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Bilevel programming and price setting problems

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Abstract

This paper is devoted to pricing optimization problems which can be modeled as bilevel programs. We present the main concepts, models and solution methods for this class of optimization problems.

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References

  • Amaldi E, Bruglieri M, Fortz B (2011) On the hazmat transport network design problem. In: Pahl J, Reiners T, Voß S (eds) Network optimization, lecture notes in computer science, vol 6701. Springer, Berlin, pp 327–338

  • Bouhtou M, Van Hoesel S, Van der Kraaij A, Lutton J (2007a) Tariff optimization in networks. INFORMS J Comput 19(3):458–469

    Google Scholar 

  • Bouhtou M, Grigoriev A, Van Hoesel S, Van der Kraaij A, Spieksma F, Uetz M (2007b) Pricing bridges to cross a river. Naval Res Logist 54:411–420

  • Bracken J, McGill J (1973) Mathematical programs with optimization problems in the constraints. Oper Res 21(1):37–44

    Article  Google Scholar 

  • Brotcorne L, Labbé M, Marcotte P, Savard G (2000) A bilevel model and solution algorithm for a freight tariff-setting problem. Transp Sci 34(3):289–302

    Article  Google Scholar 

  • Brotcorne L, Labbé M, Marcotte P, Savard G (2001) A bilevel model for toll optimization on a multicommodity transportation network. Transp Sci 35(4):345–358

    Article  Google Scholar 

  • Brotcorne L, Labbé M, Marcotte P, Savard G (2008) Joint design and pricing on a network. Oper Res 56(5):1104–1115

    Article  Google Scholar 

  • Candler W, Norton R (1977) Multilevel programming. Technical report 20, World Bank Development Research Center, Washington, DC

  • Cardinal J, Demaine E, Fiorini S, Joret G, Langerman S, Newman I, Weimann O (2011) The stackelberg minimum spanning tree game. Algorithmica 59:129–144

    Article  Google Scholar 

  • Castelli L, Labbé M, Violin A (2012) A network pricing formulation for the revenue maximization of european air navigation service providers. Transp Res Part C. doi:10.1016/j.trc.2012.04.013

  • Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR. Q J Oper Res 3:87–105

    Article  Google Scholar 

  • Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235–256

    Article  Google Scholar 

  • Dempe S (2002) Foundations of bilevel programming, nonconvex optimization and its applications, vol 61. Kluwer, Dordrecht

    Google Scholar 

  • Dewez S (2004) On the toll setting problem. PhD thesis, Université Libre de Bruxelles, Brussles

  • Dewez S, Labbé M, Marcotte P, Savard G (2008) New formulations and valid inequalities for a bilevel pricing problem. Oper Res Lett 36(2):141–149

    Article  Google Scholar 

  • Garey M, Johnson D (1979) Computers and interactability. W.H. Freeman, San Francisco

    Google Scholar 

  • Hansen P, Jaumard B, Savard G (1992) A new branch-and-bound rules for linear bilevel programming. SIAM J Sci Stat Comput 5(13):1194–1217

    Google Scholar 

  • Heilporn G, Labbé M, Marcotte P, Savard G (2010a) A parallel between two classes of pricing problems in transportation and marketing. J Rev Pricing Manage 9(1/2):110–125

    Article  Google Scholar 

  • Heilporn G, Labbé M, Marcotte P, Savard G (2010b) A polyhedral study of the network pricing problem with connected toll arcs. Networks 3(55):234–246

    Google Scholar 

  • Heilporn G, Labbé M, Marcotte P, Savard G (2011) Valid inequalities and branch-and-cut for the clique pricing problem. Discret Optimiz 8(3):393–410

    Article  Google Scholar 

  • Jeroslow R (1985) The polynomial hierarchy and a simple model for competitive analysis. Math Program 32:146–164

    Article  Google Scholar 

  • Joret G (2011) Stackelberg network pricing is hard to approximate. Networks 57(2):117–120

    Google Scholar 

  • Labbé M, Marcotte P, Savard G (1998) A bilevel model of taxation and its application to optimal highway pricing. Manage Sci 44(12):1608–1622

    Article  Google Scholar 

  • Loridan P, Morgan J (1996) Weak via strong stackelberg problem: new results. J Glob Optimiz 8:263–287

    Article  Google Scholar 

  • Migdalas A (1995) Bilevel programming in traffic planning: models, methods and challenge. J Glob Optimiz 7:381–405

    Article  Google Scholar 

  • Owen G (1968) Game theory. Emerald Group, Bingley

    Google Scholar 

  • Roch S, Marcotte P, Savard G (2005) Design and analysis of an approximation algorithm for stackelberg network pricing. Networks 46(1):57–67

    Article  Google Scholar 

  • Shioda R, Tunçel L, Myklebust T (2011) Maximum utility product pricing models and algorithms based on reservation price. Comput Optimiz Appl 48:157–198

    Article  Google Scholar 

  • Stackelberg H (1952) The theory of market economy. Oxford University Press, Oxford

    Google Scholar 

  • Van Ackere A (1993) The principal/agent paradigm: its relevance to various functional fields. Eur J Oper Res 70(1):83–103

    Article  Google Scholar 

  • Van Hoesel S (2008) An overview of stackelberg pricing in networks. Eur J Oper Res 189:1393–1402

    Article  Google Scholar 

  • Vicente L, Calamai P (1994) Bilevel and multilevel programming: a bibliography review. J Glob Optimiz 5:291–306

    Article  Google Scholar 

  • Vicente L, Savard G, Júdice J (1994) Descent approaches for quadratic bilevel programming. J Optimiz Theory Appl 81(2):379–399

    Article  Google Scholar 

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Acknowledgments

The first author acknowledges support from the “Ministerio de Ciencia e Innovacíon” through the research project MTM2009-14039-C06. The second author acknowledges support from the Belgian national scientific funding agency “Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture” (FRIA), of which she is a research fellow.

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Correspondence to Martine Labbé.

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Labbé, M., Violin, A. Bilevel programming and price setting problems. 4OR-Q J Oper Res 11, 1–30 (2013). https://doi.org/10.1007/s10288-012-0213-0

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  • DOI: https://doi.org/10.1007/s10288-012-0213-0

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