Stackelberg-Nash Equilibrium in competitive facility location game among a franchisor and two investors

Document Type : Industry Article


1 Department of Industrial engineering, Electronic Branch, Islamic Azad University, Tehran,Iran

2 Department of industrial engineering, Electronic branch, Islamic Azad University, Tehran, Iran

3 Department of e-Business, Information Technology Faculty, Iranian Research Institute for Information Science and Technology (IranDoc), Tehran, Iran.


In the competitive location problems, the matter of the optimal location of single or multiple facilities are in a condition in which competitors exist as well. This paper deals with a type of a competitive location on which a leader possibly uses the investment of other investors through concession and receives a percentage of their income. He also can place his own facilities on potential locations that are available. In fact there are three decision-makers, one as a leader, others as followers who get in the game of facility location for placing their facilities. The location of facilities is a simultaneous game in which decisions are made in two levels, in the first level they are asynchronous and sequential, in the second level decisions are synchronous, namely the leader selects places for his own facilities, where other decision-makers (investor) play the simultaneous and non-cooperative game, and according to Nash equilibrium, select the optimal location. The leader uses new locations to optimize his objective function. Eventually, a numerical example is presented to test the model and a comprehensive sensitivity analysis is carried out to extract some managerial insights.


Main Subjects

[1] F. Plastria, and L. Vanhaverbeke, “Discrete Models for Competitive Location with Foresight”, Computers & Operation Research, Vol. 35, NO. 3, 2008, pp. 683-700.
[2] مهدی بشیری، عباس حسینی‌جو و جواد حسینی‌نژاد، "طراحی سیستمهای صنعتی (مکان‌یابی و استقرار تسهیلات)"، چاپ سوم، انتشارات دانشگاه شاهد، تهران، 1392.
 [3] S. H. Owen, and M. S. Daskin, “Strategic Facility Location: A Review”, European Journal of Operational Research, Vol. 111, NO. 3, 1998, pp. 423-447.
[4] H. A. Eiselt, “Competitive Location Models: A Framework and Bibliography”, Transportation Science, Vol. 27, NO. 1, 1993, pp. 44-54.
[5] A. Hee-Kap, C. H. Siu-Wing, C. H. Otfreid, G. Mordecai, and V. O. Rene, “Competitive Facility Location: The Voronoi Game”, Theoretical Computer Science, Vol. 310, NO. 1-3, 2004, pp. 457-467.
[6] A. Groznik, and H. S. Heese, “Supply chain interactions due to store-brand introductions: The impact of retail competition”, European Journal of Operational Research, Vol. 203, NO. 3, 2010, pp. 575-582.
 [7] H. Hotelling, “Stability in Competition”, Economic Journal, Vol. 39, 1929, pp. 41-57.
[8] C. D'Aspremont, J. J. Gabszewicz, and J. F. Thisse, “On Hoteling’s Stability in Competition”, The Econometric Society, Vol. 47, 1979, pp. 1145-1150.
[9] H. A. Eiselt, “Subsidy Competition in Networks”, Computational & Mathematical Organization Theory, Vol. 6, NO. 1, 2000, pp. 99-111.
[10] A. V. Kononov, Y. A. Kochetov, and A. V. Plyasunov, “Competitive Facility Location Models”, Computational Mathematics and Mathematical physics, Vol. 49, NO. 6, 2009, pp. 994-1009.
[11] Sh. Shiode, K. Y. Yeh, and H. Ch. Hsia, “Optimal Location Policy for Three competitive facilities”, Computers & Industrial Engineering, Vol 62, NO. 3, 2012, pp. 703-707.
 [12] L. Fernandez, and M. T. E. Hendrix, “Recent Insights in Huff-like Competitive Facility Location and Design”, European Journal of Operational Research, Vol. 227, NO. 3, 2013, pp.581-584
[13] M. Hofer, and J. Cardinal, “Non-Cooperative Facility Location and Covering Game”, Theoretical Computer Science, Vol. 411, NO. 16-18, 2010, pp. 1855-1876.
[14] حامد فلاح، حمیدرضا اسکندری، سیدحسام‌الدین ذگردی، و سیدکمال چهارسوقی، " ارائه مدل دوسطحی طراحی شبکه زنجیره تامین حلقه بسته در شرایط عدم قطعیت و رقابت بین زنجیرهای: حل با رویکرد تجزیه بندرز"، مجله مدل‌سازی در مهندسی، دوره 15، شماره 49، 1396، صفحه 201-215.
[15] مهدی بشیری، و محمدرضا یعقوبی، "مدل‌سازی ریاضی مساله مکان‌یابیP مرکز با در نظر گرفتن سلسله‌مراتب لانه‌ای
و کاربرد الگوریتم بهینه‌سازی گروهی ذرات در حل آن" مجله مدل‌سازی در مهندسی، دوره 14، شماره 47، 1395، صفحه 187-197.
[16] فرشاد حکیم‌پور، سیامک طلعت‌اهری، و ابوالفضل رنجبر، "ارزیابی و مقایسه الگوریتم‌های بهینه‌سازی ژنتیک، شبیه‌سازی تبرید و فاخته‌ها در مکان‌یابی رقابتی تسهیلات (مطالعه موردی: بانک‌ها)". مجله مدل‌سازی در مهندسی، دوره 15، شماره 48، 1396، صفحه 231-246.
[17] P. Godinho, and J. Dias, “A Two-player Competitive Discrete Location Model with Simultaneous Decisions”, European Journal of Operational Research, Vol. 207, NO. 3, 2010, pp. 1419-1432
[18] K. Fisher, “Sequential discrete p-facility models for competitive location planning”, annuals of operations research, Vol. 111, NO. 1-4, 2002, pp. 253-270
[19] T. Drezner, Z. Drezner, and P. Kalczynski, “A leader–follower model for discrete competitive facility location”, Computers & Operations Research, Vol. 64, 2015, pp. 51-59.
[20] B. Biesinger, B. Hu, and G.  Raidl, “Models and algorithms for competitive facility location problems with different customer behavior”, Annals of Mathematics and Artificial Intelligence, Vol. 76, 2016, pp. 93–119.
[21] A. Konak, S. Kulturel-Konak, and L. Snyder, “A Multi-Objective Approach to the Competitive Facility Location Problem”, Procedia Computer Science, Vol. 108, 2017, pp. 1434-1442.
[22] W. Shan, Q. Yan, C. Chen, M. Zhang, B. Yao, and X. Fu, “Optimization of competitive facility location for chain stores”, Annals of Operations Research, 2017, DOI 10.1007/s10479-017-2579-z.
[23] G. Li, Y. Li, J. Shu, and D. Xu, “A Cross-Monotonic Cost-Sharing Scheme for the Concave Facility Game”, Journal of Global Optimization, Vol. 56, NO. 4, 2013, pp. 1325-1334.
[24] P. Godinho, and J. Dias, “A Two-player Simultaneous Location Game: Preferential Rights and Overbidding”, European Journal of Operational Research, Vol. 229, NO. 3, 2013, pp. 663-672.
[25] J. F. Bard, “Practical Bilevel Optimization: Applications and Algorithms”, Kluwer Academic Press, 1998.
[26] O. Ben-Ayed, D. E. Boyce, C. E. Blair, “A general bi-level linear programming formulation of the network design problem”, Transportation Research Part B, Vol. 22, NO. 4, 1988, pp. 311–318.
[27] J-S. Pang and D. Chan, “Iterative methods for variational and complementarity problems”, Mathematical Programming, Vol. 24, 1982, pp. 284–313.