Modeling the shelter site location problem using chance constraints: A case study for Istanbul

Abstract

In this work, we develop and test a new modeling framework for the shelter site location problem under demand uncertainty. In particular, we propose a maxmin probabilistic programming model that includes two types of probabilistic constraints: one concerning the utilization rate of the selected shelters and the other concerning the capacity of those shelters. By invoking the central limit theorem we are able to obtain an optimization model with a single set of non-linear constraints which, nonetheless, can be approximated using a family of piecewise linear functions. The latter, in turn, can be modeled mathematically using integer variables. Eventually, an approximate model is obtained, which is a mixed-integer linear programming model that can be tackled by an off-the-shelf solver. Using the proposed reformulation we are able to solve instances of the problem using data associated with the Kartal district in Istanbul, Turkey. We also consider a large-scale instance of the problem by making use of data for the whole Anatolian side of Istanbul. The results obtained are presented and discussed in the paper. They provide clear evidence that capturing uncertainty in the shelter site location problem by means of probabilistic constraints may lead to solutions that are much different from those obtained when a deterministic counterpart is considered. Furthermore, it is possible to observe that the probabilities embedded in the probabilistic constraints have a clear influence in the results, thus supporting the statement that a probabilistic programming modeling framework, if appropriately tuned by a decision maker, can make a full difference when it comes to find good solutions for the problem.

A Systematic Literature Review on Mathematical Models of Humanitarian Logistics

“…In addition, probabilistic constraints are used to ensure the arrival of relief items within a certain time window. Kınay [65] developed a modeling framework to find out the location of the shelter under demand uncertainty. In their probabilistic model, the utilization rate of shelters and their capacity is considered as the probabilistic constraint.…”

Approximation Schemes for Capacitated Clustering in Doubling Metrics

“…For example, consider the problem of positioning emergency supplies to support disaster relief by dispatching a certain number of emergency items, such as medicine, food, centers on a map. There has been a large body of work in this direction and it has been argued that the capacity constraint of the centers is often a hard constraint, in particular when it comes to medical centers where the cost of service increases very quickly when the capacity is violated [18] see also [24,32,25,4] for more discussions on hard capacities for prepositioning emergency resources and hard capacity constraints in plant locations.…”

On multi-criteria chance-constrained capacitated single-source discrete facility location problems

“…Kinay et al [26] also consider a single-source discrete facility location problem with stochastic demand. Again, a service level constraint is included in the optimization models proposed.…”