A branch-and-cut algorithm for the multidepot rural postman problem

This paper considers the Multidepot Rural Postman Problem, an extension of the classical Rural Postman Problem in which there are several depots instead of only one. The aim is to construct a minimum cost set of routes traversing each required edge of the graph, where each route starts and ends at the same depot. The paper makes the following scientific contributions: (i) It presents optimality conditions and a worst case analysis for the problem; (ii) It proposes a compact integer linear programming formulation containing only binary variables, as well as a polyhedral analysis; (iii) It develops a branch-and-cut algorithm that includes several new exact and heuristic separation procedures. Instances involving up to four depots, 744 vertices, and 1,315 edges are solved to optimality. These instances contain up to 140 required components and 1,000 required edges.Peer ReviewedPostprint (author's final draft

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

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

Min-Max K-vehicles Windy Rural Postman Problem

[EN] In this article the Min-Max version of the windy rural postman problem with several vehicles is introduced. For this problem, in which the objective is to minimize the length of the longest tour in order to find a set of balanced tours for the vehicles, we present here an ILP formulation and study its associated polyhedron. Based on its partial description, a branch-and-cut algorithm has been implemented and computational results on a large set of instances are finally presented. (C) 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 54(4),216-226 2009Contract grant sponsor: Ministerio de Education y Ciencia of Spain: Contract gram number: MTM2006-14961-C05-02Benavent López, E.; Corberan, A.; Plana, I.; Sanchís Llopis, JM. (2009). Min-Max K-vehicles Windy Rural Postman Problem. Networks. 54(4):216-226. https://doi.org/10.1002/net.20334S216226544D. Ahr Contributions to multiple postmen problems 2004D. Ahr G. Reinelt “New heuristics and lower bounds for the min-max k -Chinese postman problem” Algorithms-ESA 2002, 10th Annual European Symposium, Rome, Italy, 2002, Lecture Notes in Computer Science 2461 R. Möring R. Raman Springer Berlin 2002 64 74Ahr, D., & Reinelt, G. (2006). A tabu search algorithm for the min–max k-Chinese postman problem. Computers & Operations Research, 33(12), 3403-3422. doi:10.1016/j.cor.2005.02.011D. Applegate R.E. Bixby V. Chvátal W. Cook Finding cuts in the TSP 1995Barahona, F., & Grötschel, M. (1986). On the cycle polytope of a binary matroid. Journal of Combinatorial Theory, Series B, 40(1), 40-62. doi:10.1016/0095-8956(86)90063-8Belenguer, J. M., & Benavent, E. (1998). Computational Optimization and Applications, 10(2), 165-187. doi:10.1023/a:1018316919294Benavent, E., Carrotta, A., Corberán, A., Sanchis, J. M., & Vigo, D. (2007). Lower bounds and heuristics for the Windy Rural Postman Problem. European Journal of Operational Research, 176(2), 855-869. doi:10.1016/j.ejor.2005.09.021N. Christofides V. Campos A. Corberán E. Mota An algorithm for the rural postman problem 1981Christofides, N., Campos, V., Corberán, A., & Mota, E. (1986). An algorithm for the Rural Postman problem on a directed graph. Netflow at Pisa, 155-166. doi:10.1007/bfb0121091Corberán, A., Plana, I., & Sanchis, J. M. (2008). The Windy General Routing Polyhedron: A Global View of Many Known Arc Routing Polyhedra. SIAM Journal on Discrete Mathematics, 22(2), 606-628. doi:10.1137/050640886Corberán, A., Plana, I., & Sanchis, J. M. (2007). A branch & cut algorithm for the windy general routing problem and special cases. Networks, 49(4), 245-257. doi:10.1002/net.20176Eiselt, H. A., Gendreau, M., & Laporte, G. (1995). Arc Routing Problems, Part II: The Rural Postman Problem. Operations Research, 43(3), 399-414. doi:10.1287/opre.43.3.399Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017G. Ghiani D. Laganá G. Laporte R. Musmanno A branch-and-cut algorithm for the undirected capacitated arc routing problem 2007Ghiani, G., & Laporte, G. (2000). A branch-and-cut algorithm for the Undirected Rural Postman Problem. Mathematical Programming, 87(3), 467-481. doi:10.1007/s101070050007Golden, B. L., & Wong, R. T. (1981). Capacitated arc routing problems. Networks, 11(3), 305-315. doi:10.1002/net.3230110308Padberg, M. W., & Rao, M. R. (1982). Odd Minimum Cut-Sets andb-Matchings. Mathematics of Operations Research, 7(1), 67-80. doi:10.1287/moor.7.1.67Pearn, W. L. (1994). Solvable cases of the k-person Chinese postman problem. Operations Research Letters, 16(4), 241-244. doi:10.1016/0167-6377(94)90073-

