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

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.