A mathematical model for competitive location problem with product selection

Document Type : Article

Authors

1 Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

2 Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran.

3 Department of Industrial Engineering, K. N. Toosi University of Technology, Vanak, Tehran, Iran.

Abstract

In this paper, a new competitive location problem for a chain is considered. The chain’s owner can offer a variety of products. The model’s objective is to determine both the location of the new facilities and the optimal product type for each opened facility. The patronizing behavior of the customers is based on Huff rule and the location of new facilities is selected from a set of potential sites. As a result, the model is a nonlinear integer programming problem and for solving the proposed model, the problem is reformulated as a mixed integer linear programming and therefore a standard optimization solver can be used for obtaining the optimal solutions for small and medium-size problems. To cope with large-size problems, we develop two methods: 1) a heuristic method for a special case and 2) a hybrid heuristic-firefly algorithm for general cases. By using the proposed model, it is shown numerically that in multi-product industries in which owner of the facilities is able to offer different types of products, in addition to the optimal location, it is necessary to determine the best products. In the end, a real-world case study for locating a new bakery is presented.

Keywords

Main Subjects

References

References:
1. Snyder, L.V. "Facility location under uncertainty: a review", IIE Transactions, 38(7), pp. 547-564 (2006).
2. Zarandi, M.F., Davari, S., and Sisakht, S.H. "The large-scale maximal covering location problem", Scientia Iranica, 18(6), pp. 1564-1570 (2011).
3. Karimi, H. and Bashiri, M. "Hub covering location problems with different coverage types", Scientia Iranica, 18(6), pp. 1571-1578 (2011).
4. Melo, M.T., Nickel, S., and Saldanha-Da-Gama, F. "Facility location and supply chain management-A review", European Journal of Operational Research, 196(2), pp. 401-412 (2009).
5. Owen, S.H. and Daskin, M.S. "Strategic facility location: A review", European Journal of Operational Research, 111(3), pp. 423-447 (1998).
6. Hotelling, H. "Stability in competition", Economic Journal, 39, pp. 41-57 (1929).
7. Hakimi, S.L. "On locating new facilities in a competitive environment", in ISOLDE II Conference, Skodsborg, Denmark (1981).
8. Hakimi, S.L. "On locating new facilities in a competitive environment", European Journal of Operational Research, 12, pp. 29-35 (1983).
9. Hakimi, S.L. "p-median theorems for competitive location", Annals of Operations Research, 6, pp. 77-98 (1986).
10. Hakimi, S.L. "Locations with spatial interactions: Competitive locations and games", in P.B. Mirchandani and R.L. Francis, Eds., Discrete Location Theory, Wiley-Interscience, New York, NY, pp. 439-478 (1990).
11. Drezner, Z. "Competitive location strategies for two facilities", Regional Science and Urban Economics, 12, pp. 485-493 (1982).
12. Huff, D. "Defining and estimating a trading area", Journal of Marketing, 28, pp. 34-38 (1964).
13. Huff, D.L. "A programmed solution for approximating an optimum retail location", Land Economics, 42, pp. 293-303 (1966).
14. Eiselt, H.A., Laporte, G., and Thisse, J.F. "Competitive location models: A framework and bibliography", Transportation Science, 27(1), pp. 44-54 (1993).
15. Plastria, F. "Static competitive facility location: an overview of optimisation approaches", European Journal of Operational Research, 129(3), pp. 461-470 (2001).
16. Drezner, T. "A review of competitive facility location in the plane", Logistics Research, 7(1), pp. 1-12 (2014).
