An asymmetric TSP with time windows and with time-dependent travel times and costs: An exact solution through a graph transformation

[EN] In this paper we deal with an extended version of the Asymmetric Traveling Salesman Problem with Time Windows (ATSPTW) that considers time-dependent travel times and costs, for a more accurate approximation of some routing problems inside large cities, in which the time or cost of traversing certain streets (e.g. main avenues) depends on the moment of the day (for example rush-hours). Unlike other existing papers about time-dependent routing problems, we focus on an exact method for solving this new problem. For this end we first transform the problem into an Asymmetric Generalized TSP and then into a Graphical Asymmetric TSP. In this way, we can apply a known exact algorithm for the Mixed General Routing Problem, which seems to run well with our resulting instances. Computational results are presented on a set of 270 adapted instances from benchmark ATSPTW instances.This work has been partially supported by the Ministerio de Ciencia y Tecnología of Spain (project TIC2003-05982-C05-01) and the Generalitat Valenciana (Ref: GRUPOS03/189). Thanks are due to Michel Gendreau, Alain Hertz, Gilbert Laporte and Mihnea Stan for providing us the set of benchmark ATSPTW instances, and to Matteo Fischetti and Norbert Ascheuer for their suggestions and help about the computational experiments. Last we are also indebted to the three anonymous referees for their valuable comments.Albiach, J.; Sanchís Llopis, JM.; Soler Fernández, D. (2008). An asymmetric TSP with time windows and with time-dependent travel times and costs: An exact solution through a graph transformation. European Journal of Operational Research. 189(3):789-802. https://doi.org/10.1016/j.ejor.2006.09.099S789802189

New Results on the Mixed General Routing Problem

[EN] In this paper, we deal with the polyhedral description and the resolution of the Mixed General Routing Problem. This problem, in which the service activity occurs both at some of the nodes and at some of the arcs and edges of a mixed graph, contains a large number of important arc and node routing problems as special cases. Here, a large family of facet-defining inequalities, the Honeycomb inequalities, is described. Furthermore, a cutting-plane algorithm for this problem that incorporates new separation procedures for the K-C, Regular Path-Bridge, and Honeycomb inequalities is presented. Branch and bound is invoked when the final solution of the cutting-plane procedure is fractional. Extensive computational experiments over different sets of instances are included.The contribution by A. Corberán and J. M. Sanchis has been partially supported by the Ministerio de Ciencia y Tecnología of Spain (Ref: TIC2003-05982-C05-01) and by the AVCiT de la Generalitat Valenciana (Ref: GRUPOS03/189).Corberán, A.; Mejía, G.; Sanchís Llopis, JM. (2005). New Results on the Mixed General Routing Problem. Operations Research. 53(2):363-376. https://doi.org/10.1287/opre.1040.016836337653

The Windy General Routing Polyhedron: A global view of many known Arc Routing Polyhedra

[EN] 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 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.This work was supported by the Ministerio de Educación y Ciencia of Spain (project MTM2006-14961-C05-02).Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2008). The Windy General Routing Polyhedron: A global view of many known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics. 22(2):606-628. https://doi.org/10.1137/050640886S60662822

Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems

[EN] In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.The authors wish to thank the Ministerio de Ciencia y Tecnología of Spain (project TIC2003-05982-C05-01) and the Generalitat Valenciana (Ref: GRUPOS03/189) their support.Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2006). Zigzag inequalities:a new class of facet-inducing inequalities for Arc Routing Problems. Mathematical Programming. 108(1):79-96. https://doi.org/10.1007/s10107-005-0643-yS79961081Benavent, E., Carrotta, A., Corberán, A., Sanchis, J.M., Vigo, D.: Lower Bounds and Heuristics for the Windy Rural Postman Problem. Technical Report TR03-2003. Department of Statistics and OR, University of Valencia (Spain). Submitted to EJOR 2003Benavent, E., Corberán, A., Piñana, E., Plana, I., Sanchis, J.M.: New Heuristics for the Windy Rural Postman Problem. To appear in Comput. Oper. Res. 2005Chopra, S., Rinaldi, G.: The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities. SIAM J. Discrete Math. 9 (4), 602–624 (1996)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An Optimal Method for the Mixed Postman Problem. In: P. Thoft-Christensen (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences, 59. Berlin: Springer-Verlag 1984Christofides, N., Campos, V., Corberán, A., Mota, E.: An Algorithm for the Rural Postman Problem. Report IC.OR. 81.5. Imperial College, London 1981Corberán, A., Mejía, G., Sanchis, J.M.: New Results on the Mixed General Routing Problem. To appear in Oper. Res. 2005Corberán, A., Mota, E., Sanchis, J.M.: A Comparison of Two Different Formulations for Arc Routing Problems on Mixed Graphs. To appear in Comput. Oper. Res. 2005Corberán, A., Plana, I., Sanchis, J.M.: On the Windy General Routing Polyhedron. In preparation 2005Corberán, A., Romero, A., Sanchis, J.M.: The Mixed General Routing Problem Polyhedron. Math. Programming 96, 103–137 (2003)Cornuèjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Math. Programming 33, 1–27 (1985)Eiselt, H.A., Gendreau, M., Laporte, G.: Arc-Routing Problems, Part 2: the Rural Postman Problem. Oper. Res. 43, 399–414 (1995)Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press, Princeton, NJ 1962Grötschel, M., Win, Z.: On the Windy Postman Polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany 1988Grötschel, M., Win, Z.: A Cutting Plane Algorithm for the Windy Postman Problem. Math. Programming 55, 339–358 (1992)Guan, M.: On the Windy Postman Problem. Discrete Applied Mathematics 9, 41–46 (1984)Letchford, A.: New inequalities for the General Routing Problem. Eur. J. Oper. Res. 96, 317–322 (1997)Minieka, E.: The Chinese Postman Problem for Mixed Networks. Management Sci. 25, 643–648 (1979)Naddef, D., Rinaldi, G.: The Symmetric Traveling Salesman Polytope and its Graphical Relaxation: Composition of Valid Inequalities. Math. Programming 51, 359–400 (1991)Nobert, Y., Picard, J.C.: An Optimal Algorithm for the Mixed Chinese Postman Problem. Networks 27, 95–108 (1996)Ralphs, T.K.: On the Mixed Chinese Postman Problem. Oper. Res. Lett. 14, 123–127 (1993)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany 198

A comparison of two different formulations for Arc Routing Problems on Mixed graphs

[EN] Arc routing problems on mixed graphs have been modelled in the literature either using just one variable per edge or associating to each edge two variables, each one representing its traversal in the corresponding direction. In this paper, and using the mixed general routing problem as an example, we compare theoretical and computationally both formulations as well as the lower bounds obtained from them using Linear Programming based methods. Extensive computational experiments, including some big and newly generated random instances, are presented.Authors want to thank two anonymous referees for their careful reading of the manuscript. Also we thank the support given by the Ministerio de Ciencia y Tecnología of Spain through Grant TIC2003-05982-C05-01 and by the Generalitat Valenciana (Grupos 03/189).Corberán, A.; Mota, E.; Sanchís Llopis, JM. (2006). A comparison of two different formulations for Arc Routing Problems on Mixed graphs. Computers & Operations Research. 33(12):3384-3402. https://doi.org/10.1016/j.cor.2005.02.010S33843402331

The mixed general routing polyhedron

[EN] In Arc Routing Problems, ARPs, the aim is to find on a graph a minimum cost traversal satisfying some conditions related to the links of the graph. Due to restrictions to traverse some streets in a specified way, most applications of ARPs must be modeled with a mixed graph. Although several exact algorithms have been proposed, no polyhedral investigations have been done for ARPs on a mixed graph. In this paper we deal with the Mixed General Routing Problem which consists of finding a minimum cost traversal of a given link subset and a given vertex subset of a mixed graph. A formulation is given that uses only one variable for each link (edge or arc) of the graph. Some properties of the associated polyhedron and some large families of facet-inducing inequalities are described. A preliminary cutting-plane algorithm has produced very good lower bounds over a set of 100 randomly generated instances of the Mixed Rural Postman Problem. Finally, applications of this study to other known routing problems are described.The authors wish to thank the Ministerio de Innovación y Ciencia/FEDER of Spain (projects MTM2009-14039-C06-02, MTM2010-19576-C02-02 and DE2009-0057) and Junta de Andalucía/FEDER (grant number FQM-5849) for its support. They also thank two anonymous referees for their careful reading of the manuscript and for their many suggestions and comments that have helped to improve the contents and readability of the paper.S10313796

