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.

dTA ------ dn |
= | -1 ----- n ^{2} |
C - S(t) | -1 ----- n ^{2} |
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.