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

An updated annotated bibliography on arc routing problems

The number of arc routing publications has increased significantly in the last decade. Such an increase justifies a second annotated bibliography, a sequel to Corberán and Prins (Networks 56 (2010), 50–69), discussing arc routing studies from 2010 onwards. These studies are grouped into three main sections: single vehicle problems, multiple vehicle problems and applications. Each main section catalogs problems according to their specifics. Section 2 is therefore composed of four subsections, namely: the Chinese Postman Problem, the Rural Postman Problem, the General Routing Problem (GRP) and Arc Routing Problems (ARPs) with profits. Section 3, devoted to the multiple vehicle case, begins with three subsections on the Capacitated Arc Routing Problem (CARP) and then delves into several variants of multiple ARPs, ending with GRPs and problems with profits. Section 4 is devoted to applications, including distribution and collection routes, outdoor activities, post-disaster operations, road cleaning and marking. As new applications emerge and existing applications continue to be used and adapted, the future of arc routing research looks promising.info:eu-repo/semantics/publishedVersio

Multi-depot rural postman problems

The final publication is available at Springer via http://dx.doi.org/10.1007/s11750-016-0434-zThis paper studies multi-depot rural postman problems on an undirected graph. These problems extend the well-known undirected rural postman problem to the case where there are several depots instead of just one. Linear integer programming formulations that only use binary variables are proposed for the problem that minimizes the overall routing costs and for the model that minimizes the length of the longest route. An exact branch-and-cut algorithm is presented for each considered model, where violated constraints of both types are separated in polynomial time. Despite the difficulty of the problems, the numerical results from a series of computational experiments with various types of instances illustrate a quite good behavior of the algorithms. When the overall routing costs are minimized, over 43 % of the instances were optimally solved at the root node, and 95 % were solved at termination, most of them with a small additional computational effort. When the length of the longest route is minimized, over 25 % of the instances were optimally solved at the root node, and 99 % were solved at termination.Peer ReviewedPostprint (author's final draft

On the collaboration uncapacitated arc routing problem

This paper introduces a new arc routing problem for the optimization of a collaboration scheme among carriers. This yields to the study of a profitable uncapacitated arc routing problem with multiple depots, where carriers collaborate to improve the profit gained. In the first model the goal is the maximization of the total profit of the coalition of carriers, independently of the individual profit of each carrier. Then, a lower bound on the individual profit of each carrier is included. This lower bound may represent the profit of the carrier in the case no collaboration is implemented. The models are formulated as integer linear programs and solved through a branch-and-cut algorithm. Theoretical results, concerning the computational complexity, the impact of collaboration on profit and a game theoretical perspective, are provided. The models are tested on a set of 971 instances generated from 118 benchmark instances for the Privatized Rural Postman Problem, with up to 102 vertices. All the 971 instances are solved to optimality within few seconds.Peer ReviewedPostprint (author's final draft

Distribution with Quality of Service Considerations:The Capacitated Routing Problem with Profits and Service Level Requirements

Inspired by a problem arising in cash logistics, we propose the Capacitated Routing Problem with Profits and Service Level Requirements (CRPPSLR). The CRPPSLR extends the class of Routing Problems with Profits by considering customers requesting deliveries to their (possibly multiple) service points. Moreover, each customer imposes a service level requirement specifying a minimum-acceptable bound on the fraction of its service points being delivered. A customer-specific financial penalty is incurred by the logistics service provider when this requirement is not met. The CRPPSLR consists in finding vehicle routes maximizing the difference between the collected revenues and the incurred transportation and penalty costs in such a way that vehicle capacity and route duration constraints are met. A fleet of homogeneous vehicles is available for serving the customers. We design a branch-and-cut algorithm and evaluate the usefulness of valid inequalities that have been effectively used for the capacitated vehicle routing problem and, more recently, for other routing problems with profits. A real-life case study taken from the cash supply chain in the Netherlands highlights the relevance of the problem under consideration. Computational results illustrate the performance of the proposed solution approach under different input parameter settings for the synthetic instances. For instances of real-life problems, we distinguish between coin and banknote distribution, as vehicle capacities only matter when considering the former. Finally, we report on the effectiveness of the valid inequalities in closing the optimality gap at the root node for both the synthetic and the real-life instances and conclude with a sensitivity analysis on the most significant input parameters of our model

