# Handout for three day Learning Curve Workshop

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1 Handout for three day Learning Curve Workshop Unit and Cumulative Average Formulations DAUMW (Credits to Professors Steve Malashevitz, Bo Williams, and prior faculty. Blame to Dr. Roland Kankey, Introduction to Learning or Cost Improvement Curves Some of you at one time or another may have purchased unassemled furniture kits. The etter kits will often provide you with the average assemly time. This time is ased on the numer of parts, steps, and/or operations associated with that item, let s say a cainet. Now what if you had ought four cainet kits and were planning to assemle the kits one-after-another. The average assemly time is ased on uilding one cainet, not a series of cainets. If the time to uild the first cainet was one hour we would not expect to take four hours to uild all four, ut less than four hours since we would proaly get more proficient with each cainet we uilt. You might say that the average time does not take into account the effect of quantity. When we estimate the cost 1 or price of an item, whether it is ased on a detailed cost uild-up, an analogy, catalog price, or a cost estimating relationship, the cost or price may also not address the effect of quantity. One well-known approach to modeling the quantity effect has een called the learning curve, cost improvement curve, or experience curve. This technique was first discussed in the journals of the 1930 s and continues as an industry standard today oth in commercial and non-commercial (government) applications. The general learning curve theory is that people and organizations learn to do things etter and more efficiently when performing repetitive tasks, and that under certain conditions there is a usale pattern to the learning. If the conditions of the learning are different, the pattern will likely e different. For that reason a numer of different learning curve formulations have een identified and used when appropriate. In this workshop we will address two of the more widely used formulations, the unit formulation and the cumulative average formulation, and we will discuss some alternative formulations. We will also assess the impact on cost improvement due to changes in the product or the process, and the impact of an interruption or reak in the production process y using the Anderlohr and retrograde techniques. This handout deals with the unit and cumulative average formulations. In what areas of cost should we see improvement? We generally associate cost improvement or learning with repetitive actions or processes such as those directly tied to producing an item over and over. We might categorize these costs as the recurring or variale costs that are a direct result of the quantity of units eing produced. Elements of manufacturing (material, tooling, farication and assemly) are well suited for 1 Cost is used generically here to represent the expenditure of resources which may e dollars or hours.

3 Cost Improvement Curves Workshop efficiency or they will cease to exist. Suppliers may even competitively price their initial product release at a loss etting on future cost improvements in order to discourage competitors from entering even new markets. These improvements must materialize for the supplier to turn a profit and survive. What does cost improvement look like and how do we model it? The term learning curve slope and equations like y ax are proaly very familiar to you, ut where did we get these concepts? Well, as early analysts oserved production data (e.g. manufacturing laor hours) they noted that the laor hours per unit or the cumulative average laor hours decreased as illustrated in the following diagram: Unit/Cum Avg Hour Manufacturing Laor Hours Well, if the trend etween hours and quantity is of this form, where does the idea of a constant learning curve slope come from and what does the slope represent? And why do people refer to the intercept as the value at unit one when the intercept is usually the value at zero? To answer these questions we must first concern ourselves with the shape of the curve. One method of dealing with such Quantity data is to transform the data so that the transformed values are linear. Another approach is to use a model form other than the linear model (y x) such as the multiplicative model (y 0 x 1 ). We will start off with a transformation that will eventually lead us to the multiplicative model. Proaly the most flexile and convenient transformation is the LOG (ase 10) or LN (ase e) of hours and quantity. This can e accomplished either y mathematical calculation of the logs or y graphing the values on log-log paper as is seen in this graph. Unit/Cum Avg Hour Manufacturing Laor Hours Quantity We can now see a line that could e characterized as having a constant slope. We usually define the slope as the change in Y given a change in. However, in log space the slope equates to a constant percent change in Y given a constant percent change in. The unit formulation states that as the total quantity of units doules, the cost decreases y a constant percent. So we have a constant percent decrease in Y (cost) and a Unit Hours

