4. Consider a capital budgeting problem where five projects are being considered for implementation and are of three years' duration. The expected returns for each project and the yearly expenditures (both in million dollars) are shown below. The objective is to select the projects that will maximize the total returns. Expenditures Project Year 1 Year 2 Year 3 Returns 5 10 25 2 6 8 45 3 2 4 15 4 4 9 15 5 8 6 7 35 Maximum funds available 22 24 26 a. Assuming that x; is the variable which indicates whether or not project i is selected, formulate the integer linear programming model to determine which projects should be selected to maximize the total returns. (9 marks) 6 4 b. Write also the constraints needed to reflect the following additional restrictions. (i) At least two of the projects 1, 2, 4, and 5 must be selected. (3 marks) (ii) Projects 3 and 4 must be selected simultaneously or not at all. (3 marks) (iii) Project 1 or 2 must be selected, but not both. (3 marks) (iv) Project 5 cannot be selected unless both projects 3 and 4 are also selected. (3 marks) 4. Consider a capital budgeting problem where five projects are being considered for implementation and are of three years' duration. The expected returns for each project and the yearly expenditures (both in million dollars) are shown below. The objective is to select the projects that will maximize the total returns. Expenditures Project Year 1 Year 2 Year 3 Returns 5 10 25 2 6 8 45 3 2 4 15 4 4 9 15 5 8 6 7 35 Maximum funds available 22 24 26 a. Assuming that x; is the variable which indicates whether or not project i is selected, formulate the integer linear programming model to determine which projects should be selected to maximize the total returns. (9 marks) 6 4 b. Write also the constraints needed to reflect the following additional restrictions. (i) At least two of the projects 1, 2, 4, and 5 must be selected. (3 marks) (ii) Projects 3 and 4 must be selected simultaneously or not at all. (3 marks) (iii) Project 1 or 2 must be selected, but not both. (3 marks) (iv) Project 5 cannot be selected unless both projects 3 and 4 are also selected