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| Pricing Handbook
18. THE LEARNING CURVE Table of Contents 18.3 Unit and Cumulative Average Theory 18.3.1 Typical Learning Curve Computation Issues 18.3.2 Choice of Cumulative Average Versus Unit Curve Formulation 18.4 Special Learning Curve Applications 18.4.1 General Considerations in Dealing With Disruptions to the Log linear Learning Curve 18.5 Evaluation Criteria for the Learning Curve 18.5.1 Estimating Scenarios Suited to Use of the Learning Curve 18.5.2 Causes of Cost Reduction Due to Learning 18.5.4 Choice of Learning Curve Parameters 18.5.5 Labor Dollars Versus Labor Hours
THE LEARNING CURVE Chapter 15, "Quantitative Analysis Techniques", introduced the concept of the learning curve. (Cost improvement is also used to refer to learning curve theory.) Learning curves are useful both to cost estimators and analysts. While estimators may concentrate on deriving a learning curve, analysts focus efforts on critiquing a learning curve submitted by a contractor. The fundamental concepts do not differ, only the application of concepts differ. This chapter explores the theory of this learning curve in more detail and will focus on its applications. Section 18.2 explains the fundamental concepts of learning curves. Sections 18.3 and 18.4 describe both common and special learning curve applications. Section 18.5 discusses criteria that can be used to evaluate learning curves.
The learning curve was adapted from the historical observation that individuals who perform repetitive tasks exhibit an improvement in performance as the task is repeated a number of times. Empirical studies of this phenomenon (Wright, T.P.; Asher, H.; and Boston Consulting Group) yield three conclusions on which the current theory and practice are based:
Consistency in improvement has been found to exist in the form of a constant percentage reduction in time required over successively doubled quantities of units produced. The constant percentage by which the costs of doubled quantities decrease is called the rate of learning. The slope of the learning curve is 100 minus the rate of learning. For example, if the hours between doubled quantities are reduced by 20% (rate of learning), it would be described as a curve with an 80% slope. When plotted on ordinary graph paper with rectangular coordinates, such as in Figure 18-1, FIGURE Figure 18-1. 80% Learning Curve on Linear Graph Although the actual amount of decline in cost over time is a decreasing one, the amount of change in cost over the "doubling" period has been observed in empirical studies to be a constant percentage (Wright, T.P.; Boston Consulting Group.) Constants are helpful because they simplify the estimation. This type of constant percentage decline in one variable for a constant decline in another (the doubling of quantity) can be modeled using an exponential equation. Exponential equations can be solved by using logarithms. In the jargon of the cost estimating trade, it is common to speak of plotting the learning curve on "log-log" paper--a reference to bygone days when estimators plotted the learning curve by hand on graph paper with a logarithmic scale. Plotting learning curve data (such as in Figure 18-1) on a logarithmic scale causes the points to lie on a straight line, which allows for easy projection of future costs. Today, computers have replaced log-log paper.
18.3 Unit and Cumulative Average Theory Specific types (i.e., mathematical models) of learning curves have often been named after the men who proposed them or companies that first used them. They include Wright, Crawford, Boeing, and Northrup curves. All of these names refer to one of two mathematical models generally agreed to best describe how costs or labor hours decrease as the quantity of an item being produced increases. These two models are most commonly referred to as the unit curve and cumulative (cum) average curve. The equations underlying the models appear to be identical. However, because of the differences in the definition of the dependent variable, they predict different results even with identical first unit (also known as Theoretical Unit 1 (T1 for short)) and slope values. Consequently, an analyst must be aware, before evaluating the slope used to predict costs, of whether the unit or cum average curve was used to derive the slope. Learning curve theory states that as the quantity of items produced doubles, costs decrease at a predictable rate. This predictable rate is described by Equations 18-1 and 18-2. The equations have the same equation form. The two equations differ only in the definition of the Y term, but this difference can make a significant difference in the outcome of an estimate. Equation 18-1 describes the basis for what is called the unit curve. In this equation, Y represents the cost of a specified unit in a production run. If a production run has generated 200 units, the total cost can be derived by applying Equation 18-1 200 times, for units 1 to 200 and then summing the 200 values. This is cumbersome and requires the use of a computer or published tables of predetermined values.