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

Efficient routing of snow removal vehicles

This research addresses the problem of finding a minimum cost set of routes for vehicles in a road network subject to some constraints. Extensions, such as multiple service requirements, and mixed networks have been considered. Variations of this problem exist in many practical applications such as snow removal, refuse collection, mail delivery, etc. An exact algorithm was developed using integer programming to solve small size problems. Since the problem is NP-hard, a heuristic algorithm needs to be developed. An algorithm was developed based on the Greedy Randomized Adaptive Search Procedure (GRASP) heuristic, in which each replication consists of applying a construction heuristic to find feasible and good quality solutions, followed by a local search heuristic. A simulated annealing heuristic was developed to improve the solutions obtained from the construction heuristic. The best overall solution was selected from the results of several replications. The heuristic was tested on four sets of problem instances (total of 115 instances) obtained from the literature. The simulated annealing heuristic was able to achieve average improvements of up to 26.36% over the construction results on these problem instances. The results obtained with the developed heuristic were compared to the results obtained with recent heuristics developed by other authors. The developed heuristic improved the best-known solution found by other authors on 18 of the 115 instances and matched the results on 89 of those instances. It worked specially better with larger problems. The average deviations to known lower bounds for all four datasets were found to range between 0.21 and 2.61%

Models and Algorihtm for the Optimization of Real-World Routing and Logistics Problems

Logistics involves planning, managing, and organizing the flows of goods from the point of origin to the point of destination in order to meet some requirements. Logistics and transportation aspects are very important and represent a relevant costs for producing and shipping companies, but also for public administration and private citizens. The optimization of resources and the improvement in the organization of operations is crucial for all branches of logistics, from the operation management to the transportation. As we will have the chance to see in this work, optimization techniques, models, and algorithms represent important methods to solve the always new and more complex problems arising in different segments of logistics. Many operation management and transportation problems are related to the optimization class of problems called Vehicle Routing Problems (VRPs). In this work, we consider several real-world deterministic and stochastic problems that are included in the wide class of the VRPs, and we solve them by means of exact and heuristic methods. We treat three classes of real-world routing and logistics problems. We deal with one of the most important tactical problems that arises in the managing of the bike sharing systems, that is the Bike sharing Rebalancing Problem (BRP). We propose models and algorithms for real-world earthwork optimization problems. We describe the 3DP process and we highlight several optimization issues in 3DP. Among those, we define the problem related to the tool path definition in the 3DP process, the 3D Routing Problem (3DRP), which is a generalization of the arc routing problem. We present an ILP model and several heuristic algorithms to solve the 3DRP

A Branch-Price-and-Cut Algorithm for the Min-Max k -Vehicle Windy Rural Postman Problem

[EN] The min-max k -vehicles windy rural postman problem consists of minimizing the maximal distance traveled by a vehicle to find a set of balanced routes that jointly service all the required edges in a windy graph. This is a very difficult problem, for which a branch-and-cut algorithm has already been proposed, providing good results when the number of vehicles is small. In this article, we present a branch-price-and-cut method capable of obtaining optimal solutions for this problem when the number of vehicles is larger for the same set of required edges. Extensive computational results on instances from the literature are presented.Contract grant sponsor: Ministerio de Education y Ciencia of Spain: Contract gram number: MTM2006-14961-C05-02 Canadian Natural Sciences and Engineering Research Council; Contract grant number: 157935-07Benavent Lopez, E.; Corberán, A.; Desaulniers, G.; Lessard, F.; Plana, I.; Sanchís Llopis, JM. (2014). A Branch-Price-and-Cut Algorithm for the Min-Max k -Vehicle Windy Rural Postman Problem. Networks. 63(1):34-45. https://doi.org/10.1002/net.21520S3445631Baldacci, R., Mingozzi, A., & Roberti, R. (2011). New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem. Operations Research, 59(5), 1269-1283. doi:10.1287/opre.1110.0975Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P., & Vance, P. H. (1998). Branch-and-Price: Column Generation for Solving Huge Integer Programs. Operations Research, 46(3), 316-329. doi:10.1287/opre.46.3.316Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2009). Min-MaxK-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/s10287-009-0119-2Benavent, E., Corberán, A., Plana, I., & Sanchis, J. M. (2011). New facets and an enhanced branch-and-cut for the min-max K-vehicles windy rural postman problem. Networks, 58(4), 255-272. doi:10.1002/net.20469Boland, N., Dethridge, J., & Dumitrescu, I. (2006). Accelerated label setting algorithms for the elementary resource constrained shortest path problem. Operations Research Letters, 34(1), 58-68. doi:10.1016/j.orl.2004.11.011Corberán, 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:10.1137/050640886Á. Corberán I. Plana J.M. Sanchis Arc routing problems: Data instances www.uv.es/corberan/instancias.htm 2007Dantzig, G. B., & Wolfe, P. (1960). Decomposition Principle for Linear Programs. Operations Research, 8(1), 101-111. doi:10.1287/opre.8.1.101Desaulniers, G., Desrosiers, J., & Spoorendonk, S. (2011). Cutting planes for branch-and-price algorithms. Networks, 58(4), 301-310. doi:10.1002/net.20471Desaulniers, G., Lessard, F., & Hadjar, A. (2008). Tabu Search, Partial Elementarity, and Generalizedk-Path Inequalities for the Vehicle Routing Problem with Time Windows. Transportation Science, 42(3), 387-404. doi:10.1287/trsc.1070.0223Dror, M. (1994). Note on the Complexity of the Shortest Path Models for Column Generation in VRPTW. Operations Research, 42(5), 977-978. doi:10.1287/opre.42.5.977Gilmore, P. C., & Gomory, R. E. (1961). A Linear Programming Approach to the Cutting-Stock Problem. Operations Research, 9(6), 849-859. doi:10.1287/opre.9.6.849Hadjar, A., Marcotte, O., & Soumis, F. (2006). A Branch-and-Cut Algorithm for the Multiple Depot Vehicle Scheduling Problem. Operations Research, 54(1), 130-149. doi:10.1287/opre.1050.0240Hoffman, K. L., & Padberg, M. (1993). Solving Airline Crew Scheduling Problems by Branch-and-Cut. 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New dynamic programming algorithms for the resource constrained elementary shortest path problem. Networks, 51(3), 155-170. doi:10.1002/net.20212Ropke, S., & Cordeau, J.-F. (2009). Branch and Cut and Price for the Pickup and Delivery Problem with Time Windows. Transportation Science, 43(3), 267-286. doi:10.1287/trsc.1090.027

