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.
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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.
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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