Abstract
Motivated by applications in evacuation planning, we consider a problem of optimizing flow with arc reversals in which the transit time depends on the orientation of the arc. In the considered problems, the transit time on an arc may change when it is reversed, contrary to the problems considered in the existing literature. Extending the existing idea of auxiliary network construction to allow asymmetric transit time on arcs, we present strongly polynomial time algorithms for solving single-source-single-sink maximum dynamic contraflow problem and quickest contraflow problem. The results are substantiated by a computational experiment in a Kathmandu road network. An algorithm to solve the corresponding earliest arrival contraflow problem with a pseudo-polynomial-time complexity is also presented. The partial contraflow approach for the corresponding problems has also been discussed.
Highlights
E transportation network, during or after disastrous situations, becomes more congested due to large number of people and vehicles towards the safer areas on the streets
We introduce the maximum dynamic contraflow, earliest arrival, and quickest contraflow problems on asymmetric transit time network and present efficient algorithms to solve these problems in two-terminal networks
Modifying the network transformation suggested by Rebennack et al [3] in case of symmetric travel time cases, we show that the approach works well in the asymmetric travel time settings. e results are extended with partial lane reversals as well
Summary
A+i {e ∈ A: i is the tail of e}. 2.1. Given a static flow x, the residual network Nx has the same vertex set V. e arc set Ax consists of arcs constructed in the following way: For each e ∈ A directed from i to j, if x(e) < u(e), there is an arc in. A dynamic flow Φ with time horizon T consists of Lebesgue-integrable functions Φe: [0, T) ⟶ R≥0 for each arc e ∈ A such that Φe(θ) 0 for θ ≥ T − τ(e). Given a time horizon T, the dynamic flow Φ that maximizes vT(Φ) is called the maximum dynamic flow. Given a flow value Q, the dynamic flow with minimum time horizon T∗ such that vT∗ (Φ) Q is called the quickest flow, and the dynamic flow Φ which maximizes vθ(Φ) for all θ ∈ [0, T] is called the earliest arrival flow. Using the concept of natural transformations, Fleischer and Tardos [16] show the equivalence between the two problems so that the solution procedures of a problem in continuous time version can be carried to the solution procedure of the corresponding problem in the discrete version, and vice versa
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