50 Años de Arc Routing

Hace poco más de 50 años, un matemático chino, Meigu Guan, planteó el problema de encontrar un recorrido de las aristas de un grafo no dirigido que fuera el más corto posible. Guan buscaba minimizar la longitud de un camino cerrado que pasara por cada arista del grafo al menos una vez y, así, encontrar una ruta de longitud mínima para un cartero que debe recorrer cada calle de la zona que tiene asignada para repartir el correo. El artículo de Guan, sobre lo que posteriormente se conoció como el Problema del Cartero Chino, fue el primero de una larga serie de contribuciones en el área de las Rutas por Arcos (Arc Routing). Básicamente, los problemas de rutas por arcos consisten en determinar un recorrido de coste mínimo de todos o algunos arcos y/o aristas de un grafo, posiblemente sujeto a algunas condiciones adicionales. Estos problemas definen un área apasionante porque, por una parte, la mayoría de ellos son problemas que plantean un desafío desde el punto de vista de su estudio y resolución y, por otro lado, porque aparecen en muchas situaciones prácticas de la vida real como la recogida de basuras, la limpieza de las calles y el mantenimiento de carreteras o vías de tren y, puesto que el dinero involucrado en estas operaciones representa millones de euros, existe un considerable potencial para el ahorro. Esta charla es sobre el recorrido de los arcos (y aristas) de un grafo, los problemas de rutas por arcos, su historia y resolución, y sus aplicaciones a problemas de la vida real.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Profitable mixed capacitated arc routing and related problems

Mixed Capacitated Arc Routing Problems (MCARP) aim to identify a set of vehicle trips that, starting and ending at a depot node, serve a given number of links, regarding the vehicles capacity, and minimizing a cost function. If both profits and costs on arcs are considered, the Profitable Mixed Capacitated Arc Routing Problem (PMCARP) may be defined. We present compact flow based models for the PMCARP, where two types of services are tackled, mandatory and optional. Adaptations of the models to fit into some other related problems are also proposed. The models are evaluated, according to their bounds quality as well as the CPU times, over large sets of test instances. New instances have been created from benchmark ones in order to solve variants that have been introduced here for the first time. Results show the new models performance within CPLEX and compare, whenever available, the proposed models against other resolution methods.info:eu-repo/semantics/publishedVersio

Arc routing problems: A review of the past, present, and future

[EN] Arc routing problems (ARPs) are defined and introduced. Following a brief history of developments in this area of research, different types of ARPs are described that are currently relevant for study. In addition, particular features of ARPs that are important from a theoretical or practical point of view are discussed. A section on applications describes some of the changes that have occurred from early applications of ARP models to the present day and points the way to emerging topics for study. A final section provides information on libraries and instance repositories for ARPs. The review concludes with some perspectives on future research developments and opportunities for emerging applicationsThis research was supported by the Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional, Grant/Award Number: PGC2018-099428-B-I00. The Research Council of Norway, Grant/Award Numbers: 246825/O70 (DynamITe), 263031/O70 (AXIOM).Corberán, Á.; Eglese, R.; Hasle, G.; Plana, I.; Sanchís Llopis, JM. (2021). Arc routing problems: A review of the past, present, and future. Networks. 77(1):88-115. https://doi.org/10.1002/net.21965S8811577

The Chinese Postman Problem with Load-Dependent Costs

[EN] We introduce an interesting variant of the well-known Chinese postman problem (CPP). While in the CPP the cost of traversing an edge is a constant (equal to its length), in the variant we present here the cost of traversing an edge depends on its length and on the weight of the vehicle at the moment it is traversed. This problem is inspired by the perspective of minimizing pollution in transportation, since the amount of pollution emitted by a vehicle not only depends on the travel distance but also on its load, among other factors. We define the problem, study its computational complexity, provide two mathematical programming formulations, and propose two metaheuristics for its solution. Extensive computational experiments reveal the extraordinary difficulty of this problem.The work by Angel Corberan, Isaac Plana, and Jose M. Sanchis was supported by the Spanish Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER) through [project MTM2015-68097-P] (MINECO/FEDER) and by the Generalitat Valenciana [project GVPROMETEO2013-049]. Gilbert Laporte was supported by the Canadian Natural Sciences and Engineering Research Council under [Grant 2015-06189].Corberán, Á.; Erdogan, G.; Laporte, G.; Plana, I.; Sanchís Llopis, JM. (2018). The Chinese Postman Problem with Load-Dependent Costs. Transportation Science. 52(2):370-385. https://doi.org/10.1287/trsc.2017.0774S37038552

The min-max close-enough arc routing problem

[EN] Here we introduce the Min-Max Close-Enough Arc Routing Problem, where a fleet of vehicles must serve a set of customers while trying to balance the length of the routes. The vehicles do not need to visit the customers, since they can serve them from a distance by traversing arcs that are ¿close enough¿to the customers. We present two formulations of the problem and propose a branch-and-cut and a branch-and- price algorithm based on the respective formulations. A heuristic algorithm used to provide good upper bounds to the exact procedures is also presented. Extensive computational experiments to compare the performance of the algorithms are carried out.The work by Angel Corberan, Isaac Plana, Miguel Reula, and Jose M. Sanchis was supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades (MICIU) and Fondo Social Europeo (FSE) through project PGC2018-099428-B-I00. The authors want to thank the comments and suggestions done by three anonymous reviewers that have contributed to improve the content and readability of the article.Bianchessi, N.; Corberán, Á.; Plana, I.; Reula, M.; Sanchís Llopis, JM. (2022). The min-max close-enough arc routing problem. European Journal of Operational Research. 300(3):837-851. https://doi.org/10.1016/j.ejor.2021.10.047837851300

