Theoretical and Practical Issues of Evacuation Planning in Urban Areas

In December 2004, the Sumatra-Andaman earthquake occurred. It triggered tsunamis, and tragedy fell upon many people. Not only earthquakes but also diverse disasters occurred and caused serious damages in many countries. Therefore it is very important to establish crisis management systems against large-scale disasters such as big earthquakes, conflagrations and tsunamis to secure evacuation pathways and to effectively guide residents to a safe place. The problem for finding the most effective plan to evacuate people to safe place has been modelled as an evacuation problem by using dynamic network flow. In the evacuation problem, we are given a directed graph D = (V,A) which consists of a vertex set V with supply b(v) on every vertex v and an arc set A with capacity c(e) and transit time τ(e) on every arc e and a single sink s ∈ V . If we consider urban evacuation, vertices model buildings, rooms, exits and so on, and arcs model pathways or roads. For an arc e, capacity c(e) represents the number of people which can traverse e per unit time, and transit time τ(e) represents the time required to traverse e. For any vertex v, supply b(v) represents the number of people which exist at v. The evacuation problem asks to find the minimum time required to send all the supplies to a sink. Theoretically, the first polynomial time algorithm for the evacuation problem was proposed by Hoppe and Tardos [3]. However it requires to use the submodular function minimization as a subroutine. Their algorithm requires O(log(n maxe∈A τ(e) + ∑ v∈V b(v))) submodular function minimizations. Hence the running time is high-order polynomial, and the algorithm is not practical in general. Therefore it is necessary to devise a faster algorithm for a tractable and practically useful subclass of this problem. In our previous papers [4] and [5], we have presented an O(n log n) time algorithm for a √ n×√n grid network with uniform transit time and uniform arc capacity where n denotes the number of vertices in the given network and the algorithm for lager class of networks including the grid network with uniform transit time and uniform arc capacity. However, when we consider the evacuation planning in practical situations (e.g. buildings, cities, and so on), the algorithm of [3] is very complicated and hard to implement. Furthermore, in general the network which we need to treat may not belong to the above special classes. Therefore, in many applications we need to use the time-expanded network introduced by Ford and Fulkerson [2]. Although the algorithm based on the time-expanded network can be easily implemented, the size of the time-expanded network changes depending on granularity of the unit time adopted, e.g., if we set the unit time to one second, the size of the time-expanded