Fast upper and lower bounds for a large‐scale real‐world arc routing problem

This research presents a solution process for Periodic Arc Routing Problem with Refill Points (PARPRP). The Periodic Arc Routing Problem is a natural extension of Arc Routing Problem (ARP) and is classified as an NP-hard problem. The street tree watering problem is used as the case study. The study tackles the ARP requiring a specified time horizon to fulfill it. A refill point in the problem is a node in the graph at which the vehicle capacity can be recovered as long as the vehicle makes a stop on it. The strategy of solving the problem is first to transform the problem into a Vehicle Routing Problem. Secondly, an Ant Colony Optimization algorithm is developed to solve the PARPRP because of its remarkable capability in the establishment of network routes. A local search method is incorporated to reach the local minimum for further improvement. Two sets of benchmarks are used to verify the solving process and the computational results are provided. The outputs indicate that proposed methods can yield promising solutions within reasonable execution time. A real case regarding the street tree watering of Kaohsiung city, Taiwan, is investigated. The results suggest that while α = 1 and β = 3 (i.e. parameters to control the influence of the pheromone value allocated on and the desirability of the arc, respectively), the algorithm can yield the best solution. The number of refill points is analyzed as well and output result suggests that a system with two refill points can obtain the least route length.

Contributors

The adoption of electric vehicles (EVs) has gained significant momentum in recent years as a sustainable alternative to traditional internal combustion engine vehicles. However, the efficient utilization of batteries in EVs, coupled with the growing demand for sustainable transportation, has posed complex challenges for battery management in the context of electric vehicle routing problems in a broad sense, which includes vehicle routing problems, team orienteering problems, and arc routing problems, all of them using EVs. This paper presents a comprehensive review of the state-of-the-art approaches, methodologies, and strategies for battery management in each of the aforementioned optimization problems. We explore the relevant factors influencing battery performance and the interplay between routing, charging, and energy management in the context of EVs. The paper also discusses the advances in optimization algorithms, vehicle-to-grid integration, and intelligent decision-making techniques aimed at enhancing the range, reliability, and sustainability of EV operations. Key findings indicate a paradigm shift towards addressing uncertainties, dynamic conditions, and synchronization challenges inherent in large-scale and dynamic routing problems within the context of EVs that require efficient battery management.

A variable neighborhood descent heuristic for arc routing problems with time-dependent service costs

This chapter describes some of the most important models and algorithms for the classical vehicle routing problem and for several families of arc routing problems. Exact methods (mostly based on branch-and-cut) and heuristics (mostly based on tabu search) are described, and computational results are presented.

The windy postman problem consists of finding a minimum cost traversal of all of the edges of an undirected graph with two costs associated with each edge, representing the costs of traversing it in each direction. In this paper we deal with the windy general routing problem (WGRP), in which only a subset of edges must be traversed and a subset of vertices must be visited. This is also an ${\it NP}$-hard problem that generalizes many important arc routing problems (ARPs) and has some interesting real-life applications. Here we study the description of the WGRP polyhedron, for which some general properties and some large families of facet-inducing inequalities are presented. Moreover, since the WGRP contains many well-known routing problems as special cases, this paper also provides a global view of their associated polyhedra. Finally, for the first time, some polyhedral results for several ARPs defined on mixed graphs formulated by using two variables per edge are presented.

Consider an undirected graph with demands scattered over the edges and a homogeneous fleet of vehicles to service the demands. In the open capacitated arc routing problem (OCARP) the objective is to find a set of routes that collectively service all demands with the minimum cost. Each vehicle has limited capacity and it can start and finish the route at any node. The OCARP is NP-hard, and its applications include meter reading and cutting path determination problems. State-of-the-art solution methods developed for the OCARP are heuristics, which show good tradeoffs between solution quality and processing time, but do not provide optimality certificates of the obtained solutions. This work focuses on a lower bounding method for the OCARP which can be used to better assess the quality of heuristic solutions. We propose the Relaxed Flow method (RF(k)) which involves the resolution of a mixed integer linear formulation where all vehicles' capacities are modeled as flows on an augmented graph. A parameter k controls the model tightness and RF(k) is shown to be at least as tight as the well-known Belenguer and Benavent's formulation for any k⩾0. To strengthen the model, capacity cuts were included in RF(k) by means of a branch-and-cut framework. Extensive computational experiments conducted on a set of benchmark instances revealed that our method outperformed previous methods. Computational experiments also demonstrated the importance of the parameterization technique to obtain good results. The previously known lower bounds were improved substantially and optimality certificates were attained in 78.9% of the instances. As far as we know this is the first parameterized lower bounding method proposed for an arc routing problem, and we argue it can be generalized to other variants of arc routing problems and general routing problems.

