Online Order Fulfillment with Replenishment

TL;DR

A regret-based framework that quantitatively compares the value of improving replenishment versus improving fulfillment, and characterize regimes in which optimizing replenishment yields a larger revenue impact than refining the online fulfillment algorithm (and vice versa).

Abstract

In modern e-commerce and service operations, firms must jointly manage inventory replenishment and real-time order fulfillment to maximize profit under demand uncertainty. While each component has been studied extensively in isolation, their interaction remains underexplored. This paper investigates a fundamental operational question: which lever plays a more decisive role in overall system performance, replenishment or fulfillment? We model the system as a one-location online order fulfillment problem with lost sales and stochastic customer arrivals, each offering heterogeneous rewards. Replenishment follows either a base-stock or constant-order policy, while real-time fulfillment decisions are made using online algorithms. Our core performance metric is the expected average profit per replenishment cycle, evaluated across all combinations of these policies and algorithms. Our main theoretical result shows that when the replenishment cycle is long, the cumulative regret of online fulfillment remains of the same order as in a corresponding single-cycle problem, even under repeated replenishment, revealing a form of regret stability. This phenomenon also extends to a multi-location setting. We further develop a regret-based framework that quantitatively compares the value of improving replenishment versus improving fulfillment, and we characterize regimes in which optimizing replenishment yields a larger revenue impact than refining the online fulfillment algorithm (and vice versa). Motivated by examples where myopic algorithms underperform, we introduce a novel look-ahead online algorithm that anticipates future replenishment and demand. Numerical experiments verify that this algorithm outperforms myopic baselines. Overall, our results provide both theoretical and managerial insights into situations where inventory replenishment policies are more influential and vice versa.

Figures (3)

• Figure 1: Average profit versus cycle length $T$ under four policy combinations (online BS vs. online greedy) $\times$ (base-stock vs. constant-order). Parameters: $L=2$, $N=1{,}000$, $h=2$, $[r_1,r_2,r_3]=[5,8,10]$, $[\lambda_0,\lambda_1,\lambda_2,\lambda_3]=[0.3,0.2,0.3,0.2]$, and $K=10{,}000$ sample paths.
• Figure 2: Average profit under the myopic offline and online BS algorithms with and without look-ahead across different lead times and number of look-ahead cycles.
• Figure 3: Average profit under the myopic offline and online BS algorithms with and without look-ahead across different number of periods in one cycle.

Theorems & Definitions (26)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• Lemma 4: Theorem 1 in kingman1962
• Lemma 5
• Theorem 2
• Lemma 6
• Lemma 7
• Theorem 3