Goal Programming

Professor Roberts has decided to invest in off-campus student housing in a college in a town close to his retirement home in Central Florida. He will have to make 30% down payments on the apartment complexes since mortgage companies will only finance 70% of the purchase price of investment properties. The annual percentage rate is 5.75%. Monthly expenses (maintenance, insurance, and utilities) are estimated to 3% of the purchase price.

Florida laws require that each off-campus apartment houses only one student per bedroom. There is a strong demand for student housing, so we may assume that all of the apartments will be rented.

Apartment Type	Monthly Rental per Student	Monthly Rental per Apartment
One Bedroom	$1,200	$1,200
Two Bedroom	$1,000	$2,000
Three Bedroom	$600	$1,800

There are 10 apartment complexes available. The pertinent information on each apartment complex is shown in the table below:

	Apartment Complexes
	A	B	C	D	E	F	G	H	I	J
Price	$400,000	$600,000	$500,000	$700,000	$350,000	$450,000	$650,000	$550,000	$750,000	$350,000
Down Payment	Calculate for each option
Monthly Expenses	Calculate for each option
One-Bedroom	6	8	4	4	0	10	6	6	8	12
Rental Income	Calculate for each option
Two-Bedroom	4	8	6	13	10	0	8	6	8	2
Rental Income	Calculate for each option
Three-Bedroom	3	4	6	4	0	6	6	6	8	0
Rental Income	Calculate for each option
Total Apartments	13	20	16	21	10	16	20	18	24	14
Total Income	Calculate for each option
Monthly Profit	Calculate for each option

Part 1:

Professor Roberts has the following constraints that cannot be violated.

1. The total monthly rent from the one-bedroom apartments cannot exceed 45% of the total monthly rent.
2. The total monthly rent from the two-bedroom apartments cannot exceed 40% of the total monthly rent.
3. The total monthly rent from the three-bedroom apartments cannot exceed 35% of the total monthly rent.
4. The total number of apartments cannot exceed 120.

Professor Roberts initially ran a linear program with binary decision variables (1 for Buy, 0 for Dont Buy) There was an optimal solution that would satisfy all of their constraints simultaneously with all decision variables for the apartment complexes being binary.

Procedure:

Download the EXCEL spreadsheet from Carmen.

On the Linear Programming (1) tab, replicate the linear program Professor Roberts created using constraints 1 - 4 above. Remember to use binary variables! The objective function is to maximize the profit. Confirm that the model has a solution with all of the decision variables being binary.

Part 2:

After researching the rental market near the college and surveying the present students, Professor Roberts has decided to incorporate the following constraints into the model:

5. The total of the down payments cannot exceed $1,600,000.
6. No less than 30 one-bedroom apartments.
7. No more than 55 one-bedroom apartments.
8. No less than 25 two-bedroom apartments.
9. No more than 50 two-bedroom apartments.
10. No more than 30 three-bedroom apartments.
11. Total monthly profit is at least $25,000.

When constraints 5 11 are added into the linear program, the problem may be infeasible.

Procedure:

On the Linear Programming (2) tab, copy your output from the Linear Programming (1) tab. Modify the model to include constraints 5 - 11. Resolve the model. The objective function is to maximize profit. This program may or may not have a feasible solution.