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

The Emerald Investment Group Problem
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
Expected NPV ($)
Required Capital ($)
Project 1
6,000
5,000
Project 2
8,000
7,000
Project 3
3,000
2,000
Project 4
5,000
4,000
Project 5
7,000
6,000
Project 6
4,000
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.
II. Write the objective (mathematically).
Question 14 What is the objective?
a. Max Z = 5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6
b. Min Z = 6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6
c. Min Z = 5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6
d. Min Z = X1 + X2 + X3 + X4 + X5 + X6
e. Max Z = 6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6
f. Max Z = X1 + X2 + X3 + X4 + X5 + X6
III. Write the constraints (mathematically).
Question 15 At most five projects are to be selected.
a. X1 + X2 + X3 + X4 + X5 + X6 = 5
b. X1 + X2 + X3 + X4 + X5 + X6 < 5 c. X1 + X2 + X3 + X4 + X5 + X6 ≤ 5 d. X1 + X2 + X3 + X4 + X5 + X6 >= 5
e. X1 + X2 + X3 + X4 + X5 + X6 > 5
Question 16 If project 3 is selected, project 6 cannot be selected.
a. X3 ≤ X6
b. X3 ≥ X6
c. X3 > X6
d. X3 + X6 = 1
e. X3 + X6 ≤ 1
f. X3 + X6 ≥ 1
g. X3 < X6