4 Learning Curve Workshop constant percent change in (quantity, which is douling or changing y 100%). Perhaps the est way to illustrate what the slope represents is to look at some of the data that was used to generate the aove graph. Since we are talking aout the change in hours as quantity doules let s look at the change in hours from unit 1 to unit 2. The hours decreased from 1000 to 700 or a decrease of 30%. This is called the rate of learning (ROL) or the rate of improvement. Another way of looking at this is to take a ratio of the hours at some quantity (Y ) versus the hours at twice that quantity (Y 2 ). Y Y Y or 70%.70 or 70% 1000 Y The 70% is the learning curve slope. Unit 2 is 70% of the value of unit 1, unit 4 is 70% of unit 2, and so on with each douling of quantity. Keep in mind that the learning curve is a trend in the data and that in this case the 70% slope represents an average improvement (decrease) of 30%. It can also e calculated as 1.00 ROL 1.3.7, or 100% - 30% 70%. Just as we ve seen that the learning curve slope is not the usual mathematical slope, the same is true aout the intercept. The intercept, which is usually associated with the point where 0 in unit space, is the point where 1 in log space. The value of Y at 1 is the first unit cost, sometimes referred to as T 1 or as A in many learning curve formulas. Let s take another look at our graph along with its associated trend line. As you can see, the relationship can e portrayed y a multiplicative equation often written as Y A, where: Y: the cost (or avg. cost) at unit x A: the first unit (T 1 ) cost (1000) : the unit numer Unit/Cum Avg Hour Manufacturing Laor Hours y 1000x Quantity : the slope coefficient which is: Log(slope) Log(2) or Ln(slope) Ln(2) Ln(.70) Ln(2) Where do we get the slope and the intercept (first unit or T 1 )? If there have een several production lots produced of an item we could derive the slope from the trend in the data. Another approach would e to look at company history for similar efforts and calculate the slopes from those efforts. Or perhaps we could use the slope from an 4

5 Cost Improvement Curves Workshop analogous program. A similar effort would e to look at slopes for that particular industry (e.g. aircraft, electronics, and shipuilding). These slopes are sometimes reported in organizational studies, research reports, or estimating handooks. Slopes can e specific to functional areas such as manufacturing, tooling, and engineering or they may e composite slopes calculated at the system level such as an aircraft, radar, tank, or missile. The first unit cost might e arrived at through any of a variety of means such as a factor, an analogy, a cost estimating relationship, a grassroots estimate, or y fitting the actual data. In some cases you may not have the first unit cost. The work measurement standards may provide the hours for the 5 th unit. A cost estimating relationship may predict the 100 th unit cost. This is not a prolem as long as you have the value at some unit and have chosen a learning curve slope ecause you can then solve for the 1 st unit cost. The Unit Formulation The first technique we will cover is the unit formulation, which is often associated with names such as James Crawford, Stanford Research Institute, and the Boeing Company. The unit formulation states that as the quantity of units doules, the unit cost decreases y a constant percentage. It is represented y the following formula 2 : Y x A where: Y x : the cost of the th unit A: the first unit (T 1 ) cost : the unit numer Log(slope) Ln(slope) Ln(.70) : the slope coefficient where or Log(2) Ln(2) Ln(2) Example 1: If it took 25,000 manufacturing hours to uild the first APG 180 radar, how many hours should it take to uild the 50 th unit on a 90% unit curve? A 25,000 Ln(slope) Ln(.90) Ln(2) Ln(2) Y 50 A (25000)(50) hours 3 Note: We automatically convert a slope in percent to the equivalent decimal value to use in the formulas. (90% is converted to.90) 2 There is no distinction or meaning intended y using Y A versus y ax or any comination therein. 3 Answers will vary slightly due to rounding. When dealing with exponents (e.g. ) you should carry the numer to at least four decimal places. Even etter, just carry the full value in the memory of your calculator. 5

6 Learning Curve Workshop Example 2: If the 100 th Humvee cost \$70,000, what did unit one cost on an 87% unit curve? Y ,000 Ln(.87) Ln(2) Y 100 A (A)(100) A \$176,573 Example 3: What is the slope when -.074? Slope or 95% Example 4: If the first Tommachop missile cost \$127,500 and you have a 91% unit curve, which unit would cost \$78,600? A 127,500 Ln(.91) Ln(2) Y 78,600? Y A Y A th unit Example 5: If you estimated that the first GPS transceiver would require 1000 hours to manufacture and that you would expect to see a 90% unit slope, how many hours would e required to produce the first five units (CT 5 )? With the unit cost formula already presented you could estimate the hours for each unit and then sum the results as follows: 6

7 Cost Improvement Curves Workshop CT 5 A(1) + A(2) + A(3) + A(4) + A(5) or CT 5 A[ ] CT [ ] CT [ ] CT [4.3392] 4339 hours for the first 5 units Now while this provides the desired calculation, it s a somewhat tedious process. What if we wrote the expression as CT A i and had a tale for i 1 i 1 i which represents the cumulative value of the units 1 through for a given (slope coefficient)? The following is an example of a 90% cumulative progress curve tale (a.k.a. Boeing tale) where rows represent units of 10 and columns represent units of 1: 90% CUMULATIVE PROGRESS CURVE TABLE 90% To use this tale to solve the aove prolem we would select the cumulative value for units 1-5 y going to row 0 (0 * 10) and column 5 (1 * 5) to arrive at and apply as follows: CT A i 1 CT i 5 i ( ) 4,339 i 7