Equation 18-2 describes the basis for the cumulative average or cum average curve. In this equation, Y represents the average cost of different quantities (X) of units. The significance of the "cum" in cum average is that the average costs are computed for X cumulative units. Therefore, the total cost for X units is the product of X times the cum average cost. For example, to compute the total costs of units 1 to 200, an analyst could compute the cumulative average cost of unit 200 and multiply this value by 200. This is a much easier calculation than in the case of the unit curve.
There are also important similarities between the two formulations of the learning curve. Both are exponential functions. This is important information, because the solving of exponential equations requires the use of logarithms. According to Donald Stancl in Mathematics for Management and the Life and Social Sciences (page 815), "the method of solution is to take the logarithm of both sides of the equation and then use the fact that ln Xb = b* lnX to �get b out of the exponent.�" For convenience, it is typical to use the natural logarithm (usually referred to as ln in math books and electronic spreadsheets). Knowing the mechanics of how to solve the learning curve equations using logarithms will help the reader to use electronic spreadsheet tools effectively. Taking the natural logarithms of both sides of the equation reduces the equation mathematically to a straight line equation of the form ln Y = T1 + b
Both the unit and cum average learning curve equations describe and model the observation that costs decrease by a constant percentage every time the quantity doubles. This is reflected in the curves through the b value, a constant reflecting the amount of the decrease for every doubling of quantity. The b value for both curves is computed by Equation 18-4.
As an example, if the first unit cost 100 and the second unit cost 90, or 90% of unit 1, the unit curve would have a 90% slope and the S value would be 0.9. The resulting b value would be the (log 0.9)/(log 2) or -0.045758/0.30103 or -0.15200. Of course, this data set is very small and the reader is reminded that two data points will not produce a reliable estimator. This is used for example purposes only, to illustrate how b relates to the slope of the learning curve. It is more traditional, in talking about the learning curve, to discuss the slope in percent terms, such as 90%. When deriving a learning curve equation using a standard spreadsheet regression analysis tool, the tool will usually only give the b value as an output. Therefore, the analyst must understand how to transform the b value into a slope. This, of course, would be accomplished by transforming Equation 18-4 algebraically to yield Equation 18-5.
Using the T1 value (100) and slope (.90) derived from the above example of a unit curve, the cum average curve will always predict lower total costs than the unit curve because of the difference in how Yx is defined. In the example above where the first unit cost 100 and the second 90, the total cost for the two units is 190, based on use of the unit curve. Using the cum average curve with the same T1 value (100), slope value of .9, the total cost of two units is 2 times 90 or 180.
18.3.1 Typical Learning Curve Computation Issues This section describes some typical learning curve situations or issues for which cost estimates are necessary. These situations are illustrated in Figure 18-2 as cost versus quantity plots using a logarithmic scale. The discussion now turns to how to do the calculations to deal with these issues and what tools are available to assist the estimator. Each issue is discussed as a case study. The first issue involves calculating the actual learning curve parameters (T1 and slope) from a set of raw data--development of a learning curve model. This issue is presented in Case Study 18-1. All of the other problem situations illustrated in this section address applications of a known learning curve (defined by the slope and T1).
CASE STUDY 18-1. PART A--INTRODUCTION: DEVELOPMENT OF T1 AND SLOPE VALUES FROM DATA
CASE STUDY 18-1. PART B: DERIVE THE SLOPE AND T1 FROM UNIT DATA:
CASE STUDY 18-1. PART C: DERIVE THE SLOPE AND T1 FROM LOT DATA
CASE STUDY 18-2. ESTIMATE FROM A SINGLE LOG LINEAR CURVE THROUGHOUT PRODUCTION
This concludes the discussion of the basic log linear learning curve. Before proceeding to a discussion of some more complex learning curve applications, the question of which curve to use will be addressed. The following section offers some suggestions.
