Question
A machine X costs rs. 5000. its maintenance cost is Rs. 1000 in each of first four years and then it increases by Rs. 200 every year. Assuming that the machine has no salvage value and the maintenance cost incurred in the beginning of each year , determine the optimal replacement time for the machine assuming that the time value of money is 10% pa.
Answer
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Elite · Tutor for 8 years
Answer
To determine the optimal replacement time for the machine, we need to compare the present value of the costs associated with keeping the machine for a certain number of years with the present value of the costs associated with replacing the machine at different times.Let's break down the costs associated with keeping the machine for each year:- Year 1: Maintenance cost = Rs. 1000- Year 2: Maintenance cost = Rs. 1000 + Rs. 200 = Rs. 1200- Year 3: Maintenance cost = Rs. 1000 + 2 * Rs. 200 = Rs. 1400- Year 4: Maintenance cost = Rs. 1000 + 3 * Rs. 200 = Rs. 1600- Year 5: Maintenance cost = Rs. 1000 + 4 * Rs. 200 = Rs. 1800- Year 6: Maintenance cost = Rs. 1000 + 5 * Rs. 200 = Rs. 2000- ...To calculate the present value of these costs, we need to discount them using the time value of money. In this case, the discount rate is 10% per annum.Now, let's calculate the present value of the costs associated with keeping the machine for different numbers of years:- Keeping the machine for 1 year: PV = Rs. 1000 / (1 + 0.10)^1 = Rs. 909.09- Keeping the machine for 2 years: PV = (Rs. 1000 + Rs. 1200) / (1 + 0.10)^2 = Rs. 1818.18- Keeping the machine for 3 years: PV = (Rs. 1000 + Rs. 1200 + Rs. 1400) / (1 + 0.10)^3 = Rs. 2460.33- Keeping the machine for 4 years: PV = (Rs. 1000 + Rs. 1200 + Rs. 1400 + Rs. 1600) / (1 + 0.10)^4 = Rs. 3138.84- Keeping the machine for 5 years: PV = (Rs. 1000 + Rs. 1200 + Rs. 1400 + Rs. 1600 + Rs. 1800) / (1 + 0.10)^5 = Rs. 3842.07- Keeping the machine for 6 years: PV = (Rs. 1000 + Rs. 1200 + Rs. 1400 + Rs. 1600 + Rs. 1800 + Rs. 2000) / (1 + 0.10)^6 = Rs. 4570.08- ...We can continue this calculation for different numbers of years and compare the present values. The optimal replacement time for the machine is the number of years that minimizes the present value of costs.To make this comparison easier, we can create a table:| Number of Years | Present Value of Costs ||-----------------|-----------------------|| 1 | Rs. 909.09 || 2 | Rs. 1818.18 || 3 | Rs. 2460.33 || 4 | Rs. 3138.84 || 5 | Rs. 3842.07 || 6 | Rs. 4570.08 || ... | ... |From the table, we can see that the present value of costs increases as we keep the machine for more years. The optimal replacement time for the machine is the number of years that corresponds to the minimum present value of costs. In this case, it appears that the optimal replacement time is 1 year, as it has the lowest present value of costs.Therefore, based on the given information and assuming a 10% per annum discount rate, the optimal replacement time for the machine is 1 year.