Happy’s Essential Skills: Learning Theory/Learning Curves

Estimated reading time: 14 minutes

                                                               F =C2/C1                                                                                                         (1)

Where C1 is the initial average unit cost and C2 is the average unit cost for double the reference quantity.  

From equation (1) it is evident that the higher the value of F, the less change in cost is to be expected due either to process maturity (automation, optimized setup, tooling, yields), or highly customized content, as might be expected from small lot quantities of complex rigid flex assemblies.

For an initial quantity Qo and a final quantity Q the number of “doublings” or fractions thereof for the total quantity produced is given by log(Q/Qo)/log(2). Therefore, the unit cost behavior as a function of quantity can be written as:

                                            C = C1*(F/100) ^ (log(Q/Qo) / log(2))                                                        (2a)

Where C is the unit cost after quantity Q units or lots, C1 is the first unit cost, and F is the experience factor in percent.

A value of 75 for F would be typical of very steep (fast) learning curves, in which process consolidation proceeds rapidly with corresponding reductions in changeover time, improvements in yields, etc. Equation (2a) is awkward to handle since the principal variable, Q, appears in the exponent. It can be rearranged (and simplified) by noting that in general a ^ log(b) is equivalent to b ^ log(a) since either expression can be written as e ^ [log(a)*log(b)]. An alternate and better form for equation (2a) is therefore:

                                                                            C = C1*q ^ k                                                                  (2b)

 Where                                  q = Q/Qo and k = log(F/100) / log(2)

The total cost, T, to produce a quantity Q units or lots can be obtained by integrating equation (2b) over the limits q = 0 to q = Q:                                                                   

                                                       T = C1* q ^ kdq = C1*Q ^ (k+ 1)/(k+ 1)                                       (3)

The average cost, a, per unit or lot quantity is the total cost divided by the quantity:

                                                                                   A = T/Q                                                                (4)

For processes where the experience factor is accurately known, the average cost is often used to quote a lot or piece price to be effective over the entire production. Suppose, for example, that a first lot of ten pieces is produced at a cost of $20.00 by a process with a known experience factor of 80%. What would be the predicted piece cost for 1,000 units? For F = 80%, k is found to be log (0.80)/log (2) = 20.3219, and for this case the “experience” quantity Q = 1,000/10 = 100.

Therefore:

                                                                     C = 20.00*100 ^ (-0.3219) = 4.5412

So that at the end of the run the production cost has declined to $4.54 per lot. The total cost, from equation (3), becomes:

                                                      T = 20*100 ^ (0.6781)/0.6781 = 669.7274

The average production cost per unit quantity (1 lot) is therefore T/Q = $6.70 and the piece cost is about $0.67. This approach can be used to create log-log plots for various experience factors, giving unit costs as a function of quantities and initial costs. For example, a process with 80% experience factor and an initial cost of 1.00 per unit can expect unit costs to decline to about 0.11 by the time 1,024 (2 ^ 10) units have been produced. This not atypical of the semiconductor industry, where F may be 75% or even less. At the other end of the scale a complex, low volume product may be 90% or higher. One-offs with highly customized assemblies will be as high as 100%: the product lifetime is too short (one-off) and the standardized process component(s) are too limited to offer meaningful improvement opportunities.

New Technologies—the Experience Factor^[1]

To use this analysis for new technologies it is necessary to determine the experience factor. This can done using a broader experience base than the simple doubling shown in equation (1) by flipping equation (2a) around [. . .] [LL1] provided the data are available, specifically:

                                           F = 10 ^ (log(2)*log(C/C1) /log(Q))                                                          (5)

If the production cost of a metal-core type insulated metal substrate LED multichip board was 2.00 when 10,000 pieces had been produced (C1) and the cost (C) is now 0.65 when 4,000,000 have been produced  (Q = 400), what is the experience factor F?

                                               F = 10 ^ (log(2)*log(0.65/2.00)/log(400))

Or:                                                                          F = 0.878

What will be the cost for the 20,000,000th piece when Q will be effectively 2,000 (20,000,000/10,000)?

                                                   k = log(0:878)/log(2) = 20:18771

                                                  C = 2.00*2000 ^ (20.18771) = 0.4801

This example assumes a limited degree of process innovation is necessary in the introduction of a new layout for the same function/substrate. As is often the case in printed circuit manufacturing, where the emphasis is less on products and more on capabilities built on standardized processes, the experience factor may be even higher than 88%. It is important to remember that the experience factor “F” does not imply any particular degree of expertise or mastery of the technology. It is simply an index of the expected stability of processing costs over the lifetime of the design.

 References

1. Burr, W., Pearne, N., “Learning curve theory and innovation,” Circuit World, Vol. 39, Issue 4, 2003, pp 169–173.
2. Transformative Learning (Jack Mezirow)
3. www.wikipedia.org
4. The Quintessential of Generative Learning Theory
5. The Learning Curve or Experience Curve, provided by James Martin.

Happy Holden has worked in printed circuit technology since 1970 with Hewlett-Packard, NanYa/Westwood, Merix, Foxconn and Gentex. Currently, he is the co-editor, with Clyde Coombs, of the Printed Circuit Handbook, 7th Ed. To contact Holden, click here.

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