In several arc routing problems, it is necessary to take turn penalties into account when designing a solution. Traditionally, this is done through a transformation of the arc routing problem into an equivalent vertex routing problem. In this paper it is shown that a more direct approach, not resorting to such a transformation, may be more efficient.

A heuristic algorithm based on Monte Carlo methods for the Rural Postman Problem

Arc routing problems (ARPs) arise naturally in several applications where streets require maintenance, or customers located along road must be serviced. The undirected rural postman problem (URPP) is to determine a least cost tour traversing at least once each edge that requires a service. When demands are put on the edges and this total demand must be covered by a fleet of identical vehicles of capacity Q based at a depot, one gets the undirected capacitated arc routing problem (UCARP). The URPP and UCARP are known to be NP-hard. This chapter reports on recent exact and heuristic algorithms for the URPP and UCARP.

The chapter by Teodor Crainic in this book discusses the long haul truck routing problem. Some of the key differences between the long haul truck routing problem and the vehicle routing problems considered in this chapter are as follows: 1. In the long haul truck routing problem, the routes can extend over several days whereas, in the vehicle routing problem described herein, the routes are of one-day duration. 2. In the long haul truck routing problem, the locations can be scattered over a wide region, even the entire United States, whereas in the vehicle routing problems considered herein, the locations requiring service are packed in a small region.

Space Information Flow (SIF) is a new promising research area that studies network coding in geometric space, such as Euclidean space. The design of algorithms that compute the optimal SIF solutions remains one of the key open problems in SIF. This work proposes the first exact SIF algorithm and a heuristic SIF algorithm that compute min-cost multicast network coding for N (N ≥ 3) given terminal nodes in 2-D Euclidean space. Furthermore, we find that the Butterfly network in Euclidean space is the second example besides the Pentagram network where SIF is strictly better than Euclidean Steiner minimal tree. The exact algorithm design is based on two key techniques: Delaunay triangulation and linear programming. Delaunay triangulation technique helps to find practically good candidate relay nodes, after which a min-cost multicast linear programming model is solved over the terminal nodes and the candidate relay nodes, to compute the optimal multicast network topology, including the optimal relay nodes selected by linear programming from all the candidate relay nodes and the flow rates on the connection links. The heuristic algorithm design is also based on Delaunay triangulation and linear programming techniques. The exact algorithm can achieve the optimal SIF solution with an exponential computational complexity, while the heuristic algorithm can achieve the sub-optimal SIF solution with a polynomial computational complexity. We prove the correctness of the exact SIF algorithm. The simulation results show the effectiveness of the heuristic SIF algorithm.

Lower and upper bounds for location-arc routing problems with vehicle capacity constraints

The Hierarchical Windy Postman Problem (HWPP) is an arc routing problem in which an order relation is imposed on the arcs/edges of the graph, and one has to pass through each edge at least once while adhering to the hierarchical priority relations. The tour starts from and ends at a specific node and the aim is to minimize the length of the tour. We consider a variant of the HWPP in which (i) the precedence order of the edge hierarchies is linear and edges within each hierarchy are connected and (ii) the cost of serving each edge decreases with the number of times it is traversed, and we refer to it as HWPP with variable service costs. An integer non-heuristic linear mathematical formulation is proposed, and a solution approach is designed. Our solution heuristic adapts the layer algorithm of Dror et al. (Networks 17:283–294, 1987) but employs an integer mathematical formulation as a sub-procedure instead of the blossom algorithm to find the least cost path between the nodes of the graph. This choice is based on the fact that the blossom algorithm requires a symmetric cost structure while we deal here with the general case of asymmetric cost structure, which makes our problem a windy variant of the postman problem. It should be noted that our problem is not asymmetric in the sense that there are no opposite arcs with different costs but there are edges which have different costs depending on the traversal direction. In order to compare the performance of our heuristic algorithm with respect to the performance of the mathematical model that is solved by the commercial solver Gurobi, 84 test instances are generated having varying sizes and densities and with different number of hierarchies. These test instances are solved by both methods and the generated results show that the proposed heuristic method is much faster and generates better quality solutions.

We study the tractor and semi‐trailer routing problem (TSRP), a variant of the vehicle routing problem (VRP). In the TSRP model for this paper, vehicles are dispatched on a trailer‐flow network where there is only one main depot, and all tractors originate and terminate in the main depot. Two types of decisions are involved: the number of tractors and the route of each tractor. Heuristic algorithms have seen widespread application to various extensions of the VRP. However, this approach has not been applied to the TSRP. We propose a heuristic algorithm to solve the TSRP. The proposed heuristic algorithm first constructs the initial route set by the limitation of a driver’s on‐duty time. The candidate routes in the initial set are then filtered by a two‐phase approach. The computational study shows that our algorithm is feasible for the TSRP. Moreover, the algorithm takes relatively little time to obtain satisfactory solutions. The results suggest that our heuristic algorithm is competitive in solving the TSRP.