Learning Curves

1
Learning Curves

• Chapter 17

2
Learning Curve Analysis

• Developed as a tool to estimate the recurring
  costs in a production process
• recurring costs those costs incurred on each
  unit of production
• Dominant factor in learning theory is direct
  labor
• based on the common observation that as a task is
  accomplished several times, it can be be
  completed in shorter periods of time
• each time you perform a task, you become better
  at it and accomplish the task faster than the
  previous time
• Other possible factors
• management learning, production improvements such
  as tooling, engineering

3
Learning Theory
Unit Cost
Qty
4
Log-Log Plot of Linear Data
Unit Cost
Qty
5
Linear Plot of Log Data
Log Unit Cost
Log Qty
6
Learning Theory

• Two variations
• Cumulative Average Theory
• Unit Theory

7
Cumulative Average Theory

• If there is learning in the production process,
  the cumulative average cost of some doubled unit
  equals the cumulative average cost of the
  undoubled unit times the slope of the learning
  curve
• Described by T. P. Wright in 1936
• based on examination of WW I aircraft production
  costs
• Aircraft companies and DoD were interested in the
  regular and predictable nature of the reduction
  in production costs that Wright observed
• implied that a fixed amount of labor and
  facilities would produce greater and greater
  quantities in successive periods

8
Unit Theory

• If there is learning in the production process,
  the cost of some doubled unit equals the cost of
  the undoubled unit times the slope of the
  learning curve
• Credited to J. R. Crawford in 1947
• led a study of WWII airframe production
  commissioned by USAF to validate learning curve
  theory

9
Basic Concept of Unit Theory

• As the quantity of units produced doubles, the
  cost1 to produce a unit is decreased by a
  constant percentage
• For an 80 learning curve, there is a 20
  decrease in cost each time that the number of
  units produced doubles
• the cost of unit 2 is 80 of the cost of unit 1
• the cost of unit 4 is 80 of the cost of unit 2
• the cost of unit 8 is 80 of the cost of unit 4,
  etc.

1 The Cost of a unit can be expressed in dollars,
labor hours, or other units of measurement.
10
80 Unit Learning Curve
Unit Cost
Log Unit Cost
Log Qty
Qty
11
Unit Theory

• Defined by the equation Yx Axb
• where
• Yx the cost of unit x (dependent
  variable)
• A the theoretical cost of unit 1
  (a.k.a. T1)
• x the unit number (independent
  variable)
• b a constant representing the slope
  (slope 2b)

12
Learning Parameter

• In practice, -0.5 lt b lt -0.05
• corresponds roughly with learning curves between
  70 and 96
• learning parameter largely determined by the type
  of industry and the degree of automation
• for b 0, the equation simplifies to Y A which
  means any unit costs the same as the first unit.
  In this case, the learning curve is a horizontal
  line and there is no learning.
• referred to as a 100 learning curve

13
Learning Curve Slope versus the Learning Parameter

• As the number of units doubles, the unit cost
  is reduced by a constant percentage which is
  referred to as the slope of the learning curve
• Cost of unit 2n (Cost of unit n) x (Slope of
  learning curve)
• Taking the natural log of both sides
• ln (slope) b x ln (2)
• b ln(slope)/ln(2)
• For a typical 80 learning curve
• ln (.8) b x ln (2)
• b ln(.8)/ln(2)

14
Slope and 1st Unit Cost

• To use a learning curve for a cost estimate, a
  slope and 1st unit cost are required
• slope may be derived from analogous production
  situations, industry averages, historical slopes
  for the same production site, or historical data
  from previous production quantities
• 1st unit costs may be derived from engineering
  estimates, CERs, or historical data from previous
  production quantities

15
Slope and 1st Unit Costfrom Historical Data

• When historical production data is available,
  slope and 1st unit cost can be calculated by
  using the learning curve equation
• Yx Axb
• take the natural log of both sides
• ln (Yx) ln (A) b ln (x)
• rewrite as Y A b X and solve for A and b
  using simple linear regression
• A eA
• no transformation for b required

16
Example

• Given the following historical data, find the
  Unit learning curve equation which describes this
  production environment. Use this equation to
  predict the cost (in hours) of the 150th unit and
  find the slope of the curve.

