, Volume 5, Issue 4, pp 359–388 | Cite as

Probe Backtrack Search for Minimal Perturbation in Dynamic Scheduling

  • Hani El Sakkout
  • Mark Wallace


This paperdescribes an algorithm designed to minimally reconfigure schedulesin response to a changing environment. External factors havecaused an existing schedule to become invalid, perhaps due tothe withdrawal of resources, or because of changes to the setof scheduled activities. The total shift in the start and endtimes of already scheduled activities should be kept to a minimum.This optimization requirement may be captured using a linearoptimization function over linear constraints. However, the disjunctivenature of the resource constraints impairs traditional mathematicalprogramming approaches. The unimodular probing algorithm interleavesconstraint programming and linear programming. The linear programmingsolver handles only a controlled subset of the problem constraints,to guarantee that the values returned are discrete. Using probebacktracking, a complete, repair-based method for search, thesevalues are simply integrated into constraint programming. Unimodularprobing is compared with alternatives on a set of dynamic schedulingbenchmarks, demonstrating its effectiveness.

In the final discussion, we conjecture that analogous probebacktracking strategies may obtain performance improvements overconventional backtrack algorithms for a broad range of constraintsatisfaction and optimization problems.

scheduling constraint satisfaction constraint programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Abderrahmane and N. Beldiceanu. (1992). Extending CHIP in order to solve complex scheduling and placement problems. Premières Journées Francophones sur la Programation en Logique.Google Scholar
  2. 2.
    M. Bartusch, R. H. Möhring, and F. J. Radermacher. (1988). Scheduling project networks with resource constraints and time windows. Annals of Operations Research, 16: 201-240.Google Scholar
  3. 3.
    H. Beringer and B. de Backer. (1993). Satisfiability of boolean formulas over linear constraints. In IJCAI-93, pp. 296-301, Chambéry, France.Google Scholar
  4. 4.
    E. G. Coffman Jr., M. R. Garey, and D. S. Johnson. (1984). Approximation algorithms for bin-packing-an updated survey. In G. Ausiello, M. Lucertini, and P. Serafini, editors, Algorithm Design for Computer System Design, pp. 49-106. Springer-Verlag.Google Scholar
  5. 5.
    A. Davenport. (1998). Managing uncertainty in scheduling: a survey. Working Draft.Google Scholar
  6. 6.
    R. Dechter and A. Dechter. (1988). Belief maintenance in dynamic constraint networks. In Proceedings of AAAI-88, pp. 37-42.Google Scholar
  7. 7.
    R. Dechter and J. Pearl. (1988). Network-based heuristics for constraint satisfaction problems. Artificial Intelligence, 34.Google Scholar
  8. 8.
    R. Dechter, I. Meiri, and J. Pearl. (1991). Temporal constraint networks. Artificial Intelligence, 49: 61-95.Google Scholar
  9. 9.
    A. El-Kholy and B. Richards. (1996). Temporal and resource reasoning in planning: The parc PLAN approach. In Proceedings of the 11th European Conference on Artificial Intelligence, ECAI-96, pp. 614-618, Budapest, Hungary.Google Scholar
  10. 10.
    A. O. El-Kholy. (1996). Resource Feasibility in Planning. Ph.D. thesis, Imperial College, University of London.Google Scholar
  11. 11.
    H. El Sakkout, T. Richards, and M.Wallace. (1997). Unimodular probing for minimal perturbance in dynamic resource feasibility problems. In Proceedings of the CP97 workshop on Dynamic Constraint Satisfaction.Google Scholar
  12. 12.
    H. El Sakkout, T. Richards, and M. Wallace. (1998). Minimal perturbation in dynamic scheduling. In Proceedings of the 13th European Conference on Artificial Intelligence, ECAI-98, Brighton, UK, 1998.Google Scholar
  13. 13.
    M. S. Fox. (1987). Constraint-directed search: a case study of job-shop scheduling. Morgan Kaufmann Publishers Inc.Google Scholar
