Warehouse storage and retrieval optimization via clustering, dynamical systems modeling, and GPU-accelerated routing

TL;DR

The paper tackles dynamic warehouse storage and retrieval optimization under stochastic demand by framing a time-evolving graph and applying two-level clustering (order-based and picking-node-based) within a random dynamical systems (RDS) framework. A state-space model with a GPU-accelerated Bellman-Ford routing algorithm enables scalable, near-optimal routing across large graphs, while a segmentation strategy preserves memory feasibility for large-scale deployments. Empirical studies show that clustering-driven restocking concentrates storage regions (as measured by cluster centers and covariance $oldsymbol{igSigma}( ext{ell})$) and reduces route lengths, with silhouette scores indicating strong structure under low-to-moderate noise and substantial scalability to $10^5$ nodes. The work aligns with Warehousing 4.0 by delivering a principled, data-driven, and computationally scalable approach that adapts storage layouts to shifting demand, providing actionable insights for adaptive zoning and routing in modern fulfillment centers.

Abstract

This paper introduces a warehouse optimization procedure aimed at enhancing the efficiency of product storage and retrieval. By representing product locations and order flows within a time-evolving graph structure, we employ unsupervised clustering to define and refine compact order regions, effectively reducing picking distances. We describe the procedure using a dynamic mathematical model formulated using tools from random dynamical systems theory, enabling a principled analysis of the system's behavior over time even under random operational variations. For routing within this framework, we implement a parallelized Bellman-Ford algorithm, utilizing GPU acceleration to evaluate path segments efficiently. To address scalability challenges inherent in large routing graphs, we introduce a segmentation strategy that preserves performance while maintaining tractable memory requirements. Our results demonstrate significant improvements in both operational efficiency and computational feasibility for large-scale warehouse environments.

Figures (10)

• Figure 1: Each of the top panels show one of three clusters resulting from $K$-means clustering applied to an order with 200 article types given a randomized initial warehouse state $x_0$. The nodes in the warehouse have coordinates $(i,j,k)$ with $i,j,k=1,\ldots,10$. Stopping nodes are indicated with color coding describing how many picking nodes a given stopping node contains. Each of the bottom panels show one of three clusters resulting from $K$-means clustering applied to the same order being picked for the 100:th time, that is, given the warehouse state $x_{99}$ as in \ref{['eq:xn']}, obtained after 99 picking iterations. In the top panels, all three clusters are spread out over the entire warehouse with much overlap between clusters. In contrast, the clusters in the bottom panels are concentrated and much better separated.
• Figure 2: An order with 15 stopping positions clustered into 3 regions (left), and the reduced routes for each cluster represented by two boundary nodes (right). For ease of presentation, stopping nodes are labeled from 0 to 99, where node 0 corresponds to $(i,j) = (1,1)$ in Figure \ref{['fig1and2']}.
• Figure 3: Three snapshots of cluster distributions from experiment 1, illustrated by bivariate Gaussian probability density functions defined using each cluster's mean and covariance matrix $\Sigma_n(\ell)$, $\ell=1,2,3$. Each panel shows a superposition of the corresponding probability density function plots. The snapshots are taken at time $n$ (i.e., in the language of §\ref{['ss:routes']} they show the clusters computed for the $n$:th order), with $n=1$ (left), $n=10$ (middle), $n=100$ (right). The orders are the same in every iteration. The figure clearly shows that as time progresses, the separation between cluster centers increases, while the bumps become more articulated due to a decreased variance. A movie showing the evolution of the clusters for $n=1,\ldots,200$ can be found here: https://play.hb.se/media/clusters_unperturbed/0_3r1qr1w4.
• Figure 4: Three snapshots of cluster distributions from experiment 2, illustrated by bivariate Gaussian probability density functions defined using each cluster's mean and covariance matrix $\Sigma_n(\ell)$, $\ell=1,2,3$. Each panel shows a superposition of the corresponding probability density function plots. The snapshots are taken at time $n=1$ (left), $n=10$ (middle), $n=100$ (right). Two picking orders now differ by approximately $10\%$ on average. Although not as pronounced, we still clearly see an increased separation and decreased variance of the clusters as time progresses. A movie showing the evolution of the clusters for $n=1,\ldots,200$ can be found here: https://play.hb.se/media/clusters_static_perturbation/0_t9n5lbme
• Figure 5: Three snapshots of cluster distributions from experiment 3, illustrated by bivariate Gaussian probability density functions defined using each cluster's mean and covariance matrix $\Sigma_n(\ell)$, $\ell=1,2,3$. Each panel shows a superposition of the corresponding probability density function plots. The snapshots are taken at time $n=1$ (left), $n=10$ (middle), $n=100$ (right). Two picking orders now differ by approximately $19\%$ on average, and we only see a slight improvement in the separation and variance of the clusters as time progresses. A movie showing the evolution of the clusters for $n=1,\ldots,200$ can be found here: https://play.hb.se/media/clusters_random_perturbation/0_g0zx8ooj

Theorems & Definitions (5)

• Remark
• Remark
• Theorem
• proof
• Remark