Skip to main content
SpringerLink
Log in

Solving a class of stochastic mixed-integer programs with branch and price

  • Published:
Mathematical Programming Submit manuscript

Abstract

We begin this paper by identifying a class of stochastic mixed-integer programs that have column-oriented formulations suitable for solution by a branch-and-price algorithm (B&P). We then survey a number of examples, and use a stochastic facility-location problem (SFLP) for a detailed demonstration of the relevant modeling and solution techniques. Computational results with a scenario representation of uncertain costs, demands and capacities show that B&P can be orders of magnitude faster than solving the standard formulation by branch and bound. We also demonstrate how B&P can solve SFLP exactly – as exactly as a deterministic mixed-integer program – when demands and other parameters can be represented as certain types of independent, random variables, e.g., independent, normal random variables with integer means and variances.

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. Ahmed, S., Sahinidis, N.V.:. An approximation scheme for stochastic integer programs arising in capacity expansion. Oper. Res. 51, 461–471 (2003)

    MATH  Google Scholar 

  2. Ahuja, R. K., Magnanti, T.L., Orlin. J.B.: Network Flows. Prentice Hall, Englewood Cliffs, NJ, 1993

  3. Appleget, J. A., Wood, R.K.: Explicit-constraint branching for solving mixed-integer programs. M. Laguna, J.L. González-Velarde, (eds), Computing Tools for Modeling, Optimization and Simulation, Kluwer Academic Publishers, Boston, MA, 243–261 2000

  4. Appelgren, L.H.: A column generation algorithm for a ship scheduling problem. Transp. Sci. 3, 53–68 (1969)

    Google Scholar 

  5. Barcelo, J., Casanova, J.: A heuristic lagrangean algorithm for the capacitated plant location problem. Eur. J. Oper. Res. 15, 212–226 (1984)

    Article  MATH  Google Scholar 

  6. Barnhart, C., Hane, C.A., Vance, P.H.: Using branch-and-price-and-cut to solve origin-destination integer multicommodity flow problems. Oper. Res. 48, 318–326 (2000)

    Article  Google Scholar 

  7. Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: Column generation for solving huge integer programs. Oper. Res. 46, 316–329 (1998)

    MATH  MathSciNet  Google Scholar 

  8. Beale, E.M.L: On minimizing a convex function subject to linear inequalities. J. R. Stat. Soc. 17B, 173–184 (1955)

    MATH  MathSciNet  Google Scholar 

  9. Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bertsimas, D.J.: A vehicle routing problem with stochastic demand. Oper. Res. 40, 574–585 (1992)

    MATH  MathSciNet  Google Scholar 

  11. Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer-Verlag, New York, (1997)

  12. Brown, G.G., Graves, G.: Real-time dispatch of petroleum tank trucks. Manage. Sci. 27, 19–32 (1981)

    Google Scholar 

  13. Brown, G.G., Graves, G., Honczarenko, M.: Design and operation of a multicommodity production/distribution system using primal goal decomposition. Manage. Sci. 33, 1469–1480 (1987)

    Google Scholar 

  14. Butchers, E.R., Day, P.R., Goldie, A.P., Miller, S., Meyer, J.A., Ryan, D.M., Scott, A.C., Wallace, C.A.: Optimized crew scheduling at Air New Zealand. Interfaces 31, 30–56 (2001)

    Article  Google Scholar 

  15. Butler, J.C., Dyer, J.S.: Optimizing natural gas flows with linear programming and scenarios. Dec. Sci. 30, 563–580 (1999)

    Google Scholar 

  16. Camm J.D., Chorman, T.E., Dill, F.A., Evans, J.R., Sweeney, D.J., Wegryn, G.W.: Blending OR/MS, judgment, and GIS: Restructuring P&G's supply chain. Interfaces 27, 128–142 (1997)

    Google Scholar 

  17. Carøe, C.C., Tind, J.: L-shaped decomposition of two-stage stochastic programs with integer recourse. Math. Program. 83, 451–464 (1998)

    Article  MATH  Google Scholar 

  18. Chen, Z.-L., Li, S., Tirupati, D.: A scenario-based stochastic programming approach for technology and capacity planning. Comput. Oper. Res. 29, 781–806 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Chu P.C., Beasley, J.E.: A genetic algorithm for the generalized assignment problem. Comput. Oper. Res. 24, 17–23 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. COIN. 2004. http://www.coin-or.org (accessed July 2004)

