I am not even sure how to set up the objective function... Help!

Graduate student Luke currently has $400 in cash. At the beginning of months 1, 2, 3, and 4, Luke receives revenues from his research assignment, after which he pays bills (see Table). Any money left over is to be invested. If invested for one month, Luke receives the interest rate of 0.1% per month. If invested for two months, he receives an interest rate of 0.5% per month.
Month Revenues ($) Expenses ($)
1 400 600
2 800 500
3 300 500
4 300 250

A. Formulate a linear program to determine an investment strategy that maximizes cash on hand at the beginning of month 5 (or equivalently at the end of Month 4). Please clearly define all terms in your model. (Note, you are not required to solve the model, just formulate it). (15 points)

B. Luke is now able to postpone monthly payments; however, a penalty of 1% is charged on the amount postponed. He can postpone payments by only one month. Formulate a linear program to determine an investment strategy that maximizes cash on hand at the beginning of month 5.
(10 points)

A. Sure, here's a linear program formulation for Luke's investment strategy:

Let:
X1 = amount of money invested for 1 month in month 1
X2 = amount of money invested for 1 month in month 2
X3 = amount of money invested for 1 month in month 3
X4 = amount of money invested for 1 month in month 4
Y1 = amount of money invested for 2 months in month 1
Y2 = amount of money invested for 2 months in month 2
Y3 = amount of money invested for 2 months in month 3
Y4 = amount of money invested for 2 months in month 4

Objective function: Maximize cash on hand at the beginning of month 5
Maximize: 0.001(X1 + X2 + X3 + X4) + 0.005(Y1 + Y2 + Y3 + Y4)

Subject to the constraints:
1. Cash on hand before investment in month 1 = $400
400 + X1 + Y1 - 600 = 400 + X1 + Y1 - 600

2. Cash on hand before investment in month 2 = Cash on hand at the end of month 1
400 + X1 + Y1 + X2 + Y2 - 800 = 400 + X1 + Y1 - 600 + 0.001(X1 + Y1)

3. Cash on hand before investment in month 3 = Cash on hand at the end of month 2
400 + X1 + Y1 + X2 + Y2 + X3 + Y3 - 300 = 400 + X1 + Y1 + X2 + Y2 - 800 + 0.001(X1 + Y1 + X2 + Y2)

4. Cash on hand before investment in month 4 = Cash on hand at the end of month 3
400 + X1 + Y1 + X2 + Y2 + X3 + Y3 + X4 + Y4 - 300 = 400 + X1 + Y1 + X2 + Y2 + X3 + Y3 - 300 + 0.001(X1 + Y1 + X2 + Y2 + X3 + Y3)

5. Luke cannot invest negative amounts of money
X1, X2, X3, X4, Y1, Y2, Y3, Y4 ≥ 0

B. Let's incorporate Luke's ability to postpone monthly payments with a 1% penalty:

Let:
Z1 = amount of money postponed for payment in month 2 from month 1
Z2 = amount of money postponed for payment in month 3 from month 2
Z3 = amount of money postponed for payment in month 4 from month 3

Objective function: Maximize cash on hand at the beginning of month 5
Maximize: 0.001(X1 + X2 + X3 + X4) + 0.005(Y1 + Y2 + Y3 + Y4) + 0.001(Z1 + Z2 + Z3)

Subject to the constraints:
1. Cash on hand before investment in month 1 = $400
400 + X1 + Y1 - 600 = 400 + X1 + Y1 - 600

2. Cash on hand before investment in month 2 = Cash on hand at the end of month 1
400 + X1 + Y1 + X2 + Y2 - 800 = 400 + X1 + Y1 - 600 + 0.001(X1 + Y1) - Z1

3. Cash on hand before investment in month 3 = Cash on hand at the end of month 2
400 + X1 + Y1 + X2 + Y2 + X3 + Y3 - 300 = 400 + X1 + Y1 + X2 + Y2 - 800 + 0.001(X1 + Y1 + X2 + Y2) - Z1 - Z2

4. Cash on hand before investment in month 4 = Cash on hand at the end of month 3
400 + X1 + Y1 + X2 + Y2 + X3 + Y3 + X4 + Y4 - 300 = 400 + X1 + Y1 + X2 + Y2 + X3 + Y3 - 300 + 0.001(X1 + Y1 + X2 + Y2 + X3 + Y3) - Z1 - Z2 - Z3

5. Luke cannot invest negative amounts of money
X1, X2, X3, X4, Y1, Y2, Y3, Y4, Z1, Z2, Z3 ≥ 0

I hope this helps! Let me know if you have any other questions.

