Double Traversals in Boundary Subaisles: Implications for Two-Block Layouts

Abstract

The order picking problem seeks the shortest warehouse route that visits all required item locations. Strict conditions are known for single-block rectangular layouts under which optimal routes never require double traversals, while broader results show that double traversals serving cross-aisle connectivity can always be avoided. We strengthen these findings by proving that no double traversals are needed in the boundary subaisles, the uppermost and lowermost subaisle segments, of warehouses with at least two non-empty aisles. This yields a unified strict condition for all single-block layouts and for two-block layouts with more than one aisle. For these widely used layouts, exact methods such as dynamic programming and mathematical programming can therefore exclude the double-traversal configuration from every boundary subaisle, reducing the number of admissible edge configurations without loss of optimality.

Figures (4)

• Figure 1: Transformation $T \rightarrow T'$ for each non-connecting double edge state. Gray shading indicates arbitrary feasible configurations, while the red line denotes the addition of a single edge to these configurations.
• Figure 2: Boundary subaisles (shaded) in a rectangular warehouse graph. Our main result shows that no double edge in these subaisles is required for a minimal tour when there are two or more non-empty aisles.
• Figure 3: Transformation $T \rightarrow T'$ for a non-connecting double edge in a boundary subaisle with state $s=(0,1)$ (the only non-connecting state not excluded by Lemma \ref{['lem:connecting']}) and a single edge $(a_{i-1},b_{i-1})^1$ in the preceding aisle segment (the only situation in which Lemma \ref{['lem:single']} does not already reduce the double edge). Gray regions indicate portions of the tour subgraph that may take any feasible configuration (subject to even degrees and connectivity), and the dotted line indicates the warehouse border. (a) Inner-connected: the outer endpoint has no incident horizontal edges. The unlabeled vertex is the nearest pick location to $a_i$ on the boundary subaisle. If $a_i$ is not the depot, the red double segment can be deleted, yielding a strictly shorter feasible tour subgraph. (b) Outer-connected: the outer endpoint is incident to a single horizontal edge directed to the right. This incidence is forced in this state to maintain even degree at $b_i$.
• Figure 4: In single-block (left) and two-block (right) rectangular warehouses, all subaisles are boundary subaisles (shaded). Hence Theorem \ref{['thm:main']} implies that no double edges are required in a minimal tour for either layout when there are two or more non-empty aisles (Corollary \ref{['cor:unify']}).

Theorems & Definitions (6)

• Theorem 1: dunn2025double
• Lemma 1
• Lemma 2
• Theorem 2: Elimination of double edges in boundary subaisles
• proof
• Corollary 1