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 mixed capacitated general routing problem with turn penalties

In this paper we deal with the mixed capacitated general routing problem with turn penalties. This problem generalizes many important arc and node routing problems, and it takes into account turn penalties and forbidden turns, which are crucial in many real-life applications, such as mail delivery, waste collection and street maintenance operations. Through a polynomial transformation of the considered problem into a Generalized Vehicle routing problem, we suggest a new approach for solving this new problem by transforming it into an Asymmetric Capacitated Vehicle routing problem. In this way, we can solve the new problem both optimally and heuristically using existing algorithms. A powerful memetic algorithm and a set of 336 new benchmark instances are also suggested. The experimental results show that the average deviation of the suggested solution method is less than 0.05% with respect to optimum. © 2010 Elsevier Ltd. All rights reserved.This work has been partially supported by the Ministerio de Educacion y Ciencia of Spain (project TIN2008-06441-C02-01).Braysy, O.; Martínez Molada, E.; Nagata, Y.; Soler Fernández, D. (2011). The mixed capacitated general routing problem with turn penalties. Expert Systems with Applications. 38(10):12954-12966. https://doi.org/10.1016/j.eswa.2011.04.092S1295412966381

The stacker crane problem and the directed general routing problem

This is the peer reviewed version of the following article: Ávila, Thais , Corberán, Angel, Plana, Isaac, Sanchís Llopis, José María. (2015). The stacker crane problem and the directed general routing problem.Networks, 65, 1, 43-55. DOI: 10.1002/net.21591 , which has been published in final form at http://doi.org/10.1002/net.21591. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving[EN] This article deals with the polyhedral description and the resolution of the directed general routing problem (DGRP) and the stacker crane problem (SCP). The DGRP contains a large number of important arc and node routing problems as special cases, including the SCP. Large families of facet-defining inequalities for the DGRP are described and a branch-and-cut algorithm for these problems is presented. Extensive computational experiments over different sets of DGRP and SCP instances are included.Contract grant sponsor: Ministerio de Economía y Competitividad (project MTM2012-36163-C06-02) of Spain Contract grant sponsor: Generalitat Valenciana (project GVPROMETEO2013-049)Ávila, T.; Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2015). The stacker crane problem and the directed general routing problem. Networks. 65(1):43-55. https://doi.org/10.1002/net.21591S435565

An Adaptive Iterated Local Search for the Mixed Capacitated General Routing Problem

We study the mixed capacitated general routing problem (MCGRP) in which a fleet of capacitated vehicles has to serve a set of requests by traversing a mixed weighted graph. The requests may be located on nodes, edges, and arcs. The problem has theoretical interest because it is a generalization of the capacitated vehicle routing problem (CVRP), the capacitated arc routing problem (CARP), and the general routing problem. It is also of great practical interest since it is often a more accurate model for real-world cases than its widely studied specializations, particularly for so-called street routing applications. Examples are urban waste collection, snow removal, and newspaper delivery. We propose a new iterated local search metaheuristic for the problem that also includes vital mechanisms from adaptive large neighborhood search combined with further intensification through local search. The method utilizes selected, tailored, and novel local search and large neighborhood search operators, as well as a new local search strategy. Computational experiments show that the proposed metaheuristic is highly effective on five published benchmarks for the MCGRP. The metaheuristic yields excellent results also on seven standard CARP data sets, and good results on four well-known CVRP benchmarks, including improvement of the best known upper bound for one instance

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

Experiments on the Node, Edge, and Arc Routing Problem

The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004 along with the first benchmark called CBMix. The NEARP generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It is also denoted the Mixed Capacitated General Routing Problem (MCGRP). The NEARP removes the strict and unwarranted dichotomy that previously existed in the literature between arc routing and node routing. In real applications, there are many cases where the pure node or arc routing models are not adequate. In fundamentally node-based routing applications such as newspaper delivery and communal waste management that have typically been modeled as arc routing problems in the literature, the number of points is often so large that demand aggregation is necessary. Aggregation heuristics will normally give a NEARP instance, possibly with side constraints. Hence, the NEARP is a scientifically challenging problem with high industrial relevance. In this report we present experiments with Spider, SINTEF’s industrial VRP solver, on the three NEARP benchmarks that have been published so far: CBMix, BHW, and DI-NEARP. Bach, Hasle, and Wøhlk have developed a combinatorial lower bound for the NEARP and defined the two latter benchmarks. Here, we present an experimental study with Spider on the three existing NEARP benchmarks. Upper and lower bounds are given for all instances. Three of the BHW instances have been solved to optimality. SINTEF has developed a web page for NEARP results on http://www.sintef.no/NEARP

