17 Optimal lane depth for block stacking
Optimal lane depth for block stacking
Block stacking stores pallets of one stock-keeping unit (SKU) in floor lanes with a fixed stack height and access from one side. Space loss comes from honeycombing inside lanes and aisles needed for access.
Classical lane-depth models balance these losses under instantaneous storage (see, e.g., Matson 1982; Kind 1975). Finite-rate models include production or replenishment time (see, e.g., Derhami et al. 2017).
Problem description
Consider a block stacking warehouse where:
- SKUs are stored in lanes of fixed depth.
- Lanes are last-in first-out (LIFO) (i.e., the most recently added pallet is the first to be removed).
- Lanes are accessible from one side (i.e., the front).
- Batch arrivals are deterministic (i.e., known arrival times and quantities).
- Batch arrivals are either:
- instantaneous (i.e., all pallets arrive at once), or
- at finite rate (i.e., pallets arrive over time).
The goal is to determine the optimal common lane depth that minimizes average space waste across all SKUs, where average space waste is defined as the sum of honeycombing and aisle accessibility waste.
Problem formulation
Let \(N\) be the number of SKUs. For each SKU \(i \in \{1,\dots,N\}\):
- \(Q_i\): batch size in pallets.
- \(P_i\): production or storage rate in pallets per hour.
- \(\lambda_i\): depletion or demand rate in pallets per hour.
- \(Z_i\): stack height in pallets.
- \(A\): aisle width in pallet units.
The common decision variable is:
- \(Z_i*x_c\): total lane depth in pallet units.
Objective. Find the optimal common lane depth \(x_c^*\) that minimizes average space waste across all SKUs.
Optimal common lane depth
The optimal common lane depth \(x_c^*\) is derived from the average space waste function, which combines honeycombing and accessibility waste. The expression for \(x_c^*\) depends on the production and demand rates of each SKU, extending classical models to account for finite production rates.
Case 1: \(P_i > \lambda_i\). When the production rate exceeds the demand rate, the optimal lane depth is influenced by the batch size and stack height.
\[ x_c^* = \sqrt{ \left(\frac{A}{2N}\right) \sum_{i=1}^{N} \left(\frac{1}{Z_i P_i}\right) \left(Q_i\,(P_i-\lambda_i) - 2\,\lambda_i\right) }\,. \]
Case 2: \(P_i < \lambda_i\). When the production rate is less than the demand rate, the optimal lane depth is influenced by the depletion rate and stack height.
\[ x_c^* = \sqrt{ \left(\frac{A}{2N}\right) \sum_{i=1}^{N} \left(\frac{1}{Z_i P_i}\right) \left(Q_i\,(P_i-\lambda_i) - 2\,\lambda_i\right) }\,. \]
Benchmark: \(P_i = \infty\) (instantaneous storage). When the production rate is infinite, the optimal lane depth simplifies to:
\[ x_c^* = \sqrt{ \left(\frac{A}{2N}\right) \sum_{i=1}^{N} \left(\frac{Q_i}{Z_i}\right) }\,. \]
Validation under stochastic demand
A discrete-event simulation study with stochastic demand, production rates, and batch sizes shows that the analytical model remains accurate: the mean absolute percentage error for optimal depth is below about \(8\%\), and for space utilization below about \(2.6\%\) across tested scenarios (Derhami et al. 2017).
Error from the infinite-rate model
Using the instantaneous-storage model to design finite-rate manufacturing warehouses yields lane depths that are systematically too deep as \(\lambda_i/P_i\) rises. In experiments, infinite-rate prescriptions were on average about two thirds deeper than finite-rate prescriptions, leading to lower utilization and more travel because fewer aisles are created (Derhami et al. 2017).
Managerial implications
Rounding upward usually loses less space utilization than rounding downward. Space utilization degrades slowly above \(x_c^*\) and quickly below it. When exact computation is infeasible, choose the deeper nearby value for utilization. Deeper lanes can increase travel because they create fewer aisles (Derhami et al. 2017).
The design context changes the result. The model balances honeycombing and accessibility waste while a lane is occupied. It implicitly treats aisle space as available for storage when lanes are empty. This assumption can underestimate accessibility waste in layouts with fixed aisles. Use a layout-level model that reserves aisle space throughout the horizon (Derhami et al. 2019).
Practical workflow
- Estimate \(Q_i\), \(P_i\), \(\lambda_i\), \(Z_i\) for the SKU set and choose \(A\).
- Compute \(x_c^*\) using the appropriate case above.
- Round to nearby integers and evaluate waste or utilization to choose a practical depth.
- If designing a full layout, complement the depth choice with a layout model that optimizes aisle count and bay depths to capture travel and fixed aisle space (Derhami et al. 2019).