  21. Damodaran, P., Wilhelm, W.E.: Branch-and-price methods for prescribing profitable upgrades of high-technology products with stochastic demands. Dec. Sci. 35, 55–82 (2004)

    Article  Google Scholar 

  22. Dantzig, G.B., Wolfe, P.: The decomposition principle for linear programs. Oper. Res. 8, 101–111 (1960)

    MATH  Google Scholar 

  23. Day, P.R., Ryan, D.M.: Flight attendant rostering for short-haul airline operations. Oper. Res. 45, 649–661 (1997)

    MATH  Google Scholar 

  24. Desrochers, M., Desrosiers, J., Solomon, M.: 1992. A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40, 342–354.

    Google Scholar 

  25. Desrochers, M., Solomon, F.M.: A column generation approach to the urban transit crew scheduling problem. Transp. Sci. 23, 1–13 (1989)

    MATH  Google Scholar 

  26. Desrosiers, J., Dumas, Y., Solomon, M.M., Soumis, F.: Time constrained routing and scheduling. (M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser (eds), Handbooks in Operations Research and Management Science, Vol. 8, Network Routing, Elsevier, Amsterdam, 35–140 (1995)

  27. Du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discrete Math. 194, 229–237 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ehrgott, M., Ryan, D.M.: Constructing robust crew schedules with bicriteria optimization. Journal of Multicriteria Decision Analysis 11, 139–150 (2002)

    Article  MATH  Google Scholar 

  29. Ford, L.R., Fulkerson, D.R.: A suggested computation for the maximal multicommodity network flows. Manage. Sci. 5, 97–101 (1958)

    MATH  Google Scholar 

  30. Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem. Oper. Res. 9, 849–859 (1961)

    MATH  MathSciNet  Google Scholar 

  31. Girard, A., Sansó, B.: Multicommodity flow models, failure propagation, and reliable loss network design. IEEE/ACM Transactions on Networking 6, 82–93 (1998)

    Article  Google Scholar 

  32. ILOG 2002. ILOG CPLEX 8.0 Reference Manual

  33. Johnson, E.L.: Modeling and strong linear programs for mixed integer programming. S.W. Wallace (ed), Algorithms and Model Formulations in Mathematical Programming, Springer-Verlag, 1–43 (1989)

  34. Kleywegt A.J., Shapiro, A., Homem-de-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12, 479–502 (2002)

    Google Scholar 

  35. Laporte, G., Louveaux, F.V.: The integer L-shaped method for stochastic integer programs with complete resource. Oper. Res. Lett. 13, 133–142 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  36. Laporte, G., Louveaux, F.V., Van Hamme, L.: Exact solution to a location problem with stochastic demands. Transp. Sci. 28, 95–103 (1994)

    MATH  MathSciNet  Google Scholar 

  37. Linderoth, J., Wright, S.J.: Decomposition algorithms for stochastic programming on a computational grid. Optimization Technical Report 02–07, Computer Sciences Department, University of Wisconsin-Madison (2002)

  38. Lorena, L.A.N., Senne, E.L.F.: A column generation approach to capacitated p-median problems. Comput. Oper. Res. 31, 863–876 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  39. Lougee-Heimer, R.: The Common optimization interface for operations research: Promoting open-source software in the operations research community. IBM J. Res. Dev. 47, 57–66 (2003)

    Article  Google Scholar 

  40. Louveaux, F.V., Peeters, D.: A dual-based procedure for a stochastic facility location. Oper. Res. 40, 564–573 (1992)

    MATH  MathSciNet  Google Scholar 

  41. Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Les Cahiers de GERAD G-2002-64, Group for research in decision analysis, montreal, Canada. http://www.optimizationonline.org/DB_FILE/2002/12/580.pdf (accessed July 2004)

  42. Lulli, G., Sen, S.: A branch-and-price algorithm for multi-stage stochastic integer programming with application to stochastic batch-size problems. Manage. Sci. 50, 786–796 (2004)

    Article  Google Scholar 

  43. Mak, W.-K., Morton, D.P., Wood, R.K.: Monte Carlo bounding techniques for determining solution quality in stochastic programs. Oper. Res. Lett. 24, 47–56 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  44. Papagiannaki, K., Moon, S., Fraleigh, C., Thiran, P., Diot, C.: Measurement and analysis of single-hop delay on an IP backbone network. IEEE Journal on Selected Areas in Communications 21, 908–921 (2003)

