Abstract
The vector assignment p-median problem (VAPMP) (Weaver and Church in Transp Sci 19(1):58–74, 1985) was one of the first location-allocation models developed to handle split assignment of a demand to multiple facilities. The underlying construct of the VAPMP has been subsequently used in a number of reliable facility location and backup location models. Although in many applications the chance that a facility fails may vary substantially with locations, many existing models have assumed a uniform failure probability across all sites. As an improvement, this paper proposes a new model, the expected p-median problem as a generalization of existing approaches by explicitly considering site-dependent failure probabilities. Multi-level closest assignment constraints and two efficient integer linear programming (ILP) formulations are introduced. While prior research generally concludes that similar problems are not integer-friendly and cannot be solved by ILP software, computational results show that our model can be used to solve medium-sized location problems optimally using existing ILP software. Moreover, the new model can be used to formulate other reliable or expected location problems with consideration of site-dependent failure probabilities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
It should be noted that the proposed model is NP-hard since the p-median problem itself is NP-hard (when p is considered a parameter). Due to the additional complexity associated with dealing with non-uniform failures, the new problem is much harder than the PMP. While the PMP for the tested data could be solved within one tenth of a second using modern ILP solvers, the new problem would require hours of computation to be solved optimally.
The simple plant location problem (SPLP) and the p-median problem are two fundamental problems in location science. While both problems aim at minimizing the total system travel cost, the SPLP assumes knowledge of the cost associated with opening a facility (commensurate with travel cost), while the p-median problem assumes instead a fixed number of facilities to set up.
It should be noted that the model by Cui et al. (2010) and the EXPMP model in this paper are extensions of two different basic location models (the SPLP and the PMP, respectively). When different failure probability patterns are considered, the SPLP-based model may prescribe different number of facilities, whereas the EXPMP fixes the number of facilities to site in advance.
Note that the 49-city dataset used in this paper differs from the one in Snyder and Daskin (2005) in that the former consists of highest populated cities in each of the 48 states plus D.C. while the latter consists of the capital cities of each state plus D.C.
References
Bennett VL, Eaton DJ, Church RL (1982) Selecting sites for rural health workers. Soc Sci Med 16(1):63–72
Berman O, Krass D, Menezes MBC (2007) Facility reliability issues in network p-median problems: strategic centralization and co-location effects. Oper Res 55(2):332–350
Church RL, ReVelle C (1974) The maximal covering location problem. Pap Reg Sci Assoc 32(1):101–118
Church RL, ReVelle CS (1976) Theoretical and computational links between the p-median, location set-covering, and the maximal covering location problem. Geogr Anal 8(4):407–415
Church RL, Weaver JR (1986) Theoretical links between median and coverage location problems. Ann Oper Res 6(1):1–19
Church RL, Stoms DM, Davis FW (1996) Reserve selection as a maximal covering location problem. Biol Conserv 76(2):105–112
Church RL, Scaparra MP, Middleton RS (2004) Identifying critical infrastructure: the median and covering facility interdiction problems. Ann Assoc Am Geogr 94(3):491–502
Corley HW, Chang H (1974) Finding the n most vital nodes in a flow network. Manag Sci 21(3):362–364
Cui T, Ouyang Y, Shen ZJM (2010) Reliable facility location design under the risk of disruptions. Oper Res 58(4-Part-1):998–1011
Current JR, O’Kelly ME (1992) Locating emergency warning sirens. Decis Sci 23(1):221–234
Daskin MS (1983) A maximum expected covering location model: formulation, properties and heuristic solution. Transp Sci 17(1):48–70
Grubesic TH, Matisziw TC, Murray AT, Snediker D (2008) Comparative approaches for assessing network vulnerability. Int Reg Sci Rev 31(1):88–112
Hardy GH, Littlewood JE, Pólya G (1952) Inequalities, 2nd edn. Cambridge University Press, Cambridge
Hogan K, ReVelle C (1986) Concepts and applications of backup coverage. Manag Sci 32(11):1434–1444
Israeli E, Wood RK (2002) Shortest-path network interdiction. Networks 40(2):97–111
Kim H (2011) p-hub protection models for survivable hub network design. J Geogr Syst. doi:10.1007/s10109-011-0157-5
Kolli S, Evans GW (1999) A multiple objective integer programming approach for planning franchise expansion. Comput Ind Eng 37(3):543–561
Lei TL (2010) Location modeling utilizing closest and generalized closest transport/interaction assignment constructs. PhD. Dissertation, University of California, Santa Barbara, Santa Barbara
Lei TL, Church RL (2011a) Constructs for multilevel closest assignment in location modeling. Int Reg Sci Rev 34(3):339–367
Lei TL, Church RL (2011b) Locating short-term empty-container storage facilities to support port operations: a user optimal approach. Transp Res Part E: Logist Transp Rev 47(5):738–754
Lei TL, Church RL (2012) Vector assignment ordered median problem: a unified median problem. Int Reg Sci Rev. doi:10.1177/0160017612450710
Matisziw TC, Murray AT (2009) Area coverage maximization in service facility siting. J Geogr Syst 11(2):175–189
McMasters AW, Mustin TM (1970) Optimal interdiction of a supply network. Naval Res Logist Q 17(3):261–268
Murray AT, Grubesic TH (2007) Overview of reliability and vulnerability in critical infrastructure. In: Murray AT, Grubesic TH (eds) Critical infrastructure: reliability and vulnerability. Springer, Berlin, pp 1–8
Murray AT, Matisziw TC, Grubesic TH (2007) Critical network infrastructure analysis: interdiction and system flow. J Geogr Syst 9(2):103–117
Narasimhan S, Pirkul H, Schilling DA (1992) Capacitated emergency facility siting with multiple levels of backup. Ann Oper Res 40(1):323–337
Nickel S, Puerto J (1999) A unified approach to network location problems. Networks 34(4):283–290
Pirkul H, Schilling D (1989) The capacitated maximal covering location problem with backup service. Ann Oper Res 18(1):141–154
ReVelle CS, Swain RW (1970) Central facilities location. Geogr Anal 2(1):30–42
Scaparra MP, Church RL (2008) A bilevel mixed-integer program for critical infrastructure protection planning. Comput Oper Res 35(6):1905–1923
Sherali HD, Alameddine A (1992) A new reformulation-linearization technique for bilinear programming problems. J Global Optim 2(4):379–410
Shillington L, Tong D (2011) Maximizing wireless mesh network coverage. Int Reg Sci Rev 34(4):419–437
Snyder LV, Daskin MS (2005) Reliability models for facility location: the expected failure cost case. Transp Sci 39(3):400–416
Swamy C, Shmoys DB (2008) Fault-tolerant facility location. ACM Trans Algorithms (TALG) 4(4): 51:51–51:27
Tong D, Murray AT (2009) Maximizing coverage of spatial demand for service. Pap Reg Sci 88(1):85–97
Vince A (1990) A rearrangement inequality and the permutahedron. Am Math Mon 97(4):319–323
Weaver JR, Church RL (1985) A median location model with non-closest facility service. Transp Sci 19(1):58–74
Wollmer R (1964) Removing arcs from a network. Oper Res 12(6):934–940
Wood RK (1993) Deterministic network interdiction. Math Comput Model 17(2):1–18
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lei, T.L., Tong, D. Hedging against service disruptions: an expected median location problem with site-dependent failure probabilities. J Geogr Syst 15, 491–512 (2013). https://doi.org/10.1007/s10109-012-0175-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10109-012-0175-y