The generalized arc routing problem

The final publication is available at Springer via http://dx.doi.org/10.1007/s11750-017-0437-4This paper focuses on the generalized arc routing problem. This problem is stated on an undirected graph in which some clusters are defined as pairwise-disjoint connected subgraphs, and a route is sought that traverses at least one edge of each cluster. Broadly speaking, the generalized arc routing problem is the arc routing counterpart of the generalized traveling salesman problem, where the set of vertices of a given graph is partitioned into clusters and a route is sought that visits at least one vertex of each cluster. A mathematical programming formulation that exploits the structure of the problem and uses only binary variables is proposed. Facets and families of valid inequalities are presented for the polyhedron associated with the formulation and the corresponding separation problem studied. The numerical results of a series of computational experiments with an exact branch and cut algorithm are presented and analyzed.Peer ReviewedPostprint (author's final draft

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

New models and algorithms for several families of Arc Routing Problems

Some of the most common decisions to be taken within a logistic systems at an operational level are related to the design of the vehicle routes. Vehicle Routing Problems and Arc Routing Problems are well-known families of problems addressing such decisions. Their main difference is whether service demand is located at the vertices or the edges of the operating network. In this thesis we focus on the study of several arc routing problems. We concentrate on three families of problems. The first family consists of Multi Depot Rural Postman Problems, which are an extension of Rural Postman Problems where there are several depots instead of only one. The second family of problems that we study are Location-Arc Routing Problems, in which the depots are not fixed in advance, so their location becomes part of the decisions of the problem. We finally study Target-Visitation Arc Routing Problems, where the service is subject to an ordering preference among the connected components induced by demand arcs. Different models are studied for each considered family. In particular, two different Multi Depot Rural Postman Problem models are considered, which differ in the objective function: the minimization of the overall transportation cost or the minimization of the makespan. Concerning Location-Arc Routing Problems, we study six alternative models that differ from each other in their objective function, whether there is an upper bound on the number of facilities to be located, or whether there are capacity constraints on the demand that can be served from selected facilities. Finally, two Target-Visitation Arc Routing Problem models are studied, which differ from each other in whether or not it is required that all the required edges in the same component are visited consecutively. The aim in this thesis is to provide quantitative tools to the decision makers to identify the best choices for the design of the routes. To this end and for each considered problem, we first study and analyze its characteristics and properties. Based on them we develop different Integer Linear Programming formulations suitable for being solved trough branch-and-cut. Finally, all formulations are tested trough extensive computational experience. In this sense, for Multi Depot Rural Postman Problems and Location-Arc Routing Problems we propose natural modeling formulations with three-index variables, where variables are associated with edges and facilities. For some of the models we also present alternative formulations with only two-index variables, which are solely associated with edges. Finally, for the Target-Visitation Arc Routing Problems we propose three different formulations, two alternative formulations for the general case, and one for the clustered version, where all the edges in the same components are served sequentially, which exploits some optimality conditions of the problem.Algunes de les decisions més habituals que es prenen en un sistema logístic a nivell operatiu estan relacionades amb el disseny de rutes de vehicles. Els coneguts Vehicle Routing Problems i Arc Routing Problems són famílies de problemes que s'ocupen d'aquest tipus de decisions. La principal diferència entre ambdós recau en si la demanda de servei es troba localitzada als vèrtexs o a les arestes de la xarxa. Aquesta tesi es centra en l'estudi de diversos problemes de rutes per arcs. Ens centrem en tres famílies de problemes. La primera família consisteix en els Multi Depot Rural Postman Problems, que són una extensió del Rural Postman Problem on hi ha diversos dipòsits en lloc d'un de sol. La segona família de problemes que estudiem són els Location-Arc Routing Problems, en els quals els dipòsits no estan fixats amb antelació i, per tant, la seva ubicació esdevé part de les decisions a prendre en el problema. Finalment, estudiem els Target-Visitation Arc Routing Problems, on el servei està subjecte a una preferència d'ordenació entre les components connexes induïdes