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WHY REPLACEMENT?
If any equipment or machine is used for a long period of time, due to wear and tear, the item tends to worsen. A remedial action to bring the item or equipment to the original level is desired. Then the need for replacement becomes necessary. This need may be caused by a loss of efficiency in a situation leading to economic decline. By efflux of time the parts of an item are being worn out and the cost of maintenance and operation is bound to increase year after year. The resale value of the item goes on diminishing with the passage of time. The depreciation of the original equipment is a factor, which is responsible not to favour replacement because the capital is being spread over a long time leading to a lower average cost. Thus there exists an economic trade-off between increasing and decreasing cost functions. We strike a balance between the two opposing costs with the aim of obtaining a minimum cost. The problem of replacement is to determine the appropriate time at which a remedial action should be taken which minimizes some measure of effectiveness. Another factor namely technical and / or economic obsolescence may force us for replacement.
In this segment we deal with the replacement of capital equipment that deteriorates with time, group replacement and staffing problems.
REPLACEMENT OF ITEMS WITH GRADUAL DETERIORATION
As mentioned earlier the equipments, machineries and vehicles undergo wear and tear with the passage of time. The cost of operation and the maintenance are bound to increase year by year. A stage may be reached that the maintenance cost amounts prohibitively large that it is better and economical to replace the equipment with a new one. We also take into account the salvage value of the items in assessing the appropriate or opportune time to replace the item. We assume that the details regarding the costs of operation, maintenance and the salvage value of the item are already known. The problem can be analysed first without change in the value of the money and later with the value included.
If the interest rate for the money is zero the comparison can be made on an average cost basis. The total cost of the capital in owning the item and operating is accumulated for n years and this total is divided by n.
Since we have discrete values for the costs for various years, an analysis is done using the tabular method, which is simple one to use discontinuous data. There are also the classical optimization techniques using finite difference methods for discrete parameters and using the differential calculus for continuous data.
Now we take an example in which an automobile fleet owner has the following direct operation cost (Petrol and oil) and increased maintenance cost (repairs, replacement of parts etc). The initial cost of the vehicle is Rs. 70,
- The operation cost, the maintenance cost and the resale price are all given in table 1 for five years.
Table 1
Year of service
Annual operating cost (Rs.)
Annual maintenance cost (Rs.)
Resale value (Rs.) 1 10000 6000 40000
From the above 3 we conclude that the machine should be replaced at the end of the fifth year, indicated by the least average annual cost (Rs. 2700) in the last column.
Example The mill owner in the previous problem has now three machines, two of which are two years old and the third one year old. He is considering a new type of machine with 50% more capacity than one of the old ones at a unit price of Rs. 8000. He estimates the running costs and resale price for the new machine will be as follows.
Year 1 2 3 4 5 6 7 Running cost (Rs.) 1200 1500 1800 2400 3100 4000 5000 Resale Price (Rs.) 4000 2000 1000 500 300 300 300
Assuming that the loss of flexibility due to fewer machines is of no importance, and that he will continue to have sufficient work for three of the old machines, what should his policy be?
Solution: As in the previous problem we prepare a table 4 to find the average annual cost of the new type of machine.
Table 4
At the end of year
Cumulative running cost (Rs.)
Capital cost (Rs.)
Total cost (Rs.)
Average annual cost (Rs.) 1 1200 4000 5200 5200 2 2700 6000 8700 4350 3 4500 7000 11500 3833 4 6900 7500 14400 3600 5 10000 7700 17700 3540 6 14000 7700 21700 3617 7 19000 7700 26700 3814
From the above table 4 we observe that the average annual cost is least at the end of five years and it would be Rs. 3540 per machine. But the new machine can handle 50% more capacity than the old one. So in terms of the old, the new machine's annual cost is only Rs. (3540) (2/3) = Rs. 2360. This amount is less than the average annual cost for the old machine, which is Rs. 2700. If we replace the old machine with the new one, it is enough to have two new machines in place of with the new one; it is enough to have two new machines in place of three old machines. On comparing the cost of 2 new machines (Rs. 7080) with that for 3 old machines (Rs 8100), it is clear that the policy should be that the old machines have to be replaced with the new one. Still we have to decide about the time when to purchase the new machines. The new machines will be purchased when the cost for the next year of running the three old machines exceeds the average annual cost for two new types of machines. Examining the table 3 pertaining to the previous problem, we find, the total yearly cost of one small machine from the column 4. The successive difference will give the cost of running a machine for a particular year. For example, the total cost for 1 year is Rs. 4000. The total cost for 2 years is Rs. 6700. The difference of Rs. 2700 will be accounted as the cost of running a small machine during the second year. Similarly we have Rs. 2150, Rs. 2175, Rs. 2475 and Rs. 2800 as the cost of running the old machine in the third, fourth, fifth and sixth year respectively.
Now, with this information we calculate the total costs next year for the two smaller machines, which are two years old (entering the third year of service) and one smaller machine aged one year (and hence entering second year of service), which will be
2 x 2150 + 2700 = Rs. 7000
This is less than the average annual cost of two new machines, which is Rs. 7080. So the policy is not to replace right now. If we wait for the subsequent years, the total cost of running the old machines will be Rs. 6500, Rs. 7125 and Rs. 8025 etc., for years 2, 3 and 4 etc. This indicates that the cost of running the old machine exceeds the average annual cost (Rs. 7000) of the two new machines after 2 years from now. Hence the best time to purchase the new type machine will be after 2 years from now.
