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The two-dimensional vector packing problem with piecewise linear cost function

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  • Hu, Qian
  • Lim, Andrew
  • Zhu, Wenbin

Abstract

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) is a practical problem faced by a manufacturer of children׳s apparel that ships products using courier service. The manufacturer must ship a number of items using standard-sized cartons, where the cost of a carton quoted by the courier is determined by a piecewise linear function of its weight. The cost function is not necessarily convex or concave. The objective is to pack all given items into a set of cartons such that the total delivery cost is minimized while observing both the weight limit and volume capacity constraints. This problem is commonly faced by many manufacturers that ship products using courier service. We formulate the problem as an integer programming model. Since the 2DVPP-PLC generalizes the classical bin packing problem, it is more complex and challenging. Solving it directly using CPLEX is successful only for small instances. We propose a simple heuristic that is extremely fast and produces high-quality solutions for instances of practical size. We develop an iterative local search algorithm to improve the solution quality further. We generate two categories of test data that can serve as benchmark for future research.

Suggested Citation

  • Hu, Qian & Lim, Andrew & Zhu, Wenbin, 2015. "The two-dimensional vector packing problem with piecewise linear cost function," Omega, Elsevier, vol. 50(C), pages 43-53.
    • Handle: RePEc:eee:jomega:v:50:y:2015:i:c:p:43-53
    DOI: 10.1016/j.omega.2014.07.004
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    Cited by:

    1. Hu, Qian & Zhu, Wenbin & Qin, Hu & Lim, Andrew, 2017. "A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function," European Journal of Operational Research, Elsevier, vol. 260(1), pages 70-80.
    2. Wang, Ting & Hu, Qian & Lim, Andrew, 2022. "An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs," European Journal of Operational Research, Elsevier, vol. 300(1), pages 20-34.
    3. Hu, Qian & Wei, Lijun & Lim, Andrew, 2018. "The two-dimensional vector packing problem with general costs," Omega, Elsevier, vol. 74(C), pages 59-69.
    4. Alonso, M.T. & Alvarez-Valdes, R. & Iori, M. & Parreño, F. & Tamarit, J.M., 2017. "Mathematical models for multicontainer loading problems," Omega, Elsevier, vol. 66(PA), pages 106-117.
    5. Wei, Lijun & Zhu, Wenbin & Lim, Andrew & Liu, Qiang & Chen, Xin, 2018. "An adaptive selection approach for the 2D rectangle packing area minimization problem," Omega, Elsevier, vol. 80(C), pages 22-30.
    6. Zhi-Hua Hu & Yingxue Zhao & Sha Tao & Zhao-Han Sheng, 2015. "Finished-vehicle transporter routing problem solved by loading pattern discovery," Annals of Operations Research, Springer, vol. 234(1), pages 37-56, November.
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