Learning Curves technical note APPLICATION OF LEARNING CURVES # A learning curve is a line displaying the relationship between unit production time and the cumulative number of units produced. Learning (or experience) curve theory has a wide range of application in the business world. In manufacturing, it can be used to estimate the time for product design and production, as well as costs. Learning curves are important and are sometimes overlooked as one of the trade-offs in just-in-time (JIT) systems, where sequencing and short runs achieve lower inventories by forfeiting some benefits of experience from long product runs. Learning curves are also an integral part in planning corporate strategy, such as decisions concerning pricing, capital investment, and operating costs based on experience curves. Learning curves can be applied to individuals or organizations. Individual learning is improvement that results when people repeat a process and gain skill or efficiency from their own experience. That is, "practice makes perfect." Organizational learning results from practice as well, but it also comes from changes in administration, equipment, and product design. In organizational settings, we expect to see both kinds of learning occurring simultaneously and often describe the combined effect with a single learning curve. Learning curve theory is based on three assumptions: Learning curve Individual learning Organizational learning 1 The amount of time required to complete a given task or unit of a product will be less each time the task is undertaken. 2 The unit time will decrease at a decreasing rate. 3 The reduction in time will follow a predictable pattern. Each of these assumptions was found to hold true in the airplane industry, where learning curves were first applied.1 In this application, it was observed that, as output doubled, there was a 20 percent reduction in direct production worker-hours per unit between doubled units. Thus, if it took 100,000 hours for Plane 1, it would take 80,000 hours for Plane 2, 64,000 hours for Plane 4, and so forth. Because the 20 percent reduction meant that, say, Unit 4 took only 80 percent of the production time required for Unit 2, the line connecting the coordinates of output and time was referred to as an "80 percent learning curve." (By convention, the percentage learning rate is used to denote any given exponential learning curve.) A learning curve may be developed from an arithmetic tabulation, by logarithms, or by some other curve-fitting method, depending on the amount and form of the available data. There are two ways to think about the improved performance that comes with learning curves: time per unit (as in Exhibit TN2.1A) or units of output per time period (as in TN2.1B). Time per unit shows the decrease in time required for each successive unit. Cumulative average time shows the cumulative average performance times as the total x'we o Learning Curves Plotted as Times and Numbers of Units A. Cumulative average time Time per unit i ^^\T^ — Observed data Fitted line —' Unit number —>• A Progress Curve B. Output per time period 1 Average output during a time period in the future Time Industrial Learning cha06369_tn02.qxd 9/3/03 2:22 PM Page 50 50 section 1 Operations Strategy and Managing Change number of units increases. Time per unit and cumulative average times are also called progress curves or product learning, and are useful for complex products or products with a longer cycle time. Units of output per time period is also called industry learning and is generally applied to high-volume production (short cycle time). Note in Exhibit TN2.1A that the cumulative average curve does not decrease as fast as the time per unit because the time is being averaged. For example, if the time for Units 1,2, 3, and 4 were 100,80,70, and 64, they would be plotted that way on the time per unit graph, but would be plotted as 100, 90, 83.3, and 78.5 on the cumulative average time graph. PLOTTING LEARNING CURVES________________ # There are many ways to analyze past data to fit a useful trend line. We will use the simple exponential curve first as an arithmetic procedure and then by a logarithmic analysis. In an arithmetical tabulation approach, a column for units is created by doubling, row by row, as 1, 2, 4, 8, 16... The time for the first unit is multiplied by the learning percentage to obtain the time for the second unit. The second unit is multiplied by the learning percentage for the fourth unit, and so on. Thus, if we are developing an 80 percent learning curve, we would arrive at the figures listed in column 2 of Exhibit TN2.2. Because it is often desirable for planning purposes to know the cumulative direct labor hours, column 4, which lists this information, is also provided. The calculation of these figures is straightforward; for example, for Unit 4, cumulative average direct labor hours would be found by dividing cumulative direct labor hours by 