# Solving large-scale time capacitated arc routing problems: from real-time heuristics to metaheuristics

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## Abstract

This paper discusses the Time Capacitated Arc Routing Problem (TCARP) and introduces a heuristic and a metaheuristic algorithm for solving large-size instances of it. The TCARP is a realistic extension of the Capacitated Arc Routing Problem in which edge-servicing and edge-traversing costs, as well as vehicle capacities, are all time-based—*i.e.*, given in time units. Accordingly, the TCARP goal is to minimise the total time employed in servicing the required edges, for which other edges might need to be traversed too. According to the numerical experiments carried out, the proposed heuristic is able to provide real-time results of high quality even for the largest instances considered. Likewise, the proposed metaheuristic outperforms other existing approaches, both in quality as well as in computing times.

## Keywords

Capacitated arc routing problem Time-based capacities Biased randomisation Simulated annealing Metaheuristics## Notes

### Acknowledgements

This work has been partially supported by the Spanish Ministry of Economy and Competitiveness and FEDER (TRA2013-48180-C3-P, TRA2015-71883-REDT). The authors are also grateful to the College of Business, University College Dublin for supporting a research stay that contributed to the development of this work.

## References

- Aarts, E., Korst, J., & Michiels, W. (2005). Simulated annealing. In
*Search methodologies*(pp. 187–210).Google Scholar - Ahr, D., & Reinelt, G. (2014). The capacitated arc routing problem: Combinatorial lower bounds. In Á. Corberán & G. Laporte (Eds.),
*Arc routing: Problems, methods, and applications*(pp. 159–181). Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - Amberg, A., Domschke, W., & Voß, S. (2000). Multiple center capacitated arc routing problems: A tabu search algorithm using capacitated trees.
*European Journal of Operational Research*,*124*, 360–376.CrossRefGoogle Scholar - Aminua, U., & Eglese, R. (2006). A constraint programming approach to the chinese postman problem with time windows.
*Computers & Operations Research*,*33*, 3423–3431.CrossRefGoogle Scholar - Baldacci, R., & Maniezzo, V. (2006). Exact methods based on node-routing formulations for undirected arc-routing problems.
*Networks*,*47*, 52–60.CrossRefGoogle Scholar - Bartolini, E., Cordeau, J. F., & Laporte, G. (2013). An exact algorithm for the capacitated arc routing problem with deadheading demand.
*Operations Research*,*61*, 315–327.CrossRefGoogle Scholar - Belenguer, J. M., & Benavent, E. (2003). A cutting plane algorithm for the capacitated arc routing problem.
*Computers & Operations Research*,*30*, 705–728.CrossRefGoogle Scholar - Belenguer, J. M., Benavent, E., & Irnich, S. (2014). The capacitated arc routing problem: Exact algorithms. In Ángel Corberán & G. Laporte (Eds.),
*Arc routing: Problems, methods, and applications*(pp. 183–221). Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - Beltrami, E. J., & Bodin, L. D. (1974). Networks and vehicle routing for municipal waste collection.
*Networks*,*4*, 65–94.CrossRefGoogle Scholar - Benavent, E., Campos, V., Corberán, A., & Mota, E. (1992). The capacitated chinese postman problem: Lower bounds.
*Networks*,*22*, 669–690.CrossRefGoogle Scholar - Beullens, P., Muyldermans, L., Cattrysse, D., & Oudheusden, D. V. (2003). A guided local search heuristic for the capacitated arc routing problem.
*European Journal of Operational Research*,*147*, 629–643.CrossRefGoogle Scholar - Bode, C., & Irnich, S. (2015). In-depth analysis of pricing problem relaxations for the capacitated arc-routing problem.
*Transportation Science*,*49*, 369–383.CrossRefGoogle Scholar - Brandão, J., & Eglese, R. (2008). A deterministic tabu search algorithm for the capacitated arc routing problem.
