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

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

New results on the Windy Postman Problem

[EN] In this paper, we study the Windy Postman Problem (WPP). This is a well-known Arc Routing Problem that contains the Mixed Chinese Postman Problem (MCPP) as a special case. We extend to arbitrary dimension some new inequalities that complete the description of the polyhedron associated with the Windy Postman Problem over graphs with up to four vertices and ten edges. We introduce two new families of facet-inducing inequalities and prove that these inequalities, along with the already known odd zigzag inequalities, are Chvátal-Gomory inequalities of rank at most 2. Moreover, a branch-and-cut algorithm that incorporates two new separation algorithms for all the previously mentioned inequalities and a new heuristic procedure to obtain upper bounds are presented. Finally, the performance of a branch-and-cut algorithm over several sets of large WPP and MCPP instances, with up to 3,000 nodes and 9,000 edges (and arcs in the MCPP case), shows that, to our knowledge, this is the best algorithm to date for the exact resolution of the WPP and the MCPP. © 2010 Springer and Mathematical Optimization Society.The authors want to thank the three referees for their careful reading of the manuscript and for their many comments and suggestions that have contributed to improve the paper content and readability. In particular, several remarks regarding the discussion of C-G and mod-k inequalities were pointed out by one of the referees. A. Corberan, I. Plana and J.M. Sanchis wish to thank the Ministerio de Educacion y Ciencia of Spain (projects MTM2006-14961-C05-02 and MTM2009-14039-C06-02) for its support.Corberán, A.; Oswald, M.; Plana, I.; Reinelt, G.; Sanchís Llopis, JM. (2012). New results on the Windy Postman Problem. Mathematical Programming. 132(1-2):309-332. https://doi.org/10.1007/s10107-010-0399-xS3093321321-2Benavent E., Carrotta A., Corberán A., Sanchis J.M., Vigo D.: Lower bounds and heuristics for the windy rural postman problem. Eur. J. Oper. Res. 176, 855–869 (2007)Brucker P. The Chinese postman problem for mixed graphs. In Proceedings of international workshop. Lecture Notes in Computer Science 100, 354–366 (1981)Caprara A., Fischetti M.: $${\{0,\frac{1}{2}\}}$$ -Chvátal-Gomory cuts. Math. Program. 74, 221–235 (1996)Caprara A., Fischetti M., Letchford A.N.: On the separation of maximally violated mod-k cuts. Math. Program. 87, 37–56 (2000)Christof, T., Loebel, A.: PORTA—a polyhedron representation algorithm www.informatik.uni-heidelberg.de/groups/comopt/software/PORTA/ (1998)Christofides, N., Benavent, E., Campos, V., Corberán, A., Mota, E.: An optimal method for the mixed postman problem. In Thoft-Christensen, P. (ed.) System Modelling and Optimization. Lecture Notes in Control and Information Sciences 59, Springer (1984)Corberán A., Plana I., Sanchis J.M.: Zigzag inequalities: a new class of facet-inducing inequalities for arc routing problems. Math. Program. 108, 79–96 (2006)Corberán A., Plana I., Sanchis J.M.: A branch & cut algorithm for the windy general routing problem and special cases. Networks 49, 245–257 (2007)Corberán, A., Plana I., Sanchis, J.M.: Arc routing problems: data instances. www.uv.es/corberan/instancias.htm (2007)Corberán A., Plana I., Sanchis J.M.: The windy general routing polyhedron: a global view of many known arc routing polyhedra. SIAM J. Discrete Math. 22, 606–628 (2008)Grötschel, M., Win, Z.: On the windy postman polyhedron. Report No. 75, Schwerpunktprogram der Deutschen Forschungsgemeinschaft, Universität Augsburg, Germany (1988)Grötschel M., Win Z.: A cutting plane algorithm for the Windy Postman Problem. Math. Program. 55, 339–358 (1992)Guan M.: On the Windy Postman Problem. Discrete Appl. Math. 9, 41–46 (1984)Minieka E.: The Chinese postman problem for mixed networks. Manage. Sci. 25, 643–648 (1979)Naddef D., Rinaldi G.: The symmetric traveling salesman polytope and its graphical relaxation: composition of valid inequalities. Math. Program. 51, 359–400 (1991)Oswald M., Reinelt G., Seitz H.: Applying mod-k cuts for solving linear ordering problems. TOP 17, 158–170 (2009)Papadimitriou C.H.: On the complexity of edge traversing. J. Assoc. Comput. Mach. 23, 544–554 (1976)Ralphs T.K.: On the mixed Chinese postman problem. Oper. Res. Lett. 14, 123–127 (1993)Wenger, K.: Generic Cut Generation Methods for Routing Problems. PhD Dissertation, University of Heidelberg, Germany (2004)Win, Z.: Contributions to Routing Problems. PhD Dissertation, University of Augsburg, Germany (1987)Win Z.: On the Windy Postman Problem on eulerian graphs. Math. Program. 44, 97–112 (1989)Zaragoza Martínez F.J.: Series-parallel graphs are windy postman perfect. Discrete Math. 308, 1366–1374 (2008