18.3.2 Choice of Cumulative Average versus Unit Curve Formulation The unit learning curve technique has been the predominant methodology applied by both Government and contractor estimators. However, there are circumstances which may dictate the use of the cum average learning curve theory:
Considering this guidance may not remove all uncertainty about which theory would be best. However, it can provide an analyst with one or more reasons to support use of one versus the other. This section has provided an in-depth understanding of the cost improvement phenomenon and the two most common log linear mathematical models used to apply this tool. However, the learning curve phenomenon in its pure (log linear) sense does not always adequately model the estimating situation. This occurs because each new system presents a different set of conditions and program-unique aspects that may not be comprehensively duplicated in the historical database. Consequently, the analyst encounters the challenge of understanding the environment in which the new system will be manufactured and conceptually formulating the shape, slope, and behavior of the curve that best expresses the cost improvement that will occur in the anticipated environment. The next section deals with some of the "anomalies" or deviations from the straight, log linear learning curve example.
18.4 Special Learning Curve Applications Up to this point the learning curve has been discussed under the assumptions of uninterrupted production and a stable product design. Sometimes, an analyst will find that the Government sometimes buys off-the-shelf items with a slightly altered design, which might involve some learning. The Government also buys items that have a new design and require development. In both of these situations, a log linear learning curve is likely to be an appropriate estimating tool. In this type of scenario, the analyst need only determine an appropriate slope and T1 to use. The calculations are fairly straightforward after that. Often, however, numerous types of disruptions in the production process are a fact of life. There may be changes in processes that would cause an item to experience a steeper slope than when previously produced. There may be additions or deletions to an item in production, or the Government may cause a break in production or production rate change by not having enough funding to pay for items in production. These occurrences could cause a change in the cost of a unit that is not the result of learning. Therefore, the impact of these types of changes must be considered in an estimate.
18.4.1 General Considerations in Dealing With Disruptions to the Log linear Learning Curve In general, disruptions to the log linear learning curve are handled by dissecting the situation into component pieces and then approaching each piece as a separate estimate. To assist in conceptualizing an estimating situation, an analyst can create scatter diagrams of actual data. This situation could arise where the estimate involves a new lot of the same or very similar item. Alternatively, the analyst can analyze the program to be estimated and find a good analogous program to use. In any event, the analyst must understand the factors that caused data point scatter diagrams to take on various shapes and to select those historical elements that are most analogous to the current system and its estimate. Often, close analogies will not exist. At this point, the analyst must draw on personal experiences and knowledge, as well as inputs from others in the acquisition community, to adequately define the most appropriate curve. Regardless of the process employed to select the most viable depiction of the learning curve that is forecasted to evolve, the analyst must be prepared to articulate and justify the selected curve, its shape, slope, and anticipated behavior. The following sections are intended to provide some thought provoking insights that will assist in this challenging endeavor. They address two issues, each affecting the shape of the learning curve, generally moving its shape away from a pure log linear form. Issue 3 (discussed in Case Study 18-3) deals with a change in the slope of the curve also often referred to as a split learning curve. Issue 4 (Case Study 18-4) deals with a "step-down" or "step-up" from the path of the learning curve.