A. To formulate a linear program to determine the investment strategy that maximizes cash on hand at the beginning of month 5, we need to define the decision variables, objective function, and constraints.

Decision Variables:
Let x1, x2, x3, x4 be the amount of money invested (in dollars) for 1 month in months 1, 2, 3, and 4, respectively.
Let y1, y2, y3, y4 be the amount of money invested (in dollars) for 2 months in months 1, 2, 3, and 4, respectively.

Objective Function:
Maximize the cash on hand at the beginning of month 5, which is the sum of the money invested plus the interest earned:
Maximize Z = (400 + x1 + y1) * (1 + 0.001)^4 + (x2 + y2) * (1 + 0.005)^3 + (x3 + y3) * (1 + 0.001)^2 + (x4 + y4) * (1 + 0.001)^1

Constraints:
1. The total expenses in each month cannot exceed the total revenues:
400 + x1 + y1 - 600 >= 0
800 + x2 + y2 - 500 >= 0
300 + x3 + y3 - 500 >= 0
300 + x4 + y4 - 250 >= 0

2. The total amount of money invested for each month cannot exceed the available cash:
x1 + y1 <= 400
x2 + y2 <= 400 + x1 + y1
x3 + y3 <= 400 + x1 + y1 + x2 + y2
x4 + y4 <= 400 + x1 + y1 + x2 + y2 + x3 + y3

3. The amount of money invested for 2 months in months 1, 2, 3, and 4 cannot exceed the amount invested for 1 month in the same month:
y1 <= x1
y2 <= x2
y3 <= x3
y4 <= x4

B. To incorporate the penalty of 1% for postponing monthly payments by one month, we need to modify the constraints:

1. The total expenses in each month cannot exceed the total revenues, taking into account the penalty for postponed payments:
400 + x1 + y1 - 600 - 0.01 * 600 >= 0
800 + x2 + y2 - 500 - 0.01 * 500 >= 0
300 + x3 + y3 - 500 - 0.01 * 500 >= 0
300 + x4 + y4 - 250 - 0.01 * 250 >= 0

2. The total amount of money invested for each month cannot exceed the available cash:
x1 + y1 <= 400
x2 + y2 <= 400 + x1 + y1
x3 + y3 <= 400 + x1 + y1 + x2 + y2
x4 + y4 <= 400 + x1 + y1 + x2 + y2 + x3 + y3

3. The amount of money invested for 2 months in months 1, 2, 3, and 4 cannot exceed the amount invested for 1 month in the same month:
y1 <= x1
y2 <= x2
y3 <= x3
y4 <= x4

The objective function and decision variables remain the same as in part A.

Sure! Let's begin by setting up the objective function for both parts of your question.

A. In order to maximize the cash on hand at the beginning of month 5, we need to define the objective function. Let's call the amount of money invested in the first month "x1" and the amount invested in the second month "x2".

The objective function can be defined as: Maximize Cash = (400 + x1 + (1 + 0.001) * x1 + x2 + (1 + 0.005) * x2)

Here's how we arrived at this objective function:

1. The initial cash of $400 is included.

2. The revenue for month 1 is x1. Since the money is invested for one month, we add (1 + 0.001) * x1 to gain the interest.

3. The revenue for month 2 is x2. Since the money is invested for two months, we add (1 + 0.005) * x2 to gain the interest.

4. The expenses are subtracted from the cash on hand.

B. In this scenario, Luke can postpone monthly payments but incurs a 1% penalty. Let's redefine the objective function.

The objective function can be defined as: Maximize Cash = (400 + x1 + (1 + 0.001) * x1 + x2 + (1 + 0.005) * x2 - 0.01 * (600 + 500 + 500 + 250))

Here's how we arrived at this objective function:

1. The initial cash of $400 is included.

2. The revenue for month 1 is x1. Since the money is invested for one month, we add (1 + 0.001) * x1 to gain the interest.

3. The revenue for month 2 is x2. Since the money is invested for two months, we add (1 + 0.005) * x2 to gain the interest.

4. The expenses for months 1, 2, 3, and 4 are subtracted, and a 1% penalty is added for each postponed payment.

Remember, these objective functions are just the starting point for your linear programming problem. You will also have to define the constraints and decision variables to complete the formulation.