Annals of Operations Research

, Volume 118, Issue 1–4, pp 35–48 | Cite as

A Probe-Based Algorithm for Piecewise Linear Optimization in Scheduling

  • Farid Ajili
  • Hani El Sakkout


A scheduling problem with piecewise linear (PL) optimization extends conventional scheduling by imposing a conjunction of combinatorial PL constraints involving the objective function variables. To solve this problem, this paper presents a hybrid algorithm where Constraint Programming (CP) search is supported and driven by a (integer) linear programming solver running on a well-controlled subproblem which is dynamically tightened. The paper discusses and compares different ways of decomposing the problem constraints between the CP search and the solver. We show how the subproblem structure and the piecewise linearity are exploited by the search.

linear programming constraint programming scheduling hybrid algorithms 


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  1. [1]
    A. Aggoun and N. Beldiceanu, Extending CHIP in order to solve complex scheduling and placement problems, Journal of Mathematical Computer Modelling 17(7) (1993) 57–73.Google Scholar
  2. [2]
    M. Bartusch, R.H. Möhring and F.J. Radermacher, Scheduling project networks with resource constraints and time windows, Annals of Operations Research 16 (1988) 201–240.Google Scholar
  3. [3]
    E.M.L. Beale and J.A. Tomlin, Special facilities in a general mathematical system for non-convex problems using ordered sets of variables, in: Proc. of the Fifth International Conference on Operations Research (Tavistock Publications, London, 1970) pp. 447–445.Google Scholar
  4. [4]
    D. Bertsimas, Ch. Darnell and R. Soucy, Portfolio construction through mixed-integer programming at Grantham, Mayo, van Otterloo and Co, Interfaces 29(1) (1999) 49–66.Google Scholar
  5. [5]
    D. Chapman, Planning for conjunctive goals, Artificial Intelligence 32 (1987) 333–377.Google Scholar
  6. [6]
    CPLEX, CPLEX Manual (1999), URL: Scholar
  7. [7]
    G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).Google Scholar
  8. [8]
    R. Dechter, I. Meiri and J. Pearl, Temporal constraint networks, Artificial Intelligence 49 (1991) 61–95.Google Scholar
  9. [9]
    H. El Sakkout and M. Wallace, Probe backtrack search for minimal perturbation in dynamic scheduling, Constraints 5(4), Special Issue on Industrial Constraint-Directed Scheduling (2000).Google Scholar
  10. [10]
    R. Fourer, A Simplex algorithm for piecewise-linear programming III: Computational analysis and applications, Mathematical Programming 53 (1992) 213–235.Google Scholar
  11. [11]
    R. Fourer and D.M. Gay, Expressing special structures in an algebraic modeling language for mathematical programming, ORSA Journal on Computing 7 (1995) 166–190.Google Scholar
  12. [12]
    J.K. Ho, A successive linear optimization approach to the dynamic traffic assignment problem, Transportation Science 14 (1980) 295–305.Google Scholar
  13. [13]
    J.K. Ho, Relationships among linear formulations of separable convex piecewise-linear programs, Mathematical Programming Study 24 (1985) 126–140.Google Scholar
  14. [14]
    J.P. Ignizio, Goal Programming and Extensions (Lexington Books, Lexington, MA, 1976).Google Scholar
  15. [15]
    G.L. Nemhauser, E.L. Johnson and I.R. de Farias, A generalized assignment problem with special ordered sets: A polyhedral approach, to appear in the Journal of Mathematical Programming.Google Scholar
  16. [16]
    G. Ottosson, E.S. Thorsteinsson and J.N. Hooker, Mixed global constraints and inference in CLP-IP solvers, Annals of Mathematics and Artificial Intelligence 34(4) (2002) 271–290.Google Scholar
  17. [17]
    P. Refalo, Tight cooperation and its application in piecewise-linear optimization, in: Proc. of the 5th International Conference of Principles and Practice of Constraint Programming (CP'99), Alexandria, VA, October 1999, Lecture Notes in Computer Science, Vol. 1713 (Springer, New York, 1999) pp. 375–389Google Scholar
  18. [18]
    M.J. Schniederjans, Goal Programming: Methodology and Applications (Kluwer Academic, Boston, MA, 1995).Google Scholar
  19. [19]
    H.P. Williams, Model Building in Mathematical Programming, 3rd revised edn. (Wiley, New York, 1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Farid Ajili
    • 1
  • Hani El Sakkout
    • 2
  1. 1.IC-ParcImperial CollegeLondonUnited Kingdom
  2. 2.Parc Technologies LtdLondonUnited Kingdom

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