# Repalcement Of Items Whose Maintinence Cost Increase With Time

**Introduction:**

Replacement model is the part of operation research mostly used in the industries when a purchased items like machinery, buildings efficiency is reduced or wear out due to much usage.

**Replacement of Items whose Maintenance Costs Increases with Time and Value of Money also Changes with Time**

**Present worth factor: **

· Before dealing with this model, it is better to have the concept of **Present value**.

· Consider replacement of items which involve huge expenditure, both initial value (Purchase price) and maintenance expenses. For a decision maker, there may be number of alternatives for replacement.

· But he always chooses the alternative, which minimizes the annual average cost. Here manager uses the concept of present value of money to select the alternatives.

· **The present value of****number of expenditures incurred over different periods of time represents their value at the current** **time. **

Suppose a businessman borrows money for his initial investment and working capital from various sources, he has to pay interest for the money he has borrowed. The amount of interest he has to pay depends on the rate of interest and the period for which he has borrowed (that is the period in which he has repaid the amount borrowed). The borrowed money is known as **Principal **(P), and the excess amount he has paid is known as **Interest **(i). The sum of both principal and the interest is known as **Amount (A).** If P is the principal, ‘i’ is the rate of interest, and A is the amount, then the amount at the end of ‘n’ years with compound interest is:

**A ****= P (1 i )n**

**OR P = (A) / ( 1 i )n**

**OR P = A × Pwf **...

**Where Pwf is Single payment present worth factor.**

· It is represented by ‘v’ and is also known as discount rate.

· Discount rate is always less than one. In other words we can say that ‘v’ or the present worth factor (Pwf) is present value of one rupee spent after ‘n’ years from now.

· Hence P is known as the present worth of an amount A paid in ‘n’ years at interest rate ‘i’. For calculation purpose present worth factor tables are available. Similarly, if R denotes the uniform amount spent at the end of each year and S is the total expenditure at the end of ‘n’ years, then

**S ****= R { (1 i ) n – 1} / i OR**

**R ****= (S × i) / (1 i ) n – 1 = {P ( 1 i ) n × i } / { (1 i ) n – 1 } OR**

**P ****= R × {P (1 i ) n – 1} / {i ( 1 i) n } = R × (Pwf ) **...

**Pwf is known as uniform annual series present worth factor.**

In other words, suppose if we want Rs. 50,000/- for investment after 10 years, how much we save yearly, so that at the end of 10 years, we will have Rs. 50,000/- ready for investment. The discount rate to find this amount is known as Pwf.