Aerospace Cost Estimating & Operations
Crawford Unit vs Wright Learning Curve Formula
Avoid a 44% unit-cost bidding error by distinguishing J. R. Crawford’s Incremental Unit Learning Curve Y_n = T_1 * n^b from T. P. Wright’s Cumulative-Average Unit Difference T_1*(n^(1+b) - (n-1)^(1+b)).
Never confuse an "80% Crawford Unit Curve" with an "80% Wright Cumulative-Average Curve"!
Defense, aerospace, and shipbuilding cost estimators use two rival power-law learning models that share the EXACT SAME algebraic expression T_1 * n^b (where b = LN(r) / LN(2) for learning rate r in (0, 1)) — yet apply it to completely different quantities! Under J. R. Crawford’s (Boeing / Stanford) Incremental Unit Model, T_1 * n^b is the direct labor hours of the n-th INDIVIDUAL unit Y_n_Crawford = T_1 * (n^b)! Under T. P. Wright’s (1936) Cumulative-Average Model, T_1 * n^b is the CUMULATIVE AVERAGE of units 1..n (so cumulative batch hours equal T_1 * n^(1 + b)), meaning the n-th individual unit under Wright actually takes Y_n_Wright = T_1 * (n^(1 + b) - (n - 1)^(1 + b)) — substantially LOWER than Crawford at the exact same 80% slope!
Verify on First_Unit_Hours T_1 = 100.00 hr (B2), Learning_Rate r = 0.80 (80% curve, B3 -> b = LN(0.80)/LN(2) = -0.3219), and Unit_Number n = 8 (2^3 = 3 doublings, B4): Crawford Unit-8 Hours equals exact 100 * (0.80^3) = 51.20 hr! Under Wright’s 80% curve, 51.20 hr is the average across ALL 8 units (Total_8 = 8 * 51.20 = 409.60 hr, minus Total_7 = 7 * 100 * 7^b = 374.14 hr), so Wright’s incremental Unit-8 labor is only 35.46 hr — a 15.74 hr (44.4% relative!) discrepancy if a contract bid mixes the two conventions!
| Learning-Curve Convention (T_1 = 100 hr, r = 80%, Unit n = 8) | Formula | Verified Hours |
|---|---|---|
| Learning Exponent (b) | LN(r) / LN(2) = LN(0.80) / LN(2) | -0.3219 |
| Crawford Incremental Unit-8 Hours | T_1 * (n^b) = 100 * 8^(-0.3219) | 51.20 hours |
| Wright Incremental Unit-8 Hours | T_1 * (n^(1+b) - (n-1)^(1+b)) | 35.46 hours |
| Convention Mismatch Gap (Crawford - Wright) | 51.20 - 35.46 | 15.74 hours (+44.39% Higher!) |
Crawford Incremental Unit-n Labor Hours Formula
Validate T_1 > 0, learning rate 0 < r <= 1 (r = 1 means zero learning, b = 0), and integer unit number n >= 1.
=LET(t_1, B2, r_lr, B3, n_un, B4, IF(OR(t_1 <= 0, r_lr <= 0, r_lr > 1, n_un < 1, INT(n_un) <> n_un), "Invalid Learning Input", LET(b_ex, LN(r_lr) / LN(2), ROUND(t_1 * (n_un ^ b_ex), 2))))Exact Wright Incremental Unit-n Labor Hours Formula
Handle n = 1 explicitly (0^(1 + b) = 0 -> Y_1 = T_1 under both Crawford and Wright!).
=LET(t_1, B2, r_lr, B3, n_un, B4, IF(OR(t_1 <= 0, r_lr <= 0, r_lr > 1, n_un < 1, INT(n_un) <> n_un), "Invalid Learning Input", IF(n_un = 1, ROUND(t_1, 2), LET(b_ex, LN(r_lr) / LN(2), ROUND(t_1 * ((n_un ^ (1 + b_ex)) - ((n_un - 1) ^ (1 + b_ex))), 2)))))Try the example
Use the original values or change them to test your understanding.