Question

1. The admissions office at Tech wants to determine how many in-state and how many out-of-state students to accept for next fall’s entering freshman class. Tuition for an in-state student is $8,000 per year, whereas out-of-state tuition is $24,000 per year. A total of 12,000 in-state and 8,000 out-of-state freshmen have applied for next fall, and Tech does not want to accept more than 5,000 students. However, because Tech is a state institution, the state mandates that no more than 50% of the accepted students can be out-of-state students. From past experience the admissions office knows that 12% of in-state students and 24% of out-of-state students will drop out during their first year. Tech wants to maximize total tuition while limiting the total attrition to 720 first-year students.

1. Formulate a linear programming model for this problem.
2. Solve this model by using graphical analysis.

Answer 1

a.

Let the no. of in-state students be students_in

Let the no. of out-state students be students_out

Let the tuition of in-state students be tuition_in

Let the tuition of out-state students be tuition_out

According to the given situation (the linear programming model is):

tuition_in=8000 ....................(1)

tuition_out=24000 ....................(2)

students_in+students_out<=5000 ....................(3)

students_out<=2500 ....................(4)

12%*students_in+24%*students_out<=720 ....................(5)

profit=students_in*tuition_in+students_out*tuition_out ....................(6)

[student_in<=12000 and students_out<=8000 can be ignored since (3) and (4) take care of it]

b.

Graphical Analysis looks like this:

The red points are of interest.

On solving, we get values for them as:

In-State students	Out-State Students	Total Strength	Attrition	Profit
5000	0	5000	600	40000000
0	2500	2500	600	60000000
4000	1000	5000	720	56000000
1000	2500	3500	720	68000000

1st and 2nd scenario can be seen from the figure (easy calculation from equation (3)).

3rd scenario comes from solving equation (3) and (5)

4th scenario comes from solving equation (4) and (5)

The scenario with 1000 in-state students and 2500 out-state students gives the optimum result.