© 2007 Pearson Education Learning Curve Analysis Supplement G

Learning CurveAnalysis

Supplement GSupplement G

Learning Curves

0.30 0.30 –

0.25 0.25 –

0.20 0.20 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

0 0 –| | | | | |

5050 100100 150150 200200 250250 300300

Learning curveLearning curve

Cumulative units producedCumulative units produced

Learning Curves

0.30 0.30 –

0.25 –0.25 –

0.20 0.20 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

0 0 –

| | | | | |5050 100100 150150 200200 250250 300300

Learning period

Showing the learning period

Learning Curves

0.30 0.30 –

0.25 0.25 –

0.20 0.20 –

0.15 0.15 –

0.10 0.10 –

0.05 0.05 –

0 0 –| | | | | |

5050 100100 150150 200200 250250 300300

Learning Learning periodperiod

Standard time

Showing the learning period and the time when standards are calculated

Developing Learning Curves

In developing learning curves we make the following assumptions:

The direct labor required to produce the n + 1st unit will always be less than the direct time of labor required for the nth unit.

Direct labor requirements will decrease at a declining rate as cumulative production increases.

The reduction in time will follow an exponential curve.

kn = k1nb

where k1 = direct labor hours for the 1st unit

n = cumulative number of units produced b = log r / log 2

r = learning rate

80% Conversion Factors for the Cumulative Average Number of

Direct Labor Hours per Unit

90% Conversion Factors for the Cumulative Average Number of

Direct Labor Hours per Unit

Example G.1 Developing the 80% Learning Curve

Manufacturer of diesel locomotives:

Labor hours required for first unit = 50,000

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

4040 8080 120120 160160 200200 240240 280280Cumulative units producedCumulative units produced

Example G.1 Estimating Direct Labor Requirements

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Example G.1 using the formula

Labor hours required for 40th unit

k40 = 50,000(40)(log 0.8)/(log 2)

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

k40 = 50,000(40)-0.322

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

k40 = 50,000(0.30488)

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

k40 = 15,244 hours

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

k40 = 15,244 hours

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

k40 = 15,244 hours

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Cumulative average labor hours =

Learning rate = 80%n

1 1.000002 0.900003 0.83403. .. .. .

38 0.4363439 0.4330440 0.4298464 0.37382

128 0.30269

80% Learning Rate(n = cumulative production)

Example G.1 using Conversion Factors

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Cumulative average labor hours = 50,000(0.42984)

1 1.000002 0.900003 0.83403. .. .. .

38 0.4363439 0.4330440 0.4298464 0.37382

128 0.30269

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Cumulative average labor hours = 21,492 hours

1 1.000002 0.900003 0.83403. .. .. .

38 0.4363439 0.4330440 0.4298464 0.37382

128 0.30269

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Example G.1 using Unit-doublings

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Second unit = 50,000(80%)

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Second unit = 40,000 hours

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Fourth unit = 40,000(80%)

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

Learning rate = 80%

Fourth unit = 32,000 hours

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

The 80% Learning Curve forExample G.1

50 50 –

40 40 –

30 30 –

20 20 –

10 10 –

0 0 –| | | | | | |

The 80% Learning Curve forExample G.1

Application G.1 Estimating Direct Labor Requirements

The 1st unit of a new product is expected to take 1,000 hours. The learning rate is 80%, how much time should the 50th unit take?

Units perUnits per CumulativeCumulativeMonthMonth MonthMonth UnitsUnits

11 22 2222 33 5533 55 101044 88 181855 1212 3030

Example G.2Estimating Labor Requirements

11 22 2222 33 5533 55 101044 88 181855 1212 3030

Cumulative CumulativeAverage Time Total Hours

Month per Unit for All Units

1 1.000002 0.950003 0.915404 0.889055 0.86784. .. .. .

30 0.6909064 0.62043

128 0.56069

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500

1 1.000002 0.950003 0.915404 0.889055 0.86784. .. .. .

30 0.6909064 0.62043

128 0.56069

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035

1 1.000002 0.950003 0.915404 0.889055 0.86784. .. .. .

30 0.6909064 0.62043

128 0.56069

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

1 1.000002 0.950003 0.915404 0.889055 0.86784. .. .. .

30 0.6909064 0.62043

128 0.56069

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,5002 30,000(0.86784) = 26,0353 30,000(0.79945) = 23,9834 30,000(0.74080) = 22,2245 30,000(0.69090) = 20,727

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,0353 30,000(0.79945) = 23,9834 30,000(0.74080) = 22,2245 30,000(0.69090) = 20,727

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,9834 30,000(0.74080) = 22,2245 30,000(0.69090) = 20,727

11 22 2222 33 5533 55 101044 88 181855 1212 3030

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

1 2 22 3 53 5 104 8 185 12 30

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

Month 1:

1 2 22 3 53 5 104 8 185 12 30

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

Month 1: 57,000 – 0 = 57,000 hours

1 2 22 3 53 5 104 8 185 12 30

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

Month 1: 57,000 – 0 = 57,000 hoursMonth 2: 130,175 – 57,000 = 73,175 hours

1 2 22 3 53 5 104 8 185 12 30

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

Month 1: 57,000 – 0 = 57,000 hoursMonth 2: 130,175 – 57,000 = 73,175 hoursMonth 3: 239,830 – 130,175 = 109,655 hoursMonth 4: 400,032 – 239,830 = 160,202 hoursMonth 5: 621,810 – 400,032 = 221,778 hours

1 2 22 3 53 5 104 8 185 12 30

1 30,000(0.95000) = 28,500 28,500(2) = 57,0002 30,000(0.86784) = 26,035 26,035(5) = 130,1753 30,000(0.79945) = 23,983 23,983(10) = 239,8304 30,000(0.74080) = 22,224 22,224(18) = 400,0325 30,000(0.69090) = 20,727 20,727(30) = 621,810

Month 1: 57,000 – 0 = 57,000 /150 = 380 employeesMonth 2: 130,175 – 57,000 = 73,175 /150 = 488 employeesMonth 3: 239,830 – 130,175 = 109,655 /150 = 731 employeesMonth 4: 400,032 – 239,830 = 160,202 /150 = 1068 employeesMonth 5: 621,810 – 400,032 = 221,778 /150 = 1479 employees

Application G.2 Estimating Cumulative Labor Hours

An example of using the learning model to test budget constraints:

A company has a contract to make a product for the

first time. The total budget for the 38-unit job is 15,000

hours. The first unit took 1000 hours, and the rate of

learning is expected to be 80 percent.

Do you think the 38-unit job can be completed within

the 15,000-hour budget?

How many additional hours would you need for a

second job of an 26 additional units?

Application G.2First 38-unit Job

Application G.2Second additional 26-unit Job

© 2007 Pearson Education Learning Curve Analysis Supplement G

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