Nonlinear mixed integer programming models and algorithms for fair and efficient large scale evacuation planning

Abstract

Shelters are safe facilities that protect a population from possible damaging effects of a disaster. Traffic management during an evacuation and the decision of where to locate the shelters are of critical importance to the performance of an evacuation plan. From the evacuation management authority's point of view, the desirable goal is to minimize the total evacuation time by computing a system optimum (SO). However, evacuees may not be willing to take long routes enforced on them by a SO solution; but they may consent to taking routes with lengths not longer than the shortest path to the nearest shelter site by more than a tolerable factor. We develop a model that optimally locates shelters and assigns evacuees to the nearest shelter sites by assigning them to shortest paths, shortest and nearest with a given degree of tolerance, so that the total evacuation time is minimized. As the travel time on a road segment is often modeled as a nonlinear function of the ow on the segment, the resulting model is a nonlinear mixed integer programming model. We develop a solution method that can handle practical size problems using second order cone programming techniques. Using our model, we investigate the trade-of between efficiency and fairness. Disasters are uncertain events. Related studies and real-life practices show that a significant uncertainty regarding the evacuation demand and the impact of the disaster on the infrastructure exists. The second model we propose is a scenario-based two-stage stochastic evacuation planning model that optimally locates shelter sites and that assigns evacuees to shelters and paths to minimize the expected total evacuation time, under uncertainty. The model considers the uncertainty in the evacuation demand and the disruption in the road network and shelter sites. We present a case study for an impending earthquake in Istanbul, Turkey. We compare the performance of the stochastic programming solutions to solutions based on single scenarios and mean values. We also propose an exact algorithm based on Benders decomposition to solve the stochastic problem. To the best of our knowledge, ours is the first algorithm that uses duality results for second order cone programming in a Benders decomposition setting. We solve practical size problems with up to 1000 scenarios in moderate CPU times. We investigate methods such as employing a multi-cut strategy, deriving pareto-optimal cuts, using a reduced primal subproblem and preemptive priority multiobjective program to enhance the proposed algorithm. Computational results confirm the efficiency of our algorithm. This research is supported by TUBITAK, The Scientific and Technological Research Council of Turkey with project number 213M434.