pels arcs amb demanda. S'estudien diferents models per a cadascuna de les famílies considerades. En particular, es consideren dos models diferents per al Multi Depot Rural Postman Problem, que es diferencien en la funció objectiu: la minimització del cost general de transport o la minimització de la ruta més llarga. Pel que fa als Location-Arc Routing Problems, estudiem sis models alternatius que difereixen en la seva funció objectiu, considerant si hi ha un límit màxim sobre la quantitat de dipòsits a ubicar o si hi ha restriccions de capacitat sobre la demanda que es pot servir des dels dipòsits seleccionats. Finalment, s'estudien dos models de Target-Visitation Arc Routing Problem, que es diferencien en si es necessari que totes les arestes requerides en la mateixa component es visitin de forma consecutiva. L'objectiu d'aquesta tesi és proporcionar eines quantitatives als responsables, que permetin identificar les millors opcions de disseny de les rutes. Per això, i per a cadascundels problemes considerats, primer estudiem i analitzem les seves característiques i propietats. A partir d'aquestes, desenvolupem diferents formulacions de Programació Lineal Entera, adequades per a la seva solució mitjançant un branch-and-cut. Finalment, totes les formulacions són provades mitjançant un ampli testeig computacional. En aquest sentit, per als Multi Depot Rural Postman Problems i els Location-Arc Routing Problems, proposem formulacions naturals amb variables de tres índexs, on les variables estan associades a les arestes i als dipòsits. Per a alguns dels models també presentem formulacions alternatives, amb variables de només dos índexs, que només estan associades a les arestes. Finalment, per als Target-Visitation Arc Routing Problems proposem tres formulacions diferents, dues formulacions alternatives per al cas general i una per a la versió en clúster, on totes les arestes de la mateixa component es serveixen seqüencialment, cosa que explora algunes condicions d'optimització pròpies.Algunas de las decisiones más habituales que se toman en un sistema logístico a nivel operativo están relacionadas con el diseño de rutas de vehículos. Los conocidos Vehicle Routing Problems y Arc Routing Problems son familias de problemas que se ocupan de este tipo de decisiones. La principal diferencia entre ambas reside en si la demanda de servicios está localizada en los vértices o en las aristas de la red. Esta tesis se centra en el estudio de diversos problemas de rutas por arcos. Nos centramos en tres familias de problemas. La primera familia consiste en los Multi Depot Rural Postman Problems, que son una extensión del Rural Postman Problem donde hay varios depósitos en lugar de solamente uno. La segunda familia de problemas que estudiamos son los Location-Arc Routing Problems, en los que los depósitos no están fijados con antelación y, por lo tanto, su ubicación se convierte en parte de las decisiones a tomar en el problema. Finalmente, estudiamos los Target-Visitation Arc Routing Problems, donde el servicio está sujeto a una preferencia de ordenación entre las componentes conexas inducidas por los arcos con demanda. Se estudian diferentes modelos para cada una de las familias consideradas. En particular, se consideren dos modelos diferentes para el Multi Depot Rural Postman Problem que se diferencian en la función objetivo: la minimización del coste general de transporte o la minimización de la ruta más larga. En cuanto a los Location-Arc Routing Problems, estudiamos seis modelos alternativos que difieren en su función objetivo, en si hay un limite máximo sobre la cantidad de depósitos a ubicar, o en si hay restricciones de capacidad sobre la demanda que se puede servir desde los depósitos seleccionados. Finalmente, se estudian dos modelos de Target-Visitation Arc Routing Problem, que se diferencian en si es necesario que todas las aristas requeridas en la misma componente se visiten de forma consecutiva. El objetivo de esta tesis es proporcionar herramientas cuantitativas a los responsables, que permitan identificar las mejores opciones de diseño de las rutas. Por ello, y para cada uno de los problemas considerados, primero estudiamos y analizamos sus características y propiedades. A partir de estas, desarrollamos diferentes formulaciones de Programación Lineal Entera, adecuadas para su solución mediante un branch-and-cut. Finalmente, todas las formulaciones son probadas mediante un amplio testeo computacional. En este sentido, para los Multi Depot Rural Postman Problems y los Location-Arc Routing Problems, proponemos formulaciones naturales con variables de tres índices, donde las variables están asociadas a las aristas y a los depósitos. Para algunos de los modelos también presentamos formulaciones alternativas con variables de sólo dos índices, que sólo están asociadas a las aristas. Finalmente, para los Target-Visitation Arc Routing Problems proponemos tres formulaciones diferentes, dos formulaciones alternativas para el caso general y una para la versión en clúster, donde todas las aristas de la misma componente se sirven secuencialmente, lo que explora algunas condiciones de optimización propia

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