1 Learning Curve. 2 Example Consider a product with the following data about the hours of labor...

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Learning CurveLearning Curve

Cumulative Production

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ExampleConsider a product with the following data about the hours of labor required to produce a unit:

Hours required to produce 1-st unit: 100

Hours required to produce 10-th unit: 48




As more and more units are produced, the hours of labor required to produce the most recent unit is lower and lower.

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Graph for Example

1 10010 4825 3575 25200 18

Hours Required to Produce

Most Recent UnitCumulative Production

Learning Curve

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100110

0 25 50 75 100 125 150 175 200 225


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Reasons for Continual Decease in the Number of Hours Required to

Produce the Most Recent Unit

On the previous slide, we observed that, as more and more units are produced, the hours required to produce the most recent unit is lower and lower.

What are some potential reasons why this occurs?

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What happens whencumulative production doubles?

The concept of a Learning Curve is motivated by the observation (in many diverse production environments) that, each time the cumulative production doubles, the hours required to produce the most recent unit decreases by approximately the same percentage.

For example, for an 80% learning curve,

If cumulative production doubles from 50 to 100, then the hours required to produce the 100-th unit is 80% of that for the 50-th unit.

If cumulative production doubles from 100 to 200, then the hours required to produce the 200-th unit is 80% of that for the 100-th unit.

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The Functional Formof a Learning Curve

To model the behavior described in the previous slides, we proceed as follows:

Let x = cumulative production

y = hours required to produce the x-th unit

Then, y = ax-b

where a and b are parameters defined as follows:

a = hours required to produce the 1-st unit

b = a value related to the percentage associated with the Learning Curve

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An 80% Learning Curve

Assume that production of the first unit required 100 hours and that there is an 80% Learning Curve.

Again, let

x = cumulative production


Then, mathematicians can show that the Learning Curve is

y = 100x-0.322

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An 80% Learning Curve(continued)

Hours RequiredCumulative to ProduceProduction Most Recent Unit

x y = 100x -0.322

1 100.0002 80.000

--- --- 4 64.000

--- --- 8 51.200

--- --- 16 40.960 --- --- 25 35.478 --- --- 32 32.768 --- --- 50 28.383 --- --- 64 26.214 --- --- 100 22.706 --- --- 128 20.972 --- --- 200 18.165

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A 70% Learning Curve

Assume that production of the first unit required 100 hours and that there is an 70% Learning Curve.

Again, let

x = cumulative production


Then, mathematicians can show that the Learning Curve is

y= 100x-0.515

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A 70% Learning Curve(continued)


x y = 100x -0.515

1 100.0002 70.000

--- --- 4 49.000

--- --- 8 34.300

--- --- 16 24.010 --- --- 25 19.083 --- --- 32 16.807 --- --- 50 13.358 --- --- 64 11.765 --- --- 100 9.351 --- --- 128 8.235 --- --- 200 6.546

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The Relationship Betweenb and p

The table below shows the relationship between the exponent b and p, the percentage associated with the Learning Curve:

Recall that the functional form for a Learning Curve is

y = ax-b

b 0.000 0.074 0.152 0.234 0.322 0.415 0.515 0.621 0.737 0.862 1.000 1.322 1.737 2.322 3.322p 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 40% 30% 20% 10%

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The Relationship Betweenb and p (continued)

There is a direct mathematical relationship between the exponent b in the equation y = ax-b and (p/100)%, where p is the percentage associated with the learning curve:

)2ln(*)%100/()2ln()100/ln( beppb ly,equivalentor,

For example, if p=75%, then 415.0)2ln()75.0ln( b

For example, if b=0.737, then 60.0)2ln(*737.0)%100/( ep

NOTE: e=2.7183… (never ending, like ¶)

ln(x) is the exponent of e that yields x.

That is, eln(x)=x

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Operational Applicationof the Leaning Curve

Assume that production of the 1-st unit required 100 hours, and assume that there is an 80% learning curve. Then, y = 100x-0.322.

