Abstract
We consider a sequencing problem with time windows, in which a subset of a given set of jobs shall be scheduled. A scheduled job has to execute without preemption and during this time, the job needs both a common resource for a part of the execution as well as a secondary resource for the whole execution time. The common resource is shared by all jobs while a secondary resource is shared only by a subset of the jobs. Each job has one or more time windows and due to these, it is not possible to schedule all jobs. Instead, each job is associated with a prize and the task is to select a subset of jobs which yields a feasible schedule with a maximum sum of prizes. First, we argue that the problem is NP-hard. Then, we present an exact A* algorithm and derive different upper bounds for the total prize; these bounds are based on constraint and Lagrangian relaxations of a linear programming relaxation of a multidimensional knapsack problem. For comparison, a compact mixed integer programming (MIP) model and a constraint programming model are also presented. An extensive experimental evaluation on three types of problem instances shows that the A* algorithm outperforms the other approaches and is able to solve small to medium size instances with up to about 40 jobs to proven optimality. In cases where A* does not prove that an optimal solution is found, the obtained upper bounds are stronger than those of the MIP model.
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This project is partially funded by the Doctoral Program “Vienna Graduate School on Computational Optimization”, Austrian Science Foundation (FWF) Project No. W1260-N35. The work of Elina Rönnberg is supported by the Center for Industrial Information Technology (CENIIT), Project-ID 16.05. We further thank Lukas Felician Krasel for his help in the implementation and testing.
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Horn, M., Raidl, G.R. & Rönnberg, E. A* Search for Prize-Collecting Job Sequencing with One Common and Multiple Secondary Resources. Ann Oper Res 302, 477–505 (2021). https://doi.org/10.1007/s10479-020-03550-7
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DOI: https://doi.org/10.1007/s10479-020-03550-7