Learning Curve and Forgetting Factor

Laboratory Exercise 1IE 3269 – POMR. R. Lindeke, Ph.D.

General Ideas on Learning

With proper design, the learning effect can be driven together to balance out the grow phase of the product lifecycle!

Learning is usually expressed as a % -- the lower the % the faster is the learning!

Learning represents the rate of improvement, if it is 20% each time the quantity is doubled, then the learning percent would be 80% (100-20=80).

While the learning curve emphasizes time, it can be easily extended to costs as well

Modeling Learning Behavior

The learning behavior of systems and individuals is consider to follow this model:

b is a decimal value (<1) that relates the improvement seen as repetitions increase

y = axb

y - time to assemble the xth unita - time to assemble the 1st unitb - measure of "Learning Rate"x - unit number

Working with data in the Learning Experiment

When building a Learning Curve, we should perform a linearization operation and find a “least squares” regression fit to our experimental data set

To linearize our data set, compute the Log (base 10 or base e) of observed times and unit number of repetition

The ‘intercept’ of the fitted line (log data plotted on a linear axis!) is the log(a) value for the linear fit (a as defined above)

The slope of the fitted line is the b value in the learning curve equation And note: Learning Rate = 10^(b*log2)

Computing ‘b’ – by hand uses selected data when observed unit number doubles

Using the learning ideas stated above b = log(time-ratiounits double)/log(2)

This equation can be used as a spot check, real slope, and thus real learning rate, is computed by regression using all our observations!

2

2

2

(2 ) (2 ) (1)

( ) (2)

Dividing (1) by (2):

(2 ) (2 )

( )

2

Solving for b:

log

log 2

b

b

bx

bx

bx

x

x

x

f x a x

f x ax

yf x a x

f x y ax

y

y

yy

b

Learning “In the Limit”

Mathematically, we can see that if we are positively learning, the b value is a negative number (indicating a decimal value b<1)

Eventually, the time to assembly or perform any learnable task will approach an asymptotic value that is controlled by the time it takes the ideal worker to accomplish all the required steps – this is a number that can be computed using (micro) motion analysis and the various steps needed

Our learning curves will eventually fail to predict expected times for performing an activity (they will eventually underestimate actual times)

Considering the Forgetting Factor

Like all human activities (consider your math or physics courses!!?!) if we stop doing something for awhile, we are ‘rusty’ when we start up again

Same ideas hold true for manufacturing!This idea is called the forgetting factor

(mathematically)

Computing FF Effects:

Forgetting Factor studies consider Production Rate (compared to production times as seen in learning curve)

The model of the forgetting factor computes the rate loss due to forgetting between batches

This value can be used (along with learning curve values) to project batch by batch production rates (and times)

The Forgetting Factor Model:

( 1)

( 1)

(1 )*( )

:

( 0 to1)

cn n cnc n

c n

cn

n

Y Y F Y Y

where

Y Starting for the Next Batch

Y Starting for Last Batch

Y Ending for Last Batch

F Forgeting Factor range

Rate

Rate

Rate

Using Learning & Forgetting to Project Forward:

.2688301w./FF

301 ./

301 ./

Assume an 83% Learning Curve where:

67.2 minutes; .2688

25%

producing in batchs of 300 units

1 1 1Y 1 .25 *67.2 67.267.2*3000.0149 .75*.0541 0.0554 u/min

therefore:

1

w FF

w FF

a b

F

Y

y Y

301 ./

1 18.0419 min.0554w FF

Continuing:

.2688301 ./ .

301

.2688301 ./ .

301

1 1 0.069 u/min14.49267.2*301due to forgetting 0.069 0.0554 0.0136 u/min

67.2*301 14.492

due to forgetting 3.549 min

Slope offset (on lin. curve due to forge

w o FF

w o FF

Y

Y

y

y

301 ./ 301 ./ .

tting)

301 to 600: log( ) log( )

1.2562 1.1611 .0951 (longer)

note "Logs" because this is where times are linear!

w FF w o FFy y

Projecting forward to 601 w. FF:

.2688log(67.2*600 ) 0.0951 log(12.039) 0.0951600 ./

1.176600 ./

600 ./600 ./

601w.'2' FF 301 ./ 600 ./ 301 ./

601w.'2' FF

10 10

10 14.988

1 1 0.0667 u/min14.988

Y 1 .25 *

Y 0.0554 0.75* 0.0

w FF

w FF

w FFw FF

w FF w FF w FF

y

y

Y y

Y Y Y

601w.'2' FF 601 / .

667 0.0554 0.0639 u/min

1y 15.656 min vs. 12.033 min.0639 w o FFy

This iterative computational method can be continued over any number of batches using a similar technique

Computing Batch Times (for planning)

12 2

11 2

2 1

1 12 1

2 1

1

2

1 12 2

1 12 2

( ) ( )

*[( ) ( ) ]

(1 )*( 1)

:

1 ( )

( )

,

N b

N

avg

b b

avg

avg

axy

N N

a N Ny

b N N

WHERE

y Average time to produce a unit in the batch

N st unit of run batch

N Last unit of run batch

a b from Learning rate equation

T

2 1( ) *( )avgotal run batch production time y N N

Using Batch Predictors:

These are effective during interrupted processing

They are obsolete once asymptotic times are reached (unit times are a constant)

They do not hold if forgetting factors are “in play”

Summary:

The idea of the Learning Curve is universal It has a greater impact on complex than simple production

systems Organizations can plan production around this effect

To ramp up to match product life cycle curves Organization must guard against the Forgetting Factor

Effect This is one of the reasons for modern CAM to be profitable Forgetting factors are less pronounced if the ways are

remembered by design Forgetting is less important if only similar are processed

regardless of batch time and time between batches Forgetting is affected greatly by the time away for a specific

product

Summary, Typical Learning Rates: By Operation Mix

75% hand assembly/25% machining = 80% learning

50% hand assembly/50% machining = 85% learning

25% hand assembly/75% machining = 90% learning

Summary, Typical Learning Rates: By Industry

Aerospace 85% Shipbuilding 80-85% Complex machine tools for new models 75-85% Repetitive electronics manufacturing 90-95% Repetitive machining or punch-press operations 90-

95% repetitive electrical operations 75-85% Repetitive welding operations 90% Raw materials 93-96% Purchased Parts 85-88%