Also, assume that cumulative production to date is 150 units.

The learning curve can be used to provide estimates of answers to questions about the production of the next 100 units.

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Operational Application of a Leaning Curve (continued)

Question 1: To produce the next 100 units, how many hours of labor will be required?

Question 2: With a labor force of 6 workers each working 40 hours per week, how long will it take to produce then next 100 units?

Question 3: To produce 100 units in 5 weeks with each worker working 40 hours per week, what should be the size of the labor force?

Question 4: To produce 100 units in 5 weeks using a work force of 6 workers, how many hours per week should each worker work?


x y = 100x -0.322

1 100.000 Cumulative --- --- Hours Required100 22.706 from 151-st Unit --- --- through Most Recent Unit150 19.928151 19.885 19.885152 19.843 39.728153 19.801 59.529154 19.759 79.288155 19.718 99.007 --- --- --- 200 18.165 948.644 --- --- --- 246 16.994 1755.483247 16.972 1772.455248 16.950 1789.404249 16.928 1806.332250 16.906 1823.238

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Effect of Sales’Annual Growth Rate

Assume that:

Three firms have the same 80% learning curve: y=100x-0.322

During Year 1, all three firms sold 5000 units.

The three firms have respective annual growth rates in sales of 5%, 10%, and 20%.

Compare the three firms at the end of Year 4.

Conclusion?

Cummulative Production At End of Year 4Hours Required to Produce

Most Recent Unit

x y =100 x -0.322

A 5% x = [1.00+(1.05)+(1.05)2+(1.05)3](5000) = 15,764 4.453

B 10% x = [1.00+(1.05)+(1.05)2+(1.05)3](5000) = 16,551 4.384

C 20% x = [1.00+(1.05)+(1.05)2+(1.05)3](5000) = 18,202 4.252

Firm

Annual Growth Rate

in Sales

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Effect of Sales’Annual Growth Rate (continued)

Learning Curve

0

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Learning Curve Firm A Firm B Firm C

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Strategic Applicationsof a Learning Curve

Frequent Decreases in Selling Price.

Each decrease in selling price increases your market share, which in turn leads to a “faster ride” down the learning curve, which in turn makes it tougher for your competitors.

Reinvest Increased ProfitsAs the hours required to produce the most recent unit continually decreases, the cost to produce the unit continually decreases. Therefore, your profits increase. You can reinvest the incremental profit to improve the product or the production process, or you can reinvest the incremental profit in another area of the firm.

As the hours required to produce the most recent unit continually decreases, the cost to produce the unit continually decreases. Therefore, you can frequently decrease the selling price without decreasing total profit.

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How do we determine the parameters of a Learning Curve?

From previous slides, we know that, to model a learning curve, we proceed as follows:

Let x = cumulative production


Then, y = ax-b

where a and b are parameters defined as follows:

a = hours required to produce the 1-st unit

b = a value related to the percentage associated with the learning curve

For a given set of data, how do we determine the specific values of a and b?

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Example

For the Learning curve y=ax-b, how do we determine the specific values of a and b?

We begin by taking the natural logs of both sides of y=ax-b.

1 4000

7 255025 185065 1500180 1170



Most Recent Unit

Learning Curve Data

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Note the linear relationship between ln(x) and ln(y).

This suggests taking the natural logs of the data.

)*ln()ln( xbaybaxy

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Example (continued)

Natural Log Natural Log

0.000 8.2941.946 7.8443.219 7.5234.174 7.313

5.193 7.065



Most Recent Unit

Note the approximate linear relationship between ln(Cumulative Production) and ln(Hours Required).

Natural Log of Learning Curve Dataln(Cumulative Production) versus ln(Hours Required)

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ln(Cumulative Production)

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We can use the statistical technique of Regression to determine the straight line that “best fits” the data.

