We consider a resource‐constrained project scheduling problem originating in particle therapy for cancer treatment, in which the scheduling has to be done in high resolution. Traditional mixed integer linear programming techniques such as time‐indexed formulations or discrete‐event formulations are known to have severe limitations in such cases, that is, growing too fast or having weak linear programming relaxations. We suggest a relaxation based on partitioning time into so‐called time‐buckets. This relaxation is iteratively solved and serves as basis for deriving feasible solutions using heuristics. Based on these primal and dual solutions and bounds, the time‐buckets are successively refined. Combining these parts, we obtain an algorithm that provides good approximate solutions soon and eventually converges to an optimal solution. Diverse strategies for performing the time‐bucket refinement are investigated. The approach shows excellent performance in comparison to the traditional formulations and a metaheuristic.
The particle therapy patient scheduling problem (PTPSP) arises in modern cancer treatment facilities that provide particle therapy. It consists of scheduling a set of therapies within a planning horizon of several months. A particularity of PTSP compared with classical radiotherapy scheduling is that therapies need not only be assigned to days but also scheduled within each day to account for the more complicated operational scenario. In an earlier work, we introduced this novel problem setting and provided first algorithms including an iterated greedy (IG) metaheuristic. In this work, we consider an important extension to the PTPSP emerging from practice in which the therapies should be provided on treatment days roughly at the same time. To be more specific, the variation between the starting times of the therapies' individual treatments should not exceed the given limits, and needs otherwise to be minimized. This additional constraint implies that the sequencing parts within each day can no longer be treated independently. To tackle this variant of PTPSP, we revise our previous IG and exchange its main components: the part of the applied construction heuristic for scheduling within the days and the local search algorithm. The resulting metaheuristic provides promising results for the proposed extension of the PTPSP and further enhances the existing approach for the original problem.
Multivalued decision diagrams (MDD) are a powerful tool for approaching combinatorial optimization problems. Relatively compact relaxed and restricted MDDs are applied to obtain dual bounds and heuristic solutions and provide opportunities for new branching schemes. We consider a prize-collecting sequencing problem in which a subset of given jobs has to be found that is schedulable and yields maximum total prize. The primary aim of this work is to study different methods for creating relaxed MDDs for this problem. To this end, we adopt and extend the two main MDD compilation approaches found in the literature: top down construction and incremental refinement. In a series of computational experiments these methods are compared. The results indicate that for our problem the incremental refinement method produces MDDs with stronger bounds. Moreover, heuristic solutions are derived by compiling restricted MDDs and by applying a general variable neighborhood search (GVNS). Here we observe that the top down construction of restricted MDDs is able to yield better solutions as the GVNS on small to medium-sized instances.
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