4, yielding the figure given in column 4. Exhibit TN2.3 shows three curves with different learning rates: 90 percent, 80 percent, and 70 percent. Note that if the cost of the first unit was $100, the 30th unit would cost $59.63 at the 90 percent rate and $17.37 at the 70 percent rate. Differences in learning rates can have dramatic effects. In practice, learning curves are plotted using a graph with logarithmic scales. The unit curves become linear throughout their entire range and the cumulative curve becomes linear after the first few units. The property of linearity is desirable because it facilitates extrapolation and permits a more accurate reading of the cumulative curve. This type of scale is an option in Microsoft Excel. Simply generate a regular scatter plot in your spreadsheet and then select each axis and format the axis with the logarithmic option. Exhibit TN2.4 shows the 80 percent unit cost curve and average cost curve on a logarithmic scale. Note that the cumulative average cost is essentially linear after the eighth unit. Although the arithmetic tabulation approach is useful, direct logarithmic analysis of learning curve problems is generally more efficient because it does not require a complete enumeration of successive time-output combinations. Moreover, where such data are not available, an analytical model that uses logarithms may be the most convenient way of obtaining output estimates. Exhibit TN2.2 Unit, Cumulative, and Cumulative Average Direct Labor Worker-Hours Required for an 80 Percent Learning Curve ď 0° r o (1) Unit (2) Unit Direct (3) Cumulative Direct (4) Cumulative Average Number Labor Hours Labor Hours Direct Labor Hours 1 100,000 100,000 100,000 2 80,000 180,000 90,000 4 64,000 314,210 78,553 8 51,200 534591 66,824 16 40,960 892,014 55,751 32 32,768 1,467,862 45,871 64 26,214 2,392,453 37,382 128 20,972 3,874,395 30,269 256 16777 6,247,318 24,404 cha06369_tn02.qxd 9/3/03 2:22 PM Page 51 Learning Curves technical note 51 ,s TN2.3 & TN2.4 TN2.3—Arithmetic Plot of 70,80, and 90 Percent Learning Curves 90% Learning curve 80% 70% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Unit number TN2.4—Logarithmic Plot of an 80 Percent Learning Curve 20 Average cost/unit (cumulative) 4 5 6 78910 20 30 40 50 60 80 100 200 300 400600 1,000 Unit number Logarithmic Analysis The normal form of the learning curve equation is2 [TN2.1] where Yx = Kx" x = Unit number Yx = Number of direct labor hours required to produce the xth unit K = Number of direct labor hours required to produce the first unit n = log b/log 2 where b = Learning percentage We can solve this mathematically or by using a table, as shown in the next section. Mathematically, to find the labor-hour requirement for the eighth unit in our example (Exhibit TN2.2), we would substitute as follows: Using logarithms: Ys = (100,000X8)" ľ» = 100,000(8)log0'8/log2 = ioo,ooo(8r0322 100,000 100,000 (8)0-322 1.9535 51,192 Therefore, it would take 51,192 hours to make the eighth unit. (See the spreadsheet "Learning Curves.") Learning Curve Tables When the learning percentage is known, Exhibits TN2.5 and TN2.6 can be easily used to calculate estimated labor hours for a specific unit or for cumulative groups of units. We need only multiply the initial unit labor hour figure by the appropriate tabled value. To illustrate, suppose we want to double-check the figures in Exhibit TN2.2 for unit and cumulative labor hours for Unit 16. From Exhibit TN2.5, the unit improvement factor for Unit 16 at 80 percent is .4096. This multiplied by 100,000 (the hours for Unit 1) gives 40,960, the same as in Exhibit TN2.2. 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Key Terms Learning curve A line displaying the relationship between unit production time and the cumulative number of units produced. Individual learning Improvement that results when people repeat a process and gain skill or efficiency from their own experience. Organizational learning Improvement that comes both from experience and from changes in administration, equipment, and product design. Formula Review Logarithmic curve: [TN2.1] Yx = Kxn Solved Problems SOLVED PROBLEM 1 Ajob applicant is being tested for an assembly line position. Management feels that steady-state times have been approximately reached after 1,000 performances. Regular assembly line workers are expected to perform the task within four minutes. a. If the job applicant performed the first test operation in 10 minutes and the second one in 9 minutes, should this applicant be hired? b. What is the expected time that the job applicant would take to finish the 10th unit? c. What is a significant limitation of this analysis? Solution a. Learning rate = 9 minutes/10 minutes = 90% From Exhibit TN2.5, the time for the 1,000th unit is .3499 x 10 minutes = 3.499 minutes. Yes, hire the person. b. From Exhibit TN2.5, unit 10 at 90% is .7047. Therefore, the time for the 10th unit = .7047 x 10 = 7.047 minutes. c. More data