Skip to main content
SpringerLink
Log in

New constructs for the description of combinatorial optimization problems in algebraic modeling languages

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Algebraic languages are at the heart of many successful optimization modeling systems, yet they have been used with only limited success for combinatorial (or discrete) optimization. We show in this paper, through a series of examples, how an algebraic modeling language might be extended to help with a greater variety of combinatorial optimization problems. We consider specifically those problems that are readily expressed as the choice of a subset from a certain set of objects, rather than as the assignment of numerical values to variables. Since there is no practicable universal algorithm for problems of this kind, we explore a hybrid approach that employs a general-purpose subset enumeration scheme together with problem-specific directives to guide an efficient search.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E.M.L. Beale and J.A. Tomlin, “Special facilities in a general mathematical programming system for nonconvex problems using ordered sets of variables,” in OR 69: Proceedings of the Fifth International Conference on Operational Research (J. Lawrence, Ed.), Tavistock Publications: London, pp. 447–454, 1970.

    Google Scholar 

  2. J.J. Bisschop and R. Entriken, “AIMMS: The modeling system,” Paragon Decision Technology, Haarlem, The Netherlands, 1993.

    Google Scholar 

  3. J.J. Bisschop and A. Meeraus, “On the development of a general algebraic modeling system in a strategic planning environment,” Mathematical Programming Study, vol. 20, pp. 1–29, 1982.

    Google Scholar 

  4. A. Brooke, D. Kendrick, and A. Meeraus, GAMS: A User's Guide, release 2.25, Boyd & Fraser/The Scientific Press: Danvers, MA, 1992.

    Google Scholar 

  5. Chesapeake Decision Sciences, MIMI/LP User's Manual, New Providence, NJ, 1992.

  6. A. Colmerauer, “An introduction to prolog III,” Communications of the ACM, vol. 33, pp. 69–90, 1990.

    Google Scholar 

  7. CPLEX Optimization, Inc., Using the CPLEX Callable Library, version 3.0, Incline Village, NV, 1994.

  8. M. Dinchas, H. Simonis, and P.Van Hentenryck, “Solving large combinatorial problems in logic programming,” Journal of Logic Programming, vol. 8, pp. 75–93, 1990.

    Google Scholar 

  9. S.P. Dirkse, “Robust solution of mixed complementarity problems,” Mathematical Programming Technical Report 94–12, University of Wisconsin, Madison, 1994.

    Google Scholar 

  10. S.P. Dirkse, M.C. Ferris, P.V. Preckel, and T. Rutherford, “The GAMS callable program library for variational and complementarity solvers,” Mathematical Programming Technical Report 94–07, University of Wisconsin, Madison, 1994.

    Google Scholar 

  11. R. Fourer and D.M. Gay, “Expressing special structures in an algebraic modeling language for mathematical programming,” ORSA Journal on Computing, vol. 7, pp. 166–190, 1995.

    Google Scholar 

  12. R. Fourer and D.M. Gay, “Integrating programming features with an algebraic modeling language for optimization,” RD17.3, 15th International Symposium on Mathematical Programming, Ann Arbor, 1994; also MC27.1, ORSA/TIMS Joint National Meeting, Detroit, 1994.

  13. R. Fourer, D.M. Gay, and B.W. Kernighan, “A modeling language for mathematical programming,” Management Science, vol. 36, pp. 519–554, 1990.

    Google Scholar 

  14. R. Fourer, D.M. Gay, and B.W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Boyd & Fraser/The Scientific Press: Danvers, MA, 1992.

    Google Scholar 

  15. F. Glover, D. Klingman, and N.V. Phillips, “Netform modeling and applications,” Interfaces, vol. 20, no. 4, pp. 7–27, 1990.

    Google Scholar 

  16. F. Glover, D. Klingman, and N.V. Phillips, Network Models in Optimization and their Applications in Practice, John Wiley & Sons: New York, 1992.

    Google Scholar 

  17. H.J. Greenberg, “MODLER: Modeling by object-driven linear elemental relations,” Annals of Operations Research, vol. 38, pp. 239–280, 1992.

    Google Scholar 

  18. H.J. Greenberg, Modeling by Object-Driven Linear Elemental Relations: A User's Guide for MODLER, Kluwer Academic Publishers: Boston, MA, 1993.

    Google Scholar 

  19. P. Hansen and J. Hugé, “Implicit treatment of zero or range, constraints in a model for minimum cost foundry alloys,” Management Science, vol. 35, pp. 367–371, 1989.

    Google Scholar 

  20. F. Harche, J.N. Hooker, and G.L. Thompson, “A computational study of satisfiability algorithms for propositional logic,” ORSA Journal on Computing, vol. 6, pp. 423–435, 1994.

    Google Scholar 

  21. J.N. Hooker, “Logic-based methods for optimization,” Operations Research Society of America Computer Science Technical Section Newsletter, vol. 15, no. 2, pp. 4–11, 1994.

    Google Scholar 

  22. J.N. Hooker, H. Yan, I.E. Grossmann, and R. Raman, “Logic cuts for processing networks with fixed costs,” Computers and Operations Research, vol. 21, pp. 265–279, 1994.

    Google Scholar 

  23. M.S. Hung, W.O. Rom, and A.D. Waren, Optimization with IBM OSL, Boyd & Fraser Publishing Company: Danvers, MA, 1994.

    Google Scholar 

  24. T. Hürlimann, “IP, MIP, and logical modeling using LPL,” Working Paper No. 205, Institute of Informatics, University of Fribourg, March 1994, updated November 1994.