CASE STUDY 18-3. SPLIT LEARNING CURVE
Additions Whenever a new component is added to an assembly that is already in production or installation, several key factors must be considered when analyzing the impact of the addition. It is logical to assume that the addition of a new component to an assembly already in production or installation will require additional hours of effort to incorporate the new component. However, two general assumptions can be made about the nature or behavior of these additional hours in relation to effort completed prior to the addition:
From the above, it can be inferred that an addition should be treated as a new learning curve having the same slope or rate of improvement as the original unit. Deletions A deletion, in its simplest form, involves the removal of a component from an item that is already being produced. However, it may be concerned with changes other than the removal of a discrete component. For example, it could entail such things as a deletion of work, implementation of less stringent specification requirements, etc. Again, it should be expected that this type of change will have an impact upon the cost of the item being produced. However, in this instance, costs should decrease because less effort is required for each modified unit. In other words, there is not as much work to be performed on each unit as there was before the deletion occurred.
CASE STUDY 18-4. STEP FUNCTION IN CURVE
18.4.2 General Tips on Complex Learning Curve Problems
18.4.3 Production Rate Changes An area that is often overlooked in the analysis of learning curve application is that of the impact of rate changes. Normally, a specific production rate is inherent in the production schedules provided to the analyst. Therefore, when analyzing historical learning curve history, the variable of production rate is often overlooked. Various research and studies have been conducted that suggest that the rate of production is, in fact, a cost sensitive variable that should be considered. The general consensus of a Government and industry committee surveying learning curve concepts and applications (The Working Committee On Air Launched Weapon System Costs) was that production rate does affect the application of learning curves. The rationale cited in that study actually reflects basic economic theory. This theory is summarized below to serve as a foundation for understanding production rate influence on cost reduction. This economic theory suggests that every output has an equilibrium and as output is expanded beyond this equilibrium, costs will increase, in the short run, due to:
In the long run, this increased rate will develop a new equilibrium. This is due to:
If the production rate change is decreased below the original equilibrium point, the unit cost will increase, in the short run, due to:
In the long run, a new equilibrium point develops at the lower production rate. Not all of the economic phenomena can be captured in the learning curve. Much of the cost impact attendant to rate changes is caused by constraints (e.g., equipment and facilities) that represent fixed or semi-variable costs. The learning curve, however, addresses only variable costs. Therefore, the remainder of this discussion will be directed at variable costs and their impact, as a result of production rate changes (lower productivity of newly hired workers, additional labor costs associated with overtime, loss of specialization of labor), on learning curve behavior. In 1976, Larry L. Smith conducted a study of changes in direct labor requirements resulting from changes in airframe production rate. During his analysis, Smith developed a cumulative production and production rate computer program. This program contained a production rate variable, in addition to the two standard learning curve variables. Smith concluded that by incorporating the production rate variable into the basic learning curve, a significant improvement in estimating could be achieved. This was demonstrated through statistical measurements. Numerous later studies, using actual data, have been conducted to examine the impact of production rate on unit cost. The conclusions of this research have been mixed. Any estimator faced with estimating in an environment which includes production rate change(s) is advised to consult the latest research on production rate effects on cost estimating.
