ptas-11 stump_all about learning curves

Galorath Incorporated 2002

All About Learning CurvesAll About Learning Curves

Evin Stump P.E.Evin Stump P.E.SCEA Conference 2002SCEA Conference 2002

This Presentation vs. the PaperThis Presentation vs. the Paper

• Due to limitations of time, this presentation can only scratch the surface of the information provided in the paper on which it is based

• The paper develops 22 useful learning curve equations, has 41 fully worked examples, and illustrates about a dozen useful learning curve methodologies

Background-1Background-1

• “Learning” effect first noted by T.P. Wright in 1936; he created a “learning curve” math model• Used to estimate aircraft production labor in WW II,

and since then to estimate many kinds of repeated activities

• First example of true parametric estimating??• Basic idea:

• As people repeat a task again and again, the time it takes to do the task gradually decreases due to “learning”

• Rate of learning is greatest at first when “ignorance” is greatest; rate of learning decreases as ignorance decreases

Background-2Background-2

• At first, learning was attributed to increased motor skills in the workers as they repeated their tasks• Later it was realized that management also could

contribute to learning with better tools and processes

• This led to new names being applied to the curves, e.g., improvement, progress, startup, efficiency, etc.

• In this presentation we will stick with the original name: learning curves (management can learn too)

Critically Important Critically Important in Industrial Cost Analysisin Industrial Cost Analysis

• The learning effect can lead to very large reductions in cost as production progresses

• Finding ways to make learning faster can result in a huge competitive advantage• Starting production ahead of your competitors• Finding better processes and being faster to

implement them– But…any proposal to improve the learning rate usually

involves an investment– The cost of the investment should be traded off against

the savings caused by faster learning

Some Industrial UsesSome Industrial Uses

• Manufacturing labor of a repeated product

• Construction (repeated structures like spans of a bridge or tract houses)

• Creation of documents (e.g., engineering specs and drawings, manuals)

• Boring of tunnels

• Drilling of wells• Upgrades of existing

products• Purchase or raw

materials (improved yield, decreased scrap)

• Component procurement (suppliers have learning, too)

• Negotiations

Not Useful when…Not Useful when…

• …production is sporadic• Random overhauls• Small lot job shops

• …work is fully automated and there is no way to improve the production rate

• …rules & regulations limit the production rate

• …production quantities are very small • …each item produced is significantly

different from the preceding item (custom products)

Popular ModelsPopular Models

• Many learning models proposed, but only two in common use• Wright’s original model, called the unit (U) model

• A later model due to Crawford called the cumulative average (CA) model

• FAQ: which is best?• Properly used, roughly equal in accuracy

• Main difference is in difficulty of the math, but computers make this difference almost trivial

– Which one has the most difficult math depends on what you are doing

– For simple estimating, the CA model is easiest to use

Underlying Power LawUnderlying Power Law

y = axb

The two models differ in their interpretation of y (next chart)

a always represents the theoretical labor hours required to build the first unit produced (a positive number)

x always represents the number (count) of an item in the production sequence (unit #1, #2, #3, etc.)

b is called the natural slope—it represents the rate of learning [always a negative number except for (rare) “forgetting”]

The “power law” formula is the basis of both models

Interpretation of Interpretation of yy

• Unit (U) model• y is the labor hours

required to build unit #x

• Because of negative b, y decreases as x increases

• This decrease represents the learning effect

• Cumulative average (CA) model• y is the average labor

hours per unit required to build the first x units

• Because of negative b, y decreases as x increases

• This decrease represents the learning effect

Power Law PlotsPower Law Plots

• In log-log coordinates, a learning curve plots as a straight line• This is usually the best way to plot one because it

is easier to read

• The plot below is of the power law y = 100x-0.25

Power Law Plot Example

1

10

100

1 10 100 1000 10000 100000

x

y

Notation for the U ModelNotation for the U Model

• A helpful notation for the U model is:

Hn = H1nb

Hours to build unit #n

Hours to build unit #1

Unit number

Natural slope

Note: the notation T1 is commonly used for what is here designated H1. In the paper, T is reserved for “total hours.”

