In this paper, we describe a branch-and-price algorithm for the personalized nurse scheduling problem. The variants that appear in the literature involve a large number of constraints that can be hard or soft, meaning that they can be violated at the price of a penalty. We capture the diversity of the constraints on individual schedules by seven generic constraints characterized by lower and upper bounds on a given quantity. The core of the column generation procedure is in the identification of indivual schedules with minimum reduced cost. For this we solve a shortest path problem with resource constraints (SPPRC) where several generic constraints are modeled as resource constraints. We then describe dominance rules adapted to the presence of both upper and lower bounds on the resources and leverage soft constraints to improve the dominance. We also describe several acceleration techniques for the solution of the SPPRC, and branching rules that fit the specificities of the problem. Our numerical experiments are based on the instances of three benchmarks of the literature including those of the two international nurse rostering competitions (INRC-I and INRC-II). Their objective is three-fold: assess the dominance rules and the acceleration techniques, investigate the capacity of the algorithm to find provable optimal solutions of instances that are still open, and conduct a comparison to best published results. The most noticeable conclusion is that the improved solution of the SPPRC allows to solve optimally all the INRC-II instances where a 4-weeks planning horizon is considered and 40% of the 8-weeks instances.
Paru en janvier 2023 , 29 pages
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