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An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem

Author

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  • Jonathan F. Bard

    (Northeastern University, Boston, Massachusetts)

Abstract

This paper presents an algorithm using sensitivity analysis to solve a linear two-stage optimization problem. The underlying theory rests on a set of first order optimality conditions that parallel the Kuhn-Tucker conditions associated with a one-dimensional parametric linear program. The solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program. Finite convergence is established under nondegenerate assumptions. The paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements. By these measures, the algorithm presented here has an overall advantage. Finally, a comparison is drawn between bicriteria and bilevel programming, and underscored by way of an example.

Suggested Citation

  • Jonathan F. Bard, 1983. "An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem," Operations Research, INFORMS, vol. 31(4), pages 670-684, August.
  • Handle: RePEc:inm:oropre:v:31:y:1983:i:4:p:670-684
    DOI: 10.1287/opre.31.4.670
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    Cited by:

    1. Nishizaki, Ichiro & Hayashida, Tomohiro & Sekizaki, Shinya & Okabe, Junya, 2022. "Data envelopment analysis approaches for two-level production and distribution planning problems," European Journal of Operational Research, Elsevier, vol. 300(1), pages 255-268.
    2. M. Campêlo & S. Scheimberg, 2005. "A Study of Local Solutions in Linear Bilevel Programming," Journal of Optimization Theory and Applications, Springer, vol. 125(1), pages 63-84, April.
    3. Sakawa, Masatoshi & Nishizaki, Ichiro & Uemura, Yoshio, 2002. "A decentralized two-level transportation problem in a housing material manufacturer: Interactive fuzzy programming approach," European Journal of Operational Research, Elsevier, vol. 141(1), pages 167-185, August.
    4. Sakawa, Masatoshi & Nishizaki, Ichiro & Uemura, Yoshio, 2001. "Interactive fuzzy programming for two-level linear and linear fractional production and assignment problems: A case study," European Journal of Operational Research, Elsevier, vol. 135(1), pages 142-157, November.
    5. de Matta, Renato & Lowe, Timothy J. & Zhang, Dengfeng, 2017. "Competition in the multi-sided platform market channel," International Journal of Production Economics, Elsevier, vol. 189(C), pages 40-51.
    6. Steven Gabriel & Sauleh Siddiqui & Antonio Conejo & Carlos Ruiz, 2013. "Solving Discretely-Constrained Nash–Cournot Games with an Application to Power Markets," Networks and Spatial Economics, Springer, vol. 13(3), pages 307-326, September.
    7. Sakawa, Masatoshi & Nishizaki, Ichiro & Hitaka, Masatoshi, 1999. "Interactive fuzzy programming for multi-level 0-1 programming problems through genetic algorithms," European Journal of Operational Research, Elsevier, vol. 114(3), pages 580-588, May.
    8. de Matta, Renato E. & Lowe, Timothy J. & Zhang, Dengfeng, 2014. "Consignment or wholesale: Retailer and supplier preferences and incentives for compromise," Omega, Elsevier, vol. 49(C), pages 93-106.
    9. Leonardo Lozano & J. Cole Smith, 2017. "A Value-Function-Based Exact Approach for the Bilevel Mixed-Integer Programming Problem," Operations Research, INFORMS, vol. 65(3), pages 768-786, June.
    10. Mathur, Kanchan & Puri, M. C., 1995. "A bilevel bottleneck programming problem," European Journal of Operational Research, Elsevier, vol. 86(2), pages 337-344, October.
    11. Ahlatcioglu, Mehmet & Tiryaki, Fatma, 2007. "Interactive fuzzy programming for decentralized two-level linear fractional programming (DTLLFP) problems," Omega, Elsevier, vol. 35(4), pages 432-450, August.
    12. Frangioni, Antonio, 1995. "On a new class of bilevel programming problems and its use for reformulating mixed integer problems," European Journal of Operational Research, Elsevier, vol. 82(3), pages 615-646, May.
    13. I. Nishizaki & M. Sakawa, 1999. "Stackelberg Solutions to Multiobjective Two-Level Linear Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 161-182, October.
    14. S A Gabriel & Y Shim & A J Conejo & S de la Torre & R García-Bertrand, 2010. "A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(9), pages 1404-1419, September.
    15. Ashenafi Woldemariam & Semu Kassa, 2015. "Systematic evolutionary algorithm for general multilevel Stackelberg problems with bounded decision variables (SEAMSP)," Annals of Operations Research, Springer, vol. 229(1), pages 771-790, June.
    16. Gabriel, Steven A. & Leuthold, Florian U., 2010. "Solving discretely-constrained MPEC problems with applications in electric power markets," Energy Economics, Elsevier, vol. 32(1), pages 3-14, January.
    17. Tilahun, Surafel Luleseged, 2019. "Feasibility reduction approach for hierarchical decision making with multiple objectives," Operations Research Perspectives, Elsevier, vol. 6(C).
    18. Zhao, Linlin & Zha, Yong & Zhuang, Yuliang & Liang, Liang, 2019. "Data envelopment analysis for sustainability evaluation in China: Tackling the economic, environmental, and social dimensions," European Journal of Operational Research, Elsevier, vol. 275(3), pages 1083-1095.
    19. Pramanik, Surapati & Roy, Tapan Kumar, 2007. "Fuzzy goal programming approach to multilevel programming problems," European Journal of Operational Research, Elsevier, vol. 176(2), pages 1151-1166, January.
    20. Wu, Desheng Dash, 2010. "BiLevel programming Data Envelopment Analysis with constrained resource," European Journal of Operational Research, Elsevier, vol. 207(2), pages 856-864, December.

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