Robust combinatorial optimization with budget uncertainty is one of the most popular approaches for integrating uncertainty in optimization problems. The existence of a compact reformulation for (mixed-integer) linear programs and positive complexity results give the impression that these problems are relatively easy to solve. However, the practical performance of the reformulation is actually quite poor when solving robust integer problems due to its weak linear relaxation. To overcome the problems arising from the weak formulation, we propose a procedure to derive new classes of valid inequalities for robust binary optimization problems. For this, we recycle valid inequalities of the underlying deterministic problem such that the additional variables from the robust formulation are incorporated. The valid inequalities to be recycled may either be readily available model constraints or actual cutting planes, where we can benefit from decades of research on valid inequalities for classical optimization problems. We first demonstrate the strength of the inequalities theoretically, by proving that recycling yields a facet-defining inequality in surprisingly many cases, even if the original valid inequality was not facet-defining. Afterwards, we show in a computational study that using recycled inequalities leads to a significant improvement of the computation time when solving robust optimization problems.

Since its introduction in the early 2000s, robust optimization with budget uncertainty has received a lot of attention. This is due to the intuitive construction of the uncertainty sets and the existence of a compact robust reformulation for (mixed-integer) linear programs. However, despite its compactness, the reformulation performs poorly when solving robust integer problems due to its weak linear relaxation. To overcome the problems arising from the weak formulation, we propose a bilinear formulation for robust binary programming, which is as strong as theoretically possible. From this bilinear formulation, we derive strong linear formulations as well as structural properties for robust binary optimization problems, which we use within a tailored branch and bound algorithm. We test our algorithm’s performance together with other approaches from the literature on a diverse set of “robustified” real-world instances from the MIPLIB 2017. Our computational study, which is the first to compare many sophisticated approaches on a broad set of instances, shows that our algorithm outperforms existing approaches by far. Furthermore, we show that the fundamental structural properties proven in this paper can be used to substantially improve the approaches from the literature. This highlights the relevance of our findings, not only for the tested algorithms, but also for future research on robust optimization. To encourage the use of our algorithms for solving robust optimization problems and our instances for benchmarking, we make all materials freely available online.

Background: The healthcare sector poses many strategic, tactic and operational planning questions. Due to the historically grown structures, planning is often locally confined and much optimization potential is foregone. Methods: We implemented optimized decision-support systems for ambulatory care for four different real-world case studies that cover a variety of aspects in terms of planning scope and decision support tools. All are based on interactive cartographic representations and are being developed in cooperation with domain experts. The planning problems that we present are the problem of positioning centers for vaccination against Covid-19 (strategical) and emergency doctors (strategical/tactical), the out-of-hours pharmacy planning problem (tactical), and the route planning of patient transport services (operational). For each problem, we describe the planning question, give an overview of the mathematical model and present the implemented decision support application. Results: Mathematical optimization can be used to model and solve these planning problems. However, in order to convince decision-makers of an alternative solution structure, mathematical solutions must be comprehensible and tangible. Appealing and interactive decision-support tools can be used in practice to convince public health experts of the benefits of an alternative solution. The more strategic the problem and the less sensitive the data, the easier it is to put a tool into practice. Conclusions: Exploring solutions interactively is rarely supported in existing planning tools. However, in order to bring new innovative tools into productive use, many hurdles must be overcome.

Facility Location Problems with Capacities, Revenues, and Closest Assignments (FLP-CRCA) are an extension of the well known, strongly NP -complete Facility Location Problem (FLP). With this extension, we recognise that facilities have an upper capacity on the customers to be served, but also need to generate a minimum revenue to be operated economically. Furthermore, we acknowledge that customers have a strong preference towards their closest facility. We show that finding a feasible solution for FLP-CRCA is already strongly NP -complete if the underlying graph forms a star, but that the problem can be solved efficiently on paths and cycles. In the case where the number of facilities is fixed, we propose a pseudo-polynomial algorithm and show that the problem is weakly NP -complete under this condition. Our results also hold for FLPs with closest assignments and either capacities or revenues.

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 h 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. (C) 2020 Elsevier Ltd. All rights reserved.