LESSON 2: LEARNING AND EXPERIENCE CURVES

1. LESSON 2: LEARNING AND EXPERIENCE CURVES Outline • Rate of Learning • Learning Curve • Estimating Parameter Values

2. Rate of Learning • As workers gain more experience with the requirements of a particular process, or as the process is improved with time, the number of hours required to produce an additional unit declines. • The learning curve models this relationship. • Rate of learning, is defined as follows:

3. Rate of Learning • Let • Y(u) = time required for the u-th unit • Then, from the definition of rate of learning,

4. Rate of Learning • For example, using rate of learning, • Using rate of learning, • Using rate of learning,

5. Rate of Learning • Rewriting the definition of rate of learning, • Hence, • If we know the time required for the first unit, Y(1), we can find the time required for the 2nd, 4th, 8th, ….. units using the above equation iteratively.

6. Rate of Learning • The time required for the 1st unit = Y(1) [notation] • The time required for the 2nd unit, • The time required for the 4th unit, • The time required for the 8th unit,

7. Rate of Learning • An example of 80% learning rate: Suppose that it requires 100 hours to produce the first unit. Then, the 2nd unit requires (0.80)(100)=80 hours. The 4th units requires (0.80)(80)=64 hours, and so on. Unit Number of Hours Required 1st unit 100 hours 2nd unit (0.80)100=80 hours 4th unit (0.80)(80)=64 hours 8th unit (0.80)(64)=51.2 hours

8. Learning Curve • In general, for any unit u, not necessarily 1, 2, 4, 8, …, the time required can be obtained from the learning curve equation. • The learning curve is of the form Y(u)= au-b Where, a and b are parameters. a = time required for the first unit b = - ln (L)/ ln (2), where L is the rate of learning, 0.80 for 80% learning, 0.90 for 90% learning, etc. Processing time per unit, Y(u) Units produced, u Here, ln = natural log. A review on logarithms follows.

9. Learning Curve Suppose that • a = 18 hours • Learning rate = 80% • What is time for the 9th unit? Y(u)= au-b = auln(L)/ln(2) Y(9)= Here, ln = natural log. A review on logarithms follows.

10. Logarithms (Review) • Recall that if then, . • Here, p is the base. • If the base is e, “ln” (natural log) replaces “log”. So, • Here, e is a constant:

11. Logarithms (Review) • A scientific calculator usually contains 2 buttons: • log x provides log10x , logarithmic value of some number x with base 10 • ln x provides logex , logarithmic value of some number x with base e =2.71828… • To get a logarithmic value with a base other than 10 or e, use the following formula:

12. Estimating Parameter Values • Recall, that learning curve is of the form Y(u)= au-b Where, a and b are parameters. • If we observe the time required to produce various units, we can estimate parameters a and b along with the rate of learning L. • The relationship between u and Y(u), as shown on the left, is not linear. But, the relationship between ln(u) and ln(Y(u)) is linear. Processing time per unit, Y(u) Units produced, u

13. Estimating Parameter Values Y(u)= au-b(Learning Curve) or, ln(Y(u)) = ln(au-b) (Take logarithm on both sides) or, ln(Y(u)) = ln(a)+ln(u-b) or, ln(Y(u)) = ln(a) - bln(u) This equation has the form of a straight line y = c + mx(straight line, with slope m and intercept c) Thus, a plot of ln(u) vs ln(Y(u)) fits a straight line

14. Estimating Parameter Values ln(Y(u)) = ln(a) - bln(u)(Learning Curve) y = c +mx(straight line) Notice that Intercept = ln(a) Hence, a = eintercept Slope = - bHence, b = -slope Finally, Since, b = - ln (L)/ ln (2), we have L = eslope*ln(2)

15. Estimating Parameter Values • It’s an important fact that the relationship between ln(u) and ln(Y(u)) is linear. Because if the relationship between two variables is linear, we can fit a straight line that provides the relationship. • The slope and intercept of the straight line are obtained by using linear regression on ln(u) and ln(Y(u)). • The slope and intercept can then be used to get paratmeters a and b and rate of learning L. ln(Y(u)) ln(u)

16. Estimating Parameter Values • We estimate parameters as follows: • Step 1: Given a set of u and Y(u) values, compute the set of ln(u) and ln(Y(u)) values. • Step 2: Using linear regression on ln(u) and ln(Y(u)), compute slope, m and intercept, c of the straight line that best fits the set of ln(u) and ln(Y(u)) values. c m ln(Y(u)) 1 ln(u) • Step 3: Compute a, b and L using the following formula: • a = eintercept = ec • b = -slope = -m • L = eslope*ln(2) = em*ln(2)

17. Estimating Parameter Values • An interpretation of the intercept, c: • ec is an estimate of the time required for the first unit denoted by a or Y (1). • An interpretation of the slope, m: • em*ln(2) is an estimate of the rate of learning, L. • Learning is demonstrated by the negative slope. • If the slope is less, then the line is steeper, L is less and the learning is faster. c m ln(Y(u)) 1 ln(u)

18. Estimating Parameter Values: Example Consider the text example:

19. Relationship Between u and Y(u) A plot of u vs Y(u) is not linear.

20. Relationship Between ln(u) and ln(Y(u)) A plot of ln(u) vs ln(Y(u)) is linear. Hence, linear regression is used on ln(u) and ln(Y(u)).

21. Step 1 Step 1: Compute the logarithmic values. The Excel function for computing natural logarithms is LN() e.g., if a value of u is in B6, formula for ln(u) is =LN(B6)

22. Step 2 by Hand Step 2: Compute

23. Step 3 by Hand Step 3: Compute the slope and intercept:

24. Steps 2 and 3 by Excel • If Excel is used, steps 2 and 3 can be replaced by a single step. Two built-in Excel functions provides slope and intercept as shown below: • Suppose that ln(u) values are in column A rows 18-25 ln(Y(u)) values are in column B rows 18-25 • Excel formulae for Intercept is INTERCEPT(B18:B25,A18:A25) Slope is SLOPE(B18:B25,A18:A25) • Thus, intercept = 3.1301, and slope = -0.42276.

25. Step 4 Step 4: Compute the parameters a, b, L Suppose that the values of intercept and slope are in cells B29 and B30 respectively Parameter Formula Excel formula Value aeintercept = EXP(B29) b-slope = -B30 Leslope*ln(2) = EXP(B30*ln(2))

26. READING AND EXERCISES Lesson 2 Reading: Section 1.10, pp. 32-38 (4th Ed.), pp. 29-36 (5th Ed.) Exercises: 29, 30, 33, pp. 37-38 (4th Ed.), pp. 35-36 (5th Ed.)