A Stackelberg Location on a Network with Fuzzy Random Demand Quantities Using Possibility Measure

Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 15)


This paper focuses on a competitive facility location problem between leader and follower on a network with demands whose weights are given uncertainly and vaguely. By representing them as fuzzy random variables, the optimal location problem can be formulated as a fuzzy random programming problem for finding Stackelberg equilibrium. For solving the problem, it is reformulated as the problem to find the optimal solutions maximizing a degree of possibility under some chance constraint for the leader. Theorems for its complexity are shown based upon the characteristics of the facility location.


Fuzzy Number Facility Location Tree Network Fuzzy Goal Chance Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Ammar, E.E.: On fuzzy random multiobjective quadratic programming. European Journal of Operational Research 193, 329–340 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Berman, O., Krass, K.: Facility location with stochastic demands and congestion. In: Drezner, Z., Hamacher, H.W. (eds.) Facility Location: Application and Theory, pp. 329–372. Springer, Berlin (2001)Google Scholar
  3. 3.
    Drezner, Z.: Competitive location strategies for two facilities. Regional Science and Urban Economics 12, 485–493 (1982)CrossRefGoogle Scholar
  4. 4.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. Journal Systems Science. 9, 613–626 (1978)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hakimi, S.L.: On locating new facilities in a competitive environment. European Journal of Operational Research 12, 29–35 (1983)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Hakimi, S.L.: Locations with spatial interactions: Competitive locations and games. In: Mirchandani, P.B., Francis, R.L. (eds.) Discrete Location Theory. Discrete Mathematics and Optimization, pp. 439–478. Wiley Interscience (1990)Google Scholar
  7. 7.
    Hotelling, H.: Stability in competition. The Economic Journal 30, 41–57 (1929)CrossRefGoogle Scholar
  8. 8.
    Katagiri, H., Sakawa, M., Hiroaki, I.: Fuzzy random bottleneck spanning tree problems using possibility and necessity measures. European Journal of Operational Research 152, 88–95 (2004)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Katagiri, H., Sakawa, M., Kato, K., Nishizaki, I.: A fuzzy random multiobjective 0-1 programming based on the expectation optimization model using possibility and necessity measures. Mathematical and Computer Modelling 40, 411–421 (2004)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Katagiri, H., Sakawa, M., Kato, K., Nishizaki, I.: Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability. European Journal of Operational Research 188, 530–539 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Kruse, R., Meyer, K.D.: Statistics with vague data. D. Riedel Publishing Company, Dordrecht (1987)MATHCrossRefGoogle Scholar
  12. 12.
    Kwakernaak, H.: Fuzzy random variables-I. definitions and theorems. Information Sciences 15, 1–29 (1978)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Miller, T.C., Friesz, T.L., Tobin, R.L.: Equilibrium facility location on networks. Springer, Berlin (1996)MATHGoogle Scholar
  14. 14.
    Moreno Pérez, J.A., Marcos Moreno Vega, L., Verdegay, J.L.: Fuzzy location problems on networks. Fuzzy Sets and Systems 142, 393–405 (2004)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Plastria, F., Vanhaverbeke, L.: Discrete models for competitive location with foresight. Computers & Operations Research 35, 683–700 (2008)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Qiao, Z., Wang, G.: On solutions and distribution problems of the linear programming with fuzzy random variable coefficients. Fuzzy Sets and Systems 58, 120–155 (1993)MathSciNetGoogle Scholar
  17. 17.
    Shiode, S., Drezner, Z.: A competitive facility location problem on a tree network with stochastic weights. European Journal of Operational Research 149, 47–52 (2003)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Spoerhase, J., Wirth, H.C.: (r,p)-centroid problems on paths and trees. Theoretical Computer Science 410, 5128–5137 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Uno, T., Katagiri, H., Kato, K.: Competitive facility location with random demands. IAENG Transactions on Engineering Technologies 3, 83–93 (2009)Google Scholar
  20. 20.
    Uno, T., Katagiri, H., Kato, K.: Competitive facility location with fuzzy random demands. IAENG Transactions on Engineering Technologies 5, 99–108 (2010)Google Scholar
  21. 21.
    Uno, T., Katagiri, H., Kato, K.: A Stackelberg Location Problem on a Tree Network with Fuzzy Random Demands. In: Phillips-Wren, G., Jain, L.C., Nakamatsu, K., Howlett, R.J. (eds.) IDT 2010. SIST, vol. 4, pp. 581–588. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  22. 22.
    Uno, T., Katagiri, H., Kato, K.: A competitive facility location problem on a network with fuzzy random weights. Engineering Letters 19, 143–146 (2011)Google Scholar
  23. 23.
    Wagnera, M.R., Bhaduryb, J., Penga, S.: Risk management in uncapacitated facility location models with random demands. Computers & Operations Research 36, 1002–1011 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, G.Y., Qiao, Z.: Linear programming with fuzzy random variable coefficients. Fuzzy Sets and Systems 57, 295–311 (1993)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Wendell, R.E., McKelvey, R.D.: New perspective in competitive location theory. European Journal of Operational Research 6, 174–182 (1981)MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Socio-Arts and SciencesThe University of TokushimaTokushima-shiJapan
  2. 2.Graduate School of EngineeringHiroshima UniversityHigashihiroshima-shiJapan
  3. 3.Faculty of Applied Information ScienceHiroshima Institute of TechnologySaeki-kuJapan

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