Abstract
Benders’ Decomposition (BD) is a prominent technique for tackling large mixed integer programming problems having a certain structure by iteratively solving a series of smaller master and subproblem instances. We apply a generalization of this technique called Logic-Based BD, which does not restrict the subproblems to have continuous variables only, to a bi-level vehicle routing problem originating in the timely distribution of printed newspapers to subscribers. When solving all master and subproblem instances exactly by CPLEX, it turns out that the scalability of the approach is quite limited. The situation can be dramatically improved when using a meaningful metaheuristic – in our case a variable neighborhood search – for approximately solving either only the subproblems or both, the master as well as the subproblem instances. More generally, it is shown that Logic-Based BD can be a highly promising framework also for hybrid metaheuristics.
This work is supported by the Austrian Ministry for Transport, Innovation and Technology, the Ferderal State of Salzburg, Austria, and the Austrian Science Fund (FWF) under grant P24660.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)
Geoffrion, A.M.: Generalized Benders decomposition. Journal of Optimization Theory and Applications 10(4), 237–260 (1972)
Hooker, J.N., Ottosson, G.: Logic-based benders decomposition. Mathematical Programming 96, 33–60 (2003)
Hooker, J.N.: Planning and scheduling by logic-based Benders decomposition. Operations Research 55(3), 588–602 (2007)
Poojari, C.A., Beasley, J.E.: Improving Benders decomposition using a genetic algorithm. European Journal of Operational Research 199(1), 89–97 (2009)
Lai, M.C., Sohn, H.S., Tseng, T.L., Chiang, C.: A hybrid algorithm for capacitated plant location problem. Expert Systems with Applications 37(12), 8599–8605 (2010)
Lai, M.C., Sohn, H.S.: Using a genetic algorithm to solve the Benders master problem for capacitated plant location. In: Gao, S. (ed.) Bio-Inspired Computational Algorithms and Their Applications, pp. 405–420. InTech (2012)
Lai, M.C., Sohn, H.S., Tseng, T.L., Bricker, D.L.: A hybrid Benders/genetic algorithm for vehicle routing and scheduling problem. International Journal of Industrial Engineering 19(1), 33–46 (2012)
Rei, W., Cordeau, J.F., Gendreau, M., Soriano, P.: Accelerating Benders decomposition by local branching. INFORMS Journal on Computing 21(2), 333–345 (2008)
Zakeri, G., Philpott, A.B., Ryan, D.M.: Inexact cuts in Benders decomposition. SIAM Journal on Optimization 10(3), 643–657 (1999)
Perboli, G., Tadei, R., Vigo, D.: The two-echelon capacitated vehicle routing problem: Models and math-based heuristics. Transportation Science 45(3), 364–380 (2011)
Hemmelmayr, V.C., Cordeau, J.F., Crainic, T.G.: An adaptive large neighborhood search heuristic for two-echelon vehicle routing problems arising in city logistics. Computers and Operations Research 39(12), 3215–3228 (2012)
Schwengerer, M., Pirkwieser, S., Raidl, G.R.: A variable neighborhood search approach for the two-echelon location-routing problem. In: Hao, J.-K., Middendorf, M. (eds.) EvoCOP 2012. LNCS, vol. 7245, pp. 13–24. Springer, Heidelberg (2012)
Crainic, T.G., Mancini, S., Perboli, G., Tadei, R.: Multi-start heuristics for the two-echelon vehicle routing problem. In: Hao, J.-K. (ed.) EvoCOP 2011. LNCS, vol. 6622, pp. 179–190. Springer, Heidelberg (2011)
Jacobsen, S.K., Madsen, O.B.G.: A comparative study of heuristics for a two-level routing-location problem. European Journal of Operational Research 5(6), 378–387 (1980)
Boccia, M., Crainic, T., Sforza, A., Sterle, C.: A metaheuristic for a two echelon location-routing problem. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 288–301. Springer, Heidelberg (2010)
Boccia, M., Crainic, T.G., Sforza, A., Sterle, C.: Location-routing models for designing a two-echelon freight distribution system. Technical Report CIRRELT-2011-06, University of Montreal (2011)
Fisher, M.L., Jaikumar, R.: A generalized assignment heuristic for vehicle routing. Networks 11(2), 109–124 (1981)
Boschetti, M., Maniezzo, V.: Benders decomposition, Lagrangian relaxation and metaheuristic design. Journal of Heuristics 15, 283–312 (2009)
Yaman, H.: Formulations and valid inequalities for the heterogeneous vehicle routing problem. Mathematical Programming 106(2), 365–390 (2006)
Pirkwieser, S., Raidl, G.R.: A variable neighborhood search for the periodic vehicle routing problem with time windows. In: Prodhon, C., et al. (eds.) Proceedings of the 9th EU/MEeting on Metaheuristics for Logistics and Vehicle Routing, Troyes, France (2008)
Clarke, G., Wright, J.W.: Scheduling of vehicles from a central depot to a number of delivery points. Operations Research 12(4), 568–581 (1964)
Mladenović, N., Hansen, P.: Variable neighborhood search. Computers and Operations Research 24(11), 1097–1100 (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Raidl, G.R., Baumhauer, T., Hu, B. (2014). Speeding Up Logic-Based Benders’ Decomposition by a Metaheuristic for a Bi-Level Capacitated Vehicle Routing Problem. In: Blesa, M.J., Blum, C., Voß, S. (eds) Hybrid Metaheuristics. HM 2014. Lecture Notes in Computer Science, vol 8457. Springer, Cham. https://doi.org/10.1007/978-3-319-07644-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-07644-7_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07643-0
Online ISBN: 978-3-319-07644-7
eBook Packages: Computer ScienceComputer Science (R0)