Abstract
The two best studied facility location problems are the \(p\)-median problem and the uncapacitated facility location problem (Daskin, Network and discrete location: models, algorithms, and applications. Wiley, New York, 1995; Mirchandani and Francis, Discrete location theory. Wiley, New York, 1990). Both seek the location of the facilities minimizing the total cost, assuming no uncertainty in costs exists, and thus all parameters are known. In most real-world location problems the demand is not certain, because it is a long-term planning decision, and thus, together with the minimization of costs, optimizing some robustness measure is sound. In this paper we address bi-objective versions of such location problems, in which the total cost, as well as the robustness associated with the demand, are optimized. A dominating set is constructed for these bi-objective nonlinear integer problems via the \(\varepsilon \)-constraint method. Computational results on test instances are presented, showing the feasibility of our approach to approximate the Pareto-optimal set.
Similar content being viewed by others
References
Alekseeva, E., Kochetov, Y., Plyasunov, A.: Complexity of local search for the p-median problem. Eur. J. Oper. Res. 191(3), 736–752 (2008)
Avella, P., Boccia, M., Salerno, S., Vasilyev, I.: An aggregation heuristic for large scale p-median problem. Comput. Oper. Res. 39(7), 1625–1632 (2012)
Avella, P., Sassano, A., Vasilyev, I.: Computational study of large-scale p-median problems. Math. Program. 109(1), 89–114 (2007)
Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23(4), 769–805 (1998)
Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25(1), 1–13 (1999)
Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98(1–3), 49–71 (2003)
Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52(1), 35–53 (2004)
Bilde, O., Krarup, J.: Sharp lower bounds and efficient algorithms for the simple plant location problem. In: Hammer, P.L., Johnson, E.L., Korte, B.H., Nemhauser, G.L. (eds.) Studies in Integer Programming, Annals of Discrete Mathematics, vol. 1, pp. 79–97. Elsevier, Amsterdam (1977)
Blanquero, R., Carrizosa, E.: A d.c. biojective location model. J. Glob. Optim. 23(2), 139–154 (2002)
Blanquero, R., Carrizosa, E., Hendrix, E.M.T.: Locating a competitive facility in the plane with a robustness criterion. Eur. J. Oper. Res. 215(1), 21–24 (2011)
Carrizosa, E., Nickel, S.: Robust facility location. Math. Methods Oper. Res. 58(2), 331–349 (2003)
Chudak, F.: Improved approximation algorithms for uncapacitated facility location. In: Bixby, R.E., Boyd, E.A., Ros-Mercado, R.Z. (eds.) Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 1412, pp. 180–194. Springer, Berlin (1998)
Ciligot-Travain, M., Traoré, S.: On a robustness property in single-facility location in continuous space. TOP 22(1), 321–330 (2014)
Cohon, J.L.: Multiobjective Programming and Planning, Mathematics in Science and Engineering, vol. 140. Academic Press, New York (1978)
Correia, I., Saldanha da Gama, F.: Facility location under uncertainty. In: Laporte, G., Nickel, G., Saldanha da Gama, F. (eds.) Location Science, pp. 177–203. Springer International Publishing, Berlin (2015)
Daskin, M.: Network and Discrete Location: Models, Algorithms, and Applications. Wiley, New York (1995)
Daskin, M.S., Maass, K.L.: The p-median problem. In: Laporte, G., Nickel, S., Saldanha da Gama, F. (eds.) Location Science, pp. 21–45. Springer International Publishing, Berlin (2015)
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)
Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys, chap. Multiobjective Combinatorial Optimization, pp. 369–407. International Series in Operations Research & Management Science. Kluwer Academic Publishers, Dordrecht (2002)
Ehrgott, M., Gandibleux, X. (eds.): Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys. International Series in Operations Research & Management Science. Kluwer Academic Publishers (2002)
Ehrgott, M., Ryan, D.: Bicriteria robustness versus cost optimisation in tour of duty planning at air new zealand. In: Proceedings of the 35th Annual Conference of the Operational Research Society of New Zealand, pp. 31–39. ORSNZ, Auckland (2000)
Farahani, R.Z., SteadieSeifi, M., Asgari, N.: Multiple criteria facility location problems: a survey. Appl. Math. Model. 34(7), 1689–1709 (2010)
Filippi, C., Stevenato, E.: Approximation schemes for bi-objective combinatorial optimization and their application to the tsp with profits. Comput. Oper. Res. 40(10), 2418–2428 (2013)
Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: An overview. Eur. J. Oper. Res. 235(3), 471–483 (2014)
Hakimi, S.: Optimal location of switching centers and the absolute centers and medians of a graph. Oper. Res. 12(3), 450–459 (1964)
Hakimi, S.: Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper. Res. 13(3), 462–475 (1965)
