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The Storage Location Assignment and Picker Routing Problem: A Generic Branch-Cut-and-Price Algorithm

Thibault Prunet, Nabil Absi, Diego Cattaruzza

TL;DR

The paper tackles the integrated Storage Location Assignment and Picker Routing Problem (SLAPRP) in warehousing, addressing the coupling of storage decisions with optimal picker routes under dynamic, e-commerce-like conditions. It develops a generic Branch-Cut-and-Price framework built on a Dantzig-Wolfe reformulation that convexifies routing subproblems and introduces non-robust valid inequalities (SL inequalities) to tighten the master problem; a novel branching and symmetry-breaking strategy further enhances tractability. The approach is validated across variants (including popular routing policies) and benchmark sets, showing competitive performance by solving medium-sized instances and outperforming state-of-the-art methods on several tests. Practically, the framework enables integrated, exact optimization across storage and routing decisions, offering a pathway to improved order-picking efficiency in modern warehouses and guiding future matheuristic extensions for industrial-scale instances.

Abstract

The Storage Location Assignment Problem (SLAP) and the Picker Routing Problem (PRP) have received significant attention in the literature due to their pivotal role in the performance of the Order Picking (OP) activity, the most resource-intensive process of warehousing logistics. The two problems are traditionally considered at different decision-making levels: tactical for the SLAP, and operational for the PRP. However, this paradigm has been challenged by the emergence of modern practices in e-commerce warehouses, where storage decisions are more dynamic and are made at an operational level, making the integration of the SLAP and PRP pertinent to consider. Despite its practical significance, the joint optimization of both operations, called the Storage Location Assignment and Picker Routing Problem (SLAPRP), has received limited attention. Scholars have investigated several variants of the SLAPRP, including different warehouse layouts and routing policies. Nevertheless, the available computational results suggest that each variant requires an ad hoc formulation. Moreover, achieving a complete integration of the two problems, where the routing is solved optimally, remains out of reach for commercial solvers. In this paper, we propose an exact solution framework that addresses a broad class of variants of the SLAPRP, including all the previously existing ones. This paper proposes a Branch-Cut-and-Price framework based on a novel formulation with an exponential number of variables, which is strengthened with a novel family of non-robust valid inequalities. We have developed an ad-hoc branching scheme to break symmetries and maintain the size of the enumeration tree manageable. Computational experiments show that our framework can effectively solve medium-sized instances of several SLAPRP variants and outperforms the state-of-the-art methods from the literature.

The Storage Location Assignment and Picker Routing Problem: A Generic Branch-Cut-and-Price Algorithm

TL;DR

The paper tackles the integrated Storage Location Assignment and Picker Routing Problem (SLAPRP) in warehousing, addressing the coupling of storage decisions with optimal picker routes under dynamic, e-commerce-like conditions. It develops a generic Branch-Cut-and-Price framework built on a Dantzig-Wolfe reformulation that convexifies routing subproblems and introduces non-robust valid inequalities (SL inequalities) to tighten the master problem; a novel branching and symmetry-breaking strategy further enhances tractability. The approach is validated across variants (including popular routing policies) and benchmark sets, showing competitive performance by solving medium-sized instances and outperforming state-of-the-art methods on several tests. Practically, the framework enables integrated, exact optimization across storage and routing decisions, offering a pathway to improved order-picking efficiency in modern warehouses and guiding future matheuristic extensions for industrial-scale instances.

Abstract

The Storage Location Assignment Problem (SLAP) and the Picker Routing Problem (PRP) have received significant attention in the literature due to their pivotal role in the performance of the Order Picking (OP) activity, the most resource-intensive process of warehousing logistics. The two problems are traditionally considered at different decision-making levels: tactical for the SLAP, and operational for the PRP. However, this paradigm has been challenged by the emergence of modern practices in e-commerce warehouses, where storage decisions are more dynamic and are made at an operational level, making the integration of the SLAP and PRP pertinent to consider. Despite its practical significance, the joint optimization of both operations, called the Storage Location Assignment and Picker Routing Problem (SLAPRP), has received limited attention. Scholars have investigated several variants of the SLAPRP, including different warehouse layouts and routing policies. Nevertheless, the available computational results suggest that each variant requires an ad hoc formulation. Moreover, achieving a complete integration of the two problems, where the routing is solved optimally, remains out of reach for commercial solvers. In this paper, we propose an exact solution framework that addresses a broad class of variants of the SLAPRP, including all the previously existing ones. This paper proposes a Branch-Cut-and-Price framework based on a novel formulation with an exponential number of variables, which is strengthened with a novel family of non-robust valid inequalities. We have developed an ad-hoc branching scheme to break symmetries and maintain the size of the enumeration tree manageable. Computational experiments show that our framework can effectively solve medium-sized instances of several SLAPRP variants and outperforms the state-of-the-art methods from the literature.
Paper Structure (69 sections, 5 theorems, 27 equations, 6 figures, 10 tables)

This paper contains 69 sections, 5 theorems, 27 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Picker Routing Problem.
  4. Storage decisions.
  5. Interactions between the SLAP and the PRP.
  6. The SLAPRP.
  7. Paper contributions.
  8. Problem description and mathematical formulations
  9. Problem description & compact formulations
  10. Storage nodes and locations.
  11. Graph representation.
  12. Problem description.
  13. Compact formulation.
  14. Note on the subtour elimination constraints.
  15. Problem structure and decomposition
  16. ...and 54 more sections

Key Result

Theorem 1

Given $\overline{\mathcal{L}}\subset\mathcal{L}$, the following inequalities (coined SL inequalities) are valid for formulation (DW) where $\delta_r(\overline{\mathcal{L}})$ equals 1 if route $r \in \mathcal{R}_o$, $o \in \mathcal{O}$, stops in one of the locations in $\overline{\mathcal{L}} \subset \mathcal{L}$, 0 otherwise.

Figures (6)

  • Figure 1: Routing policies with route distances in brackets
  • Figure 2: Warehouse layout with storage locations (blue) and drop-off point (red).
  • Figure 3: Induced subgraph of $\mathcal{G}$ from nodes $v_l^1$, $v_l^2$, $v_{l'}^1$ and $v_{l'}^2$.
  • Figure 4: Effect of SL-1 inequalities on branching decisions
  • Figure 5: Illustrative example for heuristic routing policies.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Lemma 1