17
Example

• Or, using the Regression Add-In in Excel...

18
Example

• The equation which describes this data can be
  written
• Yx Axb
• A e4.618 101.30
• b -0.32546
• Yx 101.30(x)-0.32546
• Solving for the cost (hours) of the 150th unit
• Y150 101.30(150)-0.32546
• Y150 19.83 hours
• Slope of this learning curve 2b
• slope 2-0.32546 .7980 79.8

19
Estimating Lot Costs

• After finding the learning curve which best
  models the production situation, the estimator
  must now use this learning curve to estimate the
  cost of future units.
• Rarely are we asked to estimate the cost of just
  one unit. Rather, we usually need to estimate
  lot costs.
• This is calculated using a cumulative total cost
  equation
• where CTN the cumulative total cost of N units
• CTN may be approximated using the following
  equation

20
Estimating Lot Costs

• Compute the cost (in hours) of the first 150
  units from the previous example
• To compute the total cost of a specific lot with
  first unit F and last unit L
• approximated by

21
Estimating Lot Costs

• Compute the cost (in hours) of the lot containing
  units 26 through 75 from the previous example

-
22
Fitting a Curve Using Lot Data

• Cost data is generally reported for production
  lots (i.e., lot cost and units per lot), not
  individual units
• Lot data must be adjusted since learning curve
  calculations require a unit number and its
  associated unit cost
• Unit number and unit cost for a lot are
  represented by algebraic lot midpoint (LMP) and
  average unit cost (AUC)

23
Fitting a Curve Using Lot Data

• The Algebraic Lot Midpoint (LMP) is defined as
  the theoretical unit whose cost is equal to the
  average unit cost for that lot on the learning
  curve.
• Calculation of the exact LMP is an iterative
  process. If learning curve software is
  unavailable, solve by approximation
• For the first lot (the lot starting at unit 1)
• If lot size lt 10, then LMP Lot Size/2
• If lot size ? 10, then LMP Lot Size/3
• For all subsequent lots

24
Fitting a Curve Using Lot Data

• The LMP then becomes the independent variable (X)
  which can be transformed logarithmically and used
  in our simple linear regression equations to find
  the learning curve for our production situation.
• The dependent variable (Y) to be used is the AUC
  which can be found by
• The dependent variable (Y) must also be
  transformed logarithmically before we use it in
  the regression equations.

25
Example

• Given the following historical production data on
  a tank turret assembly, find the Unit Learning
  Curve equation which best models this production
  environment and estimate the cost (in man-hours)
  for the seventh production lot of 75 assemblies
  which are to be purchased in the next fiscal
  year.

26
Solution

• The Unit Learning Curve equation
• Yx 3533.22x-0.217

27
Solution

• To estimate the cost (in hours) of the 7th
  production lot
• the units included in the 7th lot are 216 - 290

28
Cumulative Average Theory

• As the cumulative quantity of units produced
  doubles, the average cost of all units produced
  to date is decreased by a constant percentage
• For an 80 learning curve, there is a 20
  decrease in average cost each time that the
  cumulative quantity produced doubles
• the average cost of 2 units is 80 of the cost of
  1 unit
• the average cost of 4 units is 80 of the average
  cost of 2 units
• the average cost of 8 units is 80 of the average
  cost of 4 units, etc.

29
Cumulative Average Learning Theory

• Defined by the equation YN AN b
• where
• YN the average cost of N units
• A the theoretical cost of unit 1
• N the cumulative number of units
  produced
• b a constant representing the slope
  (slope 2b)
• Used in situations where the initial production
  of an item is expected to have large variations
  in cost due to
• use of soft or prototype tooling
• inadequate supplier base established
• early design changes
• short lead times
• This theory is preferred in these situations
  because the effect of averaging the production
  costs smoothes out initial cost variations.