    Article  Google Scholar 

  45. Ralphs, T.K., Ladanyi, L.: COIN/BCP User's Manual. http://www-124.ibm.com/developerworks/opensource/coin/presentations/bcp-man.pdf (accessed July 2004) (2001)

  46. Ribeiro C.C., Soumis, F.: A column generation approach to the multiple-depot vehicle scheduling problem. Oper. Res. 42, 41–52 (1994)

    MATH  Google Scholar 

  47. Rockafellar, R.T., Wets, R.J.-B.: Stochastic convex programming: relatively complete recourse and induced feasibility. SIAM J. Control Optimization 14, 547–589 (1976)

    Google Scholar 

  48. Ryan, D.M., Foster, B.A.: An integer programming approach to scheduling. A. Wren, (ed), Computer Scheduling of Public Transport, Urban Passenger Vehicle and Crew Scheduling. North Holland, Amsterdam, 269–280 (1981)

  49. Savelsbergh, M.W.P.: A branch-and-price algorithm for the generalized assignment problem. Oper. Res. 45, 831–841 (1997)

    MATH  MathSciNet  Google Scholar 

  50. Sen, S., Higle, J.L.: The C3 theorem and a D2 algorithm for large scale stochastic integer programming: Set convexification. Stochastic Programming E-Print Series. (http://dochost.rz.hu-berlin.de/speps) (2000)

  51. Shiina, T., Birge, J.R.: Stochastic unit commitment problem. Int. Trans. Oper. Res. 11, 19–32 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  52. Silva, E.F.: Improving branch-and-price algorithms and applying them to stochastic programs. PhD Dissertation, Naval Postgraduate School, Monterey, CA (2004)

  53. Singh, K., Philpott, A., Wood, R.K.: Column generation for design of survivable networks. Working paper, Dept. of Engineering Science, University of Auckland, Auckland, New Zealand (2005)

  54. Spoerl, D., Wood, R.K.: A stochastic generalized assignment problem. INFORMS Annual Meeting, Atlanta, GA, 19–22 October (2003)

  55. Teo, C-P., Shu, J.: Warehouse-retailer network design problem. Oper. Res. 52, 396–408 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  56. Vance, P.H., Barnhart, C., Johnson, E.L., Nemhauser, G.L.: Airline crew scheduling: A new formulation and decomposition algorithm. Oper. Res. 45, 188–200 (1997)

    Article  MATH  Google Scholar 

  57. Walkup, D.W., Wets, R.J.-B.: Stochastic programs with recourse. SIAM J. Appl. Math.15, 1299–1314 (1967)

    Google Scholar 

  58. Weingartner, H.M., Ness, D.N.: Methods for the multidimensional 0/1 knapsack problem. Oper. Res. 15, 83–103 (1967)

    Google Scholar 

  59. Wets, R.J.-B. 1966. Programming under uncertainty: the complete problem. Zeitschrift für Wahr-scheinlichkeitstheorie und Verwandte Gebiete 4, 316–339 (1996)

    Article  Google Scholar 

  60. Wolsey, L.A.: Integer Programming. John Wiley& Sons, New York, 1998

  61. Zhou, J., Liu, B.: New stochastic models for capacitated location-allocation problem. Computers and Industrial Engineering 45, 111–125 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brazilian Navy Analyses Center, Rio de Janeiro, RJ, 20091-000, Brazil

    Eduardo F. Silva

  2. Operations Research Department, Naval Postgraduate School, 1749-016 Lisboa, Monterey, CA, 93943, USA

    R. Kevin Wood

Authors
  1. Eduardo F. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Kevin Wood
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Kevin Wood.

Additional information

Kevin Wood thanks the Office of Naval Research, Air Force Office of Scientific Research, the Naval Postgraduate School (NPS) and the University of Auckland for their support. Eduardo Silva thanks NPS and the Brazilian Navy for their support. Both authors are grateful to the COIN-OR team for assistance with computational issues, as well as to two anonymous referees for highly useful, constructive criticism.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Silva, E., Wood, R. Solving a class of stochastic mixed-integer programs with branch and price. Math. Program. 108, 395–418 (2006). https://doi.org/10.1007/s10107-006-0716-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0716-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation