A credit union can make three di

erent kinds of loans to its customers, while

charging the following interest rates x1: Mortgages (5 %) x2: Vehicle loans (8 %) x3: Personal loans (13 %) The credit union has a total lending capability of $100 million, all of which should be used for loan purposes. Furthermore, the use of these funds is restricted by the following federal regulations Mortgages must constitute at least 20 % of all loans issued in dollars Personal loans cannot exceed 50 % of all loans issued in dollars The average interest rate on all mortgage and personal loans must not exceed 10 %. In other words, the total interest income from mortgage and personal loans should be less than 10% of all money lent in mortgage and personal loans. The credit union would like to maximize its first year interest income while obeying the regulatory constraints (assume the duration of all loans is longer than a year). Its decision variables are the total amount of money lent in each kind of loan. 1. Write an LP formulation in standard form to obtain the optimal loan distribution. 2. Write the corresponding LP dual. 3. Explain the relevance in this problem of the sign constraints in the dual variables based on the sensitivity interpretation. 4. A primal feasible solution is (x1, x2, x3) = (30, 20, 50). Compute the corresponding slack variables for the standard LP formulation. Is this solution degenerate? 5. Show that the solution given above is actually optimal. What is the corresponding profit of the credit union? 6. Since personal loans are very profitable, the credit union will try to hire a lobbyist to encourage a one-time increase of the federal persona loan limit to 52 % from the original 50 %. The lobbyist fee is $ 40,000. Assuming that success is guaranteed, is this a good deal for the credit union?