A matheuristic for the Distance-Constrained Close-Enough Arc Routing Problem

[EN] The Close-Enough Arc Routing Problem, also called Generalized Directed Rural Postman Problem, is an arc routing problem with interesting real-life applications, such as routing for meter reading. In this application, a vehicle with a receiver travels through a series of neighborhoods. If the vehicle gets within a certain distance of a meter, the receiver is able to record the gas, water, or electricity consumption. Therefore, the vehicle does not need to traverse every street, but only a few, in order to be close enough to each meter. In this paper we deal with an extension of this problem, the Distance-Constrained Generalized Directed Rural Postman Problem or Distance-Constrained Close Enough Arc Routing Problem, in which a fleet of vehicles is available. The vehicles have to leave from and return to a common vertex, the depot, and the length of their routes must not exceed a maximum distance (or time). For solving this problem we propose a matheuristic that incorporates an effective exact procedure to optimize the routes obtained. Extensive computational experiments have been performed on a set of benchmark instances and the results are compared with those obtained with an exact procedure proposed in the literature.This work was supported by the Spanish Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER) through Project MTM2015-68097-P (MINECO/FEDER). Authors want to thank two anonymous referees for their suggestions and comments that have contributed to improve the paper.Corberán, A.; Plana, I.; Reula, M.; Sanchís Llopis, JM. (2019). A matheuristic for the Distance-Constrained Close-Enough Arc Routing Problem. Top. 27(2):312-326. https://doi.org/10.1007/s11750-019-00507-3S312326272Aráoz J, Fernández E, Franquesa C (2017) The generalized arc routing problem. TOP 25:497–525Ávila T, Corberán Á, Plana I, Sanchis JM (2016) A new branch-and-cut algorithm for the generalized directed rural postman problem. Transportation Science 50:750–761Ávila T, Corberán Á, Plana I, Sanchis JM (2017) Formulations and exact algorithms for the distance-constrained generalized directed rural postman problem. EURO Journal on Computational Optimization 5:339–365Cerrone C, Cerulli R, Golden B, Pentangelo R (2017) A flow formulation for the close-enough arc routing problem. In Sforza A. and Sterle C., editors, Optimization and Decision Science: Methodologies and Applications. ODS 2017., volume 217 of Springer Proceedings in Mathematics & Statistics, pages 539–546Corberán Á, Laporte G (editors) (2014) Arc Routing: Problems,Methods, and Applications. MOS-SIAM Series on Optimization,PhiladelphiaCorberán Á, Plana I, Sanchis J.M (2007) Arc routing problems: data instances. http://www.uv.es/~corberan/instancias.htmDrexl M (2007) On some generalized routing problems. PhD thesis, Rheinisch-Westfälische Technische Hochschule, Aachen UniversityDrexl M (2014) On the generalized directed rural postman problem. Journal of the Operational Research Society 65:1143–1154Gendreau M, Laporte G, Semet F (1997) The covering tour problem. Operations Research 45:568–576Hà M-H, Bostel N, Langevin A, Rousseau L-M (2014) Solving the close enough arc routing problem. Networks 63:107–118Mourão MC, Pinto LS (2017) An updated annotated bibliography on arc routing problems. Networks 70:144–194Renaud A, Absi N, Feillet D (2017) The stochastic close-enough arc routing problem. Networks 69:205–221Shuttleworth R, Golden BL, Smith S, Wasil EA (2008) Advances in meter reading: Heuristic solution of the close enough traveling salesman problem over a street network. In: Golden BL, Raghavan S, Wasil EA (eds) The Vehicle Routing Problem: Lastest Advances and New Challenges. Springer, pp 487–50

