All About Learning Curves

1
All About Learning Curves

• Evin Stump P.E.
• SCEA Conference 2002

2
This 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

3
Background-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

4
Background-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)

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Critically Important in 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
• Butany 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

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Some 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

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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)

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Popular Models

• Many learning models proposed, but only two in
  common use
• Wrights 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

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Underlying Power Law
The power law formula is the basis of both
models
b is called the natural slopeit represents the
rate of learning always a negative number except
for (rare) forgetting
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.)
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Interpretation of y

• 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

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Power 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

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Notation for the U Model

• A helpful notation for the U model is

Hn H1nb
Hours to build unit n
Note the notation T1 is commonly used for what
is here designated H1. In the paper, T is
reserved for total hours.
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Notation for the CA Model

• A helpful notation for the CA model is

An H1nb
Average hours per unit to build the first n units
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Natural 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

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Relationship Between S and b

• Relationship between S and b is defined as

The basic relationship
b log(S/100)/log(2) (logarithms to any base)
Solving for S yields
S 10b log(2)2 (logarithm to base 10)

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

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Understanding S

• In industry, S typically ranges from 70 to 100
• Its counterintuitive, but 100 does not mean
  furiously rapid learningit means no learning at
  all (dont blame me, I didnt 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

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Nifty 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.
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Math Form of Nifty Results

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

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

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Areas 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

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Lore 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.)

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Lore of the Slope-2

• 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.

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Lore of the Slope-3

• Guidelines (cont.)
• The following typical values assume repetitive
  operations. They are not valid if operations are
  sporadic, as in a job shop environment.

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Lore of the Slope-4

• 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 80s is typical for purchased
  parts, with 85 a reasonable average value.

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Lore of the Slope-5

• 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.

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Lore of the Slope-6

• 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

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Error 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

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Error 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 S90 when you should have
  chosen S91, and your production quantity is
  1,000, your hours estimate will be about 12 too
  low (R-1 in plot)

Curve valid for both U and CA models
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What 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

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What You Can Estimate-2

• 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

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What You Can Estimate-3

• Quantities you can estimate (cont.)
• Labor profiles given a production scenario
• Effects of breaks in production
• Trading off design and production alternatives

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Interruption 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 validwhat 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

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Interruption of Production-2

• Andelohr hypothesizes five components of learning
  that can be differently affected by various
  interruption scenariosthey are
• Personnel learning
• Supervisory learning
• Continuity of production
• Methods
• Special tooling

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Interruption of Production-3

• 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

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Interruption of Production-4

• Here is a typical result from an Andelohr
  analysis

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Fitting 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

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Fitting Learning Curves to Production Data-2

• Typical unit data

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Fitting Learning Curves to Production Data-3

• Typical block data

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Fitting Learning Curves to Production Data-4

• 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)

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Fitting Learning Curves to Production Data-5

• A learning curve cannot be fitted to one block of
  datathere 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 blockstoo complex to describe here

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Tradeoff 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
  slopesother aspects may be different as well,
  such as labor rates
• Casting vs. machining
• Aluminum vs. composites
• Make vs. buy

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Tradeoff 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

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Summary

• 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

Thank you!
Evin J. Stump Galorath Incorporated 310-414-3222
x628 estump_at_galorath.com