Abstract
The contraflow approach has been extensively considered in the literature for modeling evacuations and has been claimed, due to its lane-direction-reversal capability, as an efficient idea to speed up the evacuation process. This paper considers the contraflow evacuation model on network with prioritized capacitated vertices that allows evacuees to be held at intermediate spots too, respecting their capacities and priority order. In particular, it studies the maximum flow evacuation planning problem and proposes polynomial and pseudo-polynomial time solution algorithms for static network and dynamic multinetwork, respectively. A real dataset of Kathmandu road network with evacuation spaces is considered to implement the algorithm designed for dynamic multinetwork and to observe its computational performance.
Highlights
E first mathematical optimization model for the contraflow problem was proposed by Rebennack et al [2] that relies on the basis of the network flow model in [3]. ey have investigated analytical solutions for the maximum static contraflow (MSCF) problem and maximum dynamic contraflow (MDCF) problem with polynomial time complexities. e solution idea is based on transformation of input network into a new network for which existing network flow algorithms are applicable. e authors in [4] studied the continuous time maximum dynamic contraflow evacuation problem and proposed a polynomial time solution using the notion of natural transformation of flows suggested in [5]
Other variants that are closely related to the MDCF problem are the quickest contraflow (QCF) problem and earliest arrival contraflow (EACF) problem. e QCF problem on single-source-single-sink network has been solved polynomially in [2]. e EACF problem for the two-terminal series-parallel (TTSP) network has been studied and a polynomial time solution for this has been proposed in [6]
The solution procedures for earliest version of the problems work only for TTSP network. e contraflow approach has been incorporated in the network flow model to study facility location problem in [9], and the notion of abstract flow has been applied to network contraflow problems in [10]. e partial contraflow approach over the abstract network setting has been introduced in [11]
Summary
E non-negative flow variables f(a, t) defined by f: A × T ⟶ N0 that specify the flow over time in the network N are the number of flow units entering arc a at time step t. e number of flow units entering arc a at time step t is assumed to be bounded by the capacity of an arc, i.e., f(a, t) satisfies the capacity constraints for all a ∈ A and for all t ∈ T. at is,. E number of flow units entering arc a at time step t is assumed to be bounded by the capacity of an arc, i.e., f(a, t) satisfies the capacity constraints for all a ∈ A and for all t ∈ T. at is,. We need to ensure that the excess flow at each vertex v ∈ S over time horizon T is to be bounded by the capacity k(v), i.e., exf(v, T) ≤ k(v), for all v ∈ S. Us, the static network N, we consider here, is not a multinetwork To this end, the objective of maximum contraflow evacuation planning problem is to lexicographically maximize the vector (exf(v1, T), . E network flow problem with this objective is termed as lexicographically maximum dynamic contraflow problem and abbreviated as LexMDCF problem. E maximum contraflow problem with above objective for static network N (G, l(a), u(a), k(v), s, d) is termed as lexicographically maximum static contraflow problem and is abbreviated as LexMSCF problem
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