A New Branch-and-Cut Algorithm for the Generalized Directed Rural Postman Problem

The generalized directed rural postman problem, also known as the close-enough arc routing problem, is an arc routing problem with some interesting real-life applications, such as routing for meter reading. In this article we introduce two new formulations for this problem as well as various families of new valid inequalities that are used to design and implement a branch-and-cut algorithm. The computational results obtained on test bed instances from the literature show that this algorithm outperforms the existing exact methodsThe authors wish to thank Minh Hoang Ha, Nathalie Bostel, Andre Langevin, and Louis-Martin Rousseau for providing their instances. The authors also thank the Spanish Ministerio de Economia y Competitividad [project MTM2012-36163-C06-02] and the Generalitat Valenciana [project GVPROMETEO2013-049] for their support.Ávila, T.; Corberán, Á.; Plana, I.; Sanchís Llopis, JM. (2016). A New Branch-and-Cut Algorithm for the Generalized Directed Rural Postman Problem. Transportation Science. 50(2):750-761. https://doi.org/10.1287/trsc.2015.0588S75076150

Profitable mixed capacitated arc routing and related problems

Mixed Capacitated Arc Routing Problems (MCARP) aim to identify a set of vehicle trips that, starting and ending at a depot node, serve a given number of links, regarding the vehicles capacity, and minimizing a cost function. If both profits and costs on arcs are considered, the Profitable Mixed Capacitated Arc Routing Problem (PMCARP) may be defined. We present compact flow based models for the PMCARP, where two types of services are tackled, mandatory and optional. Adaptations of the models to fit into some other related problems are also proposed. The models are evaluated, according to their bounds quality as well as the CPU times, over large sets of test instances. New instances have been created from benchmark ones in order to solve variants that have been introduced here for the first time. Results show the new models performance within CPLEX and compare, whenever available, the proposed models against other resolution methodsinfo:eu-repo/semantics/publishedVersio

Polyhedral analysis and a new algorithm for the length constrained K-drones rural postman problem

[EN] The Length Constrained K¿Drones Rural Postman Problem (LC K¿DRPP) is a continuous optimization problem where a set of curved or straight lines of a network have to be traversed, in order to be serviced, by a fleet of homogeneous drones, with total minimum cost. Since the range and endurance of drones is limited, we consider here that the length of each route is constrained to a given limit L. Drones are not restricted to travel on the network, and they can enter and exit a line through any of its points, servicing only a portion of that line. Therefore, shorter solutions are obtained with ¿aerial¿ drones than with ¿ground¿ vehicles that are restricted to the network. If a LC K¿DRPP instance is digitized by approximating each line with a polygonal chain, and it is assumed that drones can only enter and exit a line through the points of the chain, an instance of the Length Constrained K¿vehicles Rural Postman Problem (LC K¿RPP) is obtained. This is a discrete arc routing problem, and therefore can be solved with combinatorial optimization techniques. However, when the number of points in each polygonal chain is very large, the LC K¿RPP instance can be so large that it is very difficult to solve, even for heuristic algorithms. Therefore, it is necessary to implement a procedure that generates smaller LC K¿ RPP instances by approximating each line by a few but ¿significant¿ points and segments. In this paper, we present a new formulation for the LC K¿RPP with two binary variables for each edge and each drone representing the first and second traversals of the edge, respectively. We make a polyhedral study of the set of solutions of a relaxed formulation and prove that several families of inequalities induce facets of the polyhedron. We design and implement a branch¿and¿cut algorithm for the LC K¿RPP that incorporates the separation of these inequalities. This B &C is the main routine of an iterative algorithm that, by solving a LC K¿RPP instance at each step, finds good solutions for the original LC K¿DRPP. The computational results show that the proposed method is effective in finding good solutions for LC K¿DRPP, and that the branch¿and¿cut algorithm for the LC K¿RPP outperforms the only published exact method for this problem.The work by Ángel Corberán, Isaac Plana, José M. Sanchis, and Paula Segura was supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MICIU) and Fondo Social Europeo (FSE) through project PGC2018-099428-B-I00.Campbell, J.; Corberán, Á.; Plana, I.; Sanchís Llopis, JM.; Segura-Martínez, P. (2022). Polyhedral analysis and a new algorithm for the length constrained K-drones rural postman problem. 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The capacitated arc routing problem. A heuristic algorithm

In this paper we consider the Capacitated Arc Routing Problem, in which a fleet of K vehicles, all of them based on a specific vertex (the depot) and with a known capacity Q, must service a subset of the edges of the graph, with minimum total cost and such that the load assigned to each vehicle does not exceed its capacity. A heuristic algorithm for this problem is proposed consisting of: the selection of K centers, the construction of K connected graphs with associated loads not exceeding the vehicle capacities, the resolution of a General Assignment Problem, if necessary, to get a complete assignment of edges to vehicles and finally the construction of the routes by solving heuristically a Rural Postman Problem for each vehicle. Computational results on graphs up to 50 vertices and 97 edges are included. On average, the feasible solution is within 6,4% of the best known lower bound