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Annals of Operations Research

, Volume 111, Issue 1–4, pp 253–270 | Cite as

Sequential Discrete p-Facility Models for Competitive Location Planning

  • Kathrin Fischer
Article

Abstract

Two new models for duopolistic competitive discrete location planning with sequential acting and variable delivered prices are introduced. If locations and prices are assumed to be set “once and for all” by the players, the resulting bilevel program is nonlinear. Under the assumption that further price adjustments are possible, i.e., that a Nash equilibrium in prices is reached, the model can be simplified to a linear discrete bilevel formulation. It is shown that in either situation players should not share any locations or markets if they strive for profit-maximization.

For the situation with price adjustments, a heuristic solution procedure is suggested. In addition, the bilevel models are shown to serve as a basis from which different well-known location models – as, for example, the p-median problem, the preemptive location problem and the maximum covering problem – can be derived as special cases.

competitive location discrete location models oligopoly spatial price discrimination 

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References

  1. [1]
    G. Anandalingam and T.L. Friesz, Hierarchical optimization: An introduction, Annals of Operations Research 34 (1992) 1-11.Google Scholar
  2. [2]
    S.P. Anderson, A. de Palma and J.-F. Thisse, Discrete Choice Theory of Product Differentiation (MIT Press, Cambridge, London, 1992).Google Scholar
  3. [3]
    J.F. Bard, Practical Bilevel Optimization-Algorithms and Applications (Kluwer, Dordrecht, 1998).Google Scholar
  4. [4]
    J.F. Bard and J.T. Moore, An algorithm for the discrete bilevel programming problem, Naval Research Logistics 39 (1992) 419-435.Google Scholar
  5. [5]
    M.J. Beckmann, Spatial price policies revisited, Bell Journal of Economics 7 (1976) 619-630.Google Scholar
  6. [6]
    O. Ben-Ayed and C.E. Blair, Computational difficulties of bilevel linear programming, Operations Research 38 (1990) 556-560.Google Scholar
  7. [7]
    W.F. Bialas and M.H. Karwan, Two-level linear programming, Management Science 30 (1984) 1004-1020.Google Scholar
  8. [8]
    W. Brüggemann, K. Fischer and H. Jahnke, Problems-models-complexity (2002) in preparation.Google Scholar
  9. [9]
    M. Daskin, Network and Discrete Location (Wiley, New York, 1995).Google Scholar
  10. [10]
    C. D'Aspremont, J.J. Gabszewicz and J.-F. Thisse, On Hotelling's “Stability in Competition”, Econometrica 47 (1979) 1145-1150.Google Scholar
  11. [11]
    H.A. Eiselt, Hotelling's duopoly on a tree, Annals of Operations Research 40 (1992) 195-207.Google Scholar
  12. [12]
    H.A. Eiselt and J. Bhadury, Reachability of locational Nash equilibria, OR Spektrum 20 (1998) 101-107.Google Scholar
  13. [13]
    H.A. Eiselt and G. Laporte, Sequential location problems, European Journal of Operational Research 96 (1996) 217-231.Google Scholar
  14. [14]
    H.A. Eiselt, G. Laporte and J.-F. Thisse, Competitive location models: A framework and bibliography, Transportation Science 27 (1993) 44-54.Google Scholar
  15. [15]
    K. Fischer, Standortplanung unter Berücksichtigung verschiedener Marktbedingungen (Physica, Heidelberg, German, 1997).Google Scholar
  16. [16]
    K. Fischer, Discrete sequential models for competitive location planning, Working Paper No. 41, Institut für Logistik und Transport, Universität Hamburg (2001).Google Scholar
  17. [17]
    J. Fortuny-Amat and B. McCarl, A representation and economic interpretation of a two-level programming problem, Journal of the Operational Research Society 32 (1981) 783-792.Google Scholar
  18. [18]
    J.J. Gabszewicz and J.-F. Thisse, Spatial competition and the location of firms, in: Location Theory, ed. R. Arnott, Fundamentals of Pure and Applied Economics, Vol. 5 (Harwood Academic, Chur, 1986).Google Scholar
  19. [19]
