# Replacement Of Items That Deteriorates With Time

We begin here with the simplest replacement model where the deterioration process is predictable. More complex replacement models are studied in the subsequent sections.

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.

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 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

Thus, we conclude that an item should be replaced when the average cost to date becomes equal to the current maintenance cost.

Don't be too afraid of these frightening looking formulas. Rest assured, by the time you finish this chapter, you will have learned all the details necessary to apply this approach.

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