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### 1. Examples

1. Machine A costs Rs 45,000 and its operating costs are estimated to be Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years.
Machine B costs Rs 50,000 and operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 in the second and subsequent years.
If at present we have a machine of type A, should we replace it with B? if so when?
Assume that both machines have no resale value and their future costs are not discounted.

Solution:
In order to determine the optimal time n when the machine A should be replaced, we first calculate the average cost per year during the life of the machine A,
 Yearn(1) Running CostR(n)(2) Cummulative Running CostSigma R(n)(3) Depritiation CostC-S(4) Total CostTC(5)=(3)+(4) Average Total CostATC_n(6)=(5)/(1) 1 1,000 1000 = 1000RunCost = OperatingCost 1,000 1000 = 0 + 1000(3) = Previous(3) + (2) 45,000 46,000 46000 = 1000 + 45000(5)=(3)+(4) 46,000 46000 = 46000 / 1(6)=(5)/(1) 2 11,000 11000 = 1000 + 10000RunCost = Previous RunCost + OperatingCost 12,000 12000 = 1000 + 11000(3) = Previous(3) + (2) 45,000 57,000 57000 = 12000 + 45000(5)=(3)+(4) 28,500 28500 = 57000 / 2(6)=(5)/(1) 3 21,000 21000 = 11000 + 10000RunCost = Previous RunCost + OperatingCost 33,000 33000 = 12000 + 21000(3) = Previous(3) + (2) 45,000 78,000 78000 = 33000 + 45000(5)=(3)+(4) 26,000 26000 = 78000 / 3(6)=(5)/(1) 4 31,000 31000 = 21000 + 10000RunCost = Previous RunCost + OperatingCost 64,000 64000 = 33000 + 31000(3) = Previous(3) + (2) 45,000 109,000 109000 = 64000 + 45000(5)=(3)+(4) 27,250 27250 = 109000 / 4(6)=(5)/(1) 5 41,000 41000 = 31000 + 10000RunCost = Previous RunCost + OperatingCost 105,000 105000 = 64000 + 41000(3) = Previous(3) + (2) 45,000 150,000 150000 = 105000 + 45000(5)=(3)+(4) 30,000 30000 = 150000 / 5(6)=(5)/(1) 6 51,000 51000 = 41000 + 10000RunCost = Previous RunCost + OperatingCost 156,000 156000 = 105000 + 51000(3) = Previous(3) + (2) 45,000 201,000 201000 = 156000 + 45000(5)=(3)+(4) 33,500 33500 = 201000 / 6(6)=(5)/(1)

The calculations in table show that the average cost of machine A is lowest during the 3^(rd) year (Rs 26,000).

Hence, the machine A should be replaced after every 3^(rd) years.

In order to determine the optimal time n when the machine B should be replaced, we first calculate the average cost per year during the life of the machine B,
 Yearn(1) Running CostR(n)(2) Cummulative Running CostSigma R(n)(3) Depritiation CostC-S(4) Total CostTC(5)=(3)+(4) Average Total CostATC_n(6)=(5)/(1) 1 2,000 2000 = 2000RunCost = OperatingCost 2,000 2000 = 0 + 2000(3) = Previous(3) + (2) 50,000 52,000 52000 = 2000 + 50000(5)=(3)+(4) 52,000 52000 = 52000 / 1(6)=(5)/(1) 2 6,000 6000 = 2000 + 4000RunCost = Previous RunCost + OperatingCost 8,000 8000 = 2000 + 6000(3) = Previous(3) + (2) 50,000 58,000 58000 = 8000 + 50000(5)=(3)+(4) 29,000 29000 = 58000 / 2(6)=(5)/(1) 3 10,000 10000 = 6000 + 4000RunCost = Previous RunCost + OperatingCost 18,000 18000 = 8000 + 10000(3) = Previous(3) + (2) 50,000 68,000 68000 = 18000 + 50000(5)=(3)+(4) 22,666.67 22666.67 = 68000 / 3(6)=(5)/(1) 4 14,000 14000 = 10000 + 4000RunCost = Previous RunCost + OperatingCost 32,000 32000 = 18000 + 14000(3) = Previous(3) + (2) 50,000 82,000 82000 = 32000 + 50000(5)=(3)+(4) 20,500 20500 = 82000 / 4(6)=(5)/(1) 5 18,000 18000 = 14000 + 4000RunCost = Previous RunCost + OperatingCost 50,000 50000 = 32000 + 18000(3) = Previous(3) + (2) 50,000 100,000 100000 = 50000 + 50000(5)=(3)+(4) 20,000 20000 = 100000 / 5(6)=(5)/(1) 6 22,000 22000 = 18000 + 4000RunCost = Previous RunCost + OperatingCost 72,000 72000 = 50000 + 22000(3) = Previous(3) + (2) 50,000 122,000 122000 = 72000 + 50000(5)=(3)+(4) 20,333.33 20333.33 = 122000 / 6(6)=(5)/(1)

The calculations in table show that the average cost of machine B is lowest during the 5^(th) year (Rs 20,000).

Hence, the machine B should be replaced after every 5^(th) years.

The lowest average running cost of (Rs 20,000) per year for machine B is less than the lowest average running cost of (Rs 26,000) per year for machine A.
Hence machine A should be replaced by machine B.

This material is intended as a summary. Use your textbook for detail explanation.
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