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Example (continued)

Best Linear Fit (via Regression)ln(Cumulative Production) versus ln(Hours Required)

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ln(Data) Best Linear Fit

Using Excel’s Regression Tool, we obtain

ln(y) = 8.29642 – 0.23694 ln(x)

Intercept=8.29642

Negative of Slope = 0.23694

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Example (continued)

From the previous slide, we know

ln(y) = 8.29642 – 0.23694 ln(x)

So,

eln(y) = e[8.29642 – 0.23694 ln(x)]

or, equivalently, the equation for the Learning Curve is

y = 8.29642 x-0.23694

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Example (continued)

Learning Curve

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Data Learning Curve

y = 8.29642 x-0.23694

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Example (continued)

y = 8.29642 x-0.23694

b 0.000 -0.074 -0.152 -0.234 -0.322 -0.415 -0.515 -0.621 -0.737 -0.862 -1.000 -1.322 -1.737 -2.322 -3.322p 100% 95% 90% 85% 80% 75% 70% 65% 60% 55% 50% 40% 30% 20% 10%

So, in our example, we have a Learning Curve that is close to but just below an 85% learning curve.

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Excel Templatefor a Learning Curve

A B C D E F G H I2

3 Hours Required LN of Regression4 Cumulative to Produce Cumulative LN of Estimate of5 Production Most Recent Unit Production Hours Required Hours Required6 1 4000 0.000 8.294 4009.57 7 2550 1.946 7.844 2528.48 25 1850 3.219 7.523 1870.19 65 1500 4.174 7.313 1491.2

10 180 1170 5.193 7.065 1171.4

1112131415 SUMMARY OUTPUT1617 Regression Statistics18 Multiple R 0.99987412719 R Square 0.9997482720 Adjusted R Square 0.9996643621 Standard Error 0.00876481622 Observations 52324 ANOVA25 df SS MS F Significance F26 Regression 1 0.915298765 0.915298765 11914.53938 1.69521E-0627 Residual 3 0.000230466 7.6822E-0528 Total 4 0.9155292312930 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%31 Intercept 8.296416964 0.007427526 1116.982525 1.58245E-09 8.272779238 8.320054689 8.272779238 8.32005468932 X Variable 1 -0.236941604 0.002170714 -109.1537419 1.69521E-06 -0.243849792 -0.230033415 -0.243849792 -0.23003341533 4009.48134

Rows 15-32 generated using the menu selection "Tools, Data Analysis, Regression" with Input X-Range of E6:E10 Input Y-Range of F6:F10 Output Range of A15

=LN(B8) =LN(C8) =$B$33*(B8^$B$32)

=EXP(B31)

NOTE: Regression output in cells B31 and B32 shows that LN(Hours Required) = 8.296 - 0.237*LN(Cumulative Production) or, equivalently, (Hours Required) = 4009.5*[(Cumulative Production)^(-0.237)]

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A “Not So Nice” Example

In our example, there was a very close linear relationship between

ln(Cumulative Production) and ln(Hours Required)

This is NOT the typical situation.

A more typical situation is shown on the next slide.

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A “Not So Nice” Example(continued)

1 37217 271325 174765 1593

180 1099


Hours Required to Produce Most

Recent Unit

Natural Log Natural Log0.000 8.2221.946 7.9063.219 7.4664.174 7.3735.193 7.002


Hours Required to Produce Most

Recent Unit

Learning Curve Data

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Natual Log of Learning Curve Dataln(Cumulative Production) versus ln(Hours Required)

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Although the linear relationship in this example is not as strong as in the previous example, we proceed in the same manner.

Best Linear Fit (via Regression)ln(Cumulative Production) versus ln(Hours Required)

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ln(Data) Best Linear Fit

Learning Curve

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Data Learning Curve

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An approximate linear relationship such as the one below occurs for many products and services.

Natual Log of Learning Curve Dataln(Cumulative Production) versus ln(Hours Required)

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1 Learning Curve. 2 Example Consider a product with the following data about the hours of labor...

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