AS/RS Sizing Methods

The gap between the calculated throughput number and the real production number comes from three places: single-command vs. dual-command cycle mode, physical rack geometry, and operational realities (bin digging depth in AutoStore, relocation penalties in double-deep, lift contention in shuttle systems).

Bozer & White Cycle Time Formulas

The foundational analytical model for unit-load and mini-load AS/RS, based on randomized storage and the Tchebyshev travel metric (the crane moves horizontally and vertically simultaneously; travel time is dominated by whichever dimension takes longer).

Notation

Symbol	Meaning
L	Rack length
H	Rack height
v_h, v_v	Horizontal and vertical S/R machine velocity
T	max(L/v_h, H/v_v) — time to travel to extreme corner
b	Shape factor = (H/v_v) / T; b = 1 for a “square-in-time” rack
t_pd	Pickup + deposit time (P/D time, seconds)

Single-Command (SC) Cycle

One storage or retrieval per cycle.

$$E[\text{SC}] = t_{pd} + \frac{2T}{3}\left(1 + \frac{b^2}{1+b}\right)$$

For square-in-time rack (b = 1): E[SC] = t_pd + T

Dual-Command (DC) Cycle

The crane stores one load, then travels to a retrieval location, picks, and returns. Two loads per cycle.

$$E[\text{DC}] = t_{pd} + T\left[\frac{4}{3} - \frac{b^3}{3(1+b)}\right]$$

For b = 1: E[DC] = t_pd + T × (11/12)

Critical insight: E[DC] ≈ E[SC]. The crane travels nearly the same total distance handling one or two loads — the second retrieval location is, on average, near the storage location. This makes DC roughly 2× as productive as SC per cycle.

DC throughput per aisle = (2 × 3600) / E[DC] loads/hr SC throughput per aisle = 3600 / E[SC] loads/hr

Worked Example: 6-Aisle Unit-Load AS/RS

• Rack: 60m long, 12m high
• Speeds: 3 m/s horizontal, 1 m/s vertical
• T = max(20s, 12s) = 20s; b = 12/20 = 0.60
• P/D time: t_pd = 20s

SC: E[SC] = 20 + (2×20/3)(1 + 0.36/1.60) = 36.3s → 99 loads/hr/aisle

DC: E[DC] = 20 + 20[1.333 - 0.216/(3×1.60)] = 45.76s → 157 loads/hr/aisle

6-aisle DC system: 942 loads/hr total. Against an 800-load requirement: 18% headroom — within the 15–25% design margin target.

FEM 9.851 Calculation Method

European standard for AS/RS cycle time, used globally for procurement specification and acceptance testing. Deterministic parametric method — not probabilistic like Bozer & White — making it better for writing a contractual specification both vendor and buyer can independently verify.

FEM 9.851 specifies six test cases covering different I/O point locations. Include the applicable case in the RFQ, specify the acceptance threshold (e.g., “single-cycle time shall not exceed X seconds per FEM 9.851 Case 2”), and require witnessed acceptance testing before SAT sign-off.

Critical contract language:

“Throughput guarantee shall be defined as dual-command transactions per hour per aisle, measured over a minimum 4-hour continuous run per FEM 9.851, with randomized storage assignments distributed evenly across the full rack face. Single-command-only modes shall not be used to demonstrate throughput compliance.”

Without this language, a vendor can demonstrate compliance using convenient short-cycle store/retrieve patterns that don’t represent actual randomized storage.

Double-Deep Storage: The Relocation Penalty

Double-deep racking stores two pallets per column depth, reducing aisle count for a given capacity. The tradeoff: throughput penalty whenever the target pallet is behind a different pallet that must be moved first.

Relocation probability at 50% fill rate (randomized storage):

$$P_{\text{relocation}} \approx 25%$$

At 80% fill: relocation frequency rises significantly.

Effective DC cycle time with relocation:

$$E[\text{DC}{\text{double-deep}}] = E[\text{DC}{\text{single-deep}}] + P_{\text{relocation}} \times E[\text{SC}] \times 0.5$$

Throughput impact:

• 50% fill: ~10–20% throughput reduction vs. single-deep
• 80% fill: ~25–30% throughput reduction

Size the system with the relocation penalty included. When floor space is the binding constraint, double-deep may still be the right choice — but own the decision explicitly.