Aesthetic considerations for the min-max K-Windy Rural Postman Problem

[EN] The aesthetic quality of routes is a feature of route planning that is of practical importance, but receives relatively little attention in the literature. Several practitioners have pointed out that the visual appeal of a proposed set of routes can have a strong influence on the willingness of a client to accept or reject a specific routing plan. While some work has analyzed algorithmic performance relative to traditional min-sum or min-max objective functions and aesthetic objective functions, we are not aware of any work that has considered a multi-objective approach. This work considers a multi-objective variant of the Min-Max K-Vehicles Windy Rural Postman Problem, discusses several formulations of the problem, and presents computational experiments with a heuristic algorithm. After exploring several formulations, we choose to study the problem with a bi-objective function that includes contributions from the route overlap index and average task distance aesthetic measures. The heuristic extends the cluster-first procedure presented in Lum et al. (Networks 69 (2017), 290-303) by incorporating the new objective function into the improvement phase and adding a perturbation routine. (c) 2017 Wiley Periodicals, Inc.Contract grant sponsor: Spanish Ministerio de Economia y Competitividad and Fondo Europeo de Desarrollo Regional (FEDER); Contract grant number: project MTM2015-68097-P (MINECO/FEDER). Contract grant sponsor: Generalitat Valenciana; Contract grant number: project GVPROMETEO2013-049Corberán, A.; Golden, B.; Lum, O.; Plana, I.; Sanchís Llopis, JM. (2017). Aesthetic considerations for the min-max K-Windy Rural Postman Problem. Networks. 70(3):216-232. https://doi.org/10.1002/net.21748S216232703Baños, R., Ortega, J., Gil, C., Márquez, A. L., & de Toro, F. (2013). A hybrid meta-heuristic for multi-objective vehicle routing problems with time windows. Computers & Industrial Engineering, 65(2), 286-296. doi:10.1016/j.cie.2013.01.007Attea, B. A., Khalil, E. A., & Cosar, A. (2014). Multi-objective evolutionary routing protocol for efficient coverage in mobile sensor networks. Soft Computing, 19(10), 2983-2995. doi:10.1007/s00500-014-1462-yBenavent, E., Corberán, Á., Desaulniers, G., Lessard, F., Plana, I., & Sanchis, J. M. (2013). A branch-price-and-cut algorithm for the min-maxk-vehicle windy rural postman problem. Networks, 63(1), 34-45. doi:10.1002/net.21520Benavent, E., Corberán, A., Piñana, E., Plana, I., & Sanchis, J. M. (2005). New heuristic algorithms for the windy rural postman problem. Computers & Operations Research, 32(12), 3111-3128. doi:10.1016/j.cor.2004.04.007Benavent, 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, A., Plana, I., & Sanchis, J. M. (2011). New facets and an enhanced branch-and-cut for the min-max K -vehicles windy rural postman problem. 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