A parameterized lower bounding method for the open capacitated arc routing problem

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.

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

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.

One of the recent variants of arc routing problems is the orienteering arc routing problem. In Archetti et al. [A branch and-cut algorithm for the orienteering arc routing problem, Computers and Operations Research. 2016;66:95–104] an integer programming formulation is proposed for the problem and some families of valid inequalities introduced and proven to be valid for the associated polyhedron. Here, the class of path-bridge inequalities is introduced for the problem and is proven to be facet inducing. Since the large class of path-bridge inequalities contain those classes introduced in Archetti et al. [A branch and-cut algorithm for the orienteering arc routing problem, Computers and Operations Research. 2016;66:95–104] this paper can be considered as a generalization of the results of that paper.

The Uncertain Capacited Arc Routing Problem (UCARP) is an important variant of arc routing problems that is capable of modelling uncertainties of real-world scenarios. Genetic Programming is utilised to evolve routing policies for vehicles to enable them to make real-time decisions and handle environment uncertainties. However, when the properties of a solved problem change, the trained routing policy becomes ineffective and a new routing policy is needed to be trained. The training process is time-consuming. Nevertheless, by extraction and transfer of some knowledge learned from the previous similar problem, the retraining process can be improved. Transfer learning is a challenging task that entails many aspects to decide about, which can influence the degree by which knowledge transfer can be effective. Consequently, in this paper we propose a parametric framework to formalise these details so that it can facilitate studying different aspects of using transfer learning for handling scenario changes of UCARP. Conducting a large number of experiments, we utilise this framework to analyse different transfer learning mechanisms and demonstrate how it can help with understanding dynamics of knowledge transfer for UCARP.

This paper presents heuristics that are based on a tour splitting of a general routing tour for solving the general capacitated routing problem (GCRP). This problem is a generalization of the vehicle routing problem (VRP) and the capacitated arc routing problem (CARP). For the VRP, heuristics that consist of an optimum partitioning of a TSP tour generated by Christofides are known and have a worst‐case error of 7/2 − 3/q for even q, where q is the capacity of the vehicles. If we apply a partitioning to an optimum TSP tour, the worst‐case error becomes 3 − 2/q for even q. We generalize these results to the GCRP and give also some lower bounds. © 1993 John Wiley & Sons, Inc.

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.

Arc routing problems (ARPs) are a special kind of vehicle routing problem, in which the demands are located on edges or arcs, instead of nodes. There is a huge literature on ARPs, and a variety of exact and heuristic algorithms are available. Recently, however, we encountered some real‐life ARPs with over 10 000 roads, which is much larger than those usually considered in the literature. For these problems, we develop fast upper‐ and lower‐bounding procedures. We also present extensive computational results.

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.

Constructive heuristics for the Mixed Capacity Arc Routing Problem under Time Restrictions with Intermediate Facilities

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.

Motivated by some practical applications in urban services such as water sparkling, we study a split-delivery capacitated arc-routing problem with time windows (SDCARPTW). It is a variant of arc-routing problem and is defined on an undirected graph where the demands on the arcs are splitable, and time window and capacity constraints must be satisfied. We propose a mathematical formulation for SDCARPTW and derive some nice properties of the split-delivery structure, which can help to well represent a solution of SDCARPTW. The dynamic programming, neighborhood search and perturbation process are combined to develop a tabu search algorithm. Through computational studies on CARPTW benchmark datasets, we validate the effectiveness and efficiency of our proposed algorithm. New datasets for SDCARPTW are further proposed and the impact of the split-delivery option is analyzed.

On the dynamics of a class of optimum routing problems