Cost Estimating

Cost Estimating

Learning Curves, Parametric CERs and Life-Cycle Cost

Three estimating methods

A cost estimate is built from one of three kinds of evidence: the cost of the units already made (a learning curve projects the next ones), the cost of similar systems against a physical or performance driver (a cost estimating relationship, CER), or a schedule of future expenditures to be summed over the life of the system (a life-cycle cost). The tool offers the three side by side, following the NASA Cost Estimating Handbook (CEH, 2015) and the GAO Cost Estimating and Assessment Guide (GAO-20-195G, 2020). Money is a plain number in one currency unit; nothing is converted.

Learning curves

Wright (1936) observed that as the quantity of airframes produced doubled, the cost fell by a constant percentage. The curve is Y = T₁ X^b with b = ln(slope)/ln 2: an 80 % slope gives b = −0.322, a 90 % slope b = −0.152, and 100 % means no learning. Two theories use the same equation with different meanings (GAO-20-195G Appendix VII):

• Unit theory (Crawford, 1944): Y_X is the cost of unit X itself; the cumulative total is the sum of the unit costs, which the tool adds exactly, and the cumulative average is that sum divided by X. Unit 2 costs slope × T₁, unit 4 costs slope² × T₁.
• Cumulative-average theory (Wright): Ȳ_X is the average cost of the first X units; the cumulative total is X Ȳ_X = T₁ X^1+b and the cost of unit X is the difference of successive totals, Y_X = T₁[X^1+b − (X − 1)^1+b]. With the same slope and T₁ the early units are much cheaper than under unit theory (unit 2 costs 2 × slope − 1 times T₁, 0.6 T₁ at 80 %) and the gap narrows as X grows.

GAO's advice on choosing: follow the analogous system, the industry (some space systems are modelled by cumulative-average theory), the contractor's own practice, and the production environment (cumulative averaging smooths a start with prototype tooling and early design changes; unit theory fits a well-prepared line). The tool reports both so the choice is visible.

Lot costs. A production lot of units F to L costs the exact sum of its unit costs. Estimators also use the algebraic lot midpoint, the unit Q whose cost equals the lot's average unit cost under the continuous curve, Q = [((L + ½)^1+b − (F − ½)^1+b) / ((1 + b)(L − F + 1))]^1/b, so that lot cost ≈ (L − F + 1) T₁ Q^b; the tool prints both and the error of the approximation, which is largest for a first lot of a few units (0.14 % for units 1 to 10 at 90 %) and negligible for later lots. Under cumulative-average theory the lot cost is C(L) − C(F − 1) exactly.

Fitting a curve. With unit, cost pairs the slope and T₁ come from ordinary least squares on ln Y against ln X (NASA CEH Appendix C, the power model); the fit is in logarithms, so the standard error is also given in cost units from the back-transformed fitted values (Table C-7 method). With lot data (first unit, last unit, lot cost) the unit-theory fit regresses the lot's average unit cost on its algebraic midpoint and iterates, since the midpoint depends on b; the cumulative-average fit regresses the cumulative average through each lot's last unit, which requires contiguous lots from unit 1. The two fits differ for the same data because they describe different curves; each theory's projection uses its own fit.

Not modelled: production-rate effects (Y = A X^b Q^r), breaks in production (Anderlohr learning-lost factor and the retrograde method), step-down from development to production, and end-of-production toe-up; GAO-20-195G Appendix VII describes each.

Parametric cost estimating relationships

A CER predicts cost from a driver such as weight, power or lines of code using the data of past systems (NASA CEH Appendix C). The linear form Y = b₀ + b₁ X is fitted by least squares, b₁ = S_xy/S_xx and b₀ = Ȳ − b₁ X̄; the power form Y = A X^B by the same regression on ln X and ln Y, which is the learning-curve equation with a different driver. The statistics the handbook asks for before a CER is used:

• R², the share of the cost variation explained by the driver;
• the standard error of the estimate SE = √(SSE/(n − 2)), in cost units, and the coefficient of variation CV = SE/Ȳ, which should be under 20 % and preferably under 10 %;
• the t test of the slope, t = b₁/(SE/√S_xx) on n − 2 degrees of freedom: a slope not significantly different from zero means the driver does not predict cost;
• for the power model, R² and SE in log space are not comparable with the linear model's, so the SE is recomputed in cost units from the fitted values; the model with the smaller unit-space SE is preferred (CEH C.2.2.2).

