Replacement of items which deteriorate with time without considering the change in money value
Most of the machinery and equipment having moving parts deteriorate in their performance with passage of time. The cost of maintenance and repair keeps increasing with passage of time and a stage may reach when it is more economical (in overall analysis) to replace the item with a new one. For example, a passenger car is bound to wear out with time and its repair and maintenance cost may go to such level that the owner has to replace it with a new one.
Let C = Capital cost of the item
S (t)=Scrap value of the item after t years of use,
O (t) = Operating and maintenance cost of the equipment at time t.
n = number of years the item can be used.
TC (n) = Total cost of using the equipment for n years.
Time ‘t’ in this case is a discrete variable.
In this case as long as the average TC (n) is minimum, the equipment can remain in use for that number of years. If average total cost keeps decreasing up to ith year and starts increasing from (i+1)th year then ith year may be considered as most economic year for replacement of the equipment.
The: conceptof depreciation cost also must be understood here. As the years pass by, the cost of the equipment or items keeps decreasing. How much the cost keeps decreasing can be calculated by two methods commonly used i.e. straight line depreciation method and the diminishing value method.
Example 8.1. A JCB excavator operator purchases the machine for Rs 15,00,000. The Operating cost and the resale value of the machine is given below.
Year 
1 
2 
3 
4 
5 
6 
7 
8 
Operating Cost in Rs. 
30000 
32000 
36000 
40000 
45000 
52000 
60000 
70000 
Resale value (in lakhs of Rs.) 
12 
10 
8 
5 
4.5 
4 
3 
2 
When should the machine be replaced?
Sol.
C=15,00,000
O (t) = Operating cost
S (t) = Resale value
t=Time
n = Number of years after when the machine is to replaced.
Let us draw a table showing the various variables required to make decision. This is shown in the table
Year 
O (t) (in thousand of rupees) 
Comulative O (t) 
Resale value S (t) in thousands of rupees 
Depreciation CS (t) in thousands of Rupees 
Total cost TC (n) thousands of Rupees 
Average TC (n) Thousands of Rupees 
1 
30 
30 
1200 
300 
330 
330 
2 
32 
62 
1000 
500 
562 
281 
3 
36 
98 
800 
700 
798 
266 
4 
40 
138 
500 
1000 
1138 
284.5 
5 
45 
183 
450 
1050 
1233 
246.6 
6 
52 
235 
400 
1100 
1335 
222.5 
7 
60 
295 
300 
1200 
1495 
213.6 
8 
70 
365 
200 
1300 
1665 
208 
In 3^{rd} year the minimum average cost is 2,66,000 as shown in the table above, So replacement should take place at the end of 3^{rd} year.
Example 8.2 A taxi owner estimates from his past records that the cost per year for operating a taxi whose purchase price when new is Rs 60,000 are as given below.
Age  1  2  3  4  5 
Operating Cost (Rs.)  10000  12000  15000  18000  20000 
After 5 years, the operating cost if Rs 6000k where k = 6, 7, 8, 9, 10 (k denoting age in years). If the resale value decreases 10% of purchase price per year, what is the best replacement policy? Cost of money is zero.
Sol. Since the depreciation of the taxi is 10% of Rs 60,000, it means the resale value of the taxi decreases by Rs 6000 every year. Average annual cost can be calculated as shown in the table below.
Year of Service 
Operating Cost O (t) 
Cumulative O (t) 
Resale value S (t) 
Depreciation CS (t) 
Total cost TC (n) 
Average TC (n) 
1 
10000 
10000 
54000 
6000 
16000 
16000 
2 
12000 
22000 
48000 
12000 
34000 
17000 
3 
15000 
37000 
42000 
18000 
55000 
18333 
4 
18000 
55000 
36000 
24000 
79000 
19750 
5 
20000 
75000 
30000 
30000 
105000 
21000 
6 
360000 
111000 
24000 
36000 
147000 
24500 
7 
42000 
153000 
18000 
42000 
195000 
27857 
8 
48000 
201000 
12000 
48000 
249000 
31125 
9 
54000 
255000 
6000 
54000 
309000 
34333 
10 
60000 
315000 
0 
60000 
375000 
37500 
Total cost TC (n) = operating cost O (t) + Depreciation C – S (t)
It can be seen that average Total cost is minimum after one year. Hence the taxi should be replaced after one year.
Example 8.3 A truck owner finds from his past records that the manufacturing cost of a truck (whose purchase is Rs 3,00,000) during the first 8 years of its life and the resale price at the end of each year is as follows.
Year 
Maintenance Cost (Rs.) 
Resale Price (Rs.) 
1 
36000 
200000 
2 
48000 
150000 
3 
60000 
100000 
4 
72000 
80000 
5 
84000 
70000 
6 
96000 
60000 
7 
108000 
50000 
8 
120000 
40000 
Sol. The average cost per year is calculated in the following table.
Year of Service 
Operating Cost O (t) 
Cumulative O (t) 
Resale value S (t) 
Depreciation CS (t) 
Total cost TC (n) 
Average TC (n) 
1 
36000 
36000 
200000 
100000 
136000 
136000 
2 
48000 
84000 
150000 
150000 
234000 
117000 
3 
60000 
144000 
100000 
200000 
344000 
114667 
4 
72000 
216000 
80000 
220000 
436000 
109000 
5 
84000 
300000 
70000 
230000 
530000 
106000 
6 
96000 
396000 
60000 
240000 
636000 
106000 
7 
108000 
504000 
50000 
250000 
754000 
107714 
8 
120000 
624000 
40000 
260000 
884000 
110500 
Since average total cost is minimum in the 5th & 6th year, the truck should be replaced at the end of six years. He gets no advantage by replacing it at the end of 5 years.
Example 8.4. A new tempo costs Rs 1,00,000 and may be sold at the end of year at the following prices.
Year  1  2  3  4  5  6 
Selling Price (Rs.)  60000  45000  32000  22000  10000  2000 
The corresponding annual operating costs are
Year  1  2  3  4  5  6 
Cost / Year (Rs.)  10000  12000  15000  20000  30000  45000 
It is not only possible to sell the tempo after use but also to buy a second hand tempo. It may be cheaper to do so than to buy a new tempo.
Age of tempo  0  1  2  3  4  5 
Purchase Price (Rs.)  100000  60000  45000  33000  20000  10000 
What is the age to buy and to sell so as to minimize average annual cost?
Sol. Cost of new tempo = Rs 1,00,000
Let us find out the average cost per year of the new tempo,
Year of Service (1) 
Operating Cost O (t) (2) 
Cumulative O (t) (3) 
Resale value S (t) (4) 
Depreciation CS (t) (5) 
Total cost TC (n) 6=3+5 
Average TC (n) 7=3/1 
1 
10000 
10000 
60000 
40000 
50000 
50000 
2 
12000 
22000 
45000 
55000 
77000 
38500 
3 
15000 
37000 
32000 
68000 
105000 
35000 
4 
20000 
57000 
22000 
78000 
135000 
33750 
5 
30000 
87000 
10000 
90000 
177000 
35400 
6 
45000 
132000 
2000 
98000 
230000 
35000 
Average cost is minimum at the end of 4 years, hence the new tempo should be replaced after 4 years.
Let us now find out the average total cost of second hand tempo.
Year of Service (1) 
Operating Cost O (t) (2) 
Cumulative O (t) (3) 
Resale value S (t) (4) 
Depreciation CS (t) (5) 
Total cost TC (n) 6=3+5 
Average TC (n) 7=3/1 
0 
100000 
100000 
100000 