30
80 Cumulative Average Curve
Cum Avg Cost
Cum Qty
31
Learning Curve Slope versus the Learning Parameter

• As the number of units doubles, the average
  unit cost is reduced by a constant percentage
  which is referred to as the slope of the learning
  curve
• Average Cost of units 1 thru 2n
• Slope of learning curve x Average Cost of
  units 1 thru n
• Taking the natural log of both sides
• ln (slope) b x ln (2)
• bln(slope)/ln(2)

32
Slope and 1st Unit Cost

• To use a learning curve for a cost estimate, a
  slope and 1st unit cost are required
• slope may be derived from analogous production
  situations, industry averages, historical slopes
  for the same production site, or historical data
  from previous production quantities
• 1st unit costs may be derived from engineering
  estimates, CERs, or historical data from previous
  production quantities

33
Slope and 1st Unit Costfrom Historical Data

• When historical production data is available,
  slope and 1st unit cost can be calculated by
  using the learning curve equation
• YN ANb
• take the natural log of both sides
• ln (YN) ln (A) b ln (N)
• rewrite as Y A b N and solve for A and b
  using simple linear regression
• A eA
• no transform for b required

34
Example

• The equation which best fits this data can be
  written
• Yx 12.278(N)-0.33354
• Slope of this learning curve 2b
• slope 2-0.33354 .7936

35
Example

• Or, using the Regression Add-In in Excel...

36
Estimating Lot Costs

• The learning curve which best fits the production
  data must be used to estimate the cost of future
  units
• usually must estimate the cost of units grouped
  into a production lot
• lot cost equations can be derived from the basic
  equation YN AN b which gives the average cost
  of N units
• the total cost of N units can be computed by
  multiplying the average cost of N units by the
  number of units N
• where CTN the cumulative total cost of N
  units
• the cost of unit N is approximated by (1 b)AN b

37
Estimating Lot Costs

• To compute the total cost of a specific lot with
  first unit F and last unit L

38
Example

• Given the following historical production data on
  a tank turret assembly, find the Cum Avg Learning
  Curve equation which best models this production
  and estimate the cost (in man-hours) for the
  seventh production lot of 75 assemblies which are
  to be purchased in the next fiscal year.

39
Solution
The Cum Avg Learning Curve equation YN
4359.43N-0.21048
40
Solution

• To estimate the cost of the 7th production lot
• the units included in the 7th lot are 216 - 290

41
Unit Versus Cum Avg
Cost
Cum Avg LC
Unit LC
Qty
42
Unit Versus Cum Avg
Log Cost
Unit LC
Cum Avg LC
Log Qty
43
Unit versus Cum Avg

• Since the Cum Avg curve is based on the average
  cost of a production quantity rather than the
  cost of a particular unit, it is less responsive
  to cost trends than the unit cost curve.
• A sizable change in the cost of any unit or lot
  of units is required before a change is reflected
  in the Cum Avg curve.
• Cum Avg curve is smoother and always has a higher
  r2
• Since cost generally decreases, the Unit cost
  curve will roughly parallel the Cum Avg curve but
  will always lie below it.
• Government negotiator prefers to use Unit cost
  curve since it is lower and more responsive to
  recent trends

44
Learning Curve Selection

• Type of learning curve to use is an important
  decision. Factors to consider include
• Analogous Systems
• systems which are similar in form, function, or
  development/production process may provide
  justification for choosing one theory over
  another
• Industry Standards
• certain industries gravitate toward one theory
  versus another
• Historical Experience
• some defense contractors have a history of using
  one theory versus another because it has been
  shown to best model that contractors production
  process

45
Learning Curve Selection

• Expected Production Environment
• certain production environments favor one theory
  over another
• Cum Avg contractor is starting production with
  prototype tooling, has an inadequate supplier
  base established, expects early design changes,
  or is subject to short lead-times
• Unit curve contractor is well prepared to begin
  production in terms of tooling, suppliers,
  lead-times, etc.
• Statistical Measures
• best fit, highest r2

46
Production Breaks

• Production breaks can occur in a program due to
  funding delays or technical problems.
• How much of the learning achieved has been lost
  (forgotten) due to the break in production?
• How will this lost learning impact the costs of
  future production items?
• The first question can be answered by using the
  Anderlohr Method for estimating the learning
  lost.
• The second question can then be answered by using
  the Retrograde Method to reset the learning
  curve.

