Questions:
1 The net present value (NPV) problem (10%)
A car manufacturer is looking to bring a new range of electric vehicles (EV) in the next five years. This will require an initial investment of £500M and has running
costs of £100M per year after that. The predicted incomes from this range are:
Year 2020 2021 2022 2023 2024 2025
£M 0 10 50 200 300 500
Notice that the initial expenditure takes place in 2020.
1. Assume a discount rate of 1% per year, and perform analysis of discounted cashflow. You should create a data table in R with variables Year, Outflow
and Inflow in the first three columns:
This function should modify the table by adding columns showing the net returns, the numbers of years to discount, the present value factors and the
discounted returns. Your function must be able to take different cashflow tables and discount rates as the input, print the modified cashflow table and
return the (total) net present value (NPV) for the investment. Include your code into the report. 5 marks
2. Decide whether or not this is a worthwhile investment, justifying your answer by your results. 1 mark
3. Calculate the NPV for discount rates of 2%, 3%, 4% and 5%, giving your answers to two decimal places. 1 mark
4. Plot NPV against the discount rates ranging from 0% to 5%. 1 mark
5. Use your graph to estimate the discount rate that would give an NPV of zero. What is the significance of this discount rate? (Hint: What happens
at discount rates lower and higher than this figure?) 2 marks
2 Optimisation and linear programming (20%)
2.1 Solution using analysis and graphs (10%)
A car manufacturer produces conventional cars, which generate £4,000 profit per car, and a newly developed EVs, which generate £5,000 per car. The objective
is to maximise profit by selecting the right combination of conventional cars and EVs, subject to the following constraints:
• The factory can produce up to 1,000 conventional cars and up to 500 EVs per day.
• The logistics and warehouse facilities do not allow for production of more than 750 vehicles per day in total (i.e. both conventional and EVs).
• The manufacturer has a contract to produce at least 100 conventional and 50 EVs per day.
Your task is to:
1. Write the objective function. 1 mark
2. Write the constraints for production and logistics. 2 marks
3. Plot or draw the feasible region and the isoprofit line. 3 marks
4. Find the optimal numbers of conventional cars and EVs to be produced (the solution) in order to achieve the maximum profit (the optimal value). 2 marks
5. Explain the solution analytically or using the graph. 2 marks
2.2 Solution using solver (10%)
Assume that the manufacturer adds hybrid vehicles to their range, which generate £4,500 profit per car. The factory can produce up to 1,000 hybrid cars per day,
and there are no obligations to the smallest number. The warehouse constraints remain the same.
Your task is to:
1. Write the updated objective function. 1 mark
2. Write the updated constraints for production and logistics. 2 marks
3. Use the lpSolve library in R to find the solution (an optimal combination of three types of cars) and the optimal value (the maximum profit) to this problem. Include your code into the report. 5 marks
4. What happens if the contract to supply at least 100 conventional cars is replaced by the same number of hybrid cars?
