Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deterministic Structure of Vertical Configurations in Minimal Picker Tours for Rectangular Warehouses

George Dunn, Elizabeth Stojanovski, Bishnu Lamichhane, Hadi Charkhgard, Ali Eshragh

Abstract

The picker routing problem seeks the shortest tour through a warehouse that visits every item in a given pick-list and returns to the depot. For rectangular warehouses, dynamic programming algorithms solve this problem by sequentially evaluating combinations of vertical edge configurations within subaisles and horizontal edge configurations between aisles. These methods proceed through stages one after another, but how those stages relate to each other has received limited structural analysis. Building on our recent structural result for rectangular warehouses, which shows that connecting double traversals are not required to maintain tour connectivity, we prove that for rectangular warehouses of any size, the horizontal edge structure of a minimal tour subgraph uniquely determines the required vertical edge configurations. The proof uses a case analysis on horizontal degree along each aisle and at merged-segment endpoints, showing that the admissible vertical pattern in each regime is uniquely determined by Eulerian parity and by minimizing traversal length. This deterministic relationship implies that vertical configuration stages in existing dynamic programming algorithms can be replaced by a direct inference step, reducing the combinatorial complexity of the problem and providing a structural foundation for developing more efficient exact methods for warehouse layouts of any size.

Deterministic Structure of Vertical Configurations in Minimal Picker Tours for Rectangular Warehouses

Abstract

The picker routing problem seeks the shortest tour through a warehouse that visits every item in a given pick-list and returns to the depot. For rectangular warehouses, dynamic programming algorithms solve this problem by sequentially evaluating combinations of vertical edge configurations within subaisles and horizontal edge configurations between aisles. These methods proceed through stages one after another, but how those stages relate to each other has received limited structural analysis. Building on our recent structural result for rectangular warehouses, which shows that connecting double traversals are not required to maintain tour connectivity, we prove that for rectangular warehouses of any size, the horizontal edge structure of a minimal tour subgraph uniquely determines the required vertical edge configurations. The proof uses a case analysis on horizontal degree along each aisle and at merged-segment endpoints, showing that the admissible vertical pattern in each regime is uniquely determined by Eulerian parity and by minimizing traversal length. This deterministic relationship implies that vertical configuration stages in existing dynamic programming algorithms can be replaced by a direct inference step, reducing the combinatorial complexity of the problem and providing a structural foundation for developing more efficient exact methods for warehouse layouts of any size.

Paper Structure

This paper contains 12 sections, 2 theorems, 1 equation, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Result
  4. Case 1: Vertices with odd horizontal degree.
  5. Case 2: Zero horizontal degree on the aisle.
  6. Case 3: Even horizontal degree at both endpoints.
  7. Case 4: One endpoint zero horizontal degree, the other positive even.
  8. Conclusion
  9. Funding
  10. Competing interests
  11. Data availability
  12. Author contributions

Key Result

Lemma 1

The admissible vertical configurations for a single subaisle apply to a merged subaisle segment. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Warehouse graph $G$
  • Figure 2: Tour subgraph $T$
  • Figure 4: Vertical configurations
  • Figure 5: Horizontal configurations
  • Figure 7: Illustration of subaisle merging (Lemma \ref{['lem:merging']}). Dashed lines show horizontal edges only at the top and bottom vertices; gray shaded areas illustrate the segments considered before and after merging. (a) Individual subaisles between $v_{i,0}$ and $v_{i,3}$. (b) Merged segment, where all subaisles between $v_{i,0}$ and $v_{i,3}$ are treated as a single combined segment
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 1: Subaisle Merging
  • proof : Proof of Lemma \ref{['lem:merging']}
  • Proposition 1: Deterministic Structure of Vertical Configurations
  • proof : Proof of Proposition \ref{['prop:determinacy']}