[Chapter 12] Integer Linear Optimization Models Sunrise Investment Group is considering investing in six new projects. The require capital at the present time and the expected net present value (NPV) for each proj are given in the table below. Project Required Capital ($) Expected NPV ($) 1 3,000 10,000 2. 5,000 16,000 3 2,000 6,000 4 4,000 9,000 5 7,000 20,000 6 6,000 18,000 At present, a budget of $20,000 is available for investment. Sunrise has some specific requirements (as detailed in the Constraints section below). Please help Sunrise develop an investment plan by formulating an integer linear optimization model. Follow the steps below. 1. Define the decision variables (Note: This step is done for you. Please use these decision variables hereafter.) X1 = 1 if project 1 is selected for investment; 0 otherwise. X2 = 1 if project 2 is selected for investment; 0 otherwise. X3 = 1 if project 3 is selected for investment; 0 otherwise. X4 = 1 if project 4 is selected for investment: 0 otherwise. X5 = 1 if project 5 is selected for investment; O otherwise. Xo = 1 if project 6 is selected for investment; 0 otherwise. II. Write the objective (mathematically). Question 40 (2 points) This is a problem. maximization minimization Question 41 (2 points) The objective function is 10,000 X1 + 16,000 X2 + 6,000 X3 + 9,000 X4 + 20,000 X5 + 18,000 X6 (3,000)(10,000) X1 + (5,000)(16,000) X2 + (2,000)(6,000) X3 + (4,000)(9,000) X4 + (7,000)(20,000) X5 +(6,000)(18,000) X6 X1 + X2 + X3 + X4 + X5 + X6 3,000 X1 + 5,000 X2 + 2,000 X3 + 4,000 X4 + 7,000 X5 + 6,000 X6 III. Write the constraints (mathematically). Question 42 (4 points) At least three projects must be selected. X1 + X2 + X3 + X1 + X5 + Xo 3 Question 43 (4 points) Constraint on the budget. O 3,000 X1 +5,000 X2 + 2,000 X3 + 4,000 X4 + 7,000 X5 + 6,000 X6 2 20,000 OX1 + X2 + X3 + X4 + X5 + X6 2 20,000 3,000 X1 + 5,000 X2 + 2,000 X3 + 4,000 X4 + 7,000 X5 + 6,000 X6 $ 20,000 (3,000)(10,000) X1 + (5,000)(16,000) X2 + (2.000)(6,000) X3 + (4,000)(9,000) X4 + (7,000)(20,000) X5 + (6,000)(18,000) X6 $ 20,000 OX1 + X2 + X3 + X4 + X5 + X6 $ 20,000 10,000 X1 + 16,000 X2 + 6,000 X3 + 9,000 X4 +20,000 X5 + 18,000 X6 3 20,000 (3.000)(10,000) X1 + (5,000)(16,000) X2 + (2,000)(6,000) X3 + (4,000)(9,000) X4 + (7.000)(20,000) X5 + (6,000)(18,000) X6 2 20,000 10,000 X1 + 16,000 ay, nhac0 X + 18,000 Xe ? Question 44 (4 points) If project 1 is selected, projects 5 or 6 must be selected. OX1 s X5, X1 s Xo O X1 + X5 + Xo > 1 X1 2 X5 + X6 OX13 X5 + X6 OX1 + X5 + X6 2 1 O X > Xg+Xo X1 1 Oxz> X4 Question 46 (4 points) If project 2 is selected, project 1 must be also selected. OX2 > X1 X2 + X1 > 1 OX2 + X1 s 1 O X25X1 OX2+ X1 21 O X2X1 OX2 1 O X4 X5 Question 48 (4 points) Constraints on decision variables. O X1 + X2 + X3 + X4 + X5 + Xo 2 0 OX1 + X2 + X3 + X4 + X5 + Xo so X1, X2, X3, X4, X5, X6 2 0 OX1, X2, X3, X4, X5, X6 0 or 1