*Computers & Operations Research*,*35*, 1112–1126.CrossRefGoogle Scholar - Caceres-Cruz, J., Arias, P., Guimarans, D., Riera, D., & Juan, A. A. (2014). Rich vehicle routing problem: Survey.
*ACM Computing Surveys*,*47*(32), 1–28.CrossRefGoogle Scholar - Chen, Y., Hao, J. K., & Glover, F. (2016). A hybrid metaheuristic approach for the capacitated arc routing problem.
*European Journal of Operational Research*,*253*, 25–39.CrossRefGoogle Scholar - Clarke, G., & Wright, J. (1964). Scheduling of vehicles from a central depot to a number of delivery points.
*Operations Research*,*12*, 568–581.CrossRefGoogle Scholar - Corberán, Á., & Laporte, G. (2014).
*Arc routing: Problems, methods, and applications*. Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - Corberán, A., & Prins, C. (2010). Recent results on arc routing problems: An annotated bibliography.
*Networks*,*56*, 50–69.Google Scholar - Doerner, K., Hartl, R., Maniezzo, V., & Reimann, M. (2003). An ant system metaheuristic for the capacitated arc routing problem. In
*Preprints of 5th meta-heuristics international conference*, Kyoto.Google Scholar - Dror, M. (Ed.). (2000).
*Arc routing: Theory, solutions, and applications*. Boston: Kluwer Academic.Google Scholar - Eglese, R. (1994). Routeing winter gritting vehicles.
*Discrete Applied Mathematics*,*48*, 231–244.CrossRefGoogle Scholar - Eglese, R. W., & Letchford, A. N. (2000).
*Polyhedral theory for arc routing problems*(pp. 199–230). Boston: Springer.CrossRefGoogle Scholar - Eiselt, H., Gendreau, M., & Laporte, G. (1995a). Arc routing problems, part II: The rural postman problem.
*Operations Research*,*43*, 399–414.CrossRefGoogle Scholar - Eiselt, H., Gendreau, M., & Laporte, G. (1995b). Arc routing problems, part I: The Chinese postman problem.
*Operations Research*,*43*, 231–242.CrossRefGoogle Scholar - Faulin, J., Gilibert, M., Juan, A. A., Vilajosana, X., & Ruiz, R. (2008). Sr-1: A simulation-based algorithm for the capacitated vehicle routing problem. In
*Proceedings of the 40th conference on winter simulation, winter simulation conference*(pp. 2708–2716).Google Scholar - Floyd, R. W. (1962). Algorithm 97: Shortest path.
*Communications of the ACM*,*5*, 345.CrossRefGoogle Scholar - Golden, B., Dearmon, J., & Baker, E. (1983). Computational experiments with algorithms for a class of routing problems.
*Computers & Operations Research*,*10*, 47–59.CrossRefGoogle Scholar - Golden, B. L., Raghavan, S., & Wasil, E. A. (2008).
*The vehicle routing problem: Latest advances and new challenges*(Vol. 43). Berlin: Springer.CrossRefGoogle Scholar - González-Martín, S., Juan, A. A., Riera, D., Castellà, Q., Muñoz, R., & Pérez, A. (2012). Development and assessment of the sharp and randsharp algorithms for the arc routing problem.
*Artificial Intelligence Communications*,*25*, 173–189.Google Scholar - Grasas, A., Juan, A. A., Faulin, J., de Armas, J., & Ramalhinho, H. (2017). Biased randomization of heuristics using skewed probability distributions: A survey and some applications.
*Computers & Industrial Engineering*,*110*, 216–228.CrossRefGoogle Scholar - Greistorfer, P. (2003). A tabu scatter search metaheuristic for the arc routing problem.
*Computers & Industrial Engineering*,*44*, 249–266.CrossRefGoogle Scholar - Groves, G., Le Roux, J., & Van Vuuren, J. H. (2004). Network service scheduling and routing.
*International Transactions in Operational Research*,*11*, 613–643.CrossRefGoogle Scholar - Hasle, G. (2014). Arc routing applications in newspaper delivery. In Ángel Corberán & G. Laporte (Eds.),
*Arc routing: Problems, methods, and applications*(pp. 371–395). Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - Hertz, A. (2005).
*Recent trends in arc routing*(pp. 215–236). Boston: Springer.Google Scholar - Hertz, A., Laporte, G., & Mittaz, M. (2000). A tabu search heuristic for the capacitated arc routing problem.