18.5 Evaluation Criteria for the Learning Curve To correctly evaluate learning curves, an analyst must consider several criteria. The analyst must understand in what situations it makes sense to use a learning curve to estimate costs, and this requires an understanding of the causes of learning. The analyst must also understand what factors to consider when choosing the learning curve parameters and how the parameters will affect the outcome of the learning curve. These concepts are addressed in this section. 18.5.1 Estimating Scenarios Suited to Use of the Learning Curve Since the first paper on learning curves in the aircraft industry was published in the 1930s, much has been written on the subject. Louis E. Yelle provides over 90 references published prior to 1967 in The Learning Curve: Historical Review and Comprehensive Survey. His most important finding is that costs have been observed to decrease in a predictable manner as the quantity of an item produced increased. Learning curve theory, based on numerous empirical studies, states that as the number of units produced in a repetitive process doubles, the cost to produce the doubled quantity declines by some constant percentage. This technique, then, is most suitable for estimating the reduction in cost resulting from labor and other efficiencies (like material handling or management) that come with repetition of a process. The repetitive process can involve hands-on labor or mental exercises and can range from simple to complex. The learning curve has traditionally been used to estimate the cost to manufacture items, but it can also be used to estimate equipment installation costs or the costs of any other repetitive process. Another estimating scenario question is at what level of the estimate to use the learning curve to estimate costs. The answer to this depends on the overall structure of the estimate. If a top level parametric estimate is used to estimate the cost of a developmental item, a learning curve may be used to estimate total production costs in dollars. On the other hand, if there are detailed production data available, the learning curve may be used at a lower level, such as to estimate manufacturing labor hours. Such an estimate would then have to be converted to dollars through the application of appropriate labor rates and overhead rates. The analyst must make decisions of this nature. Throughout this chapter, the data used for examples of the dependent variable are presented in labor hours, since labor hours are clearly impacted by learning and not affected by other variables not necessarily subject to learning curve behavior (e.g., overhead rates.) For Commercial-Off-The-Shelf (COTS) items, it may not be appropriate to apply the learning curve, but instead it may be appropriate to adjust for quantity discounts. This might be the case where the COTS item has already been produced many times, so that the process is so standardized that the learning effect has become minimal. In this case, the application of quantity discounts may be a better approach. A discussion of quantity discounts is in section 18.6.
18.5.2 Causes of Cost Reduction Due to Learning The cause of reduction in costs is learning on the part of individuals and entire organizations in the process of repetition. Clearly then, only recurring costs are impacted by learning. Nonrecurring costs, such as the cost of acquiring tooling, are not affected by learning. Nonrecurring costs may affect the learning rate experienced, but their acquisition cost will not be affected by learning on the process. In other words, if a manufacturer invests in process improving tooling, he might expect the learning rate in the process to be altered. However, the learning rate will not affect the cost of that initial tooling. According to Rodney Stewart, in The Cost Estimator�s Reference Manual (page 161), the factors which contribute to the cost reductions reflected in a learning curve normally include:
Upon review of these factors, it becomes evident that it is reasonable to expect a reduction in labor hours as a result of operator learning, as well as improved methods. However, other costs such as material costs can decline due to learning as well. As operators learn through repetition of a process, they may find ways to reduce scrap and material waste. There may be a reduction in personnel turnover costs due to improved processes that reduce personal fatigue or improve safety. Many factors can contribute to learning, and the learning may be on the part of either an individual or the entire organization. This brings us to the next major consideration an estimator must make in using learning curves--how to choose the key learning curve parameters. Once an analyst has decided the learning curve is an appropriate tool to model the costs of a particular estimating scenario, the next major consideration in using the learning curve is to choose the proper parameters to model the particular estimate. This subject is addressed next.
The discussion in this chapter has thus far centered on the application of various learning curve techniques and how the "learning" phenomenon reduces unit costs as more units are produced. However, there are other forces in the marketplace that will also allow the cost per unit to decrease when the quantity involved increases. For example, there is the "quantity discount" effect, a pricing strategy that consumers interact with almost daily. Examples that frequently confront consumers include "Beans 39 cents a can or three cans for a dollar"--"Buy three tires at the regular price and get the fourth free." These examples lower the average unit cost as the quantity involved increases, but not for the reason of "learning." Items that are standard and have already been produced in large quantities may have reached "standard time" for the production process, with little room left for learning. In this case, the predominant influence on cost reductions may become the quantity discount that the manufacturer is willing to "pass-along" due to:
