Replacement Models Examples : Operations Research
Constant Resale Value
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 | Running cost |
Cumulative running cost | Scrap value |
Difference between initial cost and scrap value | Average investment cost / year | Average running cost / year | Average annual total 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.
Falling Resale Value
Example
The initial cost of a machine is Rs. 6100 and resale value drops as the time passes. Cost data are given in the following table:
| Year | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Maintenance | 100 | 250 | 400 | 600 | 900 | 1200 | 1600 | 2000 |
| Resale Value | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 |
When should the machine be replaced?
Solution.
| Year | Running cost |
Cumulative running cost | Resale value |
Difference between initial cost and resale value | Average investment cost / year | Average running cost / year | Average annual total cost |
|---|---|---|---|---|---|---|---|
| 1 | 100 | 100 | 800 | 5300 | 5300 | 100 | 5400 |
| 2 | 250 | 350 | 700 | 5400 | 2700 | 175 | 2875 |
| 3 | 400 | 750 | 600 | 5500 | 1833.33 | 250 | 2083.33 |
| 4 | 600 | 1350 | 500 | 5600 | 1400 | 337.5 | 1737.50 |
| 5 | 900 | 2250 | 400 | 5700 | 1140 | 450 | 1590 |
| 6 | 1200 | 3450 | 300 | 5800 | 966.67 | 575 | 1541.67 |
| 7 | 1600 | 5050 | 200 | 5900 | 842.85 | 721.42 | 1564.27 |
| 8 | 2000 | 7050 | 100 | 6000 | 750 | 881.25 | 1631.25 |
This table shows that the average annual total cost during the sixth year is minimum. Hence, the machine should be replaced after the 6th year.
Present Worth Factor
In this method, the present value of all future expenditures and revenues
for each alternative is calculated. An item whose present worth factor
is least is preferred.
Let
P = purchase cost of an item
A = annual running cost
n = life of an item in years
S = salvage value
r = annual interest rate
The present value can be calculated as follows:
P + A (Pwf for r% interest rate for n years) - S (Pwf for r% interest rate for n years)
For an illustration, consider the following problem.
Example
The China Moon restaurant is considering to purchase a new cooling system. Cost data are given in the following table:
| Cooling system A |
Cooling system B |
Cooling system C |
|
|---|---|---|---|
| Present investment (Rs.) | 12000 | 14000 | 17000 |
| Total annual cost (Rs.) | 3000 | 2000 | 1500 |
| Life (Years) | 10 | 10 | 10 |
| Salvage value (Rs.) | 500 | 1000 | 1200 |
On the basis of above data, select the best cooling system considering
12% normal rate of return per year.
Given
Pwf (total annual cost) @ 12% for 10 years = 5.650
Pwf (salvage value) @ 12% for 10 years = 0.322
Solution.
| Cooling system A |
Cooling system B |
Cooling system C |
|
|---|---|---|---|
| Present investment (Rs.) | 12000 | 14000 | 17000 |
| Total annual cost (Rs.) | 3000 X 5.650 | 2000 X 5.650 | 1500 X 5.650 |
| Salvage value (Rs.) | 500 X 0.322 | 1000 X 0.322 | 1200 X 0.322 |
| Total Cost | 28789 | 24978 | 25088.6 |
Total Cost = Present investment + Total annual cost - Salvage value
Cooling system A = 12000 + 16950 - 161 = Rs. 28789
Cooling system B = 14000 + 11300 - 322 = Rs. 24978
Cooling system C = 17000 + 8475 - 386.4 = Rs. 25088.6
Hence, cooling system B should be purchased because it has least total cost.