  25. C.V. Jones, “An introduction to graph-based modeling systems, Part I: overview,” ORSA Journal on Computing, vol. 2, pp. 136–151, 1990.

    Google Scholar 

  26. C.V. Jones, “An introduction to graph-based modeling systems, Part II: Graph grammars and the implementation,” ORSA Journal on Computing, vol. 3, pp. 180–206, 1991.

    Google Scholar 

  27. C.V. Jones, “Attributed graphs, graph-grammars and structured modeling,” Annals of Operations Research, vol. 38, pp. 281–324, 1992.

    Google Scholar 

  28. C.V. Jones, “An integrated modeling environment based on attributed graphs and graph-grammars,” Decision Support Systems, vol. 10, pp. 255–275, 1993.

    Google Scholar 

  29. C.V. Jones and K. D'Souza, “Graph-grammars for minimum cost network flow modeling,” Technical Report, Faculty of Business Administration, Simon Fraser University, Burnaby, BC, 1992.

    Google Scholar 

  30. D.A. Kendrick, “Parallel model representations,” Expert Systems With Applications, vol. 1, pp. 383–389, 1990.

    Google Scholar 

  31. D.A. Kendrick, “A graphical interface for production and transportation system modeling: PTS,” Computer Science in Economics and Management, vol. 4, pp. 229–236, 1991.

    Google Scholar 

  32. B. Kristjansson, “MPL modelling system user manual,” Version 2.8, Maximal Software Inc., Arlington, VA, 1993.

    Google Scholar 

  33. Jean-Louis Lauriere, “A language and a program for stating and solving combinatorial problems,” Artificial Intelligence, vol. 10, pp. 29–127, 1978.

    Google Scholar 

  34. Lindo Systems Inc., “LINGO optimization modeling language,” Chicago, IL, 1994.

  35. K. McAloon and C. Tretkoff, “2LP: Linear programming and logic programming,” in Principles and Practice of Constraint Programming (V. Saraswat and P.Van Hentenryck, Eds.), The MIT Press: Cambridge MA, pp. 99–114, 1995.

    Google Scholar 

  36. R.D. McBride, “NETSYS—A generalized network modeling system,” Technical Report, University of Southern California, Los Angeles, CA, 1988.

    Google Scholar 

  37. G. Mitra, C. Lucas, S. Moody, and E. Hadjiconstantinou, “Tools for reformulating logical forms into zero-one mixed integer programs,” European Journal of Operational Research, vol. 72, pp. 262–276, 1994.

    Google Scholar 

  38. G.L. Nemhauser, M.W.P. Savelsbergh, and G.L. Sigismondi, “MINTO, a mixed integer optimizer,” Operations Research Letters, vol. 15, pp. 48–59, 1994.

    Google Scholar 

  39. W. Ogryczak, K. Studziński, and K. Zorychta, “DINAS: A computer-assisted analysis system for multiobjective transshipment problems with facility location,” Computers and Operations Research, vol. 19, pp. 637–647, 1992.

    Google Scholar 

  40. W. Ogryczak, K. Studziński, and K. Zorychta, “EDINET—A network editor for transshipment problems with facility location,” in Computer Science and Operations Research: New Developments in their Interfaces (O. Balci, R. Sharda, and S.A. Zenios, Eds.), Pergamon Press: New York, pp.197–212, 1992.

    Google Scholar 

  41. T.F. Rutherford, “Extensions of GAMS for complementarity problems arising in applied economic analysis,” Journal of Economic Dynamics and Control, vol. 19, pp. 1299–1324, 1995.

    Google Scholar 

  42. D. Steiger, R. Sharda, and B. Leclaire, “Functional description of a graph-based interface for network modeling (GIN),” in Computer Science and Operations Research: New Developments in their Interfaces (O. Balci, R. Sharda, and S.A. Zenios, Eds.), Pergamon Press: New York, pp. 213–229, 1992.

    Google Scholar 

  43. D. Steiger, R. Sharda, and B. Leclaire, “Graphical interfaces for network modeling: A model management system perspective,” ORSA Journal of Computing, vol. 5, pp. 275–291, 1993.

    Google Scholar 

  44. L. Sterling and E. Shapiro, The Art of Prolog: Advanced Programming Techniques, 2nd ed., MIT Press: Cambridge, MA, 1994.

    Google Scholar 

  45. J.A. Tomlin, “Branch and bound methods for integer and non-convex programming,” in Integer and Nonlinear Programming (J. Abadie, Ed.), American Elsevier Publishing Company: New York, pp. 437–450, 1970.

    Google Scholar 

  46. P.Van Hentenryck, Constraint Satisfaction in Logic Programming, MIT Press: Cambridge, MA, 1989.

    Google Scholar 

  47. P.Van Hentenryck, “A logic language for combinatorial optimization,” Annals of Operations Research, vol. 21, pp. 247–273, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Applied Mathematics, Twente University, P.O. Box 217, 7500 AE, Enschede, The Netherlands

    J. J. Bisschop

  2. Department of Industrial Engineering and Management Sciences, Northwestern University, 60208-3119, IL, Evanston, USA

    Robert Fourer

Authors
  1. J. J. Bisschop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert Fourer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bisschop, J.J., Fourer, R. New constructs for the description of combinatorial optimization problems in algebraic modeling languages. Comput Optim Applic 6, 83–116 (1996). https://doi.org/10.1007/BF00248011

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00248011

Keywords

Search

Navigation