should be collected on the job applicant's performance. SOLVED PROBLEM 2 Boeing Aircraft collected the following cost data on the first 8 units of their new business jet. Unit Number Cost ($ millions) Unit Number Cost ($ millions) $100 83 73 62 60 57 53 51 cha06369_tn02.qxd 9/3/03 2:22 PM Page 60 6o section 1 Operations Strategy and Managing Change a. Estimate the learning curve for the new business jet. b. Estimate the average cost for the first 1,000 units of the jet. c. Estimate the cost to produce the 1,000th jet. Solution a. First, estimate the learning curve rate by calculating the average learning rate with each doubling of production. Units 1 to 2 = 83/100 = 83% Units 2 to 4 = 62/83 = 74.7% Units 4 to 8 = 51/62 = 82.26% Average = (83 + 74.4 + 82.6)/3 = 80% b. The average cost of the first 1,000 units can be estimated using Exhibit TN2.6. The cumulative improvement factor for the 1,000th unit at 80 percent learning is 158.7. The cost to produce the first 1,000 units is $100M x 158.7 = $15,870M The average cost for each of the first 1,000 units is $15,870M/1,000 = $15.8M c. To estimate the cost to produce the 1,000th unit use Exhibit TN2.5. The unit improvement factor for the 1,000th unit at 80 percent is .1082. The cost to produce the 1,000th unit is $100M x .1082 = $10.82M Review and Discussion Questions 1 If you kept any of your old exam grades from last semester, get them out and write down the grades. Use Exhibits TN2.5 and TN2.6, use log-log graph paper, or use a spreadsheet to find whether the exponential curve fits showing that you experienced learning over the semester (insofar as your exam performance is concerned). If not, can you give some reasons why not? 2 How might the following business specialists use learning curves: accountants, marketers, financial analysts, personnel managers, and computer programmers? 3 As a manager, which learning percentage would you prefer (other things being equal), 110 percent or 60 percent? Explain. 4 What difference does it make if a customer wants a 10,000-unit order produced and delivered all at one time or in 2,500-unit batches? Problems 1 A time standard was set as 0.20 hour per unit based on the 50th unit produced. If the task has a 90 percent learning curve, what would be the expected time of the 100th, 200th, and 400th unit? 2 You have just received 10 units of a special subassembly from an electronics manufacturer at a price of $250 per unit. A new order has also just come in for your company's product that uses these subassemblies, and you wish to purchase 40 more to be shipped in lots of 10 units each. (The subassemblies are bulky, and you need only 10 a month to fill your new order.) a. Assuming a 70 percent learning curve by your supplier on a similar product last year, how much should you pay for each lot? Assume that the learning rate of 70 percent applies to each lot of 10 units, not each unit. b. Suppose you are the supplier and can produce 20 units now but cannot start production on the second 20 units for two months. What price would you negotiate for the second 20 units? 3 Johnson Industries received a contract to develop and produce four high-intensity longdistance receiver/transmitters for cellular telephones. The first took 2,000 labor hours and $39,000 worth of purchased and manufactured parts; the second took 1,500 labor hours and 1. 100th = 0.18 hr. 200th = 0.16 hr. 400th = O.15 hr. 2. a. 1st 10 units = $2,500.00 2nd 10 units = 1,750.00 3rd 10 units = 1,420.50 4th 10 units = 1,225.00 5th 10 units = 1,092.00 b. Between max. of $4,250 and min. of $2,645.50. cha06369_tn02.qxd 9/3/03 2:22 PM Page 61 Learning Curves technical note 61 $37,050 in parts; the third took 1,450 labor hours and $31,000 in parts; and the fourth took 1,275 labor hours and $31,492 in parts. Johnson was asked to bid on a follow-on contract for another dozen receiver/transmitter units. Ignoring any forgetting factor effects, what should Johnson estimate time and parts costs to be for the dozen units? (Hint: There are two learning curves—one for labor and one for parts.) Lambda Computer Products competed for and won a contract to produce two prototype units of a new type of computer that is based on laser optics rather than on electronic binary bits. The first unit produced by Lambda took 5,000 hours to produce and required $250,000 worth of material, equipment usage, and supplies. The second unit took 3,500 hours and used $200,000 worth of materials, equipment usage, and supplies. Labor is $30 per hour. a. Lambda was asked to present a bid for 10 additional units as soon as the second unit was completed. Production would start immediately. What would this bid be? b. Suppose there was a significant delay between the contracts. During this time, personnel and equipment were reassigned to other projects. Explain how this would affect the subsequent bid. You' ve just completed a pilot run of 10 units of a major product and found the processing time for each