Experiments on the Node, Edge, and Arc Routing Problem

-The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004 along with the first benchmark called CBMix. The NEARP generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It is also denoted the Mixed Capacitated General Routing Problem (MCGRP). The NEARP removes the strict and unwarranted dichotomy that previously existed in the literature between arc routing and node routing. In real applications, there are many cases where the pure node or arc routing models are not adequate. In fundamentally node-based routing applications such as newspaper delivery and communal waste management that have typically been modeled as arc routing problems in the literature, the number of points is often so large that demand aggregation is necessary. Aggregation heuristics will normally give a NEARP instance, possibly with side constraints. Hence, the NEARP is a scientifically challenging problem with high industrial relevance. In this report we present experiments with Spider, SINTEF’s industrial VRP solver, on the three NEARP benchmarks that have been published so far: CBMix, BHW, and DI-NEARP. Bach, Hasle, and Wøhlk have developed a combinatorial lower bound for the NEARP and defined the two latter benchmarks. Here, we present an experimental study with Spider on the three existing NEARP benchmarks. Upper and lower bounds are given for all instances. Three of the BHW instances have been solved to optimality. SINTEF has developed a web page for NEARP results on http://www.sintef.no/NEARP

Experiments on the Node, Edge, and Arc Routing Problem

The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004 along with the first benchmark called CBMix. The NEARP generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It is also denoted the Mixed Capacitated General Routing Problem (MCGRP). The NEARP removes the strict and unwarranted dichotomy that previously existed in the literature between arc routing and node routing. In real applications, there are many cases where the pure node or arc routing models are not adequate. In fundamentally node-based routing applications such as newspaper delivery and communal waste management that have typically been modeled as arc routing problems in the literature, the number of points is often so large that demand aggregation is necessary. Aggregation heuristics will normally give a NEARP instance, possibly with side constraints. Hence, the NEARP is a scientifically challenging problem with high industrial relevance. In this report we present experiments with Spider, SINTEF’s industrial VRP solver, on the three NEARP benchmarks that have been published so far: CBMix, BHW, and DI-NEARP. Bach, Hasle, and Wøhlk have developed a combinatorial lower bound for the NEARP and defined the two latter benchmarks. Here, we present an experimental study with Spider on the three existing NEARP benchmarks. Upper and lower bounds are given for all instances. Three of the BHW instances have been solved to optimality. SINTEF has developed a web page for NEARP results on http://www.sintef.no/NEARP

Minimum Makespan Multi-vehicle Dial-a-Ride

Dial a ride problems consist of a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs, where each object requires to be moved from its source to destination vertex. We consider the multi-vehicle Dial a ride problem, with each vehicle having capacity k and its own depot-vertex, where the objective is to minimize the maximum completion time (makespan) of the vehicles. We study the "preemptive" version of the problem, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log^3 n)-approximation algorithm for preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint. We also show that the approximation ratios improve by a log-factor when the underlying metric is induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200

A PTAS for Bounded-Capacity Vehicle Routing in Planar Graphs

The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot

A new exact algorithm for the multi-depot vehicle routing problem under capacity and route length constraints

This article presents an exact algorithm for the multi-depot vehicle routing problem (MDVRP) under capacity and route length constraints. The MDVRP is formulated using a vehicle-flow and a set-partitioning formulation, both of which are exploited at different stages of the algorithm. The lower bound computed with the vehicle-flow formulation is used to eliminate non-promising edges, thus reducing the complexity of the pricing subproblem used to solve the set-partitioning formulation. Several classes of valid inequalities are added to strengthen both formulations, including a new family of valid inequalities used to forbid cycles of an arbitrary length. To validate our approach, we also consider the capacitated vehicle routing problem (CVRP) as a particular case of the MDVRP, and conduct extensive computational experiments on several instances from the literature to show its effectiveness. The computational results show that the proposed algorithm is competitive against stateof-the-art methods for these two classes of vehicle routing problems, and is able to solve to optimality some previously open instances. Moreover, for the instances that cannot be solved by the proposed algorithm, the final lower bounds prove stronger than those obtained by earlier methods

Locating Depots for Capacitated Vehicle Routing

We study a location-routing problem in the context of capacitated vehicle routing. The input is a set of demand locations in a metric space and a fleet of k vehicles each of capacity Q. The objective is to locate k depots, one for each vehicle, and compute routes for the vehicles so that all demands are satisfied and the total cost is minimized. Our main result is a constant-factor approximation algorithm for this problem. To achieve this result, we reduce to the k-median-forest problem, which generalizes both k-median and minimum spanning tree, and which might be of independent interest. We give a (3+c)-approximation algorithm for k-median-forest, which leads to a (12+c)-approximation algorithm for the above location-routing problem, for any constant c>0. The algorithm for k-median-forest is just t-swap local search, and we prove that it has locality gap 3+2/t; this generalizes the corresponding result known for k-median. Finally we consider the "non-uniform" k-median-forest problem which has different cost functions for the MST and k-median parts. We show that the locality gap for this problem is unbounded even under multi-swaps, which contrasts with the uniform case. Nevertheless, we obtain a constant-factor approximation algorithm, using an LP based approach.Comment: 12 pages, 1 figur