# Comparing Of Replacement Alternatives

**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.

**Comparing of replacement alternative:**

Sometimes a business man may come across a problem, when he wants to purchase a machine or a vehicle for his business. He may have different models with comparatively equal price and with different maintenance and other expenses. In such cases, he can use **present worth concept **to select the right type of machine or a vehicle. **Here the present value of all future expenditures and** **revenues is calculated for each alternative and the one for which the present value is minimum** **is preferred. **Let

*Q *= Annual cost or purchase price in Rs.

*i *= Annual interest rate.

*t *= Period in years.

*P *= Principal amount in Rs.

*S *= Scrap value or salvage value in Rs.

The present value of total cost is given by:

*P *** Q (Pwfs for i % interest rate for n years) – S (Pwf for i % interest for n years)**

Similarly, if the annual operating costs vary for different years then the present value of these costs will be calculated on the basis of time period for which these expenditures are made. *i.e. *say for example, if *Q*1, *Q*2, *Q*3…….*Qn *is the operating costs for different years, then the present value of the operating cost during '*n*' years will be given by:

*Q*_{1}(Pwfs at *i *% interest for 1 year) *Q*_{2} (*Pwfs *at *i *% for 2 years) ……….. *Q _{n} *(

*Pwfs*at

*i*% interest for

*n*years).

Now present value of total cost will be:

*P *** Q1 (Pwfs at i % interest for 1 year) Q2 (Pwfs at i % interest rate for 2 years) Q3 (Pwfs at i % rate of interest for 3 years) ………… Qn (Pwfs at i% interest for n years) – S (Pwf at i % interest for n years).**

**Example .**

**An entrepreneur is considering purchasing a machine for his factory. The related data for alternative machines are as **

As an advisor of the company, you have been asked to select the best machine considering 12 % normal rate of return per year. Given that: Present worth factor series @ 12 % for 10 years is 5.650 Present worth factor @ 12 % for 10 th year is 0.322

**Solution**

Machine *B *is having less present value of total cost. Hence to purchase the machine *B*.