*Operations Research*,*48*, 129–135.CrossRefGoogle Scholar - Hertz, A., & Mittaz, M. (2001). A variable neighborhood descent algorithm for the undirected capacitated arc routing problem.
*Transportation Science*,*35*, 425–434.CrossRefGoogle Scholar - Hirabayashi, R., Nishida, N., & Saruwatari, Y. (1992a). Node duplication lower bounds for the capacitated arc routing problems.
*Journal of the Operations Research Society of Japan*,*35*, 119–133.CrossRefGoogle Scholar - Hirabayashi, R., Nishida, N., & Saruwatari, Y. (1992b). Tour construction algorithm for the capacitated arc routing problem.
*Asia-Pacific Journal of Operational Research*,*9*, 155–175.Google Scholar - Irnich, S. (2008). Solution of real-world postman problems.
*European Journal of Operational Research*,*190*, 52–67.CrossRefGoogle Scholar - Juan, A., Faulin, J., Jorba, J., Caceres, J., & Marquès, J. (2013). Using parallel & distributed computing for real-time solving of vehicle routing problems with stochastic demands.
*Annals of Operations Research*,*207*, 43–65.CrossRefGoogle Scholar - Juan, A. A., Faulin, J., Ferrer, A., Lourenço, H. R., & Barrios, B. (2011). Mirha: Multi-start biased randomization of heuristics with adaptive local search for solving non-smooth routing problems.
*Top*,*21*, 109–132.CrossRefGoogle Scholar - Juan, A. A., Lourenço, H. R., Mateo, M., Luo, R., & Castella, Q. (2014). Using iterated local search for solving the flow-shop problem: Parallelization, parametrization, and randomization issues.
*International Transactions in Operational Research*,*21*, 103–126.CrossRefGoogle Scholar - Keenan, P., & Naughton, M. (1996). Arc routing for rural irish networks. In Doležal, J., & Fidler, J. (Eds.)
*System modelling and optimization: Proceedings of the seventeenth IFIP TC7 conference on system modelling and optimization, 1995*(pp. 599–606). Boston: Springer.Google Scholar - Keenan, P. B. (2001). Spatial decision support systems for large arc routing problems. Ph.D. Thesis. Faculty of Commerce, University College Dublin, Dublin, Ireland.Google Scholar
- Keenan, P. B. (2005). Lower bounds for the time capacitated arc routing problem. Technical Report. UCD Business School, University College Dublin. http://mis.ucd.ie/Members/pkeenan/Working.
- Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing.
*Science*,*220*, 671–680.CrossRefGoogle Scholar - Kirlik, G., & Sipahioglu, A. (2012). Capacitated arc routing problem with deadheading demands.
*Computers & Operations Research*,*39*, 2380–2394.CrossRefGoogle Scholar - Kuan, M. K. (1962). Graphic programming using odd or even points.
*Chinese Mathematics*,*1*, 273–276.Google Scholar - Lacomme, P., Prins, C., Ramdane-Cherif, W. (2001). Competitive genetic algorithms for the capacitated arc routing problem and its extensions. In
*Proceedings of the 4th European conference on genetic programming*(pp. 473–483).Google Scholar - Lacomme, P., Prins, C., & Ramdane-Cherif, W. (2004). Competitive memetic algorithms for arc routing problems.
*Annals of Operations Research*,*131*, 159–185.CrossRefGoogle Scholar - Letchford, A. N., & Oukil, A. (2009). Exploiting sparsity in pricing routines for the capacitated arc routing problem.
*Computers & Operations Research*,*36*, 2320–2327.CrossRefGoogle Scholar - Li, L. (1992). Vehicle routeing for winter gritting. Ph.D. Thesis. Department of Management Science, Lancaster University.Google Scholar
- Li, L. Y. O., & Eglese, R. W. (1996). An interactive algorithm for vehicle routeing for winter—gritting.
*The Journal of the Operational Research Society*,*47*, 217–228.CrossRefGoogle Scholar - Lin, Y., & Zhao, Y. (1988). A new algorithm for the directed chinese postman problem.
*Computers & Operations Research*,*15*, 577–584.CrossRefGoogle Scholar - Liu, T., Jiang, Z., & Geng, N. (2013). A memetic algorithm with iterated local search for the capacitated arc routing problem.
*International Journal of Production Research*,*51*, 3075–3084.CrossRefGoogle Scholar - Longo, H., de Aragão, M. P., & Uchoa, E. (2006). Solving capacitated arc routing problems using a transformation to the CVRP.
