Annals of Operations Research

, Volume 122, Issue 1–4, pp 43–58 | Cite as

An Improved Branch & Bound Method for the Uncapacitated Competitive Location Problem

  • Stefano Benati


In this paper, the problem of locating new facilities in a competitive environment is considered. The problem is formulated as the firm expected profit maximization and a set of nodes is selected in a graph representing the geographical zone. Profit depends on fixed and deterministic location costs and, since customers are independent decision-makers, on the expected market share. The problem is an instance of nonlinear integer programming, because the objective function is concave and submodular. Due to this complexity a branch & bound method is developed for solving small size problems (that is, when the number of nodes is less than 50), while a heuristic is necessary for larger problems. The branch & bound is called data-correcting method, while the approximate solutions are obtained using the heuristic-concentration method.

competitive location models random utility theory submodular functions heuristic concentration data-correcting method 


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  1. Benati, S. (1997). “Submodularity in Competitive Location Problems.” Ricerca Operativa 26, 3–34.Google Scholar
  2. Benati, S. (1999). “The Maximum Capture Problem with Heterogeneous Customers.” Computers and Operations Research 26, 1351–1367.Google Scholar
  3. Benati, S. (2000). “NP-Hardness of Some Competitive Location Models with Random Utilities.” Studies in Locational Analysis 14, 211–232.Google Scholar
  4. Benati, S. and P. Hansen. (2002). “The Maximum Capture Problem with Random Utilities: Problem Formulation and Algorithms.” European Journal of Operational Research 143, 518–530.Google Scholar
  5. Berman, O. and K. Dmitry. (1998). “Flow Intercepting Spatial Interaction Model: A New Approach to Optimal Location of Competitive Facilities.” Location Science 6, 41–65.Google Scholar
  6. Brimberg, J., P. Hansen, N. Mladenović, and E.D. Taillard. (2000). “Improvements and Comparison of Heuristics for Solving the Multisource Weber Problem.” Operations Research 48, 440–460.Google Scholar
  7. Choi, D.S., W.S. Desarbo and P.T. Harker. (1990). “Product Positioning Under Price Competition.” Management Science 38, 175–199.Google Scholar
  8. Drezner, T. and Z. Drezner. (1996). “Competitive Facilities: Market Share and Location with Random Utility.” Journal of Regional Science 36, 1–15.Google Scholar
  9. Eiselt, H.A. and G. Laporte. (1997). “Demand Allocation Functions.” Location Science.Google Scholar
  10. Fotheringham, A.S. and M.E. O'Kelly. (1989). Spatial Interaction Models: Formulations and Applications. Dordrecht: Kluwer.Google Scholar
  11. Goldengorin, B. (1995). “Requirements of Standard: Optimization Models and Algorithms.” Russian Operations Research. Hoogezand, The Netherlands.Google Scholar
  12. Goldengorin, B., G. Sierksma, G.A. Tijssen, and M. Tso. (1999). “The Data-Correcting Algorithm for the Minimization of Supermodular Functions.” Management Science 45, 1359–1551.Google Scholar
  13. Hakimi, S.L. (1990). “Locations with Spatial Interactions: Competitive Location and Games.” In Mirchandani and Francis (eds.), Discrete Location Theory. New York: Wiley.Google Scholar
  14. Hansen, P. and N. Mladenović. (1998). “Variable Neighbourhood Search for the p-Median Problem.” Location Science 5(4), 207–226.Google Scholar
  15. Hansen, P. and N. Mladenović. (1999). “An Introduction to Variable Neighborhood Search.” In S. Voss et al. (eds.), Metaheuristics, Advances and Trends in Local Search Paradigms for Optimization. Dordrecht: Kluwer, pp. 433–458.Google Scholar
  16. Hansen, P., N. Mladenović, and D. Perez-Brito. (2001). “Variable Neighborhood Decomposition Search.” Journal of Heuristics 7, 335–350.Google Scholar
  17. Leonardi, G. (1983). “The Use of Random Utility Theory in Building Location–Allocation Models.” In J.F. Thisse and H.G. Zoller (eds.), Locational Analysis of Public Facilities. Amsterdam: North-Holland.Google Scholar
  18. Nemhauser, G.L. and L.A. Wolsey. (1981). “Maximizing Submodular Set Functions: Formulations and Analysis of Algorithms.” Annals of Discrete Mathematics 11, 279–301.Google Scholar
  19. Nemhauser, G.L. and L.A. Wolsey. (1988). Integer and Combinatorial Optimization. New York: Wiley Interscience.Google Scholar
  20. O'Kelly, M.E. (1999). “Trade-Area and Choice Based Samples: Methods.” Environment and Planning A 31, 613–627.Google Scholar
  21. Plastria, F. (2001). “Static Competitive Facility Location: An Overview of Optimization Approaches.” European Journal of Operational Research 129, 461–470.Google Scholar
  22. ReVelle, C. and D. Serra. (1995). “Competitive Location in Discrete Space.” In Z. Drezner (ed.), Facility Location: A Survey of Applications and Methods. Berlin: Springer.Google Scholar
  23. Rosing, K.E. and C.S. ReVelle (1997). “Heuristic Concentration: Two Stage Solution Construction.” European Journal of Operational Research 104, 93–99.Google Scholar
  24. Rosing, K.E., C.S. ReVelle, and D.A. Shilling. (1999). “A Gamma-Heuristic for the p-Median Problem.” European Journal of Operational Research 117, 522–532.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Stefano Benati
    • 1
  1. 1.Dipartimento di Informatica e Studi AziendaliUniversitá di TrentoTrentoItaly

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