Applications of robust optimization in disaster management

Abstract

Practical implementation of optimization models always faces the challenge of obtaining precise inputs, e.g., the joint distribution of random variables, on which the performance of the resulting solutions relies on. This difficulty is much more prominent in disaster management since the time, location, magnitude and damages of a disaster are very hard, if not impossible, to predict accurately. This study is thus motivated to develop robust optimization approaches in disaster management, which significantly alleviate the dependence on inputs and return strongly performing solutions. Three important problems in disaster management are studied in this thesis, which are elaborated as follows.

The inventory pre-positioning problem decides how to pre-position supplies of an emergency commodity in a set of facilities so that, in the case a disaster occurs, the shortage of this commodity in certain affected areas could be significantly reduced. This problem is non-trivial because not only the demands of the commodity but also the amounts of pre-positioned supplies that would remain usable in disaster operations are highly uncertain. To tackle the uncertainties, the robust approaches with data and distribution ambiguities, respectively, are applied to this problem. These approaches only require the most likely values, the lower bounds, and the upper bounds of the uncertain parameters. For the robust model with data ambiguity, the optimal solution can be obtained by solving a linear program. A closed-form optimal solution of this model is also derived under certain conditions. For the robust model with distribution ambiguity, the closed-form representation of the distribution that yields the highest expected shortage transforms the model into a linear stochastic program. Numerical experiments reveal that both robust models outperform the stochastic counterparts.

The location and inventory pre-positioning problem is considered to optimize the decisions of facility location, inventory pre-positioning, and relief delivery operations simultaneously within a single-commodity disaster relief network. A min-max robust model with data ambiguity is proposed to capture the uncertainties in both the left- and right-hand-side parameters in the constraints. The former corresponds to the proportions of the pre-positioned inventories that remain usable after a disaster attack, while the latter represents the demands of the inventories and the road capacities in the disaster-affected areas. We study how to solve the robust model efficiently. The advantage of the min-max robust model is demonstrated through the implementations of the 2010 Yushu earthquake and its variants with significantly larger scales.

The inventory management problem with multiple demand sources is studied to optimize the amount of emergency supplies that should be prepared for potential disaster relief operations. Two robust models with distribution ambiguity, namely, the mean-support and mean-variance models, are developed to optimize the worst-case expected social profit. The worst-case distribution of the mean-support model is obtained in closed form, which reduces the mean-support model to a linear program. The mean-variance model can be equivalently formulated as a second-order cone program and the closed-form worst-case distributions are derived in special cases. The good performance of these two robust models is corroborated by numerical experiments.