8 Learning Curve Workshop If we wanted the total hours for units 1 through 37, we would go to row 3 and column 7: CT A i 1 i CT i i ( ) 24,824 Example 6: How many hours would e required to uild units 16 through 50? Solve y taking the total hours for 50 units (units 1-50) minus the total hours for 15 units (units 1-15). A 1000 Slope 90% TC 16,50 [CT 50 - CT 15 ] A 50 i 1 i - A 15 i 1 i A 50 i 1 i - 15 i 1 i ,758 [ ] Or, as an alternative, an approximation formula that does not require the use of tales: 8

9 Cost Improvement Curves Workshop TC F,L + 1 ( L +.5) - ( F -.5) T ( 50.5) - ( ) + ( 27.82) - ( 10.22) ,755 (or 20,758 if you don' t round) 9

10 Learning Curve Workshop Using Historical Data to find the Slope and the First Unit Cost If the historical production data was availale in the following format then we could take the Log or Ln of the units and hours, perform regression, and arrive at the learning curve equation: Unit Hours Log(Unit) Log(Hours) Log(Hours) [Log(Unit)] Or Hours 1000(Unit) However, in most production situations, cost data is not tracked or maintained y the cost of the individual units produced. Instead, manufacturers maintain cost data y production lot. The task of the analyst is then to use this lot data to develop a learning curve. To do this we must find some way to represent the lot cost as the cost of a specific unit within that lot. The specific unit we choose to represent is known as the Lot Mid-point (LMP). The LMP is defined as the theoretical unit whose cost is equal to the Average Unit Cost (AUC) or Lot Average Cost (LAC) for that lot on the learning curve. The difficulty in finding the LMP is that you must know the slope of the learning curve in order to determine the LMP, ut you need all of the LMPs to find the slope of the learning curve. The solution to this prolem then ecomes an iterative process est done y a computer program. Most learning curve software in use today solves for the LMPs in an iterative manner. However, if a learning curve software package is not accessile, a cost analyst can approximate the LMP for a given production lot using the following rules: For the first lot (the lot starting at unit 1): If Lot Size < 10, then If Lot Size > 10, then LMP LMP Lot Size 2 Lot Size 3 For all other lots: F + L + 2 LMP 4 FL where: F the first unit # in a lot, L the last unit # in a lot 10

11 Cost Improvement Curves Workshop The LMP then ecomes our independent variale () which can e transformed logarithmically and used in our simple linear regression equations to find the learning curve for our production situation. The dependent variale (Y) to e used is the AUC (or LAC) which can e found y: Lot Cost AUC Lot Size Again, the dependent variale (Y) must e transformed logarithmically efore use in the simple linear regression equations. Example 7: Given the following historical production data on the turret assemly for a particular tank system, find the Unit learning curve equation which est models this production environment. Given Calculated Lot Size Units Cost (man-hrs) LMP AUC , , , , , The LMP and AUC now ecome our independent and dependent variales respectively. These variales can now e transformed logarithmically and used in our simple linear regression equations to solve for A and, as follows: LMP AUC Log (LMP) Log (AUC) Y Now using the simple linear regression equations, we otain: and, slope or 86.05% Log (A) , and therefore A We can now write the learning curve equation that models this production environment as: Yx () Cumulative Average Formulation 11

12 Learning Curve Workshop The second learning curve formulation that we are going to discuss is the cumulative average formulation commonly associated with T.P. Wright and his 1936 article Factors Affecting the Cost of Airplanes. The theory is stated that as the total quantity of units produced doules, the cumulative average cost decreases y a constant percentage. This approach uses the same functional form as the unit formulation, ut it is interpreted differently. Y x A where: Y x : the average cost of units A: the first unit (T 1 ) cost : the cumulative numer of units : the slope coefficient where Log(slope) Log(2) or Ln(slope) Ln(2) The difference etween the unit and cumulative average formulations can e illustrated with this tale: Unit Formulation Cum Avg Form. Unit(s) Cum Avg Cost Y Unit Cost Y Both columns were generated with the equation: Cost 1000(Unit) (70% slope) Unit Formulation Y x A Cum Avg Y x A Cost of Unit 1000(Unit) Average Cost of units 1000(Unit) ( the 10 th unit cost 305.8) ( the average cost of 10 units is 305.8) So a 70% unit curve a 70% cum avg curve 4. This is a good reason to specify oth the slope and the formulation when you are documenting the use of a learning curve. 4 Additionally, the Cum Avg formulation does not make use of the cumulative progress curve tales as in the Unit formulation. 12

