High-Bay Warehouse Automation, AS/RS Stacker Cranes & Bozer-White Chebyshev Travel
AS/RS Crane Bozer-White Single vs Dual Command Formula
Calculate AS/RS Rack Time-Shape Factor b, Expected Single-Command Cycle Time E[SC] (s), Interleaved Dual-Command Cycle Time E[DC] (s), and Dual-Command Throughput (pallets/h).
Stacker cranes move horizontally AND vertically at the same time—and interleaving putaway + retrieval boosts pallet throughput 26%!
High-bay Automated Storage & Retrieval System (AS/RS) stacker cranes drive aisle travel `v_x` and mast hoist `v_y` simultaneously under the Chebyshev (`L_infinity`) metric `t = MAX(x / v_x, y / v_y)`, not Manhattan sum `x/v_x + y/v_y`! Yavuz Bozer and John White (1984) normalized rack aisle travel `t_x = L / v_x` and mast lift `t_y = H / v_y` by Scaling Time `T = MAX(t_x, t_y)` and Dimensionless Rack Shape Factor `b = MIN(t_x, t_y) / T` (`0 < b <= 1`, with `b = 1` representing a "Square-in-Time" rack!). Under randomized uniform storage with shuttle Pickup/Deposit time `t_PD` (s), Expected Single-Command Time (1 putaway OR 1 retrieval) is `E[SC] = T * (1 + b^2 / 3) + 2 * t_PD`, whereas Dual-Command Interleaving (storing 1 inbound pallet AND retrieving 1 outbound pallet on the same aisle excursion!) takes `E[DC] = T * (4/3 + 0.5*b^2 - b^3 / 30) + 4 * t_PD`—moving `2` pallets per `E[DC]` cycle!
Verify on a 90 m x 24 m High-Bay Unit-Load Stacker Crane—Rack_Length `L` = 90.0 m (B2), Rack_Height `H` = 24.0 m (B3), Horizontal_Speed `v_x` = 2.50 m/s (`t_x = 36` s, B4), Vertical_Hoist_Speed `v_y` = 0.80 m/s (`t_y = 30` s, B5 -> Shape Factor `b = 30 / 36 = 0.833` in E2), and Telescoping_Fork_P/D `t_PD` = 12.0 s (B6): Single-Command `E[SC]` equals `68.33` s (`52.68` plts/h, E3), Dual-Command `E[DC]` equals `107.81` s (E4), and Interleaved Throughput hits `66.79` pallets/hour (`+26.8%` more moves per crane!, E5)!
| Bozer-White AS/RS Crane KPI (L=90 m @ 2.5 m/s, H=24 m @ 0.8 m/s, t_PD=12 s) | Formula | Verified Output |
|---|---|---|
| Rack Time-Shape Factor b (0..1) | MIN(L/vx, H/vy) / MAX(L/vx, H/vy) | 0.833 (0.556 @ faster 1.2 m/s hoist) |
| Single-Command Cycle E[SC] (sec) | T * (1 + (b^2)/3) + 2 * t_PD | 68.33 s (63.70 s @ 1.2 m/s hoist) |
| Dual-Command Cycle E[DC] (sec) | T * (4/3 + 0.5*b^2 - (b^3)/30) + 4*t_PD | 107.81 s (101.35 s @ 1.2 m/s hoist) |
| Dual-Command Pallets / Hour | 2 * (3600 / E_DC_sec) | 66.79 plts/h (71.04 plts/h @ 1.2 m/s) |
Bozer-White Chebyshev Single-Command & Dual-Command AS/RS Throughput Formula
Validate `Rack_Length > 0` m, `Rack_Height > 0` m, `Horiz_Speed > 0` m/s, `Vert_Speed > 0` m/s, and `Shuttle_PD_sec >= 0`!
=LET(ln_x, B2, ht_y, B3, vx_s, B4, vy_s, B5, pd_s, B6, IF(OR(ln_x <= 0, ht_y <= 0, vx_s <= 0, vy_s <= 0, pd_s < 0), "Invalid ASRS Input", LET(tx_s, ln_x / vx_s, ty_s, ht_y / vy_s, tm_x, MAX(tx_s, ty_s), bf_r, MIN(tx_s, ty_s) / tm_x, dc_s, tm_x * ((4 / 3) + 0.5 * (bf_r ^ 2) - ((bf_r ^ 3) / 30)) + 4 * pd_s, ROUND(2 * (3600 / dc_s), 2))))High-Speed 1.20 m/s Hoist Motor Upgrade Check (Set Vertical Speed vy = 1.20 m/s)
Upgrade the mast hoist drive from `0.80` to `1.20` m/s (`B5 = 1.2`): Vertical Full-Mast Time drops from `30` s to `20` s (`b` E2 drops to `0.556` because `36` s horizontal travel now dominates Chebyshev time!), trimming `E[SC]` E3 to `63.70` s, `E[DC]` E4 to `101.35` s, and lifting Dual-Command Capacity E5 to `71.04` pallets/h!
=IF(ISTEXT(E5), E5, LET(tx_s, B2 / B4, ty_s, B3 / B5, tm_x, MAX(tx_s, ty_s), bf_r, MIN(tx_s, ty_s) / tm_x, ROUND(tm_x * ((4 / 3) + 0.5 * (bf_r ^ 2) - ((bf_r ^ 3) / 30)) + 4 * B6, 2)))Try the example
Use the original values or change them to test your understanding.