Decomposition algorithms for the design of a nonsimultaneous capacitated evacuation tree network

Abstract

In this article, we examine the design of an evacuation tree, in which evacuation is subject to capacity restrictions on arcs. The cost of evacuating people in the network is determined by the sum of penalties incurred on arcs on which they travel, where penalties are determined according to a nondecreasing function of time. Given a discrete set of disaster scenarios affecting network population, arc capacities, transit times, and penalty functions, we seek to establish an optimal a priori evacuation tree that minimizes the expected evacuation penalty. The solution strategy is based on Benders decomposition, in which the master problem is a mixed-integer program and each subproblem is a time-expanded network flow problem. We provide efficient methods for obtaining primal and dual subproblem solutions, and analyze techniques for improving the strength of the master problem formulation, thus reducing the number of master problem solutions required for the algorithm's convergence. We provide computational results to compare the efficiency of our methods on a set of randomly generated test instances.

A proposed pedestrian waiting-time model for improving space–time use efficiency in stadium evacuation scenarios

“…One avenue of such research focuses on network flow theory [3,24,45], which addressed three major tasks: determining the evacuation route for an evacuee [3,8,25,34,44], generating an evacuation plan based on the dynamic nature of a network [31], and simulating evacuation routings in fine or coarse networks [15,57]. These studies seldom considered the behaviour of individual evacuees (congestion-induced waiting), especially in non-vehicular evacuation scenarios.…”

Decomposition algorithms for solving the minimum weight maximal matching problem

“…Theorem 1 (Gallai-Edmonds decomposition [22]). If G is a graph and D(G), A(G) and C(G) are defined as above, then: 1. the connected components of the subgraph induced by D(G) are factor-critical, 2. the subgraph induced by C(G) has a perfect matching, 3. the bipartite graph obtained from G by deleting the vertices of C(G) and the edges induced by A(G), and by contracting each connected component of D(G) to a single vertex has positive surplus (as viewed from A(G)).…”

“…It then solves the master problem and the subproblem iteratively, adding cuts derived from linear programming duality theory to the master problem in each iteration. We refer the reader to [2] and [30] for details on Benders decomposition, and to [1,6,7,25] for some applications of Benders decomposition within the context of optimization on graphs and networks. Our solution procedure decomposes the formulation [31] into a master problem, which seeks a vertex cover on the graph, and a subproblem, which seeks a perfect matching in the subgraph induced by the vertex cover selected by the master problem.…”

Constraint generation methods for robust optimization in radiation therapy

“…Decomposition methods, which have a long history [15,16], represent a wide range of methods that can be used to solve large-scale optimization problems. Constraint generation is one type of decomposition method that has been used extensively to solve large-scale optimization problems in applications such as timetable scheduling [17], network reliability [18], network design [19], facility location [20], and network interdiction [21,22]. Oskoorouchi et al [23] developed an interior point constraint generation algorithm for semi-infinite problems that was applied to radiation therapy.…”