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.
Similar content being viewed by others
References
Amaldi, E., Bruglieri, M., & Fortz, B. (2011). On the hazmat transport network design problem. In J. Pahl, T. Reiners, & S. Voß (Eds.), Network Optimization Lecture Notes in Computer Science (Vol. 6701, pp. 327–338)., Berlin/Heidelberg: Springer.
Bouhtou, M., Grigoriev, A., Van Hoesel, S., Van der Kraaij, A., Spieksma, F., & Uetz, M. (2007a). Pricing bridges to cross a river. Naval Research Logistics, 54, 411–420.
Bouhtou, M., Van Hoesel, S., Van der Kraaij, A., & Lutton, J. (2007b). Tariff optimization in networks. INFORMS Journal of Computing, 19(3), 458–469.
Bracken, J., & McGill, J. (1973). Mathematical programs with optimization problems in the constraints. Operations Research, 21(1), 37–44.
Brotcorne, L., Labbé, M., Marcotte, P., & Savard, G. (2000). A bilevel model and solution algorithm for a freight tariff-setting problem. Transportation Science, 34(3), 289–302.
Brotcorne, L., Labbé, M., Marcotte, P., & Savard, G. (2001). A bilevel model for toll optimization on a multicommodity transportation network. Transportation Science, 35(4), 345–358.
Brotcorne, L., Labbé, M., Marcotte, P., & Savard, G. (2008). Joint design and pricing on a network. Operations Research, 56(5), 1104–1115.
Candler, W., Norton, R. (1977). Multilevel programming. Tech. Rep. 20, World Bank Development Research Center, Washington DC, USA
Cardinal, J., Demaine, E., Fiorini, S., Joret, G., Langerman, S., Newman, I., et al. (2011). The stackelberg minimum spanning tree game. Algorithmica, 59, 129–144.
Castelli, L., Labbé, M., & Violin, A. (2013). A network pricing formulation for the revenue maximization of european air navigation service providers. Transportation Research Part C: Emerging Technologies, 33, 214–226.
Colson, B., Marcotte, P., & Savard, G. (2005). Bilevel programming: A survey. 4OR: A Quarterly Journal of Operations Research, 3, 87–105.
Colson, B., Marcotte, P., & Savard, G. (2007). An overview of bilevel optimization. Annals of Operations Research, 153, 235–256.
Dempe, S. (2002). Foundations of bilevel programming, nonconvex optimization and its applications (Vol. 61). Dordrecht: Kluwer Academic Publishers.
Dewez, S. (2004). On the toll setting problem. PhD thesis, Université Libre de Bruxelles.
Dewez, S., Labbé, M., Marcotte, P., & Savard, G. (2008). New formulations and valid inequalities for a bilevel pricing problem. Operation Research Letters, 36(2), 141–149.
Fortz, B., Labbé, M., & Violin, A. (2013). Dantzig–Wolfe reformulation for the network pricing problem with connected toll arcs. Electronic Notes in Discrete Mathematics, 41, 117–124.
Garey, M., & Johnson, D. (1979). Computers and interactability. San Francisco: W.H. Freeman.
Hansen, P., Jaumard, B., & Savard, G. (1992). A new branch-and-bound rules for linear bilevel programming. SIAM Journal on Scientific and Statistical Computing, 5(13), 1194–1217.
Heilporn, G., Labbé, M., Marcotte, P., & Savard, G. (2010a). A parallel between two classes of pricing problems in transportation and marketing. Journal of Revenue and Pricing Management, 9(1/2), 110–125.
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.
Heilporn, G., Labbé, M., Marcotte, P., & Savard, G. (2011). Valid inequalities and branch-and-cut for the clique pricing problem. Discrete Optimization, 8(3), 393–410.
Jeroslow, R. (1985). The polynomial hierarchy and a simple model for competitive analysis. Mathematical Programming, 32, 146–164.
Joret, G. (2011). Stackelberg network pricing is hard to approximate. Networks, 57(2), 117–120.
Kara, B. Y., & Verter, V. (2004). Designing a road network for hazardous materials transportation. Transportation Science, 38(2), 188–196.
Labbé, M., Marcotte, P., & Savard, G. (1998). A bilevel model of taxation and its application to optimal highway pricing. Management Science, 44(12), 1608–1622.
Labbé, M., & Violin, A. (2013). Bilevel programming and price setting problems. 4OR: A Quarterly Journal of Operations Research, 11(1), 1–30.
Loridan, P., & Morgan, J. (1996). Weak via strong stackelberg problem: New results. Journal of Global Optimization, 8, 263–287.
Marcotte, P., Mercier, A., Savard, G., & Verter, V. (2009). Toll policies for mitigating hazardous materials transport risk. Transportation Science, 43(2), 228–243.
Migdalas, A. (1995). Bilevel programming in traffic planning: Models, methods and challenge. Journal of Global Optimization, 7, 381–405.
Owen, G. (1968). Game theory. Bingley: Emerald Group Publishing Limited.
Roch, S., Marcotte, P., & Savard, G. (2005). Design and analysis of an approximation algorithm for stackelberg network pricing. Networks, 46(1), 57–67.
Shioda, R., Tunçel, L., & Myklebust, T. (2011). Maximum utility product pricing models and algorithms based on reservation price. Computational Optimization and Applications, 48, 157–198.
Stackelberg, H. (1952). The theory of market economy. Oxford: Oxford University Press.
Van Ackere, A. (1993). The principal/agent paradigm: Its relevance to various functional fields. European Journal of Operational Research, 70(1), 83–103.
Van Hoesel, S. (2008). An overview of stackelberg pricing in networks. European Journal of Operational Research, 189, 1393–1402.
Vicente, L., & Calamai, P. (1994). Bilevel and multilevel programming: A bibliography review. Journal of Global Optimization, 5, 291–306.
Vicente, L., Savard, G., & Júdice, J. (1994). Descent approaches for quadratic bilevel programming. Journal of Optimization Theory and Applications, 81(2), 379–399.
Violin, A. (2014). Mathematical programming approaches to pricing problems. PhD thesis, Université Libre de Bruxelles and Università degli Studi di Trieste.
Acknowledgments
The research of the first author is supported by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. 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 was a research fellow, and from the “Fonds David et Alice Van Buuren”.
Author information
Authors and Affiliations
Corresponding author
Additional information
This is an updated version of the paper that appeared in 4OR, 11(1), 1–30 (2013).
Rights and permissions
About this article
Cite this article
Labbé, M., Violin, A. Bilevel programming and price setting problems. Ann Oper Res 240, 141–169 (2016). https://doi.org/10.1007/s10479-015-2016-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-015-2016-0