Shuttle System Sizing

Shuttle-based AS/RS (SBS/RS) uses autonomous shuttles per rack level plus vertical lifts at aisle ends. Throughput bottleneck is whichever is slower: the shuttle or the lift.

Shuttle cycle time (single-command, per tier): t_shuttle,SC = 2 × t_ride + t_transfer

Lift interarrival time per tier: t_A = t_lift × n_tiers

One lift serving 12 tiers means each tier receives service once every 12 × lift cycle.

Bottleneck test:

• Shuttle utilization ρ = t_shuttle / t_A
• ρ < 1: lift determines throughput
• ρ ≥ 1: shuttle determines throughput

Adding a second lift to a 12-tier aisle typically increases throughput 60–80% — because the lift is almost always the bottleneck, not the shuttle. Adding a second shuttle per tier only helps when shuttle utilization already exceeds ~85%. Fix the binding constraint first.

AutoStore: Robot-to-Port-to-Bin Sizing

AutoStore robots drive across the top of a dense grid, lifting bins through a stack (typically 5–16 bins deep) to deliver to workstation ports.

Core Sizing Rules

Parameter	Rule
Robot delivery rate	~30 bin deliveries/hr/robot (for a mature operating grid)
Port processing rate	250–350 picks/hr/port (midpoint: 300)
Robot-to-port ratio	2–5 robots per port; target 3–4
Minimum density	1 robot per 65 bins (1 robot per ~15–25 grid cells)
Grid density	~4.7 bins/m² at standard 449-mm cell pitch

Worked Example: 2,000-Line/Hr System

Design: 2,000 order lines/hr. Average picks per bin presentation: 3.0.

1. Bin deliveries/hr: 2,000 / 3.0 = 667/hr
2. Robots: 667 / 30 = 22.2 → 23 robots (+ 1 spare for maintenance = 24 budget)
3. Ports: 2,000 / 300 = 6.7 → 7 ports
4. Robot-to-port ratio: 24 / 7 = 3.4 robots/port ✓ (within 2–5 range)
5. Grid cells: 24 robots × 25 cells/robot = 600 cells minimum → 24×25 = 600-cell grid
6. Grid footprint: 24 × 0.449m = 10.8m wide; 25 × 0.449m = 11.2m long → ~121 m²
7. Storage capacity: 600 cells × 12 bins high = 7,200 bins

Bin Digging Penalty

The 30 bin/hr average delivery rate already incorporates the digging depth for a mature operating grid. It becomes a significant deduction above 90% fill, when hot bins may be deeply buried under slow-moving inventory.

Density Comparison

At 16-bin height: 300 m² × 4.7 × 16 = 22,560 bins

4.7 bins/m² × 16 high = 75 bin-locations per square meter of floor. Conventional 4-deep shelving with 6 levels: ~3–4 SKU positions/m². The 4× density advantage is real for the right SKU profile.

Storage Capacity Calculation (Unit-Load AS/RS)

$$\text{Positions per aisle side} = \left\lfloor \frac{L_{\text{rack}}}{\text{horizontal pitch}} \right\rfloor \times \left\lfloor \frac{H_{\text{rack}}}{\text{vertical pitch}} \right\rfloor$$

Typical pitches: horizontal = 58 in (48-in pallet + 10-in clearance); vertical = 74 in (66-in load height + 8-in clearance).

Example — 300-ft rack, 40-ft high:

• Horizontal: (300×12)/58 = 62 positions/row
• Vertical: (40×12)/74 = 6 levels
• Per aisle (2 sides, single-deep): 2 × 62 × 6 = 744 locations
• 6-aisle system: 4,464 gross positions
• At 85% operational fill: 3,794 effective pallet positions

The 85% fill ceiling is an operational reality: above 85%, WMS search for compliant put-away locations slows, DC-mode pairing options decrease, and average travel time grows. Design for 85% maximum operating fill on gross capacity.

Source: 2.6-advanced-automation-design