Notation for the CA ModelNotation for the CA Model

• A helpful notation for the CA model is:

An = H1nb

Average hours per unit to build the first n units

Hours to build unit #1

Unit number

Natural slope

Natural vs. Percentage SlopeNatural vs. Percentage Slope

• Learning curve calculations generally (but not always) require a value for b• b is typically a negative number between 0 and –1

• b is a mathematically appropriate but non-intuitive number for describing slope

• For convenience, analysts universally use (in conversations as opposed to calculations) another expression called “percentage slope,” where slope is a number between 0 and 100 (except that if “forgetting” occurs, percentage slope can exceed 100)

• We use S as a symbol for percentage slope

Relationship Between S and bRelationship Between S and b

• Relationship between S and b is defined as:

• Although these relationships appear a bit unfriendly, there is at least one good reason for them (as you will soon see)

b = log(S/100)/log(2) (logarithms to any base)

S =10b log(2)+2

(logarithm to base 10)

The basic relationship:

Solving for S yields:

Understanding SUnderstanding S

• In industry, S typically ranges from 70% to 100%

• It’s counterintuitive, but 100% does not mean furiously rapid learning—it means no learning at all (don’t blame me, I didn’t do it!)

• The highest rate of learning achieved in most industrial situations is about 70%

• A later chart will show some typical percentage values realized in practice

Nifty ResultsNifty Results

• Nifty results of the relationships defined on the previous chart (see paper for proof):

U Model: If the slope is S%, any doubling of the production quantity from some unit #n to another unit #2n results in a reduction in labor hours from Hn to S% of Hn.

CA Model: If the slope is S%, the average hours for units 1 through 2n are S% of the average hours for units 1 through n.

Math Form of Nifty ResultsMath Form of Nifty Results

• Nifty results on the previous chart can be expressed mathematically as follows:

H2n/Hn = S/100 U Model

A2n/An = S/100 CA Model

These relationships are very useful in fitting learning curves to unit historical production data.

What You Can Do with Learning CurvesWhat You Can Do with Learning Curves

• Subsequent charts will illustrate important considerations in using learning curves and valuable uses of learning curve relationships

• Due to limitations of time, these cannot be fully explored in this presentation, but all are explored in detail in the paper

• All illustrated uses can be done with either the U or the CA model, as is demonstrated in the paper

Areas We Will Review HereAreas We Will Review Here

• We will look briefly at each of the following areas that are discussed in detail in the paper (the paper looks at a few more)• Lore of the slope

• Error analysis

• What you can estimate

• Interruption of production

• Fitting learning curves to production data

• Tradeoff analysis with learning curves

Lore of the Slope-1Lore of the Slope-1

• Fit learning curves to historical data when available• This is usually the best source, but not always

• Guidelines for use when historical data are not available:• Operations that are fully automated tend to have

slopes of 100%, or a value very close to that (no learning can happen).

• Operations that are entirely manual tend to have slopes in the vicinity of 70% (maximum learning can happen). (cont.)


• Guidelines (cont.)• If an operation is 75% manual and 25% automated,

slopes in the vicinity of 80% are common.

• If it is 50% manual and 50% automated, expect about 85%.

• If it is 25% manual and 75% automated, expect about 90%.

• The average slope for the aircraft industry is about 85%. But there are departments in a typical aircraft factory that may depart substantially from that value.

• Shipbuilding slopes tend to run between 80 and 85%.

Manufacturing Activity Typical Slope %

Electronics 90-95

Machining 90-95

Electrical 75-85

Welding 88-92


• Guidelines (cont.)• The following typical values assume

repetitive operations. They are not valid if operations are sporadic, as in a job shop environment.


• Guidelines (cont.)• A slope of 93-96% is often applied to raw

materials, based on increasing procurement efficiencies, higher yields, and lower scrap rates as manufacturing progresses.

• A slope in the 80’s is typical for purchased parts, with 85% a reasonable average value.


• Guidelines (cont.)• When very large quantities will be built, slopes tend

to flatten, because manufacturing planners depend on economies of scale to build better tooling and use more automation.

• The flattening of slopes for large quantities is typically accompanied by a reduction in first unit hours. This effect has been used to estimate the amount that can be spent on automation. The answer sometimes comes out in favor of automation, but that is not always the case.


• Guidelines (cont.)• Slopes tend to be flatter if a project is closely

similar to a previous project, the time gap between them is not too large, and many of the same people will be involved. This is sometimes called the “heritage” effect.

• Experienced crews tend to have lower first unit costs than inexperienced crews, and since they are already knowledgeable, their learning rate tends to be less. Inexperienced crews tend to have higher first unit costs, and higher learning rates.