17. Ashtiani, M. "Competitive location: a state-of-art review", International Journal of Industrial Engineering Computations, 7(1), pp. 1-18 (2016).
18. Beresnev, V. and Melnikov, A. "Exact method for the capacitated competitive facility location problem", Computers & Operations Research, 95, pp. 73-82 (2018).
19. Plastria, F. and Carrizosa, E. "Optimal location and design of a competitive facility", Mathematical Programming, 100(2), pp. 247-265 (2004).
20. Hendrix, E.M. "On competition in a Stackelberg location-design model with deterministic supplier choice", Annals of Operations Research, 246(1-2), pp. 19-30 (2016).
21. Fernandez, J., Pelegrin, B., Plastria, F., et al. "Planar location and design of a new facility with inner and outer competition: an interval lexicographical-like solution procedure", Networks and Spatial Economics, 7(1), pp. 19-44 (2007).
22. Wang, X. and Ouyang, Y. "A continuum approximation approach to competitive facility location design under facility disruption risks", Transportation Research Part B: Methodological, 50, pp. 90-103 (2013). 
23. Fernandez, J., Boglarka, G., Redondo, J.L., et al. "A planar single-facility competitive location and design problem under the multi-deterministic choice rule". Computers & Operations Research, 78, pp. 305-315 (2017).
24. Aboolian, R., Berman, O., and Krass, D. "Competitive acility location and design problem", European Journal of Operational Research, 182(1), pp. 40-62 (2007).
25. Arrondo, A.G., Redondo, J.L., Fernandez, J., et al. "Solving a leader-follower facility problem via parallel evolutionary approaches", The Journal of Supercomputing, 70(2), pp. 600-611 (2014).
26. Fernandez, J., Pelegri, B., Plastria, F., et al. "Solving a Huff-like competitive location and design model for profit maximization in the plane", European Journal of Operational Research, 179(3), pp. 1274-1287 (2007).
27. Kucukaydin, H., Aras, N., and Altnel, I.K. "Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution", uropean Journal of Operational Research, 208(3), pp. 206-220 (2011).
28. Redondo, J.L., Arrondo, A.G., Fernandez, J., et al. "A two-level evolutionary algorithm for solving the facility location and design (1j 1)-centroid problem on the plane with variable demand", Journal of Global Optimization, 56(3), pp. 983-1005 (2013).
29. Redondo, J.L., Fernandez, J., Garcia, I., et al. "Parallel algorithms for continuous competitive location problems", Optimisation Methods & Software, 23(5), pp. 779-791 (2008).
30. Redondo, J.L., Fernandez, J., Garcia, I., et al.  Sensitivity analysis of a continuous multifacility competitive location and design problem", Top, 17(2), p. 347 (2009).
31. Redondo, J.L., Fernandez, J., Garcia, I., et al. "Heuristics for the facility location and design (1j1)-centroid problem on the plane", Computational Optimization and Applications, 45(1), pp. 111-141 (2010).
32. Saidani, N., Chu, F., and Chen, H. "Competitive facility location and design with reactions of competitors already in the market", European Journal of Operational Research, 219(1), pp. 9-17 (2012).
33. Saiz, M.E., Hendrix, E.M., and Pelegrin, B. "On Nash equilibria of a competitive location-design problem", European Journal of Operational Research, 210(3), pp. 588-593 (2011).
34. Arrondo, A.G., Fernandez, J., Redondo, J.L., et al. "An approach for solving competitive location problems with variable demand using multicore systems", Optimization Letters, 8(2), pp. 555-567 (2014).
35. Arrondo, A.G., Redondo, J.L., Fernandez, J., et al.  Parallelization of a non-linear multi-objective optimization algorithm: application to a location problem", Applied Mathematics and Computation, 255, pp. 114-124 (2015).
