This study proposes a mixed-integer nonconvex programming (MINP) model for the winner determination problem (WDP) considering two discount functions in a combinatorial auction to save shipper’s transportation cost. For the WDP, the shipper allows carriers to submit bids for a bundle of lanes. Then the winning carries are selected by solving the WDP. Specifically, this study considers the shipment distance-based and volume-based discounts for transportation cost, simultaneously. The state-of-the-art linearization technique is available to convert the MINP model into a mixed-integer linear program (MILP) to obtain a global optimum, but the solution time becomes inefficient when the problem size becomes large. To find efficient and effective linearization techniques for large-scale WDP, this study (1) proposes a novel WDP model with discount policies, (2) utilizes superior encoding formulation to avoid the unbalanced branch-and-bound trees in solving MILP, and (3) reduces big-M constraints to speed up the solving time. The proposed method leads to significant savings in computational efforts. Numerical experiments with real-world-sized truckload service procurement problems are solved by the proposed method and further confirmed the drastic reduction in computational time for solving the large-size WDP.

本研究提出了一种混合整数非凸规划 (MINP) 模型，用于在组合拍卖中考虑两个折扣函数的获胜者确定问题 (WDP)，以节省托运人的运输成本。对于 WDP，托运人允许承运人提交一组航线的投标。然后通过解决 WDP 来选择获胜的进位。具体而言，本研究同时考虑了基于运输距离和基于数量的运输成本折扣。最先进的线性化技术可用于将 MINP 模型转换为混合整数线性规划 (MILP) 以获得全局最优，但当问题规模变大时，求解时间变得效率低下。为了为大规模 WDP 寻找高效且有效的线性化技术，本研究 (1) 提出了一种具有折扣政策的新型 WDP 模型，(2) 在求解 MILP 时利用优越的编码公式来避免不平衡的分支定界树，以及 (3) 减少 big-M 约束以加快求解时间。所提出的方法导致计算工作量的显着节省。所提出的方法解决了真实世界大小的整车服务采购问题的数值实验，并进一步证实了解决大型 WDP 的计算时间显着减少。