• For more “lore of the slope,” see the paper

Error Analysis-1Error Analysis-1

• Most common learning curve analysis errors• Choosing wrong value of H1

• Choosing wrong value of S (with resultant wrong value of b)

• H1 is always a simple multiplier• The percentage error in the hours estimate is the

same as the percentage error in H1

• b is an exponent; it can create a much larger % error in hours than the % error in b, especially at large production quantities

Error Analysis-2Error Analysis-2

• It can be shown (see paper) that the hours estimate error due to a one percentage point error in S is given approximately by the following curve:

• Example: if you choose S=90% when you should have chosen S=91%, and your production quantity is 1,000, your hours estimate will be about 12% too low (R-1 in plot)

R Versus N

1.00

1.05

1.10

1.15

1.20

1.25

1 10 100 1000 10000 100000

N

R

Curve valid for both U and CA models

What You Can Estimate-1What You Can Estimate-1

• The quantities listed here can all be estimated with either the U or the CA model

• They assume the slope is known or can be determined


• Quantities you can estimate• Labor hours for any unit given hours for any other

unit

• Labor hours for any contiguous block of units given the hours for any single unit

• Labor costs given labor hours and a labor rate

• Material costs if they follow a learning curve


• Quantities you can estimate (cont.)• Labor profiles given a production scenario

• Effects of breaks in production

• Trading off design and production alternatives

Interruption of Production-1Interruption of Production-1

• Production is interrupted for many reasons• When this happens, learning can be “lost”• The learning curve that was being followed

is no longer valid—what to do?• An answer has been provided by George

Andelohr, an industrial engineer• His structured process provides a realistic

basis for negotiations about the cost of interruption


• Andelohr hypothesizes five components of learning that can be differently affected by various interruption scenarios—they are:• Personnel learning

• Supervisory learning

• Continuity of production

• Methods

• Special tooling


• Each of these components is assigned a weight, and a loss of learning in hours is computed based on both objective fact and subjective opinion

• The lost hours are added to the first unit after the interruption, and the learning is “backed up” the curve to the unit that had that number of hours

• Production follows the new curve thus defined


• Here is a typical result from an Andelohr analysis:

Example Plot of Interrupted Learning

1000

10000

1 10 100

Units Produced

Ho

urs

Production break

Learning starts again at unit determined by amount of learning lost

Fitting Learning Curves to Production Data-1Fitting Learning Curves to Production Data-1

• If previous production data are available it is often the best source for learning slope for future projects

• Two types of data are encountered in practice• Unit data

• Block data

• Either a U or a CA model can be fitted to either type of data


• Typical unit data

Unit Hours

1 200

2 185

3 176

4 165

5 150

Etc.


• Typical block data

Units Hours

1-50 55,260

51-150 98,320

151-200 37,360


• Unit data is best but not always available• A simple technique for unit data that works

for both U and CA models is looking at doublings of productions quantity, and averaging

• This technique relies on two equations previously shown (see paper for details):

H2n/Hn = S/100 U Model

A2n/An = S/100 CA Model


• A learning curve cannot be fitted to one block of data—there must be at least two

• Fits for only two blocks are relatively simple, using derived formulas for total hours (see paper)

• Fits for three or more blocks generally employ regression analysis• A special technique is demonstrated in the paper

for the fitting the U model to three or more production blocks—too complex to describe here

Tradeoff Analysis with Learning Curves-1Tradeoff Analysis with Learning Curves-1

• Tradeoff analysis is a powerful way to use learning curves, probably not used as much as it should be

• An obvious application is to trade off manufacturing methods and materials• Different methods may have significantly different

first unit costs and learning slopes—other aspects may be different as well, such as labor rates

– Casting vs. machining– Aluminum vs. composites– Make vs. buy

Tradeoff Analysis with Learning Curves-2Tradeoff Analysis with Learning Curves-2

• More sophisticated tradeoffs involving learning curves• Target cost

• Time to market

• Labor skill mix

• Introduction of product changes

• All of these can be done using formulas from the paper

SummarySummary

• Background• Importance & uses• Models (U & CA)• Power law• Typical slopes• Error analysis• Things you can estimate• Interruption of production• Fitting to production data• Tradeoffs using learning curves

Evin J. Stump

Galorath Incorporated 310-414-3222 x628

[email protected]

ptas-11 stump_all about learning curves

Documents

learning rate of learning

learning curve math

learning curve plots

production sequence

later model

learning curves evin

unit number natural

y decreases