Handler, G.Y., Mirchandani, P.B.: Location on Networks: Theory and Algorithms, chap. Multiobjective and Other Location Problems on Network, pp. 165–195. MIT Press, Cambridge (1979)
Hatefi, S.M., Jolai, F.: Robust and reliable forwardreverse logistics network design under demand uncertainty and facility disruptions. Appl. Math. Model. 38(9–10), 2630–2647 (2014)
Kalcsics, J., Nickel, S., Pozo, M.A., Puerto, J., Rodrguez-Cha, A.M.: The multicriteria p-facility median location problem on networks. Eur. J. Oper. Res. 235(3), 484–493 (2014)
Kariv, O., Hakimi, S.: An algorithmic approach to network location problems; part 2. the \(p\)-medians. SIAM J. Appl. Math. 37(3), 539–560 (1979)
Kochetov, Y., Ivanenko, D.: Computationally difficult instances for the uncapacitated facility location problem. In: Ibaraki, T., Nonobe, K., Yagiura, M. (eds.) Metaheuristics: Progress as Real Problem Solvers, pp. 351–367. Springer, New York (2005)
Kolokolov, A.A., Zaozerskaya, L.A.: Solving a bicriteria problem of optimal service centers location. JMMA 12(2), 105–116 (2013)
Körkel, M.: On the exact solution of large-scale simple plant location problems. Eur. J. Oper. Res. 39(2), 157–173 (1989)
Krarup, J., Pruzan, P.M.: The simple plant location problem: survey and synthesis. Eur. J. Oper. Res. 12(1), 36–81 (1983)
Labbé, M., Peeters, D., Thisse, J.F.: Location on networks. In: Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.) Network Routing, Handbooks in Operations Research and Management Science, vol. 8, pp. 551–624. Elsevier, Amsterdam (1995)
Letchford, A.N., Miller, S.J.: An aggressive reduction scheme for the simple plant location problem. Eur. J. Oper. Res. 234(3), 674–682 (2014)
Manne, A.S.: Plant location under economies-of-scale—decentralization and computation. Manage. Sci. 11(2), 213–235 (1964)
Mavrotas, G.: Effective implementation of the \(\varepsilon \)-constraint method in multi-objective mathematical programming problems. Appl. Math. Comput. 213(2), 455–465 (2009)
Mavrotas, G., Figueira, J.R., Siskos, E.: Robustness analysis methodology for multi-objective combinatorial optimization problems and application to project selection. Omega 52, 142–155 (2015)
Melamed, I., Sigal, I.: A computational investigation of linear parametrization of criteria in multicriteria discrete programming. Comput. Math. Phys. 36(10), 1341–1343 (1996)
Melamed, I., Sigal, I.: The linear convolution of criteria in the bicriteria traveling salesman problem. Comput. Math. Phys. 37(8), 902–905 (1997)
Melamed, I., Sigal, I.: Numerical analysis of tricriteria tree and assignment problems. Comput. Math. Phys. 38(10), 1704–1707 (1998)
Melamed, I., Sigal, I.: Combinatorial optimization problems with two and three criteria. Dokl. Math. 59(3), 490–493 (1999)
Mirchandani, P., Francis, R. (eds.): Discrete Location Theory. Wiley, New York (1990)
Mladenović, N., Brimberg, J., Hansen, P., Moreno-Pérez, J.: The \(p\)-median problem: a survey of metaheuristic approaches. Eur. J. Oper. Res. 179(3), 927–939 (2007)
Mulvey, J.M., Vanderbei, R.J., Zenios, S.A.: Robust optimization of large-scale systems. Oper. Res. 43(2), 264–281 (1995)
Razmi, J., Zahedi-Anaraki, A.H., Zakerinia, M.S.: A bi-objective stochastic optimization model for reliable warehouse network redesign. Math. Comput. Model. 58(11–12), 1804–1813 (2013)
Reese, J.: Solution methods for the p-median problem: an annotated bibliography. Networks 28(3), 125–142 (2006)
ReVelle, C., Swain, R.: Central facilities location. Geograph. Anal. 2(1), 30–42 (1970)
Romero-Morales, M., Carrizosa, E., Conde, E.: Semi-obnoxious location models: a global optimization approach. Eur. J. Oper. Res. 102(2), 295–301 (1997)
Snyder, L.: Facility location under uncertainty: a review. IIE Trans. 38(7), 547–564 (2006)
Srinivasan, V., Thompson, G.: Algorithms for minimizing total cost, bottleneck time and bottleneck shipment in transportation problems. Nav. Res. Logist. 23(4), 567–595 (1976)
Tricoire, F., Graf, A., Gutjahr, W.: The bi-objective stochastic covering tour problem. Comput. Oper. Res. 39(7), 1582–1592 (2012)
Verter, V.: Uncapacitated and capacitated facility location problems. In: Eiselt, H.A., Marianov, V. (eds.) Foundations of Location Analysis, International Series in Operations Research & Management Science, vol. 155, pp. 25–37. Springer, New York (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the E. Carrizosa is supported by Grants MTM2012-36163, Spain, P11-FQM-7603 and FQM-329, Andalucía, all financed in part with EU ERD Funds. The study of the A. Ushakov and I. Vasilyev is partially supported by RFBR, research project NO. 14–07–00382-a.
Rights and permissions
About this article
Cite this article
Carrizosa, E., Ushakov, A. & Vasilyev, I. Threshold robustness in discrete facility location problems: a bi-objective approach. Optim Lett 9, 1297–1314 (2015). https://doi.org/10.1007/s11590-015-0892-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-015-0892-5