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A note on the complexity of the picker routing problem in multi-block warehouses and related problems

Thibault Prunet, Nabil Absi, Diego Cattaruzza

TL;DR

The paper proves that the Picker Routing Problem (PRP) in rectangular, multi-block warehouses with parallel aisles is strongly NP-hard. It achieves this via a polynomial reduction from the Hamiltonian Cycle Problem on grid graphs (HCPGG) to PRP, including a connectivity-preserving construction that bounds instance values by length. The hardness result extends to related problems such as Order Batching (OBP) and prize-collecting PRP, situating PRP as a structured TSP with Steiner points and clarifying the intractability of many integrated order-picking problems. Practically, the findings justify the use of heuristics and fixed-parameter approaches in real-world warehouse planning and decomposition methods that rely on picker routing decisions.

Abstract

The Picker Routing Problem (PRP), which consists of finding a minimum-length tour between a set of storage locations in a warehouse, is one of the most important problems in the warehousing logistics literature. Despite its popularity, the tractability of the PRP in multi-block warehouses remains an open question. This technical note aims to fill this research gap by establishing that the problem is strongly NP-hard. As a corollary, the complexity status of other related problems is settled.

A note on the complexity of the picker routing problem in multi-block warehouses and related problems

TL;DR

The paper proves that the Picker Routing Problem (PRP) in rectangular, multi-block warehouses with parallel aisles is strongly NP-hard. It achieves this via a polynomial reduction from the Hamiltonian Cycle Problem on grid graphs (HCPGG) to PRP, including a connectivity-preserving construction that bounds instance values by length. The hardness result extends to related problems such as Order Batching (OBP) and prize-collecting PRP, situating PRP as a structured TSP with Steiner points and clarifying the intractability of many integrated order-picking problems. Practically, the findings justify the use of heuristics and fixed-parameter approaches in real-world warehouse planning and decomposition methods that rely on picker routing decisions.

Abstract

The Picker Routing Problem (PRP), which consists of finding a minimum-length tour between a set of storage locations in a warehouse, is one of the most important problems in the warehousing logistics literature. Despite its popularity, the tractability of the PRP in multi-block warehouses remains an open question. This technical note aims to fill this research gap by establishing that the problem is strongly NP-hard. As a corollary, the complexity status of other related problems is settled.
Paper Structure (15 sections, 6 theorems, 4 figures)

This paper contains 15 sections, 6 theorems, 4 figures.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Previous complexity results
  4. Known complexity results for PRP variants.
  5. Complexity results for related TSP variants.
  6. Complexity results for related Hamiltonian cycle problems.
  7. Complexity of the PRP in conventional warehouses
  8. Proof of Lemma \ref{['lemma:1']}.
  9. Proof of Lemma \ref{['lemma:2']}.
  10. Proof of Theorem \ref{['theorem:complexity']}.
  11. Complexity results for related problems
  12. The Order Batching Problem (OBP)
  13. The prize collecting PRP
  14. Other integrated order picking problems
  15. Conclusion

Key Result

Theorem 1

The Picker Routing Problem in multi-block warehouses is strongly NP-hard.

Figures (4)

  • Figure 1: Example of a two-block layout. Blue dots represent locations, and red diamonds represent dummy points corresponding to intersections between aisles and cross aisles.
  • Figure 2: Complexity map of the problems related to the PRP. Round nodes represent NP-hard problems (or NP-complete problems for decision problems), while square nodes represent polynomially solvable problems. Diamonds represent problems whose tractability is undetermined. An arrow represents a link of polynomial reducibility between two problems, where the problem at its destination is reducible to the problem at its origin. For instance, the PRP is reducible to the TSP, as TSP is a generalization of the PRP. Note that the polynomial reducibility is a transitive relationship. The red arrow is the reduction we use in the proof of Theorem \ref{['theorem:complexity']}.
  • Figure 3: Example of a grid graph.
  • Figure 4: PRP instance after reduction (only blue locations need to be visited).

Theorems & Definitions (6)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Corollary 2
  • Corollary 3