unit was as follows: Unit Number Time (hours) 1 970 2 640 3 420 4 380 5 320 6 250 7 220 8 207 9 190 10 190 a. According to the pilot run, what would you estimate the learning rate to be? b. Based on a, how much time would it take for the next 190 units, assuming no loss of learning? c. How much time would it take to make the 1,000th unit? Lazer Technologies Inc. (LTI) has produced a total of 20 high-power laser systems that could be used to destroy any approaching enemy missiles or aircraft. The 20 units have been produced, funded in part as private research within the research and development arm of LTI, but the bulk of the funding came from a contract with the U.S. Department of Defense (DoD). Testing of the laser units has shown that they are effective defense weapons, and through redesign to add portability and easier field maintenance, the units could be truck-mounted. DoD has asked LTI to submit a bid for 100 units. The 20 units that LTI has built so far cost the following amounts and are listed in the order in which they were produced: 3. LR parts, 90%, LR labor, 80% Labor: 11,556 hours. Parts: $330,876. 4. a. Labor, $570,150. Materials, 1,356,750 plus something for profit. b. Need to consider forgetting and relearning. Time and cost could be much higher. 5. a. From log-log plot, LR = 60% b. About 8,029 hours. c. 6.0 hours. a. 70% b. $145,956,000. C. Cost .0851 X $12 = $1,021,200. Unit Cost Unit Cost Number ($ millions) Number ($ millions) 1 $12 11 $3-9 2 10 12 3-5 3 6 13 3-0 4 6-5 14 2.8 5 5-8 15 2.7 6 6 16 2.7 7 5 17 2.3 8 3-6 18 3-0 9 3-6 19 2.9 10 4.1 20 2.6 cha06369_tn02.qxd 9/3/03 2:22 PM Page 62 62 section 1 Operations Strategy and Managing Change 7. 4,710 hours. 8. Learning rate = 70%; unreasonable to ask for 4.5 hours. After 25, average repetitions time is about 3 hours. 9. a. Cost of 22nd unit = $32,732.40. b. 1,886 hours. c. Average cost = $43,126.50. 10. a. If first unit is 100 minutes, the learning rate needs to be 75%, not 80% (80/100). Do not hire. b. See ISM. 11. a. 3rd = 35.1 hrs. b. Average = 7.9 hrs. each; well worth it. 12. 11th 2.4476/.9 = $2.7196 million 12th 2.3953/.g = $2.6615 million $5.3811 million total 13. a. 42.165 minutes, k. 35.56 minutes. 10 11 12 13 a. Based on past experience, what is the learning rate? b. What bid should LTI submit for the total order of 100 units, assuming that learning continues? c. What is the cost expected to be for the last unit under the learning rate you estimated? Jack Simpson, contract negotiator for Nebula Airframe Company, is currently involved in bidding on a follow-up government contract. In gathering cost data from the first three units, which Nebula produced under a research and development contract, he found that the first unit took 2,000 labor hours, the second took 1,800 labor hours, and the third took 1,692 hours. In a contract for three more units, how many labor hours should Simpson plan for? Honda Motor Company has discovered a problem in the exhaust system of one of its automobile lines and has voluntarily agreed to make the necessary modifications to conform with government safety requirements. Standard procedure is for the firm to pay a flat fee to dealers for each modification completed. Honda is trying to establish a fair amount of compensation to pay dealers and has decided to choose a number of randomly selected mechanics and observe their performance and learning rate. Analysis demonstrated that the average learning rate was 90 percent, and Honda then decided to pay a $60 fee for each repair (3 hours x $20 per flat-rate hour). Southwest Honda, Inc., has complained to Honda Motor Company about the fee. Six mechanics, working independently, have completed two modifications each. All took 9 hours on the average to do the first unit and 6.3 hours to do the second. Southwest refuses to do any more unless Honda allows at least 4.5 hours. The dealership expects to perform the modification to approximately 300 vehicles. What is your opinion of Honda's allowed rate and the mechanics' performance? United Research Associates (URA) had received a contract to produce two units of a new cruise missile guidance control. The first unit took 4,000 hours to complete and cost $30,000 in materials and equipment usage. The second took 3,200 hours and cost $21,000 in materials and equipment usage. Labor cost is charged at $18 per hour. The prime contractor has now approached URA and asked to submit a bid for the cost of producing another 20 guidance controls. a. What will the last unit cost to build? b. What will be the average time for the 20 missile guidance controls? c. What will the average cost be for guidance control for the 20 in the contract? United Assembly Products (UAP) has a personnel screening process for job applicants to test their ability to perform at the department's long-term average rate. UAP has asked you to modify the test by incorporating learning theory. From the company's data, you discovered that if people