Productivity increases with time. This improvement is commonly associated with improvements in efficiency brought about by increased experience and skill levels. Figure 1.1 shows the direct workhours per pound of airplane against the cumulative number of planes produced for eight types of fighters produced by four manufacturers in World War II. If the same data is plotted on log-log paper, a straight line results as shown in Fig. 1.2. This relationship can be used to generate the equation
EN+KNs (1.14)
where EN is the effort per unit of production required to produce the Nth unit.
K is a constant, derived from the data at hand, that represents the amount of
theoretical effort required to produce the first unit. And S is the slope constant, which will always be negative since increasing experience and efficiency leads to reduced effort on a given task. Note that:
and
Figure 1.1 Industry average unit curve for century series aircraft. (From K.K. Humphreys, Jelen’s Cost and Optimization Engineering, 3rd ed., 1991, McGraw-Hill, Inc., New York, p. 212. Reprinted by permission.)
and so on. Every time cumulative production is doubled, the effort per unit required is a constant 2S of what it had been. It is common to express the learning curve as a function of the gain for double the production. Thus a 90% learning curve function means it requires only 90% of the effort to produce the (2N)th unit as it did the Nth unit.
If we designate the percentage learning ratio as Lp the relationship between it and the slope is
(1.15) For the total effort required for N units from 1 through N, the cumulative effort, ET, is
(1.16) An approximation of this summation is given by
(1.17) This approximation improves as N increases. If N is very large in comparison to 0.5, (1.17) reduces to
(1.18)
Figure 1.2 Log-log plot of industry average unit curve for century series aircraft.
(From K.K.Humphreys, Jelen’s Cost and Optimization Engineering, 3nd ed., 1991, McGraw-Hill, Inc., New York, p. 213. Reprinted by permission.)
The following examples show some practical applications for the learning curve.
Example 1.4*
If 846.2 workhours are required for the third production unit and 783.0 for the fifth unit, find the percentage learning ratio and the workhours required for the second, fourth, tenth, and twentieth units.
Solution: By Eq. (1.14)
E3=846.2=K(3s)
E5=783.0=K(5s) By division we obtain
By Eq. (1.15)
LP=90 percent=percentage learning ratio.
Using the data for N=3 and S=-0.1520 in Eq. (1.14), 846.2=K(3-0.1520)
log 846.2=log K-0.1520 log 3 2.9278=log K-0.07252
log K=3.0000
K=1,000
so that the learning curve function is
EN=1000N-0.1520
Source: K.K.Humphreys, ed., Jelen’s Cost and Optimization Engineering, 3rd ed. Copyright 1991 by
The effort required for any unit can now be calculated directly. Thus, for the twentieth unit,
E20 = 1000(20-0.1520)
log E20 = log 1000-0.1520 log 20 = 3.0000-0.1520(1.30103) = 2.80224
E20= 634.2
A tabulation for other units is:
E1 = 1000.0 E2 = 900.0 E4 = 810.0 E5 = 783.0 E10= 704.7 E20= 634.2 Note that: Example 1.5*
Every time the production is tripled, the unit workhours required are reduced by 20 percent. Find the percentage learning ratio.
Solution: It will require 80 percent, or 0.80, for the ratio of effort per unit for tripled production.
By Eq. (1.14)
*Source: K.K.Humphreys, ed., Jelen’s Cost and Optimization Engineering, 3rd ed. Copyright 1991 by
By Eq. (1.15) for doubled production
log Lp=1.93886
LP=86.86 percent=percentage learning ratio 1.8 PROFITABILITY
The six most common criteria for profitability are: 1. Payout time
2. Payout time with interest
3. Return on original investment (ROI) 4. Return on average investment (RAI)
5. Discounted cash flow rate of return (DCFRR) 6. Net present value (NPV)
Consider a project having an income after taxes but before depreciation (cash flow) as follows:
and consider the profitability as measured by these six criteria.
1. Payout time is the time to get the investment back. Working capital does not enter payout time calculations since working capital is always available as such. (The project has no stated working capital anyway; working capital is discussed in the next section.) A tabulation gives
The initial investment is returned during the third year; by interpolation, the payout time is 2.38 years.
2. Payout time with interest allows for a return on the varying investment. The interest charge is on the fixed investment remaining only; another variant applies an interest charge on working capital. The project leads to the following tabulation with 10% interest:
By interpolation the payout time is 2.95 years, allowing for a 10% return on the varying investment.
3. The return on original investment (ROI) method regards the investment as fixed. Working capital should be included in the investment figure. In this example working capital is not given and therefore is zero. Using straight- line depreciation and a 5-year project life we get:
Thus for an original investment of $1000 and no working capital,
4. The return on average investment (RAI) is similar to the ROI except that the divisor is the average outstanding investment plus working capital. (Again, working capital in this example is considered to be zero.) The average investment for the project with straight-line depreciation is
5. The discounted cash flow rate of return takes the timing of all cash flows into consideration. This method finds the rate of return that makes the present value of all of the receipts equal to the present value of all of the expenses. For this project we have
By solving the above expression iteratively (i.e., by trial and error) to find a value of r that yields zero, we have
r=DCFRR=23.9%
6. The net present value (NPV) is calculated without the need for an iterative solution. Rather, a rate of return is chosen, ordinarily the minimum acceptable rate of return, and the NPV is calculated on that basis. If the minimum acceptable rate of return is 10%,
Table 1.11 gives a simple format for calculating cash flow and the DCFRR.
Depreciation is not an out-of-pocket expense, but must be considered in calculating the taxable income (tax base); otherwise taxes would be overpaid and net income would drop. Table 1.11 is for a venture that requires $1 million up front and has end-of-year receipts of $600,000, $700,000, and $340,000, for years 1, 2, and 3, respectively. (For simplicity’s sake, consider these receipts to be the difference between gross income and expenses, but before taxes and depreciation.) Taxes are 34%, and sum-of-the-years-digits depreciation is used. As shown, for zero net present value for the cash flow, the DCFRR is 22.45%.