Strategies for LP-based solving a general class of scheduling problems

  • L. F. Escudero
  • G. Pérez


In this work we describe some strategies that-have been proved to be very efficient for solving the following type of schedling problems: Assume a set of jobs is to be performed along a planning horizon by selecting one from several alternatives for doing so. Besides selecting the alternative for each job, the target consists of choosing the periods at which each component of the job will be done, such that a set of scheduling and technological constraints is satisfied. The problem is formulated as a large-scale pure 0–1 model. Three facts are observed: (1) The number of variables with nonzero value at each feasible solution is much more smaller that the total number of variables; (2) For some types of objectives (e.g., makespan minimizing), each incumbent solution allows for a problem’s reduction without eliminating any better solution; (3) Initial feasible solutions can be found, by means of an heuristic procedure, without great difficulty. The three above characteristics allow for a modification on the traditional using of the branch-and-bound methods and, hence, increase the problem’s dimensions that could be dealt with at a reasonable computing effort. Computational experience on a broad set of real-life problems is reported.

Key words

scheduling large-scale 0–1 model variable fixing coefficient reduction special ordered sets 


  1. [1]
    K. R. BAKER (1974):Introduction to sequencing and scheduling, Wiley, N.Y.Google Scholar
  2. [2]
    E. M. L. BEALE and J. J. H. FORREST (1976): «Global optimization using special ordered sets»,Mathematical Programming, 10, 52–69.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    E. M. L. BEALE and J. A. TOMLIN (1970): «Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables», in: J. LAWRENCE (ed.),Operational Research’69, Tavistock Publishing, London, 447–454.Google Scholar
  4. [4]
    H. CROWDER, E. L. JOHNSON and M. PADBERG (1983): «Solving large-scale zero-one linear programming problems»,Operations Research, 31, 803–834.MATHGoogle Scholar
  5. [5]
    L. F. ESCUDERO (1979): «Special sets in mathematical programming, State-of-the-art survey», in: E. ALARCON and C. BREBBIA (eds.),Applied Numerical Modeling, Pentech Press, London, 535–551.Google Scholar
  6. [6]
    L. F. ESCUDERO (1981):On energy generators maintenance and operations scheduling, IBM Scientific Center report G320-3419, Palo Alto, California.Google Scholar
  7. [7]
    L. F. ESCUDERO (1982): «On maintenance scheduling of production units»,European J. of Operational Research, 9, 264–274.MATHCrossRefGoogle Scholar
  8. [8]
    L. F. ESCUDERO (1985):On sequencing and scheduling problems, School on Combinatorial Programming, Rio de Janeiro.Google Scholar
  9. [9]
    L. F. ESCUDERO (1989): «A production planning problem in FMS»,Annals of Operations Research, 17, 69–104.MATHCrossRefGoogle Scholar
  10. [10]
    J. J. H. FORREST, F. P. H. HIRST and J. A. TOMLIN (1974): «A practical solution of large and complex integer programming problems with Umpire»,Management Science, 20, 736–773.MATHMathSciNetGoogle Scholar
  11. [11]
    M. GROTSCHEL and M. W. PADBERG (1985): «Polyhedral theory and Polyhedral computations», in: E. L. LAWLER, J. K. LENSTRA, A. H. G. RINNOOY-KAN and D. B. SHMOYS (eds.),The Traveling Salesman Problem, A guided tour of combinatorial optimization, Wiley, N.Y., 251–360.Google Scholar
  12. [12]
    M. GUIGNARD and K. SPIELBERG (1977): «Propagation, penalty improvement and use of logical inequalities»,Mathematics of Operations Research, 25, 157–171.Google Scholar
  13. [13]
    K. HOFFMAN and M. W. PADBERG (1986): «LP-based combinatorial problem solving»,Annals of Operations Research, 5, 145–194.MathSciNetGoogle Scholar
  14. [14]
    IBM (1988). MPSX/370, Mathematical Programming System Extended/370; Ref. manuals. SH19-6553.Google Scholar
  15. [15]
    E. L. JOHNSON, M. M. KOSTREVA and U. SUHL (1985): «Solving 0–1 integer programming problems arising from large-scale planning models»,Operations Research, 33, 803–819.MATHGoogle Scholar
  16. [16]
    A. LAND and S. POWELL (1979): «Computer codes for problems of integer programming», in: P. HAMMER, E. L. JOHNSON and B. H. KORTE (eds.),Discrete Optimization 2 (North-Holland, The Netherlands), 221–269.Google Scholar
  17. [17]
    E. L. LAWLER, J. K. LENSTRA and A. H. G. RINNOOY KAN (1982): «Recent developments in deterministic sequencing and scheduling: a survey», in: M. A. H. DEMPSTER, J. K. LENSTRA and A. H. G. RINNOOY KAN (eds.),Deterministic and stochastic scheduling, Reidel, Dordrecht, 35–73.Google Scholar
  18. [18]
    M. W. PADBERG and G. RINALDI (1987): «Optimization of a 532-city symmetric traveling salesman problem by branch and cut»,Operations Research Letters, 6, 1–7.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    A. H. G. RINNOOY KAN (1976):Machine scheduling problems: classification, complexity and computation, Nijhoff, The Hague.Google Scholar
  20. [20]
    J. A. TOMLIN (1988): «Special Ordered Sets and an application to gas supply operations planning».Mathematical Programming, 42, 69–84.CrossRefMathSciNetGoogle Scholar
  21. [21]
    T. J. VAN ROY AND L. A. WOLSEY (1987): «Solving mixed integer programming problems using automatic reformulation»,Operations Research, 35, 45–57.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© SEIO 1990

Authors and Affiliations

  • L. F. Escudero
    • 1
  • G. Pérez
    • 2
  1. 1.IBM T. J. Watson Research CenterYorktown HeightsUSA
  2. 2.Fac. de Ciencias Univ. del País Vasco (Lejona)País Vasco (Lejona)Spain

Personalised recommendations