*Computers & Operations Research*,*33*, 1823–1837.CrossRefGoogle Scholar - Marzolf, F., Trépanier, M., & Langevin, A. (2006). Road network monitoring: Algorithms and a case study.
*Computers & Operations Research*,*33*, 3494–3507. (Part special issue: Recent algorithmic advances for arc routing problems).CrossRefGoogle Scholar - Mourão, M., & Pinto, L. S. (2017). An updated annotated bibliography on arc routing problems.
*Networks*,*70*(3), 144–194.CrossRefGoogle Scholar - Nouraniy, Y., & Andresen, B. (1998). A comparison of simulated annealing cooling strategies.
*Journal of Physics A: Mathematical and General*,*31*, 8373–8385.CrossRefGoogle Scholar - Pearn, W. L. (1988). New lower bounds for the capacitated arc routing problem.
*Networks*,*18*, 181–191.CrossRefGoogle Scholar - Pearn, W. L. (1989). Approximate solutions for the capacitated arc routing problem.
*Computers & Operations Research*,*16*, 589–600.CrossRefGoogle Scholar - Pearn, W. L. (1991). Augment-insert algorithms for the capacitated arc routing problem.
*Computers & Operations Research*,*18*, 189–198.CrossRefGoogle Scholar - Pearn, W. L., Assad, A., & Golden, B. L. (1987). Transforming arc routing into node routing problems.
*Computers & Operations Research*,*14*, 285–288.CrossRefGoogle Scholar - Prins, C. (2014). The capacitated arc routing problem: Heuristics. In Ángel Corberán & G. Laporte (Eds.),
*Arc routing: Problems, methods, and applications*(pp. 131–157). Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar - Shaw, P. (1997).
*A new local search algorithm providing high quality solutions to vehicle routing problems*. Glasgow: APES Group, Department of Computer Science, University of Strathclyde.Google Scholar - Stern, H. I., & Dror, M. (1979). Routing electric meter readers.
*Computers & Operations Research*,*6*, 209–223.CrossRefGoogle Scholar - Tagmouti, M., Gendreau, M., & Potvin, J. Y. (2011). A dynamic capacitated arc routing problem with time-dependent service costs.
*Transportation Research Part C: Emerging Technologies*,*19*, 20–28.CrossRefGoogle Scholar - Usberti, F. L., França, P. M., & França, A. L. M. (2013). Grasp with evolutionary path-relinking for the capacitated arc routing problem.
*Computers & Operations Research*,*40*, 3206–3217.CrossRefGoogle Scholar - Vansteenwegen, P., Souffriau, W., & Sörensen, K. (2010). Solving the mobile mapping van problem: A hybrid metaheuristic for capacitated arc routing with soft time windows.
*Computers & Operations Research*,*37*, 1870–1876.CrossRefGoogle Scholar - Vidal, T. (2017). Node, edge, arc routing and turn penalties: Multiple problems–one neighborhood extension.
*Operations Research*,*65*, 992–1010.CrossRefGoogle Scholar - Welz, S. (1994). Optimal solutions for the capacitated arc routing problem using integer programming. Ph.D. Thesis. Department of QT and OM, University of Cincinnati.Google Scholar
- Willemse, E., & Joubert, J. (2012). Applying min–max k postmen problems to the routing of security guards.
*The Journal of the Operational Research Society*,*63*, 245–260.CrossRefGoogle Scholar - Win, Z. (1988). Contributions to routing problems. Ph.D. Thesis. Universität Augsburg, Germany.Google Scholar
- Wøhlk, S. (2005). Contributions to arc routing. Ph.D. Thesis. University Southern Denmark.Google Scholar
- Wøhlk, S. (2006). New lower bound for the capacitated arc routing problem.
*Computers & Operations Research*,*33*, 3458–3472.CrossRefGoogle Scholar - Wøhlk, S. (2008a). A decade of capacitated arc routing. In B. Golden, S. Raghavan, & E. Wasil (Eds.),
*The vehicle routing problem: Latest advances and new challenges*(pp. 29–48). Berlin: Springer.CrossRefGoogle Scholar - Wøhlk, S. (2008b). An approximation algorithm for the capacitated arc routing problem.
*Open Operational Research Journal*,*2*, 8–12.CrossRefGoogle Scholar