  14. 14.
    S. French. (1982). Sequencing and Scheduling: An introduction to the Mathematics of the Job-Shop. Ellis Horwood, England.Google Scholar
  15. 15.
    M. R. Garey and D. S. Johnson. (1979). Computers and Intractability: A Guide to the Theory of NPCompleteness. Bell Telephone Laboratories, Inc.Google Scholar
  16. 16.
    R. S. Garfinkel and G. L. Nemhauser. (1972). Integer Programming. John Wiley & Sons.Google Scholar
  17. 17.
    I. Heller and C.B.Tompkins. (1956). An extension of a theorem of Dantzig's. In Kuhn and Tucker, editors, Linear Inequalities and Related Systems, pp. 247-254. Princeton University Press.Google Scholar
  18. 18.
    D.W. Hildum. (1994). Flexibility in a Knowledge-Based System for Solving Dynamic Resource-Constrained Scheduling Problems. Ph.D. thesis, Dept. of Computer Science, University of Massachusetts, Amherst. 19. J. N. Hooker and M. A. Osorio. (1996). Mixed logical/linear programming. Discrete Applied Mathematics (to appear). Electronic copy available from first author's home page: <>.Google Scholar
  19. 20.
    N. Karmarkar. (1984). A new polynomial-time algrithm for linear programming. Combinatorica, 4(4): 373-395.Google Scholar
  20. 21.
    L. G. Khachian. (1979).Apolynomial algorithm in linear programming. Soviet Math. Dokl., 20(1): 191-194.Google Scholar
  21. 22.
    C. Le Pape, P. Couronné, D. Vergamini, and V. Gosselin. (1994). Time-versus-capacity compromises in project scheduling. In Proceedings of the 13th Workshop of the UK Planning and Scheduling SIG, Glasgow, Scotland.Google Scholar
  22. 23.
    V. J. Leon, S. D. Wu, and R. H. Storer. (1994). Robustness measures and robust scheduling for job shop. IIE Transactions, 26(5): 32-43.Google Scholar
  23. 24.
    Olivier Lhomme. (1993). Consistency techniques for numeric csps. In Proceedings of the 13th International Joint Conference on Artificial Intelligence, IJCAI-93, pp. 232-238, Chambéry, France.Google Scholar
  24. 25.
    V. Liatsos. (1998). Short term scheduling. Presentation at the DIMACS Workshop on Constraint Programming and Large-Scale Discrete Optimization, Rutgers University, NJ.Google Scholar
  25. 26.
    S. Minton, M. D. Johnston, A. B. Philips, and P. Laird. (1992). Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems. Artificial Intelligence, 58: 161-205.Google Scholar
  26. 27.
    W. Nuijten and E. Aarts. (1994). Constraint satisfaction for multiple capacitated job shop scheduling. In Proceedings of the 11th European Conference on Artificial Intelligence, ECAI-94. JohnWiley & Sons, Ltd.Google Scholar
  27. 28.
    Wim Nuijten. (1994). Time and Resource Constrained Scheduling: A constraint satisfaction approach. PhD thesis, Eindhoven University of Technology.Google Scholar
  28. 29.
    D. Pothos. (1997). A constraint-based approach to the british airways schedule re-timing problem. Technical Report 97/04-01, IC-Parc, Imperial College.Google Scholar
  29. 30.
    P. W. Purdom, Jr. and G. N. Haven. (1997). Probe order backtracking. Siam Journal of Computing, 26: 456-483.Google Scholar
  30. 31.
    M. Queyranne and Y. Wang. (1991). Single-machine scheduling polyhedra with precedence constraints. Mathematics of Operations Research, pp. 1-20.Google Scholar
  31. 32.
    N. Sadeh. (1994). Micro-opportunistic scheduling: The micro-boss factory scheduler. In M. Zweben and M. Fox, editors, Intelligent Scheduling, chapter 4, pp. 99-136. Morgan Kaufman.Google Scholar
  32. 33.
    G. Verfaillie and T. Schiex. (1994). Solution reuse in dynamic constraint satisfaction problems. In AAAI-94, pp. 307-312, Seattle, WA.Google Scholar
  33. 34.
    M. Yokoo. (1994). Weak-commitment search for solving constraint satisfaction problems. In AAAI-94, pp. 313-318, Seattle, WA.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Hani El Sakkout
    • 1
  • Mark Wallace
    • 1
  1. 1.IC-Parc, Imperial CollegeLondon

Personalised recommendations