The prediction at a new driver value X₀ carries a prediction interval for a single new system, Ŷ ± t_{1−α/2, n−2} SE √(1 + 1/n + (X₀ − X̄)²/S_xx); for the power model the interval is formed in log space and exponentiated, so it is asymmetric. A driver value outside the range of the data is an extrapolation, flagged in the output (CEH C.2.1.2); a CER's intercept usually has no physical meaning for the same reason. Rounding matters: the handbook's example (Table C-5, storage tanks) prints the intercept as 0.26 after rounding the slope to 7.75; the unrounded value is 0.254.

Life-cycle cost

The life-cycle cost is the sum of acquisition (research, development, investment), operating and support, and disposal costs over the period of analysis (NASA CEH Appendix N). The year table takes those three columns for each year, with an optional saving (benefit) column or a constant annual saving; a salvage value is a negative disposal cost. Two dollar conventions are kept apart (CEH main body Table 6):

• Constant-year (base-year) dollars carry the purchasing power of one year and are the right basis for comparing alternatives; the table is entered in them.
• Then-year (as-spent, real-year) dollars include the inflation of each year and are what a budget must provide: cost_TY = cost_CY (1 + g)^t with g the escalation rate and t = year − base year.

Discounting recognises that a dollar now is worth more than a dollar later. The present value of a flow at time t is its amount divided by (1 + r)^t; with constant-year flows r is the real discount rate (OMB Circular A-94 for federal analyses). The equivalent nominal rate (1 + r)(1 + g) − 1 applied to the then-year flows gives the same present value, which is why escalation does not enter the NPV. The timing option sets where a year's flow sits: the end of the year, the middle of the year, or the NASA Appendix N convention of investment and salvage at the end of the year with recurring costs at mid-year. A flow at t ≤ 0 sits at time zero and is not discounted.

Measures of merit (CEH N.5.3.5):

• Net present value, PV of savings less PV of costs, the preferred measure when the alternatives have equal lives; the savings-to-investment ratio PV_S/PV_C and the return on investment NPV/PV_C.
• Equivalent uniform annual cost, the present value spread over n years by the capital recovery factor CRF = r(1 + r)ⁿ/((1 + r)ⁿ − 1) (CEH eq. N-7); mid-year flows use (1 + r)^n−½ in the numerator (eq. N-8). It is the measure for alternatives of unequal lives: the handbook's lease-versus-buy example has NPVs of −277,242 over 6 years and −375,739 over 8 years, which the EUAW turns into 50,000 and 52,449 a year. The period n is the last year index of the table.
• Payback, the time at which the cumulative saving first equals the cumulative cost, interpolated within the year; the discounted form uses the present values. It ignores everything after the payback year, so it ranks alternatives badly and is reported as a supplement.

Reading the results

The learning-curve section gives the cost of unit N, the cumulative total and average by both theories, the lot table and a log-log chart on which both theories are straight lines (the cumulative-average theory's unit curve bends at the start). The CER section gives both fits with their statistics, the prediction with its interval and the data against the fitted curves. The life-cycle section gives the year table in constant-year and then-year dollars with the discount factors and present values, the measures of merit and the cumulative discounted cost chart, whose net curve crosses zero at the discounted payback.

Default case

A 90 % unit-theory curve with T₁ = 1,000 over 100 units in lots 1–10, 11–50 and 51–100 (unit 100 costs 496.59, the 100 units 58,141); the storage-tank CER of NASA CEH Appendix C Table C-5, whose linear fit Y = 0.26 + 7.75 X predicts 17.31 ($K) at 2.2 tons with R² = 0.90, SE = 2.77 and CV = 0.14; and the 10-year program of Appendix N Table N-7 at 7 %, with PV of costs 106.40, PV of benefits 142.41, NPV +36.01 and SIR 1.34 (the handbook's total row prints the benefits as 270, the column it discounts sums to 230). The curve-data box holds the handbook's Table C-6 unit costs, whose power fit is Y = 99.98 X^−0.3222, an 80 % curve, when the fit option is chosen.