An exact analysis and comparison of manual picker routing heuristics

TL;DR

This work derives exact expressions for the first two moments of total picking time $T$ under random storage for four manual routing heuristics, using the order-size PGF $P_M(x)$ to capture arbitrary order-size distributions. It provides a comprehensive moment-based framework, including $k^+$, $A_i$, $D_i$, and occupancy-derived variables, and yields explicit formulas for each routing policy (Return, Midpoint, Largest Gap, and S-shaped). The paper also connects these results to end-to-end performance by modeling the warehouse as an $M/G/c$ queue to approximate the average order-lead time $\mathbb{E}[R]$, showing how routing choice and warehouse layout interact with distributional characteristics of orders. Numerical results illustrate that routing decisions and layout optimization depend on the performance metric (picking time vs. lead time) and on the order-size distribution, with, for example, S-shaped routing excelling for large orders and even-numbered aisle layouts often favored for this policy. Overall, the work provides exact, moment-based tools for comparing routing heuristics and designing warehouse layouts under stochastic demand and storage randomness.

Abstract

This paper presents exact derivations of the first two moments of the total picking time in a warehouse for four routing heuristics, under the assumption of random storage. The analysis is done for general order size distributions and provides formulas in terms of the probability generating function of the order size distribution. These results are used to investigate differences between routing heuristics, order size distributions and warehouse layouts. In specific, we model a warehouse with $c$ pickers as an M/G/c queue to estimate the average order-lead time.

Figures (9)

• Figure 1: Comparison of different routing heuristics for the same example.
• Figure 2: Scheme of dependencies in the warehouse model
• Figure 3: Moments of $k^+$ for various distributions and $k$.
• Figure 4: Moments of $A_i$ for choice of the distribution and of $k$.
• Figure 5: Moments of $1-D_i$ for choice of the distribution and of $k$.

Theorems & Definitions (48)

• Lemma 4.1
• Example 4.1.1: $M \sim \mathrm{Poi}(\lambda) + 1$
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6
• Example 4.2.1: $M \sim \mathrm{Poi}(\lambda) + 1$
• Corollary 4.6.1
• Lemma 4.7