36. Fernandez, J., Toth, B., Plastria, F., et al. "Reconciling franchisor and franchisee: a planar biobjective competitive location and design model", In Recent Advances in Optimization, pp. 375-398, Springer, Berlin, Heidelberg (2006).
37. Kucukaydin, H., Aras, N., and Altinel, _I.K. "A discrete competitive facility location model with variable attractiveness", Journal of the Operational Research Society, 62(9), pp. 1726-1741 (2011).
38. Kucukaydin, H., Aras, N., and Altinel, _I.K. "A leaderfollower game in competitive facility location", Computers & Operations Research, 39(2), pp. 437-448 (2012).
39. Lancinskas, A., Ortigosa, P.M., and Zilinskas, J. "Parallel shared-memory multi-objective stochastic search for competitive facility location", In European Conference on Parallel Processing, pp. 71-82, Springer, Cham (2014, August).
40. Redondo, J.L., Fernandez, J., Arrondo, A.G., et al. "Fixed or variable demand? Does it matter when locating a facility?", Omega, 40(1), pp. 9-20 (2012).
41. Redondo, J.L., Fernandez, J., Garcia, I., et al. "A robust and efficient algorithm for planar competitive location problems", Annals of Operations Research, 167(1), pp. 87-105 (2009).
42. Redondo, J.L., Fernandez, J., Garcia, I., et al. "Solving the multiple competitive facilities location and design problem on the plane", Evolutionary Computation, 17(1), pp. 21-53 (2009).
43. Redondo, J.L., Fernandez, J., Hervas, J.D. A., et al. "Approximating the Pareto-front of a planar biobjective competitive facility location and design problem", Computers & Operations Research, 62, pp. 337- 349 (2015).
44. Toth, B., Fernandez, J., Pelegrin, B., et al. "Sequential versus simultaneous approach in the location and design of two new facilities using planar Huff-like models", Computers & Operations Research, 36(5), pp. 1393-1405 (2009).
45. Bagherinejad, J. and Niknam, A. "Solving the competitive facility location problem considering the reactions of competitor with a hybrid algorithm including tabu search and exact method", Journal of Industrial Engineering International, 14(1), pp. 171-183 (2018).
46. Fernandez, J., Boglarka, G., Redondo, J.L., et al. "The probabilistic customer's choice rule with a threshold attraction value: Effect on the location of competitive facilities in the plane", Computers & Operations Research, 101, pp. 234-249 (Jan. 2019).
47. Aboolian, R., Berman, O., and Krass, D. "Optimizing pricing and location decisions for competitive service facilities charging uniform price", Journal of the Operational Research Society, 59(11), pp. 1506- 1519 (2008).
48. Fernandez, J., Salhi, S., and Boglarka, G.  Location equilibria for a continuous competitive facility location problem under delivered pricing", Computers & Operations Research, 41, pp. 185-195 (2014).
49. Fischer, K. "Sequential discrete p-facility models for competitive location planning", Annals of Operations Research, 111(1-4), pp. 253-270 (2002).
50. Perez, M.D.G., Hernandez, P.F., and Pelegrin, B.P. "On price competition in location-price models with spatially separated markets", Top, 12(2), pp. 351-374 (2004).
51. Panin, A.A., Pashchenko, M.G., and Plyasunov, A.V. "Bilevel competitive facility location and pricing problems", Automation and Remote Control, 75(4), pp. 715-727 (2014).
52. Pelegrin, B., Fernandez, P., Perez, M.D.G., et al. "On the location of new facilities for chain expansion under delivered pricing", Omega, 40(2), pp. 149-158 (2012).
53. Rohaninejad, M., Navidi, H., Nouri, B.V., et al. "A new approach to cooperative competition in facility location problems: Mathematical formulations and an approximation algorithm", Computers & Operations Research, 83, pp. 45-53 (2017).
54. Meagher, K.J. and Zauner, K.G. "Product differentiation and location decisions under demand uncertainty", Journal of Economic Theory, 117(2), pp. 201- 216 (2004).
55. Meagher, K.J. "Optimal product variety in a Hotelling model", Economics Letters, 117(1), pp. 71-73 (2012).
56. Kress, D. and Pesch, E. "Competitive location and pricing on networks with random utilities", Networks and Spatial Economics, 16(3), pp. 837-863 (2016).
57. Konur, D. and Geunes, J. "Competitive multi-facility location games with non-identical firms and convex traffic congestion costs", Transportation Research Part E: Logistics and Transportation Review, 48(1), pp. 373-385 (2012).
58. Kwasnica, A.M. and Stavrulaki, E. "Competitive location and capacity decisions for firms serving timesensitive customers", Naval Research Logistics (NRL), 55(7), pp. 704-721 (2008).
59. Zhang, L. and Rushton, G. "Optimizing the size and locations of facilities in competitive multi-site service systems", Computers & Operations Research, 35(2), pp. 327-338 (2008).
60. Bozkaya, B., Yanik, S., and Balcisoy, S. "A GIS-based optimization framework for competitive multi-facility location-routing problem", Networks and Spatial Economics, 10(3), pp. 297-320 (2010).
61. Lancaster, K. "The economics of product variety: A survey", Marketing Science, 9(3), pp. 189-206 (1990).
62. Rhim, H., Ho, T.H., and Karmarkar, U.S. "Competitive location, production, and market selection", European Journal of Operational Research, 149(1), pp. 211-228 (2003).
63. Zhang, R. "Product market competition, competitive strategy, and analyst coverage", Review of Quantitative Finance and Accounting, 50(1), pp. 239-260 (2018).
64. Brander, J.A. and Eaton, J. "Product line rivalry", The American Economic Review, 74(3), pp. 323-334 (1984).
65. Economides, N. "Minimal and maximal product differentiation in Hotelling's duopoly", Economics Letters, 21(1), pp. 67-71 (1986).
66. Beresnev, V.L. and Suslov, V.I. "A mathematical model of market competition", Journal of Applied and Industrial Mathematics, 4(2), pp. 147-157 (2010).
67. Sadjadi, S.J., Ashtiani, M.G., Ramezanian, R., et al. "A fire y algorithm for solving competitive locationdesign problem: a case study", Journal of Industrial Engineering International, 12(4), pp. 517-527 (2016).
68. Hansen, P., De Aragao, M.V.P., and Ribeiro, C.C. "Boolean query optimization and the 0-1 hyperbolic sum problem", Annals of Mathematics and Artificial Intelligence, 1(1-4), pp. 97-109 (1990).
69. Benati, S. and Hansen, P. "The maximum capture problem with random utilities: Problem formulation and algorithms", European Journal of Operational Research, 143(3), pp. 518-530 (2002).
70. Benati, S. "The maximum capture problem with heterogeneous customers", Computers & Operations Research, 26(14), pp. 1351-1367 (1999).
71. Yang, X.S. "Fire y algorithms for multimodal optimization", In Stochastic Algorithms: Foundations and Applications, pp. 169-178, Springer Berlin Heidelberg (2009).
72. Yang, X.S. "Fire y algorithms for multimodal optimization", In International Symposium on Stochastic Algorithms, pp. 169-178, Springer Berlin Heidelberg (2009, October).
73. Yang, X.S., Nature-Inspired Metaheuristic Algorithms, Luniver press (2010).
74. Sayadi, M., Ramezanian, R., and Ghaffari-Nasab, N. "A discrete fire y meta-heuristic with local search for makespan minimization in permutation  flow shop scheduling problems", International Journal of Industrial Engineering Computations, 1(1), pp. 1-10 (2010).
75. Parasuraman, A., Zeithaml, V.A., and Berry, L.L. "Servqual: A multiple-item scale for measuring consumer perc", Journal of Retailing, 64(1), p. 12 (1988).
76. Mostaghel, R. and Albadvi, A. "A cross-cultural comparison of service quality prioritization", Scientia Iranica, Transactions E, Industrial Engineering, 16(1), p. 65 (2009).
Scientia Iranica
Volume 27, Issue 4
Transactions on Industrial Engineering (E)
July and August 2020
Pages 2157-2176

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