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 ($)	Required Capital ($)
1	6,000	5,000
2	8,000	7,000
3	3,000	2,000
4	5,000	4,000
5	7,000	6,000
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 A: What is the objective?

Question options:

	Max Z = X1 + X2 + X3 + X4 + X5 + X6
	Min Z = 6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6
	Max Z = 5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6
	Min Z = X1 + X2 + X3 + X4 + X5 + X6
	Min Z = 5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6
	Max Z = 6,000 X1 + 8,000 X2 + 3,000 X3 + 5,000 X4 + 7,000 X5 + 4,000 X6

III. Write the constraints (mathematically).

Question B: Among projects 1, 3, and 4, two of them must be selected.

Question options:

	X1 + X3 + X4 2
	X1 + X2 + X3 + X4 + X5 + X6 = 2
	X1 + X3 + X4 > 2
	X1 + X3 + X4 2
	X1 + X3 + X4 < 2
	X1 + X3 + X4 = 2

Question C: Constraint on the budget.

Question options:

	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
	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
	5,000 X1 + 7,000 X2 + 2,000 X3 + 4,000 X4 + 6,000 X5 + 3,000 X6 21,000

Question D: If project 3 is selected, project 6 cannot be selected.

Question options:

	X3 X6
	X3 < X6
	X3 + X6 1
	X3 + X6 = 1
	X3 + X6 1
	X3 X6
	X3 > X6

Question E: At least three projects must be selected.

Question options:

	X1 + X2 + X3 + X4 + X5 + X6 = 3
	X1 + X2 + X3 + X4 + X5 + X6 < 3
	X1 + X2 + X3 + X4 + X5 + X6 3
	X1 + X2 + X3 + X4 + X5 + X6 > 3
	X1 + X2 + X3 + X4 + X5 + X6 3

Question F: If project 2 is selected, project 5 must be also selected.

Question options:

	X2 < X5
	X2 X5
	X2 X5
	X2 + X5 1
	X2 + X5 1
	X2 + X5 = 1
	X2 > X5

Question G: At most five projects are to be selected.

Question options:

	X1 + X2 + X3 + X4 + X5 + X6 5
	X1 + X2 + X3 + X4 + X5 + X6 5
	X1 + X2 + X3 + X4 + X5 + X6 = 5
	X1 + X2 + X3 + X4 + X5 + X6 > 5
	X1 + X2 + X3 + X4 + X5 + X6 < 5

Question H: Constraints on decision variables.

Question options:

	X1, X2, X3, X4, X5, X6 0
	X1 + X2 + X3 + X4 + X5 + X6 0
	X1, X2, X3, X4, X5, X6 0 or 1
	X1, X2, X3, X4, X5, X6 = 0 or 1
	X1 + X2 + X3 + X4 + X5 + X6 1