(a) The Dean of Physical and Mathematical Sciences of ABX University is drawing up her plan for the full time professor needs for the next four years. After looking at the projected total credit hours offered in each year, she finds out that the required number of professors should be: Year 2020 2021 2022 2023 no. of Profs required 7294 69 103 From past experience, she knows that professors prefer long term job contracts and that they would accept a lower salary if the contract is longer The table below displays professorial salaries based on contract length. Length of Contract in yrs: Salary in 2000's 1 120 2 110 3 4 100 90 For example, if a professor is needed for two years, it is better to sign a two year contract and pay 220,000 over the duration of the appointment, instead of 240,000 for two one-year contracts Suppose that the Dean is starting afresh, that is no contracts exist with any professor Formulate a linear program to minimise the total salaries paid to professors at this Faculty while ensuring that there are enough professors signed up for each year. Clearly state the decision variables, the objective function and all the constraints. [12 marks] (b) The Asset Capital managing group is going to invest 10,000,000 on behalf of a large pension fund. The group has identified six mutual funds with returns Mutual Fund 1 2 3 4 5 6 Price(/share) 50 75 80 30 40 20 Expected retur () 30 25 20 15 10 5 Risk Category High High High Medium Medium Low The pension fund requires Asset Capital to reduce the risk of the portfolio To make sure of this it set out the following requests: the total amount invested in high risk funds must be between 50% and 75% of the total portfolio, the total amount invested in low risk funds must be at least 5% of the total portfolio (0) Formulate a linear program to maximise the expected total retum from the funds. 15 marks (ii) Diversification is a key strategy in managing the risk of a portfolio Asset Capital's diversification strategy is as follows: the investment into Funds 1 and Fund 2 should be equal, and the investment into Fund 4 should be twice the investment into Fund Modify your formulation in (i) to reflect this diversification strategy (a) The Dean of Physical and Mathematical Sciences of ABX University is drawing up her plan for the full time professor needs for the next four years. After looking at the projected total credit hours offered in each year, she finds out that the required number of professors should be: Year 2020 2021 2022 2023 no. of Profs required 7294 69 103 From past experience, she knows that professors prefer long term job contracts and that they would accept a lower salary if the contract is longer The table below displays professorial salaries based on contract length. Length of Contract in yrs: Salary in 2000's 1 120 2 110 3 4 100 90 For example, if a professor is needed for two years, it is better to sign a two year contract and pay 220,000 over the duration of the appointment, instead of 240,000 for two one-year contracts Suppose that the Dean is starting afresh, that is no contracts exist with any professor Formulate a linear program to minimise the total salaries paid to professors at this Faculty while ensuring that there are enough professors signed up for each year. Clearly state the decision variables, the objective function and all the constraints. [12 marks] (b) The Asset Capital managing group is going to invest 10,000,000 on behalf of a large pension fund. The group has identified six mutual funds with returns Mutual Fund 1 2 3 4 5 6 Price(/share) 50 75 80 30 40 20 Expected retur () 30 25 20 15 10 5 Risk Category High High High Medium Medium Low The pension fund requires Asset Capital to reduce the risk of the portfolio To make sure of this it set out the following requests: the total amount invested in high risk funds must be between 50% and 75% of the total portfolio, the total amount invested in low risk funds must be at least 5% of the total portfolio (0) Formulate a linear program to maximise the expected total retum from the funds. 15 marks (ii) Diversification is a key strategy in managing the risk of a portfolio Asset Capital's diversification strategy is as follows: the investment into Funds 1 and Fund 2 should be equal, and the investment into Fund 4 should be twice the investment into Fund Modify your formulation in (i) to reflect this diversification strategy