Advertisement

Obtaining Sharp Lower and Upper Bounds for Two-Stage Capacitated Facility Location Problems

  • Andreas Klose
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 460)

Abstract

The Two-Stage Capacitated Facility Location Problem (TSCFLP) is to find the optimal locations of depots to serve customers with a given demand, the optimal assignment of customers to depots and the optimal product flow from plants to depots. To compute an optimal solution to the problem, Benders’ decomposition has been the preferred technique. In this paper, a Lagrangean heuristic is proposed to produce good suboptimal solutions together with a lower bound. Lower bounds are computed from the Lagrangean relaxation of the capacity constraints. The Lagrangean subproblem is an Uncapacitated Facility Location Problem (UFLP) with an additional knapsack constraint. From an optimal solution of this subproblem, a heuristic solution to the TSCFLP is computed by reassigning customers until the capacity constraints are met and by solving the transportation problem for the first distribution stage. The Lagrangean dual is solved by a variant of Dantzig-Wolfe decomposition, and elements of cross decomposition are used to get a good initial set of dual cuts.

Keywords

Transportation Problem Master Problem Cover Inequality Optimal Dual Solution Capacitate Facility Location Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aardal, K. (1994): Capacitated Facility Location: Separation Algorithm and Computational Experience. CentER Discussion Paper 9480, Tilburg. (available via ftp on ftp://ftp.cs.ruu.nl/pub/ruu/cs/techreps).Google Scholar
  2. Aardal, K./Labbé, M./Leung, J./Queyranne, M. (1994): On the Two-Level Uncapacitated Facility Location Problem. CentER Discussion Paper 9486, Tilburg. (available via ftp on ftp://ftp.cs.ruu.nl/pub/ruu/cs/techreps).Google Scholar
  3. Aardal, K./Pochet, Y./Wolsey, L. A. (1993): Capacitated Facility Location: Valid Inequalities and Facets. Mathematics of Operations Research, 20:552–582.Google Scholar
  4. Balas, E./Martin, R. (1980): Pivot and Complement: A Heuristic for 0–1 Programming. Management Science, 26:86–96.CrossRefGoogle Scholar
  5. Barcelo, J./Casanovas, J. (1984): A Heuristic Lagrangean Algorithm for the Capacitated Plant Location Problem. European Journal of Operational Research, 15:212–226.CrossRefGoogle Scholar
  6. Barcelo, J./Fernandez, E./Jörnsten, K. (1991): Computational Results from a New Lagrangean Relaxation Algorithm for the Capacitated Plant Location Problem. European Journal of Operational Research, 52:38–45.CrossRefGoogle Scholar
  7. Barcelo, J./Hallefjord, Å./Fernandez, E./Jörnsten, K. (1990): Lagrangean Relaxation and Constraint Generation Procedures for Capacitated Plant Location Problems with Single Sourcing. Operations Research-Spektrum, 12:79–88.CrossRefGoogle Scholar
  8. Benders, J. F. (1962): Partitioning Procedures for Solving Mixed-Variables Programming Problems. Numerische Mathematik, 4:238–252.CrossRefGoogle Scholar
  9. Cho, D. C./Johnson, E. L./Padberg, M. W./Rao, M. R. (1983a): On the Uncapacitated Plant Location Problem I: Valid Inequalities and Facets. Mathematics of Operations Research, 8:579–589.CrossRefGoogle Scholar
  10. Cho, D. С./Johnson, E. L./Padberg, M. W./Rao, M. R. (1983b): Google Scholar
  11. On the Uncapacitated Plant Location Problem II: Facets and Lifting Theorems. Mathematics of Operations Research, 8:590–612.Google Scholar
  12. Cornuejols, G./Fisher, M. L./Nemhauser, G. L. (1977): On the Uncapacitated Location Problem. Annals of Discrete Mathematics, 1:163–177.CrossRefGoogle Scholar
  13. Cornuejols, G./Thizy, J.-M. (1982): Some Facets of the Simple Plant Location Polytope. Mathematical Programming, 23:50–74.CrossRefGoogle Scholar
  14. Francis, R. L./McGinnis, L. F./White, J. A. (1992): Facility Layout and Location: An Analytical Approach. Prentice-Hall, Englewood Cliffs NJ.Google Scholar
  15. Geoffrion, A. M./Graves, G. W. (1974): Multicommodity Distribution System Design by Benders Decomposition. Management Science, 20:822–844.CrossRefGoogle Scholar
  16. Gu, Z./Nemhauser, G. L./Savelsbergh, M. W. P. (1995): Lifted Cover Inequalities for 0–1 Linear Programs: Computation. Report 94–09, Logistic Optimization Center, Georgia Institute of Technology (available on http://akula.isye.gatech.edu:80/~mwps/).Google Scholar
  17. Guignard, M. (1980): Fractional Vertices, Cuts and Facets of the Simple Plant Location Problem. Mathematical Programming Study, 12:150–162.CrossRefGoogle Scholar
  18. Guignard, M./Opaswongkarn, K. (1990): Lagrangean Dual Ascent Algorithms for Computing Bounds in Capacitated Plant Location Problems. European Journal of Operational Research, 46:73–83.CrossRefGoogle Scholar