47
Anderlohr Method

• To assess the impact on cost of a production
  break, it is first necessary to quantify how much
  learning was achieved prior to the break, and
  then quantify how much of that learning was lost
  due to the break.
• George Anderlohr, a Defense Contract
  Administration Services (DCAS) employee in the
  1960s, divided the learning lost due to a
  production break into five categories
• Personnel learning
• Supervisory learning
• Continuity of productivity
• Methods
• Special tooling.

48
Anderlohr Method

• Each production situation must be examined and a
  weight assigned to each category. An example
  weighting scheme for a helicopter production line
  might be

49
Anderlohr Method

• To find the percentage of learning lost (Learning
  Lost Factor or LLF) we must find the learning
  lost in each category, and then calculate a
  weighted average based upon the previous weights.
• Example A contractor who assembles helicopters
  experiences a six-month break in production due
  to the delayed issuance of a follow-on production
  contract. The resident Defense Plant
  Representative Office (DPRO) conducted a survey
  of the contractor and provided the following
  information.
• During the break in production, the contractor
  transferred many of his resources to commercial
  and other defense programs. As a result the
  following can be expected when production resumes
  on the helicopter program

50
Anderlohr Method Example

• 75 of the production personnel are expected to
  return to this program, the remaining 25 will be
  new hires or transfers from other programs.
• 90 of the supervisors are expected to return to
  this program, the remainder will be recent
  promotes and transfers.
• During the production break, two of the four
  assembly lines were torn down and converted to
  other uses, these assembly lines will have to be
  reassembled for the follow-on contract.
• An inventory of tools revealed that 5 of the
  tooling will have to be replaced due to loss,
  wear and breakage.
• Also during the break, the contractor upgraded
  some capital equipment on the assembly lines
  requiring modifications to 7 of the shop
  instructions
• Finally, it is estimated that the assembly line
  workers will lose 35 of their skill and
  dexterity, and the supervisors will lose 10 of
  the skills needed for this program during the
  production break.

51
Anderlohr Method Example
52
Retrograde Method

• Once the Learning Lost Factor (LLF) has been
  estimated, we use the LLF to estimate the impact
  of the cost on future production using the
  Retrograde Method.

53
Retrograde Method

• The theory is that because you lose hours of
  learning, the LLF should be applied to the hours
  of learning that you achieved prior to the break.
• The result of the Anderlohr Method gives you the
  number of hours of learning lost.
• These hours can then be added to the cost of the
  first unit after the break on the original curve
  to yield an estimate of the cost (in hours) of
  that unit due to the break in production.
• Finally, we can then back up the curve
  (retrograde) to the point where production costs
  were equal to our new estimate.

54
Retrograde Example

• Continuing with our previous example
• Assume 10 helicopters were produced prior to the
  six month production break.
• The first helicopter required 10,000 man-hours to
  complete and the learning slope is estimated at
  88. Using the LLF from the previous example,
  estimate the cost of the next ten units which are
  to be produced in the next fiscal year.

55
Retrograde Example

• Step 1 - Find the amount of learning achieved to
  date.

56
Retrograde Example

• Step 2 - Estimate the number of hours of learning
  lost.
• In this case we achieved 3,460 hours of learning,
  but we lost 31 of that, or 1,073 hours, due to
  the break in production.

57
Retrograde Example

• Step 3 - Estimate the cost of the first unit
  after the break.
• The cost, in hours, of unit 11 is estimated by
  adding the cost of unit 11 on the original curve
  to the hours of learning lost found in the
  previous step.

58
Retrograde Example

• Step 4 - Find the unit on the original curve
  which is approximately the same as the estimated
  cost, in hours, of the unit after the break.
• This can be done using actual data, but since the
  actual data contains some random error, it is
  best to use the unit cost equation to solve for
  X. In this case, X 5.

