Capacity provisioning for evacuation on path networks

Abstract

Designing an appropriate evacuation plan in the event of large scale disasters is extremely important. A considerable research effort has been invested in effective mathematical models and algorithms for the evacuation of densely populated areas. Dynamic networks and flows, and the closely related sink location and evacuation problems are research directions that are being actively pursued at the moment. In this work, we propose an extension of the sink location problem in dynamic networks that is suitable for planning the evacuation of remote and sparsely populated communities. We exploit the connection between the capacity of a transportation network and the amount of resources allocated to the rescue operation, and we introduce a new objective function for the problem of evacuation in a dynamic network. Given a network with known edge lengths and with a known number of evacuees located at the vertices, we would like to assign capacities to the edges of the network, from a fixed capacity budget, in such a way that the evacuation time of all evacuees to the set of fixed sink nodes in the network is minimized. We consider a simple path network with one fixed sink node and observe that the solution can be obtained numerically from a non‐linear program. However, by exploiting two useful properties of the optimal capacity allocation, we propose a combinatorial algorithm that returns a parametric solution, that is, a solution that depends on a single parameter, namely the capacity assigned to one edge of the path. Our algorithm runs in time Ofalse(nlogn+nlogfalse(cfalse/ξfalse)false)$$ O\left(n\log n+n\log \left(c/\xi \right)\right) $$ where c$$ c $$ is the budgeted capacity, ξ$$ \xi $$ is the precision to which the solution is computed, and n$$ n $$ is the number of vertices occupied by evacuees in an false(n+1false)$$ \left(n+1\right) $$‐vertex path. This work is the first to consider the capacity allocation problem in the context of sink location problems.