Experiments on the Node, Edge, and Arc Routing Problem

-The Node, Edge, and Arc Routing Problem (NEARP) was defined by Prins and Bouchenoua in 2004 along with the first benchmark called CBMix. The NEARP generalizes the classical Capacitated Vehicle Routing Problem (CVRP), the Capacitated Arc Routing Problem (CARP), and the General Routing Problem. It is also denoted the Mixed Capacitated General Routing Problem (MCGRP). The NEARP removes the strict and unwarranted dichotomy that previously existed in the literature between arc routing and node routing. In real applications, there are many cases where the pure node or arc routing models are not adequate. In fundamentally node-based routing applications such as newspaper delivery and communal waste management that have typically been modeled as arc routing problems in the literature, the number of points is often so large that demand aggregation is necessary. Aggregation heuristics will normally give a NEARP instance, possibly with side constraints. Hence, the NEARP is a scientifically challenging problem with high industrial relevance. In this report we present experiments with Spider, SINTEF’s industrial VRP solver, on the three NEARP benchmarks that have been published so far: CBMix, BHW, and DI-NEARP. Bach, Hasle, and Wøhlk have developed a combinatorial lower bound for the NEARP and defined the two latter benchmarks. Here, we present an experimental study with Spider on the three existing NEARP benchmarks. Upper and lower bounds are given for all instances. Three of the BHW instances have been solved to optimality. SINTEF has developed a web page for NEARP results on http://www.sintef.no/NEARP

New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem

[EN] The min-max windy rural postman problem is a multiple vehicle version of the windy rural postman problem, WRPP, which consists of minimizing the length of the longest route to find a set of balanced routes for the vehicles. In a previous paper, an ILP formulation and a partial polyhedral study were presented, and a preliminary branch-and-cut algorithm that produced some promising computational results was implemented. In this article, we present further results for this problem. We describe several new facet-inducing inequalities obtained from the WRPP, as well as some inequalities that have to be satisfied by any optimal solution. We present an enhanced branch-and-cut algorithm that takes advantage of both these new inequalities and high quality min-max K-WRPP feasible solutions obtained by a metaheuristic. Computational results on a large set of instances are also reported. © 2011 Wiley Periodicals, Inc.Contract grant sponsor: Ministerio de Ciencia e Innovacion of Spain; Contract grant numbers: MTM2006-14961-C05-02, MTM2009-14039-C06-02Benavent López, E.; Corberán, A.; Plana, I.; Sanchís Llopis, JM. (2011). New Facets and an Enhanced Branch-and-Cut for the Min-Max K-Windy Rural Postman Problem. Networks. 58(4):255-272. https://doi.org/10.1002/net.20469S255272584D. Ahr Contributions to multiple postmen problems 2004Ahr, D., & Reinelt, G. (2002). New Heuristics and Lower Bounds for the Min-Max k-Chinese Postman Problem. Lecture Notes in Computer Science, 64-74. doi:10.1007/3-540-45749-6_10Ahr, 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 1995Benavent, 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.2005.09.021Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2009). Min-Max K -vehicles windy rural postman problem. Networks, 54(4), 216-226. doi:10.1002/net.20334Benavent, E., Corberán, Á., & Sanchis, J. M. (2009). A metaheuristic for the min–max windy rural postman problem with K vehicles. Computational Management Science, 7(3), 269-287. doi:10.1007/s10287-009-0119-2Corberáan, A., Letchford, A. N., & Sanchis, J. M. (2001). A cutting plane algorithm for the General Routing Problem. Mathematical Programming, 90(2), 291-316. doi:10.1007/pl00011426Corberán, 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.20176Corberán, 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:10.1137/050640886Frederickson, G. N., Hecht, M. S., & Kim, C. E. (1978). Approximation Algorithms for Some Routing Problems. SIAM Journal on Computing, 7(2), 178-193. doi:10.1137/0207017Pearn, 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)90073-6I. Plana The windy general routing problem 200

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