Supplementary information files for Bid price controls for car rental network revenue management

Supplementary files for article Bid price controls for car rental network revenue management We consider a car rental network revenue management (RM) problem, accounting for the key operational characteristics of car rental services such as the varying length of rentals and mobility of inventories which imply the inter-temporal and spatial correlations of rental demands for inventories across different locations and days. The problem is formulated as an infinite-horizon cyclic stochastic dynamic program to account for the time-varying and cyclic nature of car rental businesses. To tackle the curse of dimensionality, we propose a Lagrangian relaxation (LR) approach with product- and time-dependent Lagrangian multipliers to decomposing the dynamic network problem into multiple singlestation single-day sub-problems. We show that the Lagrangian dual problem is a convex program and then develop a subgradient-based algorithm to solve the dual problem and derive an LR-based bid price policy. To improve the scalability of the LR approach, we further propose three simpler LR-based bid price policy variants with either location-dependent or leadtime-dependent Lagrangian multipliers, or both. Our numerical study indicates that the LR-based bid price policies can outperform some commonly used heuristics. Using a set of real-world booking data, we provide a case study in which we empirically demonstrate the operational characteristics of car rental services, calibrate the arrival process of booking requests using a Poisson regression model and demonstrate that the LR-based bid price policies indeed outperform other heuristics consistently in both in-sample and out-of-sample horizons.论文《租车网络收益管理中的竞价定价控制》补充材料 我们研究一类租车网络收益管理（Revenue Management，RM）问题，充分考量租车服务的核心运营特征：租赁时长的差异性与库存的流动性，这意味着不同地点、不同日期的库存租赁需求存在跨时段与跨空间的相关性。 该问题被建模为无限时域周期性随机动态规划，以适配租车业务随时间动态变化且具有周期性的本质属性。 为应对维度灾难问题，我们提出一种基于产品与时域相关拉格朗日乘子的拉格朗日松弛（Lagrangian Relaxation，LR）方法，将动态网络问题分解为多个单站点单日的子问题。 我们证明拉格朗日对偶问题为凸规划，随后设计基于次梯度的算法求解该对偶问题，并推导出基于LR的竞价定价策略。 为提升LR方法的可扩展性，我们进一步提出三种更简化的LR基竞价定价策略变体，分别采用仅与站点相关、仅与提前期相关，或同时兼顾两者的拉格朗日乘子。 我们的数值实验结果表明，基于LR的竞价定价策略性能优于若干常用启发式算法。 我们采用一组真实预订数据集开展案例研究，实证验证了租车服务的核心运营特征；通过泊松回归模型校准预订请求的到达过程，并证实基于LR的竞价定价策略在样本内与样本外评估区间中，均持续优于其他启发式算法。