The price of oil has been dropping significantly recently. This urges fund managers to adjust their investment portfolios. In particular, many fund managers in Canada used to have a significant portion of their investments in the energy sector, but now they need to look into other investment opportunity. The fund manager of XYZ investment corp. estimates that in the past the rate-of-return for the investment made in the energy sector is about 3%, i.e. 100 dollars of investment can generate 3 dollars of profit, but given the trend of oil price the manager also predicts that the rate of return might drop to -5% in the worst-case scenario. The manager read the market reports of the other sectors and learned that in average the past rate-of-return for non- energy sectors is around 1.5%. Given that investing in other sectors appears to be less risky, the manager predicts that even in the worst-case scenario the rate-of-return for non-energy sectors can still be around 1%.
The manager needs to determine how much money she should invest in the energy sector and how much she should invest in the non-energy sectors so that the total profit estimated based on the past rate-of-return can be maximized. However, she cannot invest more than 5 million dollars and has to make sure that the overall investment cannot lose more than $25,000 in the worst-case scenario estimated based on the predicted rate-of-return.
(a) Formulate algebraically a linear programming model for the above problem. Define the decision variables, objective function, and constraints.