This model is represented by:
- Increasing maintenance cost.
- Decreasing salvage value.
Assumption
- Increased age reduces efficiency
Generally, the criteria for measuring efficiency is the discounted value of all future costs associated with each policy.
Let
C = the capital cost of a certain item, say a machine
S(t) = the selling or scrap value of the item after t years.
F(t) = operating cost of the item at time t
n = optimal replacement period of the time
Now, the annual cost of the machine at time t is given by C – S(t) + F(t) and since the total maintenance cost incurred on the machine during n years is F(t) dt, the total cost T, incurred on the machine during n years is given by:
T = C – S(t) + F(t) dt
Thus, the average annual total cost incurred on the machine per year during n years is given by
| TA = | 1 —– n |
C – S(t) + F(t) dt |
To determine the optimal period for replacing the machine, the above function is differentiated with respect to n and equated to zero.
| dTA —— dn |
= | -1 —– n2 |
C – S(t) | -1 —– n2 |
F(t) dt | + | F(n) —— n |
| Equating | dTA —— dn |
= 0, we get |
| F(n) = | 1 —– n |
C – S(t) + F(t) dt |
That is, F(n) = TA
Example
The initial cost of a machine is Rs. 7100 and scrap value is Rs. 100. The maintenance costs found from experience are as follows:
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Maintenance | 200 | 350 | 500 | 700 | 1000 | 1300 | 1700 | 2100 |
When should the machine be replaced?
Solution.
| Year | Running cost |
Cumulative running cost | Scrap value |
Difference between initial cost and scrap value | Average investment cost / year | Average running cost / year | Average annual total cost |
||
| A | B | C | D | E | F = E/A | G = C/A | H = F + G | ||
| 1 | 200 | 200 | 100 | 7000 | 7000 | 200 | 7200 | ||
| 2 | 350 | 200 + 350 = 550 | 100 | 7000 | 3500 | 225 | 3775 | ||
| 3 | 500 | 550 + 500 = 1050 | 100 | 7000 | 2333.33 | 350 | 2683.33 | ||
| 4 | 700 | 1050 + 700 = 1750 | 100 | 7000 | 1750 | 437.5 | 2187.50 | ||
| 5 | 1000 | 1750 + 1000 = 2750 | 100 | 7000 | 1400 | 550 | 1950 | ||
| 6 | 1300 | 2750 + 1300 = 4050 | 100 | 7000 | 1166.67 | 675 | 1841.67 | ||
| 7 | 1700 | 4050 + 1700 = 5750 | 100 | 7000 | 1000 | 821.42 | 1821.42 | ||
| 8 | 2100 | 5750 + 2100 = 7850 | 100 | 7000 | 875 | 981.25 | 1856.25 | ||
This table shows that the average annual total cost during the seventh year is minimum. Hence, the machine should be replaced after the 7th year.
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