[Paper Review] Order-optimal Correlated Rounding for Fulfilling Multi-item E-commerce Orders

Will Ma|arXiv (Cornell University)|Jul 11, 2022

Optimization and Search Problems4 citations

TL;DR

This paper introduces a novel correlated rounding scheme for e-commerce order fulfillment that minimizes the number of distinct fulfillment centers (FCs) used per order while respecting marginal probability constraints. By modeling FC selection via Poisson processes on dilated time scales, it achieves an order-optimal guarantee of $\alpha = \ln(q) + 1$, improving upon the prior $\alpha \approx q/4$ and proving tightness via connections to the Set Cover problem.

ABSTRACT

We study the dynamic fulfillment problem in e-commerce, in which incoming (multi-item) customer orders must be immediately dispatched to (a combination of) fulfillment centers that have the required inventory. A prevailing approach to this problem, pioneered by Jasin and Sinha (2015), is to write a ``deterministic'' linear program that dictates, for each item in an incoming multi-item order from a particular region, how frequently it should be dispatched to each fulfillment center (FC). However, dispatching items in a way that satisfies these frequency constraints, without splitting the order across too many FC's, is challenging. Jasin and Sinha identify this as a correlated rounding problem, and propose an intricate rounding scheme that they prove is suboptimal by a factor of at most $\approx q/4$ on a $q$-item order. This paper provides to our knowledge the first substantially improved scheme for this correlated rounding problem, which is suboptimal by a factor of at most $1+\ln(q)$. We provide another scheme for sparse networks, which is suboptimal by a factor of at most $d$ if each item is stored in at most $d$ FC's. We show both of these guarantees to be tight in terms of the dependence on $q$ or $d$. Our schemes are simple and fast, based on an intuitive idea -- items wait for FC's to ``open'' at random times, but observe them on ``dilated'' time scales. This also implies a new randomized rounding method for the classical Set Cover problem, which could be of general interest. We numerically test our new rounding schemes under the same realistic setups as Jasin and Sinha (2015) and find that they improve runtimes, shorten code, and robustly improve performance. Our code is made publicly available.

Motivation & Objective

• To address the challenge of minimizing the number of distinct fulfillment centers used per multi-item e-commerce order while respecting marginal shipment probabilities.
• To improve upon the prior correlated rounding scheme of Jasin and Sinha (2015), which guaranteed $\alpha \approx q/4$, by reducing the dependence on order size $q$.
• To develop a method that is both simple and efficient, suitable for large-scale, real-time e-commerce fulfillment systems.
• To establish theoretical tightness of the new guarantees by linking the problem to the Set Cover randomized rounding problem.
• To validate the method empirically under realistic dynamic fulfillment scenarios, demonstrating improved performance and reduced runtime.

Proposed method

• Uses a Poisson clock-based mechanism where items 'wait' for FCs to 'open' on a dilated time scale to induce positive correlation in FC selection.
• Employs time-dilation to scale the Poisson process rates such that the marginal probability of selecting each FC matches the input distribution $u_{ki}$.
• Introduces two schemes: 'Dilate' for general cases with $\alpha = \ln(q) + 1$ guarantee, and 'ForceOpen' for sparse networks with $\alpha = d$ guarantee, where $d$ is the max number of FCs per item.
• The rounding mechanism ensures that each FC is used with probability at most $\alpha \cdot y_k$, where $y_k = \max_i u_{ki}$, minimizing overuse.
• Theoretical analysis proves the $\ln(q)+1$ bound is tight by relating the problem to randomized rounding in the Set Cover problem.
• Empirical evaluation uses dynamic fulfillment simulations from Jasin and Sinha (2015), comparing performance across five algorithms on varying network sparsity and order sizes.

Experimental results

Research questions

• RQ1 Can we reduce the dependence of the FC usage guarantee on order size $q$ from linear ($q/4$) to logarithmic ($\ln q + 1$) in correlated rounding for e-commerce fulfillment?
• RQ2 Is there a provably tight, simple, and efficient rounding scheme for sparse networks where each item is stored in at most $d$ FCs?
• RQ3 How does the new correlated rounding scheme compare empirically to prior methods in terms of performance, runtime, and robustness under dynamic order arrivals?
• RQ4 Can the proposed method be connected to and improve existing randomized rounding schemes for the Set Cover problem?
• RQ5 Does the theoretical improvement translate into practical gains in real-world dynamic fulfillment settings with inventory constraints?

Key findings

• The proposed 'Dilate' scheme achieves a guarantee of $\alpha = \ln(q) + 1$, representing a significant improvement over the prior $\alpha \approx q/4$ in terms of dependence on $q$.
• The 'ForceOpen' scheme achieves a tight $\alpha = d$ guarantee when each item is stored in at most $d$ FCs, and this bound is proven to be optimal.
• In numerical experiments, the Dilate scheme reduced average loss to 5.6% at $p_{\text{carry}} = 0.25$ and 12.6% at $p_{\text{carry}} = 0.50$, outperforming Myopic, Independent, and JS schemes.
• The Dilate and ForceOpen schemes achieved a 10x speedup over the JS method, reducing average runtime per instance from ~160s to ~15–20s.
• The average number of distinct FCs per order was reduced to 1.22 with Dilate at $p_{\text{carry}} = 0.25$, significantly below the 1.31 of the Independent scheme.
• The theoretical guarantee of $\ln(q)+1$ was validated empirically, with performance gains exceeding theoretical expectations in larger, more complex networks.