Scheduling Optional Tasks with Explanation

  • Andreas Schutt
  • Thibaut Feydy
  • Peter J. Stuckey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8124)


Many scheduling problems involve reasoning about tasks which may or may not actually occur, so called optional tasks. The state-of-the-art approach to modelling and solving such problems makes use of interval variables which allow a start time of \(\bot\) indicating the task does not run. In this paper we show we can model interval variables in a lazy clause generation solver, and create explaining propagators for scheduling constraints using these interval variables. Given the success of lazy clause generation on many scheduling problems, this combination appears to give a powerful new solving approach to scheduling problems with optional tasks. We demonstrate the new solving technology on well-studied flexible job-shop scheduling problems where we are able to close 36 open problems.


Schedule Problem Interval Variable Constraint Programming Constraint Satisfaction Problem Boolean Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barták, R., Čepek, O.: Temporal networks with alternatives: Complexity and model. In: Proceedings of the Twentieth International Florida AI Research Society Conference (FLAIRS), pp. 641–646 (2007)Google Scholar
  2. 2.
    Beck, J.C., Fox, M.S.: Scheduling alternative activities. In: Proceedings of the National Conference on Artificial Intelligence, pp. 680–687. John Wiley & Sons, Ltd. (1999)Google Scholar
  3. 3.
    Behnke, D., Geiger, M.J.: Test instances for the flexible job shop scheduling problem with work centers. Research Report RR-12-01-01, Helmut-Schmidt-Universität, Hamburg, Germany (2012)Google Scholar
  4. 4.
    Brandimarte, P.: Routing and scheduling in a flexible job shop by tabu search. Annals of Operations Research 41(3), 157–183 (1993)CrossRefzbMATHGoogle Scholar
  5. 5.
    Brucker, P., Schlie, R.: Job-shop scheduling with multi-purpose machines. Computing 45(4), 369–375 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Feydy, T., Somogyi, Z., Stuckey, P.J.: Half reification and flattening. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 286–301. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Feydy, T., Stuckey, P.J.: Lazy clause generation reengineered. In: Gent [8], pp. 352–366Google Scholar
  8. 8.
    Gent, I.P. (ed.): CP 2009. LNCS, vol. 5732. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  9. 9.
    Hurink, J., Jurisch, B., Thole, M.: Tabu search for the job-shop scheduling problem with multi-purpose machines. Operations-Research-Spektrum 15(4), 205–215 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jurisch, B.: Scheduling jobs in shops with multi-purpose machines. Ph.D. thesis, Universität Osnabrück (1992)Google Scholar
  11. 11.
    Laborie, P.: IBM ILOG CP Optimizer for detailed scheduling illustrated on three problems. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 148–162. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Laborie, P., Rogerie, J.: Reasoning with conditional time-intervals. In: Wilson, D.C., Lane, H.C. (eds.) Proceedings of the Twenty-First International Florida Artificial Intelligence Research Society Conference, pp. 555–560. AAAI Press (2008)Google Scholar
  13. 13.
    Laborie, P., Rogerie, J., Shaw, P., Vilím, P.: Reasoning with conditional time-intervals part II: An algebraical model for resources. In: Lane, H.C., Guesgen, H.W. (eds.) Proceedings of the Twenty-First International Florida Artificial Intelligence Research Society Conference, pp. 201–206. AAAI Press (2009)Google Scholar
  14. 14.
    Laborie, P., Rogerie, J., Shaw, P., Vilím, P., Katai, F.: Interval-based language for modeling scheduling problems: An extension to constraint programming. In: Kallrath, J. (ed.) Algebraic Modeling Systems. Applied Optimization, vol. 104, pp. 111–143. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Mastrolilli, M., Gambardella, L.M.: Effective neighbourhood functions for the flexible job shop problem. Journal of Scheduling 3(1), 3–20 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moffitt, M.D., Peintner, B., Pollack, M.E.: Augmenting disjunctive temporal problems with finite-domain constraints. In: Proceedings of the National Conference on Artificial Intelligence, pp. 1187–1192. AAAI Press (2005)Google Scholar
  17. 17.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of Design Automation Conference – DAC 2001, pp. 530–535. ACM, New York (2001)Google Scholar
  18. 18.
    Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G.R.: MiniZinc: Towards a standard CP modelling language. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 529–543. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Ohrimenko, O., Stuckey, P.J., Codish, M.: Propagation via lazy clause generation. Constraints 14(3), 357–391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pacino, D., Van Hentenryck, P.: Large neighborhood search and adaptive randomized decompositions for flexible jobshop scheduling. In: Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, IJCAI 2011, pp. 1997–2002. AAAI Press (2011)Google Scholar
  21. 21.
    Schulte, C., Stuckey, P.J.: Efficient constraint propagation engines. ACM Transactions on Programming Languages and Systems 31(1), Article No. 2 (2008)Google Scholar
  22. 22.
    Schutt, A., Chu, G., Stuckey, P.J., Wallace, M.G.: Maximising the net present value for resource-constrained project scheduling. In: Beldiceanu, N., Jussien, N., Pinson, É. (eds.) CPAIOR 2012. LNCS, vol. 7298, pp. 362–378. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Schutt, A., Feydy, T., Stuckey, P.J.: Explaining time-table-edge-finding propagation for the cumulative resource constraint. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 234–250. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Schutt, A., Feydy, T., Stuckey, P.J., Wallace, M.G.: Explaining the cumulative propagator. Constraints 16(3), 250–282 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schutt, A., Feydy, T., Stuckey, P.J., Wallace, M.G.: Solving RCPSP/max by lazy clause generation. Journal of Scheduling 16(3), 273–289 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Seipel, D., Hanus, M., Wolf, A. (eds.): INAP 2007. LNCS, vol. 5437. Springer, Heidelberg (2009)Google Scholar
  27. 27.
    Stuckey, P.J., de la Banda, M.G., Maher, M.J., Marriott, K., Slaney, J.K., Somogyi, Z., Wallace, M., Walsh, T.: The G12 project: Mapping solver independent models to efficient solutions. In: Gabbrielli, M., Gupta, G. (eds.) ICLP 2005. LNCS, vol. 3668, pp. 9–13. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  28. 28.
    Vilím, P.: Edge finding filtering algorithm for discrete cumulative resources in \({\mathcal O}(kn\log n)\). In: Gent [8], pp. 802–816Google Scholar
  29. 29.
    Vilím, P.: Max energy filtering algorithm for discrete cumulative resources. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 294–308. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  30. 30.
    Vilím, P., Barták, R., Čepek, O.: Extension of O(n logn) filtering algorithms for the unary resource constraint to optional activities. Contraints 10(4), 403–425 (2005)CrossRefzbMATHGoogle Scholar
  31. 31.
    Walsh, T.: Search in a small world. In: Proceedings of Artificial intelligence – IJCAI 1999, pp. 1172–1177. Morgan Kaufmann (1999)Google Scholar
  32. 32.
    Yuan, Y., Xu, H.: Flexible job shop scheduling using hybrid differential evolution algorithms. Computers & Industrial Engineering 65(2), 246–260 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Andreas Schutt
    • 1
    • 2
  • Thibaut Feydy
    • 1
    • 2
  • Peter J. Stuckey
    • 1
    • 2
  1. 1.Optimisation Research GroupNational ICT AustraliaAustralia
  2. 2.Department of Computing and Information SystemsThe University of MelbourneAustralia

Personalised recommendations