# Replacement of Assets that Deteriorate with Time

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 Runningcost Cumulative running cost Scrapvalue Difference between initial cost and scrap value Average investment cost / year Average running cost / year Average annualtotal 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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