Cambridge University is not within the central university exam system, but processing freshman applications for the upcoming academic year on its own. The applications fall into three categories: in-city, out-of-city, and international. The university does not provide Enlish preparatory classes, so Toefl score is an important factor in accepting new students. The statistics gathered by the university indicate that the average Toefl scores for in-city, out-of-city, and international students are 96, 91, and 105, respectively. The committee on admissions has established the following desirable goals for the new freshman class:

(a) The incoming class is at least 1200 freshmen.

(b) The average Toefl score for all incoming students is at least 94.

(c) International students constitute at least 10% of the incoming class.

(d) In-city students constitute at most 40% of the incoming class.

From these goals, the lower limit on the total number of students (goal a) is strict. But the rest can be treated as flexible. The penalties for not achieving these goals are:

a. 4 for each point of average Toefl score under 94.

b. 8 for each international student less than the goal of 10%.

c. 5 for in-city student above the goal of 40 %.

i. Formulate the problem as a goal programming model to find the number of students to accept from each group of in-city, out-of-city, and international students. Please make sure that you clearly define your decisions variables, the objective function, and all relevant constraints.

ii. The male-female ratios for in-city and out-of-city applicants are 1:1 and 3:2, respectively. For

international students, the corresponding ratio is 7:3. The university administration sets a new goal for the admissions such that

(e) The male-female ratio is at most 3:4.

If the penalty for each female student below this ratio is 7, how can you add this goal to the above formulation? Note a constraint will be added and the objective function will change.

iii. If the strict constraint is also relaxed such that there can be less than 1200 students accepted. There is no penalty for each student less than 1200, but the amount of deviation below 1200 is restricted to 100. How would your formulation change?