The effect of quantity discount is not isolated to retail outlets, but can also be found in the acquisition of other items. For instance, an electronics manufacturer might quote a radio's price at $20,000 for a buy quantity of ten units or less and $18,000 for a buy quantity of eleven units or more. Raw material suppliers quote prices similarly, based on the total volume of each order. Once a manufacturer's or vendor's discount rate is known, unit costs can be computed for several quantities. The estimator should become particularly sensitive to the influence that the quantity discount effect may have on historical data. Historical data are typically reported and analyzed in the chronological order of manufacture or purchase and is, therefore, aligned properly to portray the "learning" phenomenon. The application of the quantity discount curve differs from that of a learning curve. With a learning curve, efficiencies accumulated from prior experiences influence the efficiencies (number of hours/dollars) with which a succeeding task can be accomplished. Consequently, as unit costs or lot buy values are calculated from a learning curve, credit is taken for the efficiencies accumulated from prior units or lot buys. In other words, if the first lot buy was for 20 and the second for 50, the second lot buy would be calculated by reading the values from unit 21-70 giving credit for the cost reducing efficiencies acquired during the manufacture of units 1-20. However, when a quantity discount is involved, credit is not given for prior activity. Using the preceding example, the cost of the first 20 units would be calculated by moving out the curve from unit 1 to unit 20 and multiplying the discounted unit cost times 20. For the next 50 unit lot buy the analyst would again depart from unit 1, move out the curve to unit 50 and multiply this discounted unit cost times 50. In most cases, it is simply a matter of inquiry to the manufacturer or vendor to determine if a quantity discount is the underlying influence of price quotes. Cataloged prices are more straightforward since they clearly depict the quantity discount that can be applied for volume purchases. When quantity discounts are the predominant factor affecting unit cost reductions, they should be applied using the procedure cited above rather than using the learning curve treatment. Estimates using quantity discount theory can be projected using Equation 18-11 rather than the traditional learning curve equation.
18.5.4 Choice of Learning Curve Parameters The learning curve has two key parameters that will determine the outcome of the estimate: 1.) the slope of the learning curve (which represents the constant percentage decline in costs due to learning), and 2.) the first unit value of the learning curve. Learning Curve Parameter 1: The Slope The amount of learning expected to occur in a process is clearly a key consideration to an analyst. If the analyst assumes an 80% rate of improvement instead of a 70% slope, the difference in cost could be significant. If unit 100, for instance, actually cost $100,000 to produce, unit 200 would cost $80,000, assuming an 80% learning curve. With a 70% learning curve, it would cost $70,000. Therefore, the choice of slope must clearly be well-justified. There are a number of sources of learning curve slopes: actual producer data, experience on analogous processes, and industry standard learning experience (Handbook of Electronics Industry Cost Estimating Data, 1985.) Learning curves exist for all sorts of industries, from software development to electronics. If there are actual producer data available from a process and that same process is to be used again, the analyst will use this data as the basis for the estimate. This is because actual data are generally superior to analogous data or industry standards. The producer may provide the historically-experienced slope, or the analyst may use the actual data to develop the slope. This chapter explains how to develop a slope from actual data using the least squares regression analysis described in section 18.2. If there is not actual data available for the same process, the next best choice is to find a similar process, an analogy. Engineering input on how to adjust the slope to account for the differences in the process will complete the analysis. For example, if the amount of automation in the process increases, the learning curve slope will tend to be flatter (have a higher slope value). The reason for this is that machines cannot learn--any learning related to machine design is captured in the cost of those machines or in systems engineering costs. Specifically, this cost would typically show up in the theoretical first unit cost, which is the second key parameter in a learning curve equation. Finally, an analyst may opt to use an industry standard learning curve. There are industry standards for repetitive electronics manufacturing, repetitive machining operations, even repetitive clerical operations. Some general observations about learning curve slopes include: 1) normal turnover of personnel has very little effect on the slope of the learning curve; 2) in general, the learning curve slope will increase (move further below 100%) as product complexity increases; and 3) commercial off-the-shelf items often have flatter slopes than fabrication labor. The first observation is explained by the fact that most people will learn at roughly the same rate. The second observation is explained by reasoning that a more complex product involves more opportunities for learning. One of the reasons for the third observation is that vendors supplying off-the-shelf items are usually on the shallow portion of their learning curves (i.e., at high quantity values), therefore, the purchaser does not experience large changes in costs. Another reason for observation 3 is that off-the-shelf items also have loadings in their prices that do not follow the learning curve effect and therefore appear to flatten the slope. In addition to choice of the slope, the analyst needs a theoretical first unit cost (commonly referred to as T1) from which to start projecting cost reductions. Just knowing the slope is not enough information to make a cost estimate--one must also have a starting cost from which to project what the reduced costs will be.
