Potential-Based Greedy Matching for Dynamic Delivery Pooling

TL;DR

This paper demonstrates the effectiveness of using the delivery distance as a proxy for opportunity cost via a potential-based greedy algorithm (PB), and shows that the potential approximates the true opportunity cost of dispatching a job, in a stochastic setting with sufficient density.

Abstract

We study the dynamic pooling of multiple orders into a single trip, a strategy widely adopted by online delivery platforms. When an order has to be dispatched, the platform must determine which (if any) of the available orders to pool it with, weighing the immediate efficiency gains against the uncertain, differential benefits of holding each order for future pooling opportunities. In this paper, we demonstrate the effectiveness of using the delivery distance as a proxy for opportunity cost via a potential-based greedy algorithm (PB). The algorithm is simple, pooling each departing job with the available job that maximizes the immediate savings in travel distance minus "half its delivery distance", which we call the potential of the available job. Theoretically, we show that PB achieves vanishing worst-case regret per job as market density increases, whereas a naive greedy policy suffers constant regret. We further show that the potential approximates the true opportunity cost of dispatching a job, in a stochastic setting with sufficient density. Finally, we conduct extensive numerical experiments on both synthetic data and real-world data from the Meituan platform. Despite being forecast-agnostic, PB consistently outperforms greedy heuristics that rely on historical data. Moreover, PB achieves performance comparable to computationally-intensive batching heuristics, which themselves also benefit from incorporating the potential to further improve their performance or drastically reduce computational costs.

Figures (26)

• Figure 1: Fraction of pooled orders by hour-of-week in Meituan data. The dataset includes $8$ days of order-level data from one city (see \ref{['sec:sim_meituan']} for more details). The gray shade indicates peak lunch hours (10:30am-1:30pm), while the green shade indicates peak dinner hours (5pm-8pm).
• Figure 2: Illustration of the construction of the modified offline instance, with $d=4$. In \ref{['fig:thm3_exm_original']}, job $5t+1$ chooses job $5t+3$ in the online execution of $\mathsf{PB}$, because only jobs up to $5t+5$ would have arrived when job $5t+1$ becomes critical. However, in the offline execution of $\mathsf{PB}$, job $5t+1$ would choose job $5t+6$, because all jobs are available. Our modified instance in \ref{['fig:thm3_exm_phantom']} corrects the inconsistent decision.
• Figure 3: Average regret and reward ratio, comparing with alternative opportunity costs, in random 1D instances.
• Figure 4: Average regret and reward ratio, comparing with batching-based heuristics, in random 1D instances.
• Figure 5: Average regret and reward ratio, comparing with alternative opportunity costs, in Meituan data.

Theorems & Definitions (18)

• Remark 1
• Remark 2
• Proposition 1: proof in \ref{['pf:greedy_linear_lower_bound']}
• Theorem 1
• Proposition 2: proof in \ref{['pf:loglowerboundOffline']}
• Proposition 3: proof in \ref{['pf:greedy_linear_lower_bound_dynamic']}
• Theorem 2
• Proposition 4: proof in \ref{['pf:loglowerboundOnline']}
• Remark 3
• Proposition 5