13 Cost Improvement Curves Workshop Example 8: If under the cum avg formulation the represents the average cost of 10 units, then what is the cost of the 10 th unit given a 70% cum avg curve? To answer that question let s start out with the asic formula. If Y x A is the average cost of units, then multiplying y should give us the total cost of units: CT A * or CT A +1 Given the first unit cost 1000 and we have a 70% curve, the total cost of 10 units is: CT 10 (1000)(10) (1000)(10) Since we can calculate the total cost of units, then we can solve for the cost of the 10 th unit y taking the total cost of 10 units and sutracting the total cost of 9 units. Y 10 CT 10 CT 9 A +1 A(-1) +1 A[ +1 (-1) +1 ] 1000[ ] Example 9: What is the lot cost for units 16 to 30? We can make use of the unit formula and replace the s with L (last unit in the lot) and F (first unit in the lot). Starting with the unit cost formula: Y A[ +1 (-1) +1 ] we could generalize and say that the total cost (TC) of a lot containing units F through L can e calculated y: TC F,L A[L +1 (F-1) +1 ] 1000[ (16-1) ]

14 Learning Curve Workshop Using Historical Data to find the Slope and the First Unit Cost As we discussed earlier, under the unit formulation, when your data is in the form of the cost of production lots, you need some means of representing each lot y a specific point. In the unit form we did this y using the AUC and the LMP for each lot. Under the cum avg form this is accomplished y using the cumulative average cost and the cumulative units 5 as follows: Units Lot Cost Cum Cost Cum Units Cum Avg Cost ,960 5, ,130 11, ,160 15, ,825 20, Example 10: Given the following historical production data on the turret assemly for a particular tank system, find the Cum Avg learning curve equation which est models this production environment. Lot Size Cost (man-hrs) Cum Units Cum Avg Cost , , , , , The Cum Units and Cum Avg Cost now ecome our independent and dependent variales respectively. These variales can now e transformed logarithmically and used in our simple linear regression equations to solve for A and, as follows: LMP AUC Log (LPP) Log (AUC) Y Now using the simple linear regression equations, we otain: slope % Log (A) A The learning curve equation that models this data is: Y ( ) Choosing Between Unit and Cum Avg 5 The cumulative units are sometimes referred to as the lot plot point (LPP). 14

15 Cost Improvement Curves Workshop Which formulation should you use? Choosing etween the Unit and the Cum Average Formulations is not so much a science as it is an art. There are no firm rules that would cause you to select one formulation over the other, ut there are some factors that we can look at to decide which might est model the actual production environment. Some of the factors to e considered when determining the formulation to use are: - Analogous systems - Industry standards - Historical experience - Expected production environment Analogous Systems - Systems that are similar in form, function, development, or production process may provide justification for choosing one formulation over another. For example, if a service is looking to uy a modified version of a commercial jet and on previous purchases of modified commercial jets the Unit curve was proven to est model the production environment, the estimator may choose the Unit formulation ased upon this analogy. Industry Standards - Certain industries sometimes gravitate toward one formulation versus another. For example, certain types of space systems have een shown to est fit the Cum Average formulation. If an estimator were estimating the same type of space system, the Cum Average formulation might e selected ased upon the industry standard. Historical Experience - Some defense contractors have a history of using one formulation versus another. This formulation may e used ecause it has een shown to est model the production environment for that particular contractor. An estimator may choose to also use that formulation ased on that historical experience. Expected Production Environment - Certain production environments tend to favor one formulation versus another. For example, the Cum Average formulation has een found to est model production environments where the contractor is starting production with the use of soft or prototype tooling, has an inadequate supplier ase estalished, expects early design changes, or is suject to short lead-times. This might e the result of a high degree of concurrency or overlap etween the development and production phases. This formulation is sometimes preferred in these situations ecause the effect of averaging the production costs tends to smooth out the initial cost variations and provides a etter fit to the data. Conversely, the Unit formulation tends to fit est in production environments where the contractor is well prepared to egin production in terms of tooling, suppliers, lead-times, etc. Less smoothing means changes are easier to see. There are no firm rules as to choosing one formulation versus another. However, an estimator should not choose a formulation solely ecause it generates the lowest cost estimate. A formulation should e chosen ased on the estimator s aility to determine which would est model the costs for the system eing produced. The estimator should also e aware that the smoothed (averaged) data used in the cumulative average formulation will most likely have a etter fit statistic (example, R 2 ) than the unit formulation. Beware of using a fit statistic as your asis for formulation selection. 15

16 Learning Curve Workshop Conclusion The general learning curve theory is that people and organizations learn (ecome more efficient) when performing repetitive tasks. Under certain conditions there is a usale pattern to this learning. This pattern can e very useful to the analyst performing cost or price analysis. Learning or Improvement Curves have een used for most of a century, and remain an important estimating tool. As Jack Hale indicated, we need to recognize that different situations and different environments will drive different yet usale learning curve patterns. The Unit and Cumulative Average formulations are the most common, ut other forms clearly exist The Learning Curve analyst should ecome familiar with these differing formulations, when they most likely would apply, their limitations and advantages, and what to expect when the situation, product, or process changes. 16

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