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

A branch-and-cut algorithm for the maximum benefit Chinese postman problem

[EN] The Maximum Benefit Chinese Postman Problem (MBCPP) is an NP-hard problem that considers several benefits associated with each edge, one for each time the edge is traversed with a service. The objective is to find a closed walk with maximum benefit.We propose an IP formulation for the undirected MBCPP and, based on the description of its associated polyhedron, we propose a branch-and-cut algorithm and present computational results on instances with up to 1,000 vertices and 3,000 edges.The authors wish to thank the Ministerio de Innovacion y Ciencia/FEDER of Spain (projects MTM2009-14039-C06-02, MTM2010-19576-C02-02 and DE2009-0057) and Junta de Andalucia/FEDER (grant number FQM-5849) for its support. They also thank two anonymous referees for their careful reading of the manuscript and for their many suggestions and comments that have helped to improve the contents and readability of the paper.Corberán, A.; Plana, I.; Rodríguez-Chía, AM.; Sanchís Llopis, JM. (2013). A branch-and-cut algorithm for the maximum benefit Chinese postman problem. Mathematical Programming. 141(1-2):21-48. https://doi.org/10.1007/s10107-011-0507-6S21481411-2Aráoz J., Fernández E., Franquesa C.: The clustered price-collecting arc-routing problem. Transp. Sci. 43, 287–300 (2009)Aráoz J., Fernández E., Meza O.: Solving the prize-collecting rural postman problem. Eur. J. Oper. Res. 196, 886–896 (2009)Aráoz J., Fernández E., Zoltan C.: Privatized rural postman problems. Comput. Oper. Res. 33, 3432–3449 (2006)Archetti C., Feillet D., Hertz A., Speranza M.G.: The undirected capacitated arc routing problem with profits. Comput. Oper. Res. 37, 1860–1869 (2010)Barahona F., Grötschel M.: On the cycle polytope of a binary matroid. J. Comb. Theory B 40, 40–62 (1986)Fernández E., Fernández E., Franquesa C., Sanchis J.M.: The windy clustered prize-collecting problem. Transp. Sci. 45, 317–334 (2011)Letchford A.N., Letchford A.N., Sanchis J.M.: A cutting-plane algorithm for the general routing problem. Math. Progr. 90, 291–316 (2001)Plana I., Plana I., Sanchis J.M.: A branch & cut algorithm for the windy general routing problem and special cases. Networks 49, 245–257 (2007)Corberán, Á., Plana, I., Sanchis, J.M.: Arc Routing Problems: Data Instances. http://www.uv.es/corberan/instancias.htmSanchis J.M., Sanchis J.M.: A polyhedral approach to the rural postman problem. Eur. J. Oper. Res. 79, 95–114 (1994)Feillet D., Dejax P., Gendreau M.: The profitable arc tour problem: solution with a branch-and-price algorithm. Transp. Sci. 39, 539–552 (2005)Franquesa, C.: The Clustered Prize-collecting Arc Routing Problem. PhD Thesis, Technical University of Catalonia, Barcelona (2008)Ghiani G., Laporte G.: A branch-and-cut algorithm for the undirected rural postman problem. Math. Progr. 87, 467–481 (2000)Lenstra J.K., Rinnooy Kan A.H.G.: On general routing problems. Networks 6, 593–597 (1976)Letchford A.N., Reinelt G., Theis D.O.: Odd minimum cut-sets and b-matchings revisited. SIAM J. Discret. Math. 22, 1480–1487 (2008)Malandraki C., Daskin M.S.: The maximum benefit chinese postman problem and the maximum benefit traveling salesman problem. Eur. J. Oper. Res. 65, 218–234 (1993)Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, New York (1988)Orloff C.S.: A fundamental problem in vehicle routing. Networks 4, 35–64 (1974)Pearn W.L., Chiu W.C.: Approximate solutions for the maximum benefit Chinese postman problem. Int. J. Syst. Sci. 36, 815–822 (2005)Pearn W.L., Wang K.H.: On the maximum benefit Chinese postman problem. OMEGA 31, 269–273 (2003)Reinelt G., Theis D.O.: Transformation of facets of the general routing problem polytope. SIAM J. Optim. 16, 220–234 (2005