Learning Curves Learning Curve Introduction Learning Curves are

Learning Curves

Learning Curve Introduction • Learning Curves are used to estimate the time saving that is experienced when performing the same task multiple times • The greater the amount of labour involved in the task, the greater the saving experienced with each iteration

Learning Curve Introduction • Clearly manufacturing a ship or airline has a steeper learning curve (greater savings per iteration) than an automated process of making widgets (e. g. previous figure which is steep - ships) • Two predominant learning curve theories (though there are more) – Unit Theory – Cumulative Average Theory • Both will be explained using the Canadian Frigate program from the late 1980 s and early 1990 s

Unit Theory Learning Curve • The theory states that the cost decreases by a fixed percentage as the quantity produced doubles – The following table reflects an 80% learning curve which means the cost decreases by 20% for each doubling of production (multiply by. 8) Unit Number Cost 1 $100 2 $80 4 $64 8 $51. 2 16 $40. 96 Note: Examples from Cost Estimation by Mislick and Nussbaum

Unit Theory Learning Curve • Unit Theory equation – Yx = A * xb where: » Yx = the cost of unit x » A = cost of the first unit » x = the unit number » s = the slope of the learning curve = 2 b » Cost of 2 x = (cost of x)*s » Substituting A*xb for cost & rearrange » s = A*(2 x)b/A*xb = 2 b » Using logarithms: b = ln(s)/ln(2)

Unit Theory Learning Curve • From the previous example: – A=100 – Slope of learning curve = 0. 8 – 0. 8 = 2 b – Using logarithms: » ln(0. 8) = b * ln(2) » b = ln(0. 8)/ln(2) = -0. 3219 – Therefore, cost unit x = 100 * x -0. 3219 » For x = 4, cost = 64. 0 » For x =3, cost = 70. 2

Unit Theory Learning Curve • Canadian Frigate Example Ship number Number of hours to build ln(ship number) ln(hours) 1 5, 228, 212 0. 00000 15. 46958 2 4, 651, 262 0. 69315 15. 35265 3 3, 892, 713 1. 09861 15. 17462 4 3, 366, 301 1. 38629 15. 02933 5 2, 959, 776 1. 60944 14. 90062 6 2, 538, 396 1. 79176 14. 74704 7 2, 326, 265 1. 94591 14. 65977 8 2, 218, 333 2. 07944 14. 61227 9 2, 142, 963 2. 19722 14. 5777

Unit Theory Learning Curve • Yx = A * xb • To determine “b”, use logarithms, i. e. – ln(Yx) = ln(A) + b*ln(x) – “b” slope of this equation (learning slope = 2 b) • Graph using Excel Scatter with Markers after converting your costs (hours in this case) and number of units using natural logs • Under Chart Tools, select Trendline – more trendline options – Then select display equation on chart and display R-squared value on chart

Unit Theory Learning Curve • From the chart equation, the slope is -0. 4482 which = b • From before, if s = 2 b, then s = 0. 7330 or a 73. 3% unit learning curve which is steeper than typical but reasonable given SJSL’s inexperience

Cumulative Average Theory • Subtly different from Unit Theory • Cumulative average theory is average cost of groups of units while unit theory is based on individual unit cost • Therefore, for cumulative average theory (using the previous 80% example): – The average cost of 2 units is 80% of 1 unit – The average cost of 4 units is 80% of 2 units – The average cost of 50 units is 80% of 25 units • Doubling your production reduces your average cost by your learning curve (80% for this example)

Cumulative Average Theory • Cumulative Average Theory equation – YN = A * Nb where: » YN = the cumulative average cost of N units » A = cost of the first unit » N = the cumulative number of units produced » As before: » s = the slope of the learning curve = 2 b » Using logarithms: b = ln(s)/ln(2) • Cumulative Average Theory usually used for production lots where batches of units are made

Cumulative Average Theory • Canadian Frigate example again: Ship number Number of hours to build Cumulative Quantity Cumulative average hours ln(ship number) ln(cum avg hours) 1 5, 228, 212 0. 00000 15. 46958 2 4, 651, 262 2 4, 939, 737 0. 69315 15. 41282 3 3, 892, 713 3 4, 590, 729 1. 09861 15. 33955 4 3, 366, 301 4 4, 284, 622 1. 38629 15. 27054 5 2, 959, 776 5 4, 019, 653 1. 60944 15. 20671 6 2, 538, 396 6 3, 772, 777 1. 79176 15. 14332 7 2, 326, 265 7 3, 566, 132 1. 94591 15. 08699 8 2, 218, 333 8 3, 397, 657 2. 07944 15. 0386 9 2, 142, 963 9 3, 258, 247 2. 19722 14. 9967

Cumulative Average Theory • From the chart equation, the slope is -0. 2244 which = b • From before, if s = 2 b, then s = 0. 8560 or a 85. 6% unit learning curve which is less steep than unit theory as expected

Unit vs Cumulative Average Theory • Cumulative average theory is a less steep curve such that unit learning is always below it • Cumulative average is less responsive to variation in costs from unit to unit since it is based on averages • Use cumulative if initial production is expected to have large cost variation unit to unit

Unit vs Cumulative Average Theory • For Frigate example, which learning curve is better? • Hard to tell visually, I used stats, went with unit • R 2 for unit = 0. 9562 vs cumulative = 0. 9355 • P-value (not shown) unit=5. 2 x 10 -6 cum=2. 0 x 10 -5