can perform a given task in 30 minutes or less on the 20th unit, they achieve the group long-run average. Obviously, all job applicants cannot be subjected to 20 performances of such a task, so you are to determine whether they will likely achieve the desired rate based on only 2 performances. a. Suppose a person took 100 minutes on the first unit and 80 minutes on the second. Should this person be hired? b. What procedure might you establish for hiring (i.e., how to evaluate the job applicant's two performances)? c. What is a significant limitation of this analysis? A potentially large customer offered to subcontract assembly work that is profitable only if you can perform the operations at an average time of less than 20 hours each. The contract is for 1,000 units. You run a test and do the first one in 50 hours and the second one in 40 hours. a. How long would you expect the third one to take? b. Would you take the contract? Explain. Western Turbine, Inc., has just completed the production of the 10th unit of a new high-efficiency turbine/generator. Its analysis showed that a learning rate of 85 percent existed over the production of the 10 units. If the 10th unit contained labor costs of $2.5 million, what price should Western Turbine charge for labor on the units 11 and 12tomakeaprofitof lOper-cent of the selling price? FES Auto has recently hired Meg the mechanic to specialize in front-end alignments. Although she is a trained auto mechanic, she has not used FES's brand of equipment before taking this job. The standard time allocated for a front-end alignment is 30 minutes. Her first front-end alignment took 50 minutes and her second 47.5 minutes. a. What is the expected time for Meg's 10th front-end alignment? b. What is the expected time for Meg's 100th front-end alignment? cha06369_tn02.qxd 9/3/03 2:22 PM Page 63 Learning Curves technical note 63 14 An initial pilot run of 10 units produces the following times: Unit Number Time (minutes) 1 39 2 29 3 23 4 19 5 17 6 16 7 15 8 13 9 13 10 12 14. a. 70% b. 501.5 minutes. c. .78 minute. a. According to this pilot run, what is your estimate of the learning rate? b. How much time will it take for the next 90 units? c. How much time will it take to make the 2,000th unit? 15 A new bank clerk needed an hour to encode his first 500 checks, 51 minutes for the second 500, and 46 minutes for the third 500. After how many batches of 500 checks will he be able to work at the standard rate of 1,000 checks per hour? 16 A fast-food trainee takes an hour to prepare his first 20 sandwiches, 45 minutes for the second 20, and 38 minutes for the third 20. What will his production rate be after 24 hours of experience? 2? 15. 85% learning, .4930 hour after 20 batches. 16. 75% learning, 100 batches in 24 hours, .1479 hour/batch, or 135 sandwiches per hour. Selected Bibliography Argote, L. "Organizational Learning Curves: Persistence, Transfer and Turnover." International Journal of Technology Management 11, nos. 7/8 (1996), pp. 759-69. Argote, L., and D. Epple. "Learning Curves in Manufacturing." Science 247 (February 1990), pp. 920-24. Bailey, C. D. "Forgetting and the Learning Curve: A Laboratory Study." Management Science 35, no. 3 (March 1989), pp. 340-52, Globerson, S. "The Influence of Job-Related Variables on the Predictability Power of Three Learning Curve Models." AIIE Transactions 12, no. 1 (March 1980), pp. 64-69. Kostiuk, P. F, and D. A. Follmann. "Learning Curves, Personal Characteristics, and Job Performance." Journal of'Labor Economics 7, no. 2 (April 1989), pp. 129^16. Smith, D. B., and J. L. Larsson. "The Impact of Learning on Cost: The Case of Heart Transplantation." Hospital and Health Services Administration 34, no. 1 (Spring 1989), pp. 85-97, Smunt, T L. "A Comparison of Learning Curve Analysis and Moving Average Ratio Analysis for Detailed Operational Planning." Decision Sciences 17, no. 4 (Fall 1986), pp. 475-95. Wright, T P. "Factors Affecting the Cost of Airplanes." Journal of Aeronautical Sciences, February 1936, pp. 122-28, Yelle, L. E. "The Learning Curves: Historical Review and Comprehensive Survey." Decision Sciences 10, no. 2 (April 1979), pp. 302-28, Footnotes 1 See the classic paper by T P. Wright, "Factors Affecting the Cost of Airplanes," Journal of the Aeronautical Sciences, February 1936, pp. 122-28. 2 This equation says that the number of direct labor hours required for any given unit is reduced exponentially as more units are produced, 3 Rodney D. Stewart, Richard M. Wyskida, and James D. Johannes (eds.), Cost Estimator's Reference Manual, 2nd ed. (New York: John Wiley & Sons, 1995). 4 See L. Argote, "Organizational Learning Curves: Persistence, Transfer and Turnover," International Journal of Technology Management 11, no. 7, 8 (1996), pp. 759-69. 5 D. B. Smith, and J. L. Larsson, "The Impact of Learning on Cost: The Case of Heart Transplantation," Hospital and Health Sciences Administration 34, no. 1 (Spring 1989), pp. 85-97,