shifts on the pharmacists is crucial. In this paper, we present a model for the assignment of out-of-hours services to pharmacies, which arises from a collaboration with the Chamber of Pharmacists North Rhine. Since the problem, which we formulate as an MILP, is very hard to solve for large-scale instances, we propose several tailored solution approaches. We aggregate mathematically equivalent pharmacies in order to reduce the size of the MILP and to break symmetries. Furthermore, we use a rolling horizon heuristic in which we decompose the planning horizon into a number of intervals on which we iteratively solve subproblems. The rolling horizon algorithm is also extended by an intermediate step in which we discard specific decisions made in the last iteration. A case study based on real data reveals that our approaches provide nearly optimal solutions. The model is evaluated by a detailed analysis of the obtained out-of-hours plans.
Preprint Lehrstuhl II für Mathematik, RWTH Aachen University