    A. Ghosh, S. McLafferty and C.S. Craig, Multifacility retail networks, in: Facility Location: A Survey of Applications and Methods, ed. Z. Drezner (Springer, New York, 1995).Google Scholar
  20. [20]
    S.L. Hakimi, On locating new facilities in a competitive environment, European Journal of Operational Research 12 (1983) 29–35.Google Scholar
  21. [21]
    S.L. Hakimi, Locations with spatial interactions: Competitive locations and games, in: Discrete Location Theory, eds. P.B. Mirchandani and R.L. Francis (Wiley, New York, 1990).Google Scholar
  22. [22]
    [22] P. Hanjoul, P. Hansen, D. Peeters and J.-F. Thisse, Uncapacitated plant location under alternative spatial price policies, Management Science 36 (1990) 41-57.Google Scholar
  23. [23]
    P. Hansen, M. Labbé, D. Peeters and J.-F. Thisse, Facility location analysis, in: Systems of Cities and Facility Location, ed. R. Arnott, Fundamentals of Pure and Applied Economics, Vol. 22 (Harwood Academic, Chur, 1987).Google Scholar
  24. [24]
    P. Hansen and J.-F. Thisse, Multiplant location for profit maximisation, Environment and Planning A 9 (1977) 63-73.Google Scholar
  25. [25]
    H. Hotelling, Stability in competition, Economic Journal 39 (1929) 41-57.Google Scholar
  26. [26]
    E. Kohlberg and W. Novshek, Equilibrium in a simple price-location model, Economics Letters 9 (1982) 7-15.Google Scholar
  27. [27]
    M. Labbé and S.L. Hakimi, Market and locational equilibrium for two competitors, Operations Research 39 (1991) 749-756.Google Scholar
  28. [28]
    M. Labbé, D. Peeters and J.-F. Thisse, Location on networks, in: Network Routing, eds. M.O. Ball, T.L. Magnanti, C.L. Monma and G.L. Nemhauser, Handbooks in Operations Research and Management Science, Vol. 8 (Elsevier, Amsterdam, 1995).Google Scholar
  29. [29]
    P.J. Lederer and J.-F. Thisse, Competitive location on networks under delivered pricing, Operations Research Letters 9 (1990) 147-153.Google Scholar
  30. [30]
    T.C. Miller, T.L. Friesz and R.L. Tobin, Heuristic algorithms for delivered price spatially competitive network facility location problems, Annals of Operations Research 34 (1992) 177-202.Google Scholar
  31. [31]
    T.C. Miller, T.L. Friesz and R.L. Tobin, Equilibrium Facility Location on Networks (Springer, Berlin, 1996).Google Scholar
  32. [32]
    P.B. Mirchandani, The p-median problem and generalizations, in: Discrete Location Theory, eds. P.B. Mirchandani and R.L. Francis (Wiley, New York, 1990).Google Scholar
  33. [33]
    J.T. Moore and J.F. Bard, The mixed integer linear bilevel programming problem, Operations Research 38 (1990) 911-921.Google Scholar
  34. [34]
    G. Owen, Game Theory, 2nd ed. (New York, 1982).Google Scholar
  35. [35]
    D. Peeters and J.-F. Thisse, Economic models of firm location, in: Facility Location: A Survey of Applications and Methods, ed. Z. Drezner (Springer, New York, 1995).Google Scholar
  36. [36]
    E.C. Prescott and M. Visscher, Sequential location among firms with foresight, Bell Journal of Economics 8 (1977) 378-393.Google Scholar
  37. [37]
    C. ReVelle, The maximum capture or “Sphere of influence” location problem: Hotelling revisited on a network, Journal of Regional Science 26 (1986) 343-358.Google Scholar
  38. [38]
    R. Selten, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4 (1975) 25-55.Google Scholar
  39. [39]
    D. Serra and C. ReVelle, Market capture by two competitors: The preemptive location problem, Journal of Regional Science 34 (1994) 549-561.Google Scholar
  40. [40]
    D. Serra and C. ReVelle, Competitive location in discrete space, in: Facility Location: A Survey of Applications and Methods, ed. Z. Drezner (Springer, New York, 1995).Google Scholar
  41. [41]
    J. Tirole, The Theory of Industrial Organization (MIT Press, Cambridge, London, 1988).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Kathrin Fischer
    • 1
  1. 1.Institut für Logistik und TransportUniversität HamburgHamburgGermany

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