59
Retrograde Example

• Step 5 - Find the number of units of retrograde.
• The number of units of retrograde is how many
  units you need to back up the curve to reach the
  unit found in step 4. In this example, since the
  estimated cost of unit 11 is approximately the
  same as unit 5 on the original curve you need to
  back up 6 units to estimate the cost of unit 11
  and all subsequent units.

60
Retrograde Example

• Step 6 - Estimate lot costs after the break.
• This can be done by applying the retrograde
  number to our standard lot cost equation.
• To estimate the cost, in hours, of units 11
  through 20, we subtract the units of retrograde
  from the units in question and instead solve for
  the cost of units 5 through 14.
• Therefore, our estimate of cost for the next 10
  helicopters is 66,753 man-hours.

61
Step-Down Functions

• A Step-Down Function is a method of estimating
  the theoretical first unit production cost based
  upon prototype (development) cost data.
• It has been found, in general, that the unit cost
  of a prototype is more expensive than the first
  unit cost of a corresponding production model.
• The ratio of production first unit cost to
  prototype average unit cost is known as a
  Step-Down factor.
• An estimate for the Step-Down factor for a given
  weapon system can be found by examining
  historical similar weapon systems and developing
  a cost estimating relationship, with prototype
  average unit cost as the independent variable.
• Once an appropriate CER is developed, it can be
  used with actual or estimated prototype costs to
  estimate the first unit production cost.

62
Step-Down Example

• We desire to estimate the first unit production
  cost for a new missile radar system (APGX-99).
  The slope of the system is expected to be a 95
  unit curve. The estimated average prototype cost
  is expected to be 3.5M for 8 prototype radars.
• The following historical data on similar radar
  systems has been collected

63
Step-Down Example

• Because the data gives production costs for unit
  150, we can develop our CER based on unit 150 and
  then back up the curve to the first unit.
• Using Simple Linear Regression, we find the
  relationship between development average cost
  (DAC) and the unit 150 production cost (PR150) to
  be

64
Step-Down Example

• We can now estimate our first unit cost using our
  unit cost equation and the (assumed) 95 slope as
  follows
• This result gives an estimate for the first unit
  production cost of 905,766. We have stepped
  down from a development cost estimate to a
  production cost estimate.

65
Production Rate

• A variation of the Unit Learning Curve Model
• Adds production rate as a second variable
• unit quantity costs should decrease when the
  rate of production increases as well as when the
  quantity produced increases
• two independent effects
• Model Yx AxbQc
• where Yx the cost of unit x (dependent
  variable)
• A the theoretical cost of 1st unit
• x the unit number
• b a constant representing the slope (slope
  2b)
• Q rate of production (quantity per period
  or lot)
• c rate coefficient (rate slope 2c)

66
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67
Rate Model Advantages

• It directly models cost reductions which are
  achieved through economies of scale
• quantity discounts received when ordering bulk
  quantities
• reduced ordering and processing costs
• reduced shipping, receiving, and inspection costs

68
Rate Model Weaknesses

• Appropriate production rate (i.e., annual,
  quarterly, monthly) is not always clear
• If Q is always increasing, there tends to be a
  high degree of collinearity between the quantity
  and rate variables
• Always estimates decreasing unit costs for
  increasing production rates
• when a manufacturers capacity is exceeded, unit
  costs generally increase due to costs of
  overtime, hiring/training new workers, purchase
  of new capital, etc.

69
When to Consider a Rate Model

• Production involves relatively simple components
  for which lot size is a much greater cost driver
  than cumulative quantity
• When production is taking place well down the
  learning curve where it flattens out
• When there is a major change in production rate

70
Model Selection Example

• Using the historical airframe data below for a
  Navy aircraft program, estimate the learning
  curve equations using
• Unit theory
• Cumulative Average Theory
• Rate Theory

71
Model Selection Example

• Unit Theory results...

72
Model Selection Example

• Cumulative Average Theory results...

73
Model Selection Example

• Production Rate Theory results...

74
Model Selection Example

• Which model should we use?
• From a purely statistical point of view, we
  prefer the Cumulative Average Theory model, since
  it gives the best statistics.
• The real answer may depend on other issues.