1 
10000 
10000 
60000 
40000 
50000 
50000 
2 
12000 
22000 
45000 
55000 
77000 
38500 
3 
15000 
37000 
32000 
68000 
105000 
35000 
4 
20000 
57000 
20000 
78000 
137000 
33750 
5 
30000 
87000 
10000 
90000 
177000 
35400 
The tempo may be replaced by second hand tempo at the end of third year and the owner can save Rs (3500034666) i.e. Rs 334 instead of buying a new one.
Example 8.5. A machine type A costs Rs 50,000 and the operating costs are estimated at Rs 1200 for the first year and increased by Rs 8000 in second and subsequent years. Another machine type B costs Rs 60,000 and operating costs are Rs 1500 for the first year and increasing by Rs 5000 per year. Should machine A be replaced by machine B assuming that both machines have no resale value and cost of money does not change with time?
Machine A
Year of Service

Operating Cost O (t) (1) 
Cumulative O (t) (2) 
Depreciation CS (t) (3) 
Total cost TC (n) 4=2+3 
Average TC (n)

1 
1200 
1200 
50000 
51200 
51200 
2 
9200 
10400 
50000 
60400 
30200 
3 
17200 
27600 
50000 
77600 
25866 
4 
25200 
52800 
50000 
102800 
25700 
5 
33200 
86000 
50000 
136000 
27200 
6 
41200 
127200 
50000 
177000 
29500 
Machine B
Year of Service