Learning Curve Parameter 2: Theoretical First Unit Cost (T1) Just as is the case with the slope, the T1 value can be derived in a number of ways. They include deriving the T1 from actual data, estimating the T1 from an analogy or Cost Estimating Relationship (CER), or estimating the T1 by using a detailed engineering build-up estimate. An example of how to derive a T1 from actual data is presented in Case Study 18-1. For a detailed discussion of CERs, analogies, or detailed estimates, the reader is referred to the chapters on those subjects. The slope and T1 are the key parameters in the learning curve equation. Both parameters are sensitive--a small difference in the slope can, for instance, make a large difference in the total costs, depending on the size of the acquisition. Table 18-1 illustrates how total costs change for a two percent increase in the slope, from 85 to 87%. This is the same as a relative slope increase of 2/85 or 2.35%. Table 18-1 was developed using the cum average curve theory and an assumed T1 value of 1.0. A quick look at the table shows that a small slope change or error in the slope can significantly change a total cost estimate. Therefore, analysts should perform a sensitivity analysis on the curve slope values, especially if the values used are likely to be questioned. Table 18-1. Sensitivity Analysis Results for Slope Increase from 85 to 87 Percent
In estimating total recurring costs with a learning curve, when the T1 value changes a given percent, it is easy to see that the total cost changes by the same percent. However, this is not true for changes in the slope of the learning curve. For large quantities the total cost percent changes are much larger than the slope percent change. In addition, for a given percent slope change the percent total cost change varies with both the original slope and the quantity of units to be procured.
Goodness of Fit The usefulness of the learning curve as a predictor is determined as it is with any parametric estimator. Goodness of fit measures, like R2 and standard error of the estimate, are tools that tell the estimator how much confidence can be placed in a particular learning curve. Generally, a R2 of .95 or higher is desired, and the slower the standard error of the estimate the better.
18.5.5 Labor Dollars Versus Labor Hours When estimating labor costs, it is--often better to use learning curves based on hours rather than dollars, since the latter contains the additional variable of inflation or deflation and possibly overhead rates which do not follow the same learning curve. If an analyst chooses to use labor dollars as the basis to construct and express improvement rates from historical data, he or she should ensure that the data are normalized for economic influences such as inflation. Even with this precaution,
18.5.6 The Impact of Technology and Automation Over time, systems and manufacturing processes have become more complex. The introduction of such advancements as micro-circuitry and robotics into system design and the manufacturing environment influenced learning curves. However, it has not changed the application of them in the estimating process. More specifically, the influence technology and automation have on "learning" is inherent to the database that evolves from systems that incorporate advanced technologies and are manufactured in a highly automated facility. However, the curves calculated from this data employ the same methods described in earlier sections of this chapter. There are no simple rules-of-thumb A description of typical estimating issues that arise with respect to learning curves follows.
18.5.7 Reliability of Data Underlying Learning Curve Sections 6.3 and 6.4 of Chapter 6 "Gathering and Evaluating Data for Price Analysis includes a discussion of factors that affect the reliability of data. This discussion can be extended to learning curves. Accordingly, an analyst should assess the reliability of data used to develop a learning prior to applying the model. The general considerations for data are: timing, quality, and use of historical data. To be useful, data must be collected in a systemic and timely manner. In addition, the data must be of reasonable quality with respect to: accuracy and relevancy, completeness, and currency. Finally, historical data must be adjusted for any changes that occurred in the industry, inflation etc.
