Multi-depot rural postman problems
Tóm tắt
This 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.
Tài liệu tham khảo
Ahr D (2004) Contributions to multiple postmen problems. Technical report, PhD dissertation, Departament of Computer Science, Heidelberg University, Heidelberg, Germany
Amberg A, Domschke W, Voß S (2000) Multiple center capacitated arc routing problems: a tabu search algorithm using capacitated trees. Eur J Oper Res 124(2):360–376
Aráoz J, Fernández E, Franquesa C (2009) The clustered prize-collecting arc routing problem. Transp Sci 43(3):287–300
Aráoz J, Fernández E, Meza O (2009) Solving the prize-collecting rural postman problem. Eur J Oper Res 196(3):886–896
Aráoz J, Fernández E, Zoltan C (2006) Privatized rural postman problems. Comput Oper Res 33(12):3432–3449
Barahona F, Grötschel M (1986) On the cycle polytope of a binary matroid. J Comb Theory Ser B 40(1):40–62
Belenguer J-M, Benavent E (1998) The capacitated arc routing problem: valid inequalities and facets. Comput Optim Appl 10(2):165–187
Benavent E, Corberán A, Desaulniers G, Lessard F, Plana I, Sanchis J (2013) A branch-and-cut for the maximum benefit chinese postman problem. Networks 63:34–45
Benavent E, Corberán A, Plana I, Sanchis J (2009) Min–max \(k\)-vehicles windy rural postman problem. Networks 54:216–226
Benavent E, Corberán A, Plana I, Sanchis J (2011) New facets and enhanced branch-and-cut for the min–max \(k\)-vehicles windy rural postman problem. Networks 58:255–272
Benavent E, Corberán A, Plana I, Sanchis J (2014) Arc routing problems with min-max objectives. In: Corberán A, Laporte G (eds) Arc routing: problems., methods and applications MOS-SIAM series on optimization, Philadelphia, pp 255–280
Benavent E, Corberán A, Sanchis J (2010) A metaheuristic for the min–max windy rural postman problem with \(k\)-vehicles. Comput Manag Sci 7:269–287
Butsch A, Kalcsics J, Laporte G (2014) Districting for arc routing. INFORMS J Comput 26(4):809–824
Christofides N, Campos V, Corberán A, Mota E (1981) An algorithm for the rural postman problem. Technical report, Imperial College Report IC.O.R.81.5,
Corberán A, Fernández E, Franquesa C, Sanchis J (2011) The windy clustered prize-collecting arc routing problem. Transp Sci 45(3):317–344
Corberán A, Plana I, Rodríguez-Chía A, Sanchis J (2013) A branch-and-cut for the maximum benefit chinese postman problem. Math Program 141(1):21–48
Corberán A, Sanchis J (1994) A polyhedral approach to the rural postman problem. Eur J Oper Res 79(1):95–114
Corberán A, Sanchis J (1998) The general routing problem polyhedron: facets from the rpp and gtsp polyhedra. Eur J Oper Res 108(3):538–550
Fernández E, Fontana D, Speranza G (2016) On the collaboration uncapacitated arc routing problem. Comput Oper Res 67:120–131
Fernández E, Meza O, Garfinkel R, Ortega M (2003) On the undirected rural postman problem: tight bounds based on a new formulation. Oper Res 51(2):281–291
García Ayala G, González-Velarde J, Ríos-Mercado R, Fernández. E (2015) A novel model for arc territory design: promoting eulerian districts. Int Trans Oper Res 23:433–458
Ghiani G, Laporte G (2000) A branch-and-cut algorithm for the undirected rural postman problem. Math Program 87(3):467–481
Golden BL, Wong RT (1981) Capacitated arc routing problems. Networks 11(3):305–315
Gusfield D (1993) Very simple methods for all pairs network flow analysis. SIAM J Appl Math 19(1):143–555
Hertz A, Laporte G, Nanchen P (1999) Improvement procedure for the undirected rural postman problem. INFORMS J Comput 11:53–62
Hu H, Liu T, Zhao N, Zhou Y, Min D (2013) A hybrid genetic algorithm with perturbation for the multi-depot capacitated arc routing problem. J Appl Sci 13(16):32–39
Kansou A, Yassine A (2009) A two ant colony approaches for the multi-depot capacitated arc routing problem. In: IEEE International Conference on Computers and Industrial Engineering, 2009. CIE 2009, pp 1040–1045
Krushinsky D, van Woensel T (2015) An approach to the asymmetric multi-depot capacitated arc routing problem. Eur J Oper Res 244(1):100–109
Letchford AN, Salazar-González J-J (2015) Stronger multi-commodity flow formulations of the capacitated vehicle routing problem. Eur J Oper Res 244(3):730–738
Muyldermans L (2003) Routing, districting and location for arc traversal problems. Technical report, PhD dissertation, Catholic University of Leuven, Leuven, Belgium
Muyldermans L, Cattrysse D, Oudheusden DV, Lotan T (2002) Districting for salt spreading operations. Eur J Oper Res 139(3):521–532
Muyldermans L, Cattrysse D, Van Oudheusden D (2003) Districting for arc-routing applications. J Oper Res Soc 54:1209–1221
Muyldermans L, Pang G (2014) Variants of the capacitated arc routing problem. In: Corberán A, Laporte G (eds) Arc routing: problems, methods and applications. MOS-SIAM Series on Optimization, Philadelphia, pp 223–254
Orloff CS (1974) A fundamental problem in vehicle routing. Networks 4(1):35–64
Orloff CS (1976) On general routing problems: comments. Networks 6(3):281–284
Willemse EJ, Joubert J (2012) Applying min–max \(k\) postmen problems to the routing of security guards. J Oper Res Soc 63:245–260
Wøhlk S (2004) Contributions to arc routing. Technical report, PhD dissertation, University of Southern Denmark, Denmark
Xing L, Rohlfshagen P, Chen Y, Yao X (2010) An evolutionary approach to the multidepot capacitated arc routing problem. IEEE Trans Evol Comput 14(3):356–374