Abstract
Accurate Vessel Capacity Models (VCMs) expressing the trade-off between different container types that can be stowed on container vessels are required in core liner shipping functions such as uptake-, capacity-, and network management. Today, simple models based on volume, weight, and refrigerated container capacity are used for these tasks, which causes overestimations that hamper decision making. Though previous work on stowage planning optimization in principle provide fine-grained linear Vessel Stowage Models (VSMs), these are too complex to be used in the mentioned functions. As an alternative, this paper contributes a novel framework based on Fourier-Motzkin Elimination that automatically derives VCMs from VSMs by projecting unneeded variables. Our results show that the projected VCMs are reduced by an order of magnitude and can be solved 20–34 times faster than their corresponding VSMs with only a negligible loss in accuracy. Our framework is applicable to LP models in general, but are particularly effective on block-angular structured problems such as VSMs. We show similar results for a multi-commodity flow problem.
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References
Ajspur, M.L., Jensen, R.M.: Using Fourier-Motzkin-elimination to derive capacity models of container vessels. Technical Report TR-2017-197, IT University of Copenhagen (2017)
Ambrosino, D., Sciomachen, A., Tanfani, E.: Stowing a containership: the master bay plan problem. Transp. Res. Part A: Policy Pract. 38(2), 81–99 (2004)
Andersen, E.D., Andersen, K.D.: Presolving in linear programming. Math. Program. 71(2), 221–245 (1995)
Benoy, F., King, A., Mesnard, F.: Computing convex hulls with a linear solver. Theory Pract. Logic Program. 5(1–2), 259–271 (2005)
Delgado, A.: Models and Algorithms for Container Vessel Stowage Optimization. Ph.D. thesis, IT University of Copenhagen (2013)
Duffin, R.J.: On Fourier’s analysis of linear inequality systems, pp. 71–95. Springer, Heidelberg (1974). https://doi.org/10.1007/BFb0121242
Fordan, A., Yap, R.H.C.: Early projection in CLP(R). In: Maher, M., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 177–191. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-49481-2_14
Huynh, T., Lassez, C., Lassez, J.L.: Practical issues on the projection of polyhedral sets. Ann. Math. Artif. Intell. 6(4), 295–315 (1992)
Jensen, R.M., Ajspur, M.L.: The standard capacity model: towards a polyhedron representation of container vessel capacity. In: Cerulli, R., Raiconi, A., Voß, S. (eds.) ICCL 2018. LNCS, vol. 11184, pp. 175–190. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00898-7_11
Jensen, R.M., Pacino, D., Ajspur, M.L., Vesterdal, C.: Container Vessel Stowage Planning. Weilbach (2018)
Jones, C., Kerrigan, E.C., Maciejowski, J.: Equality set projection: a new algorithm for the projection of polytopes in halfspace representation. Technical report, Cambridge University Engineering Dept (2004)
Jones, K., Lustig, I., Farwolden, J., Powell, W.: Multicommodity network flows: the impact of formulation on decomposition. Math. Program. 62, 95–117 (1993)
Kang, J.G., Kim, Y.D.: Stowage planning in maritime container transportation. J. Oper. Res. Soc. 53(4), 415–426 (2002)
Lassez, J.L.: Querying constraints. In: Proceedings of the Ninth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (PODS), pp. 288–298. ACM (1990)
Lukatskii, A.M., Shapot, D.V.: A constructive algorithm for folding large-scale systems of linear inequalities. Comput. Math. Math. Phys. 48(7), 1100–1112 (2008)
Pacino, D., Delgado, A., Jensen, R.M., Bebbington, T.: Fast generation of near-optimal plans for eco-efficient stowage of large container vessels. In: Böse, J.W., Hu, H., Jahn, C., Shi, X., Stahlbock, R., Voß, S. (eds.) ICCL 2011. LNCS, vol. 6971, pp. 286–301. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-24264-9_22
Shapot, D.V., Lukatskii, A.M.: Solution building for arbitrary system of linear inequalities in an explicit form. Am. J. Comput. Math. 2(01), 1 (2012)
Simon, A., King, A.: Exploiting sparsity in polyhedral analysis. In: Hankin, C., Siveroni, I. (eds.) SAS 2005. LNCS, vol. 3672, pp. 336–351. Springer, Heidelberg (2005). https://doi.org/10.1007/11547662_23
Williams, H.P.: Model Building in Mathematical Programming. Wiley, London (2007)
Wilson, I.D., Roach, P.A.: Container stowage planning: a methodology for generating computerised solutions. J. Oper. Res. Soc. 51(11), 1248–1255 (2000)
Ziegler, G.M.: Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152. Springer, New York (1995). https://doi.org/10.1007/978-1-4613-8431-1
Acknowledgements
We would like to thank Stefan Røpke, Thomas Stidsen, and David Pisinger for discussions on applications of the FME framework beyond container vessel capacity models. This research is supported by the Danish Maritime Fund, Grant No. 2016-064.
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Ajspur, M.L., Jensen, R.M., Andersen, K.H. (2019). A Decomposed Fourier-Motzkin Elimination Framework to Derive Vessel Capacity Models. In: Paternina-Arboleda, C., Voß, S. (eds) Computational Logistics. ICCL 2019. Lecture Notes in Computer Science(), vol 11756. Springer, Cham. https://doi.org/10.1007/978-3-030-31140-7_6
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