Operating Cost O (t) (1) 
Cumulative O (t) (2) 
Depreciation CS (t) (3) 
Total cost TC (n) 4=2+3 
Average TC (n)

1 
1500 
1500 
60000 
61500 
61500 
2 
6500 
8100 
60000 
68100 
34050 
3 
11500 
19600 
60000 
796000 
26533 
4 
16500 
36100 
60000 
96100 
24025 
5 
21500 
57600 
60000 
117600 
23520 
6 
26500 
84100 
60000 
144100 
24016 
It can be exactly seen that the lowest average cost of machine B (Rs. 23520) is lesser than that of machine A (Rs. 25700), machine A should be replaced by machine B at the end of 4th year.
Example 8.6. A firm is considering replacement of a machine, whose cost price is Rs 12.200 and the scrap value Rs 200. The running (maintenance and operation) costs in Rs, are found from experience to be as follows:
Year  1  2  3  4  5  6  7  8 
Running Cost  200  500  800  1200  1800  2500  3200  4000 
When should the machine be replaced?
Sol. We are given the following data.
C = Rs 12,200
S (t) = Rs200
O (t) is also given.
Let n be the number of years the item can be used.
Average total cost per year during the life of the machine is shown in the table below.
The average total cost per year is minimum in 6th year i.e Rs 3167 And the average cost in 7th year is Rs 3171 which is more than the cost in 6th year. Hence, the machine should be replaced after 6 years.
Example 8.7 (a) Machine A costs Rs 9000. Annual operating costs are Rs 200 for the first year and then increases Rs 2000 every year. Determine the best age at which to replace the machine. If the optimum replacement policy is followed, what will be the average yearly cost of owning and operating the machine?
(b) Machine B costs Rs 10.000. Annual operating cost are Rs 400 for the first year and then increases by Rs 800 every year. You now have a machine of type A which is one year old. Should you replace it with B, if so when?
Sol. (a) Let us assume that there is no scrap value of the machine Average total cost can be computed as
Year (n) 
Operating Cost O (t) 
Cumulative 12Î£ O(t)”> 
Depreciation CS(t) 
Total Cost 
Average Cost 
1 2 3 4 5 
200 2200 4200 6200 8200 
200 2400 6600 12800 21000 
9000 for all years 
9200 11400 15600 21800 30000 
9200 5700 5200 5450 6000 
It can be seen that the best age of replacement is 3^{rd} year.
(b) For machine B, the average cost be calculated as follows.
Year (n) 
Operating Cost O (t) 
Cumulative 12Î£ O(t)”> 
Depreciation CS(t) 
Total Cost 
Average Cost 
1 2 3 4 5 6 
40 1200 2000 2800 3600 4400 
400 1600 3600 6400 10000 14400 
10000 for all years 
10400 11600 13600 16400 20000 24400 
10400 5800 4533 4100 4000 4066 
Since the minimum average cost for machine B is lower than for machine A, machine B should be replaced by machine A Minimum average cost is (Rs 4,000), it should be replaced when it exceeds Rs 4000. In case of one year old machine Rs 2200/ will be spent next year and Rs 4200 the following year. We should keep machine A for one year.
Replacement policy of an equipment/item whose operating cost increases with time and money value also changes with time.
In previous examples, we assumed that the money value does not change and remains constant but it is well known that as the equipment deteriorates and operating costs keep increasing, the money value keeps decreasing with time. Hence we must calculate the Net Present Value (NPV) of the money to be spent a few years hence. Otherwise the resale value, the operating costs, which are to take place in future, will not be realistic and management will not be able to take optimal decisions.
Let C = initial cost of item/equipment
OC = operating cost
r = rate of interest
A rupee invested at present will be equivalent to (1 + r) a year after (1 + r)^{2} two years hence and (1 + r)^{n} in 11 years time. It means that making a payment of one rupee after n years is equivalent to paying (1 + r)^{n} now. The quantity (1 + r)^{n} is called the present worth or present value of one rupee spent n years from now.
Present value of a rupee V = (1 + r)^{1} = 1/1 + r is called discount rate and is always less than 1.
Then year wise present value of expenditure in future years can be calculated as
Present value (n) = (c + oc_{1}) + oc_{2} v + oc_{3} i + + oc_{n} v^{n}1
+ (c + oc_{1}) v^{n} + oc_{2} v^{n+1}+ oc_{3} v^{n+2} + …. + oc_{n} v2^{n1}
+ (c + oc_{1}) v^{2n} + oc_{2 v}^{2n }^{+ 1} …. + oc_{n} v^{3n1}
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