[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. Management Science, 39(6), 657-682. doi:10.1287/mnsc.39.6.657Jepsen, M., Petersen, B., Spoorendonk, S., & Pisinger, D. (2008). Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows. Operations Research, 56(2), 497-511. doi:10.1287/opre.1070.0449Lübbecke, M. E., & Desrosiers, J. (2005). Selected Topics in Column Generation. Operations Research, 53(6), 1007-1023. doi:10.1287/opre.1050.0234Padberg, 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)90073-6Righini, G., & Salani, M. (2006). Symmetry helps: Bounded bi-directional dynamic programming for the elementary shortest path problem with resource constraints. Discrete Optimization, 3(3), 255-273. doi:10.1016/j.disopt.2006.05.007Righini, G., & Salani, M. (2008). 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
