Modified Dynamic Programming Algorithms for Order Picking in Single-Block and Two-Block Rectangular Warehouses

TL;DR

The paper addresses the order-picking problem in rectangular warehouses by exploiting structural results that eliminate unnecessary double traversals and deterministically relate vertical and horizontal decisions. It introduces a merged-stage dynamic programming approach for single-block warehouses and two successive modifications for two-block warehouses, preserving optimality while reducing the number of state-action evaluations; both maintain linear-time complexity in the number of aisles $m$. Theoretical analysis yields reductions of about $1.81$ for single-block and $1.18$ and $1.31$ for the two-block modifications, confirmed by extensive computational experiments showing substantial practical speedups. The findings highlight how targeted structural refinements can translate into meaningful performance gains in real-world picker-routing systems, with future work exploring applicability to larger layouts and alternative cross-aisle configurations.

Abstract

Recent research has shown that optimal picker tours in rectangular warehouses exhibit deterministic travel patterns within each aisle, and that certain previously considered traversals are unnecessary. Using these insights, this paper proposes modifications to dynamic programming algorithms that improve computational efficiency without affecting optimality. For layouts with and without a central cross-aisle, the modifications preserve linear-time complexity in the number of aisles while reducing the number of state-action evaluations per stage. The proposed modifications reduce computational effort by factors up to 1.81, confirmed by numerical experiments. These findings are encouraging and highlight how structural refinements can yield significant improvements in practical performance of algorithms.

Figures (9)

• Figure 1: Rectangular warehouse examples where the black squares represent the locations of items to be picked and the gray circle shows the depot.
• Figure 2: Valid vertical edge configurations within aisle $j$. Unlabeled nodes represent item vertices within the aisle.
• Figure 3: Valid horizontal edge configurations between aisle $j$ and $j+1$ of a single-block warehouse.
• Figure 4: Example of a minimal tour subgraph in a single-block rectangular warehouse. The solution has an optimal sequence of states ($(0,0,0C)$,$(0,E,1C)$,$(0,E,1C)$,$(U,U,1C)$, $(U,U,1C)$,$(E,E,1C)$,$(0,0,1C)$) and actions $(bottom, 02, 1pass, 11, 1pass, 00)$.
• Figure 5: State space of the original (top) and modified (bottom) algorithms for the warehouse in Figure \ref{['fig:warehouse']}. The minimal sequence of transitions and actions are highlighted in red.