We consider an e-commerce retailer operating a supply chain that consists of middle- and last-mile transportation, and study its ability to deliver products stored in warehouses within a day from customer's order time. Successful next-day delivery requires inventory availability and timely truck schedules in the middle-mile and in this paper we assume a fixed inventory position and focus on optimizing the middle-mile. We formulate a novel optimization problem which decides the departure of the last middle-mile truck at each (potential) network connection in order to maximize the number of next-day deliveries. We show that the respective \emph{next-day delivery optimization} is a combinatorial problem that is $NP$-hard to approximate within $(1-1/e)\cdot\texttt{opt}\approx 0.632\cdot\texttt{opt}$, hence every retailer that offers one-day deliveries has to deal with this complexity barrier. We study three variants of the problem motivated by operational constraints that different retailers encounter, and propose solutions schemes tailored to each problem's properties. To that end, we rely on greedy submodular maximization, pipage rounding techniques, and Lagrangian heuristics. The algorithms are scalable, offer optimality gap guarantees, and evaluated in realistic datasets and network scenarios were found to achieve near-optimal results.

翻译：我们考虑一家运营包含中程与末端运输供应链的电商零售商,研究其将仓库存储的产品在客户下单后24小时内送达的能力。实现次日送达需要中程物流的库存可用性和准时卡车调度安排,本文假设库存位置固定,重点优化中程物流环节。我们提出一个新型优化问题,该问题通过决定每个(潜在)网络连接点最后一班中程卡车的发车时间,最大化可实现次日送达的订单数量。我们证明相应的次日送达优化是组合优化问题,其逼近难度为$NP$-hard,近似比不超过$(1-1/e)\cdot\texttt{opt}\approx 0.632\cdot\texttt{opt}$,因此每个提供次日达服务的零售商都必须应对这一复杂性壁垒。我们研究了受不同零售商运营约束驱动的三种问题变体,并针对各问题的特性提出了定制化解法方案。为此,我们采用了贪婪子模最大化、管道舍入技术和拉格朗日启发式算法。这些算法具有可扩展性,能保证最优性差距,在真实数据集和网络场景中的评估表明其能达到接近最优的效果。