Job Sequencing with One Common and Multiple Secondary Resources: A Problem Motivated from Particle Therapy for Cancer Treatment

  • Matthias HornEmail author
  • Günther Raidl
  • Christian Blum
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10710)


We consider in this work the problem of scheduling a set of jobs without preemption, where each job requires two resources: (1) a common resource, shared by all jobs, is required during a part of the job’s processing period, while (2) a secondary resource, which is shared with only a subset of the other jobs, is required during the job’s whole processing period. This problem models, for example, the scheduling of patients during one day in a particle therapy facility for cancer treatment. First, we show that the tackled problem is NP-hard. We then present a construction heuristic and a novel A* algorithm, both on the basis of an effective lower bound calculation. For comparison, we also model the problem as a mixed-integer linear program (MILP). An extensive experimental evaluation on three types of problem instances shows that A* typically works extremely well, even in the context of large instances with up to 1000 jobs. When our A* does not terminate with proven optimality, which might happen due to excessive memory requirements, it still returns an approximate solution with a usually small optimality gap. In contrast, solving the MILP model with the MILP solver CPLEX is not competitive except for very small problem instances.


  1. 1.
    Allahverdi, A.: A survey of scheduling problems with no-wait in process. Eur. J. Oper. Res. 255(3), 665–686 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Conforti, D., Guerriero, F., Guido, R.: Optimization models for radiotherapy patient scheduling. 4OR 6(3), 263–278 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Co., New York (1979)zbMATHGoogle Scholar
  4. 4.
    Gilmore, P.C., Gomory, R.E.: Sequencing a one-state variable machine: a solvable case of the traveling salesman problem. Oper. Res. 12(5), 655–679 (1964)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hart, P., Nilsson, N., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern. 4(2), 100–107 (1968)CrossRefGoogle Scholar
  6. 6.
    Hartmann, S., Briskorn, D.: A survey of variants and extensions of the resource-constrained project scheduling problem. Eur. J. Oper. Res. 207(1), 1–14 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kapamara, T., Sheibani, K., Haas, O., Petrovic, D., Reeves, C.: A review of scheduling problems in radiotherapy. In: Proceedings of the International Control Systems Engineering Conference, pp. 207–211. Coventry University Publishing, Coventry (2006)Google Scholar
  8. 8.
    Maschler, J., Riedler, M., Stock, M., Raidl, G.R.: Particle therapy patient scheduling: first heuristic approaches. In: PATAT 2016: Proceedings of the 11th International Conference of the Practice and Theory of Automated Timetabling, Udine, Italy, pp. 223–244 (2016)Google Scholar
  9. 9.
    Röck, H.: The three-machine no-wait flow shop is NP-complete. J. ACM 31(2), 336–345 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Van der Veen, J.A.A., Wöginger, G.J., Zhang, S.: Sequencing jobs that require common resources on a single machine: a solvable case of the TSP. Math. Program. 82(1–2), 235–254 (1998)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Computer Graphics and AlgorithmsTU WienViennaAustria
  2. 2.Artificial Intelligence Research Institute (IIIA-CSIC)BellaterraSpain

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