  19. Guignard, M./Rosenwein, M. В. (1989): An Application-Oriented Guide for Designing Lagrangean Dual Ascent Algorithms. European Journal of Operational Research, 43:197–205.CrossRefGoogle Scholar
  20. Hillier, F. S. (1969): Efficient Heuristic Procedures for Integer Linear Programming. Operations Research, 17:600–637.CrossRefGoogle Scholar
  21. Hindi, K. S./Basta, T. (1994): Computationally Efficient Solution of a Multiproduct, Two-Stage Distribution-Location Problem. The Journal of the Operational Research Society, 45:1316–1323.Google Scholar
  22. Holmberg, K. (1990): On the Convergence of Cross Decomposition. Mathematical Programming, 47:269–296.CrossRefGoogle Scholar
  23. Holmberg, K. (1992a): Generalized Cross Decomposition Applied to Nonlinear Integer Programming Problems: Duality Gaps and Convexification in Parts. Optimization, 23:341–356.CrossRefGoogle Scholar
  24. Holmberg, K. (1992b): Linear Mean Value Cross Decomposition: A Generalization of the Kornai-Liptak Method. European Journal of Operational Research, 62:55–73.CrossRefGoogle Scholar
  25. Holmberg, K. (1994): Cross Decomposition Applied to Integer programming Problems: Duality Gaps and Convexification in Parts. Operations Research, 42:657–668.CrossRefGoogle Scholar
  26. Kuncewicz, J. G./Luss, H. (1986): A Lagrangian Relaxation Heuristic for Capacitated Facility Location with Single Source Constraints. The Journal of the Operational Research Society, 37:495–500.Google Scholar
  27. Klose, A. (1994): A Branch and Bound Algorithm for an Uncapacitated Facility Location Problem with a Side Constraint. Working paper, Institut für Unternehmensforschung (Operations Research), Hochschule St. Gallen.Google Scholar
  28. Krarup, J./Pruzan, P. M. (1983): The Simple Plant Location Problem: Survey and Synthesis. European Journal of Operational Research, 12:36–81.CrossRefGoogle Scholar
  29. Leung, J. M. Y./Magnanti, T. L. (1989): Valid Inequalities and Facets of the Capacitated Plant Location Problem. Mathematical Programming, 44:271–291.CrossRefGoogle Scholar
  30. Magnanti, T. L./Wong, R. T. (1981): Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria. Operations Research, 29:464–484.CrossRefGoogle Scholar
  31. Magnanti, T. L./Wong, R. T. (1989): Decomposition Methods for Facility Location Problems. In: P. B. Mirchandani and R. L. Francis, editors, Discrete Location Theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, pages 209–262. John Wiley & Sons, Chichester New York.Google Scholar
  32. Martello, S./Toth, P. (1990): Knapsack Problems — Algorithms and Computer Implementations. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Chichester New York.Google Scholar
  33. McDaniel, D./Devine, M. (1977): A Modified Benders’ Partitioning Algorithm for Mixed Integer Programming. Management Science, 24:312–319.CrossRefGoogle Scholar
  34. Nemhauser, G. L./Wolsey, L. A. (1988): Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, Chichester New York.Google Scholar
  35. Osman, I. H. (1995): Heuristics for the Generalized Assignment Problem: Simulated Annealing and Tabu Search Approaches. Operations Research-Spektrum, 17:211–225.CrossRefGoogle Scholar
  36. Sridharan, R. (1993): A Lagrangian Heuristic for the Capacitated Plant Location Problem with Single Source Constraints. European Journal of Operational Research, 66:305–312.CrossRefGoogle Scholar
  37. Van Roy, T. J. (1980): Cross Decomposition for Large-Scale, Mixed Integer Linear Programming with Application to Facility Location on Distribution Networks. PhD thesis, Katholieke Universiteit Leuven, Leuven.Google Scholar
  38. Van Roy, T. J. (1983): Cross Decomposition for Mixed Integer Programming. Mathematical Programming, 25:46–63.CrossRefGoogle Scholar
  39. Van Roy, T. J. (1986): A Cross Decomposition Algorithm for Capacitated Facility Location. Operations Research, 34:145–163.CrossRefGoogle Scholar
  40. Wentges, P. (1994): Standortprobleme mit Berücksichtigung von Kapazitätsrestriktionen: Modellierung und Lösungsverfahren. Dissertation Nr. 1620, Hochschule St. Gallen.Google Scholar
  41. Wentges, P. (1996): Accelerating Benders’ Decomposition for the Capacitated Facility Location Problem. Mathematical Methods of Operations Research, 44:267–290.CrossRefGoogle Scholar
  42. Wentges, P. (1997): Weighted Dantzig-Wolfe Decomposition for Linear Mixed-Integer Programming. International Transactions in Operational Research, 4:151–162.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Andreas Klose
    • 1
  1. 1.Universität St. GallenSt. GallenSwitzerland

Personalised recommendations