The Emerald Investment Group is considering investing in six projects. The expected net present…

The Emerald Investment Group is considering investing in six projects. The expected net present value (NPV) and the required capital at the present time for each project are given in the table below. Project 1: Expected NPV ($): 6,000 – Required Capital ($): 5,000 Project 2: Expected NPV ($): 8,000 – Required Capital ($): 7,000 Project 3: Expected NPV ($): 3,000 – Required Capital ($): 2,000 Project 4: Expected NPV ($): 5,000 – Required Capital ($): 4,000 Project 5: Expected NPV ($): 7,000 – Required Capital ($): 6,000 Project 6: Expected NPV ($): 4,000 – Required Capital ($): 3,000 At present, a budget of $21,000 is available for investment. Emerald has specific requirements for the investment, as detailed in the Constraints section given below. Please help Emerald develop an investment plan by formulating an integer linear optimization model.
I. 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; 0 otherwise.
X6 = 1 if project 6 is selected for investment; 0 otherwise.
Question 20 Constraint on the budget.
6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6 ≤ 21,000
5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 = 21,000
5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 ≤ 21,000
5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 ≥ 21,000
6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6 ≥ 21,000
5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 < 21,000 5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 > 21,000
6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6 = 21,000
Question 21 Constraints on decision variables.
X1, X2, X3, X4, X5, X6 ≥ 0 or 1
X1 + X2 + X3 + X4 + X5 + X6 ≥ 1
X1, X2, X3, X4, X5, X6 ≥ 0
X1, X2, X3, X4, X5, X6 = 0 or 1
X1 + X2 + X3 + X4 + X5 + X6 ≥ 0
IV. Construct an Excel Model and solve the optimization problem with Excel Solver. Answer the following questions based on the results from Excel Solver. Note: You are NOT required to submit your Excel Model.
Question 22 According to the optimal solution, Emerald should NOT invest in __________.
projects 2 and 3
projects 5 and 6
projects 2 and 5
project 1 only
project 2 only
projects 3 and 4
projects 4 and 6
Question 23 Which statement is CORRECT about the optimal objective function value Z?
$19,000 < Z ≤ $22,000 $28,000 < Z ≤ $31,000 $22,000 < Z ≤ $25,000 $25,000 < Z ≤ $28,000 Z > $31,000
Z ≤ $19,000