Lower bounds and heuristics for the Windy Rural Postman Problem

Abstract

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains a big number of important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.

Min‐Max K‐vehicles windy rural postman problem

“…The set of WRPP instances was presented in Benavent et al [8] and generated from the 24 RPP instances proposed in Christofides et al [9]. They can be found in http://www.uv.es/corberan.…”

“…Note: If δ R (S) = {e} the aggregate R-odd cut inequality (8), k x k (δ(S)) ≥ 2, is not facetinducing because it is the sum of K parity inequalities (9), x k (δ(S)) ≥ 2y k e (taking F = {e}), that are showed to be facet-inducing in what follows. Finally, for each edge e ∈ E R \ F let w i be the tour on set {w 1 , w 2 , .…”

“…R-odd cut, K-C, Honeycomb and other inequalities for the WRPP described in Corberán et al [11] are then valid inequalities for the MM K-WRPP. In particular, the aggregate R-odd cut inequalities are: (8) and are based on the fact that any K-WRPP solution must cross any given edge cutset an even number of times. On the other hand, when we consider a single vehicle k, all the edges e ∈ E R can not be considered as 'required' for vehicle k because it could either not serve nor traverse e. The actual required edges for a given vehicle k are those serviced by it, which are determined by the vector y k .…”

“…For example, in Benavent et al [8] the problem of finding the traversal of 3-dimensional structures such as bridges or building skeletons by a climbing robot is described. The problem is modelled as a WRPP because the cost (energy consumption) of traversing an edge (either inspecting a side of a beam or moving from it to another) in each of its two directions can be different.…”

A branch-and-price algorithm for the windy rural postman problem

“…By using a stochastic method to duplicate the edges or changing their costs in a given range, Benavent et al then present a multi-start scatter search algorithm. They also detail a cutting-plane procedure to exactly solve or to obtain lower bounds of the WRPP [1]. In our work, during a field analysis for a waste collection project, the WRPP was used as a key component in order to find an optimal rotation of a truck.…”