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

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

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 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

The Team Orienteering Arc Routing Problem

The team orienteering arc routing problem (TOARP) is the extension to the arc routing setting of the team orienteering problem. In the TOARP, in addition to a possible set of regular customers that have to be serviced, another set of potential customers is available. Each customer is associated with an arc of a directed graph. Each potential customer has a profit that is collected when it is serviced, that is, when the associated arc is traversed. A fleet of vehicles with a given maximum traveling time is available. The profit from a customer can be collected by one vehicle at most. The objective is to identify the customers that maximize the total profit collected while satisfying the given time limit for each vehicle. In this paper we propose a formulation for this problem and study a relaxation of its associated polyhedron. We present some families of valid and facet-inducing inequalities that we use in the implementation of a branch-and-cut algorithm for the resolution of the problem. Computational experiments are run on a large set of benchmark instances.The authors thank the reviewers for their comments that helped to provide an improved and clearer version of this paper. Angel Corberan, Isaac Plana, and Jose M. Sanchis wish to thank the Ministerio de Ciencia e Innovacion [Project MTM2009-14039-C06-02] and the Ministerio of Economia y Competitividad [Project MTM2012-36163-C06-02] of Spain for their support.Archetti, C.; Speranza, MG.; Corberan, A.; Sanchís Llopis, JM.; Plana, I. (2014). The Team Orienteering Arc Routing Problem. Transportation Science. 48(3):442-457. https://doi.org/10.1287/trsc.2013.0484S44245748

A matheuristic for the Team Orienteering Arc Routing Problem

In the Team OrienteeringArc Routing Problem (TOARP) the potential customers are located on the arcs of a directed graph and are to be chosen on the basis of an associated profit. A limited fleet of vehicles is available to serve the chosen customers. Each vehicle has to satisfy a maximum route duration constraint. The goal is to maximize the profit of the served customers. We propose a matheuristic for the TOARP and test it on a set of benchmark instances for which the optimal solution or an upper bound is known. The matheuristic finds the optimal solutions on all, except one, instances of one of the four classes of tested instances (with up to 27 vertices and 296 arcs). The average error on all instances fo rwhich the optimal solution is available is 0.67 percent.Angel Corberan, Isaac Plana and Jose M. Sanchis wish to thank the Ministerio de Economia y Competitividad (project MTM2012-36163-C06-02) of Spain and the Generalitat Valenciana (project GVPROMETEO2013-049) for their support.Archetti, C.; Corberan, A.; Plana, I.; Sanchís Llopis, JM.; Speranza, MG. (2015). A matheuristic for the Team Orienteering Arc Routing Problem. European Journal of Operational Research. 245(2):392-401. https://doi.org/10.1016/j.ejor.2015.03.022S392401245