Learning curves are useful estimation and analytical tools. Knowledge of fundamental concepts and evaluation criteria enables analysts to assess the validity of learning curve estimates provided to them. In addition to unit and cumulative average theory, analyst should consider special situations such as split learning curves and production rate changes. When critiquing learning curves that have been supplied, analysts should examine the estimating scenario, causes of cost reduction, choice of curve parameters, impact of technology, and the reliability of data underlying the curve.
Bibliography Asher, H., Cost Quantity Relationships in the Airframe Industry, Santa Monica, CA, Rand Corporation, 1956. Report 291. Baloff, Nicholas. "Extensions of the Learning Curve�Some Empirical Results." Operations Research Quarterly, 22.44 (1971). Boston Consulting Group. Perspectives on Experience, Boston, MA, 1970. Wright, T.P., "Factors Affecting the Cost of Airplanes," Journal of Aeronautical Sciences, 3.4 (1936): 122 -128. |
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the line becomes hyperbolic. This is because the amount of cost reduction is not constant. Rather, the amount of decline continually diminishes as the quantities double. With very large quantities, the decline trickles off to a very insignificant amount, as depicted by the tapering off of the graph in Figure 18-1. This happens as workers approach a "standard cost" for a process. With continual change in the process or product, this point can be delayed. Several research studies of manufacturing operations in the steel, auto assembly, apparel, and musical instruments industries have described this phenomenon as "plateauing" (Baloff.)
lnX or more commonly Y=aX +b. Straight lines are useful for analysts because a straight line is easy to extend beyond the range of the data--if you fit a least squares regression line to a data set, it is easy to extend that straight line and estimate from that line. A curved line, by contrast, is tougher to extend because the precise angle of the curve is unclear outside the range of the data set. To summarize, the cum average learning curve equation can be rewritten as Equation 18-3.
before entering the data into a software tool. Some learning curve programs will do that automatically, calculating what are called "true lot plot points". The analyst should consult the software user manual to use the tool correctly in this regard. The following discussion will provide the reader with an understanding of the data conversion that needs to occur with lot data.
= 103.45
a very good approximation of the total cost for a lot of N units is provided by Equation 18-9.
the estimator should be sensitive to the fact that most review authorities have, over time, become familiar with various slopes for different functional cost categories based on labor hour data (e.g., manufacturing labor--85%, engineering labor--75%). If an estimate incorporates learning curves based on labor dollars rather than hours, the slopes may depart from those typically associated with specific functional cost categories. This departure will most likely be in the direction of steeper slopes due to the influence of declining average labor rates as the work force builds toward rate production. This decline in average labor rate occurs because senior labor is initially brought to the job with junior labor being introduced as build-up of the labor force occurs, thus reducing the average labor rate. Labor rates tend to level off as the work force stabilizes at rate production. This effect plus normal learning will steepen labor curves calculated on dollars rather than hours. To avoid confusion, the estimator should always demonstrate to review authorities the basis for curve calculations and the historical data that supports the selected slope.
or mathematical equations that will allow curves to be directly adjusted to reflect the influence of technology and automation. One intuitively expects that a system manufactured in a highly automated environment will demonstrate a flatter learning curve slope than the same system manufactured with a high degree of "touch" labor. This is due to the significant level of initial efficiency and reduced opportunity for "learning" (learning improvement on automated equipment is limited to such things as more efficient loading/unloading and improved software programs) that accompany automated manufacturing facilities. Even with this intuitive knowledge, however, the degree of influence will depend on the degree of automation. Because each system's technology and manufacturing scheme are unique, the estimator's challenge is to select from the historical database those systems whose technology and manufacture closely align to those of the system being estimated. This may require analysis of several systems and the evolution of a composite measure of influence that incorporates various aspects of the systems analyzed. The end result will be a judgment as to what adjustment of T1 and slope for the system is appropriate, given the technology that it will incorporate and the manufacturing environment planned for its production.