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Exact results for the order picking time distribution under return routing

Tim Engels, Ivo Adan, Onno Boxma, Jacques Resing

TL;DR

The paper addresses the distribution of order picking time under return routing in single- and 2-block warehouses with Poisson order sizes. It develops exact Laplace-Stieltjes transform expressions by decomposing the problem into independent per-aisle contributions and deriving per-aisle transforms. The results are specialized to random and class-based storage policies, and numerical inversion is used to compare layouts and policies. The work provides distributional insights enabling tail-probability analysis and lead-time approximations, and lays groundwork for extending to multi-block warehouses and other routing policies.

Abstract

This paper derives exact expressions for the Laplace-Stieltjes transform of the order picking time in single- and 2-block warehouses. We consider manual warehouses that deploy return routing and assume that order sizes follow a Poisson distribution. The results in this paper apply to a wide range of storage policies, including but not restricted to class-based and random storage. Furthermore, we compare the performance of the storage policies and warehouse lay-outs by using numerical inversion of the Laplace-Stieltjes transforms.

Exact results for the order picking time distribution under return routing

TL;DR

The paper addresses the distribution of order picking time under return routing in single- and 2-block warehouses with Poisson order sizes. It develops exact Laplace-Stieltjes transform expressions by decomposing the problem into independent per-aisle contributions and deriving per-aisle transforms. The results are specialized to random and class-based storage policies, and numerical inversion is used to compare layouts and policies. The work provides distributional insights enabling tail-probability analysis and lead-time approximations, and lays groundwork for extending to multi-block warehouses and other routing policies.

Abstract

This paper derives exact expressions for the Laplace-Stieltjes transform of the order picking time in single- and 2-block warehouses. We consider manual warehouses that deploy return routing and assume that order sizes follow a Poisson distribution. The results in this paper apply to a wide range of storage policies, including but not restricted to class-based and random storage. Furthermore, we compare the performance of the storage policies and warehouse lay-outs by using numerical inversion of the Laplace-Stieltjes transforms.
Paper Structure (9 sections, 4 theorems, 22 equations, 3 figures)

This paper contains 9 sections, 4 theorems, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Warehousing Model
  3. Routing Policy
  4. Results
  5. Single-block warehouse
  6. 2-block warehouse
  7. Storage policy examples
  8. Numerical Illustration
  9. Conclusion

Key Result

Lemma 3.1

The following equations hold:

Figures (3)

  • Figure 1: Examples of the return routing policy in both single- and 2-block warehouses.
  • Figure 2: Illustrations of the class-based storage areas for the numerical example.
  • Figure 3: Illustration of the density of the total order picking time under the Class-Based Storage and Random Storage policies.

Theorems & Definitions (7)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof