Question

Ozark University admission office is processing freshman applications for the upcoming academic year. The applications fall into three categories: in state, out-of-state, and international. The male-female ratios for in-state and out-of-state applicants are 1:1 and 3:2, respectively. For international students, the corresponding ratio is 8:1. The American College Test (ACI) score is an important factor in accepting new students. The statistics gathered by the university indicate that the average ACT scores for in-state, out-of-state, and international students are 27, 26, and 23, 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 ACT score for all incoming students is at least 25. (c) Intemational students constitute at least 10% of the incoming class. (d) The female-male ratio is at least 3:4. (e) Out-of-state students constitute at least 20% of the incoming class. Let xi=No. of in-state freshmen. x2=No. of out-of-state freshmen, X-No. of international freshmen, and use deviational yariables s7,(i-1,2,3,4,5) Formulate the problem as a goal programming model (you do not need to solve it)

Answer 1

We assume that each goal has equal weight.

Minimize s₁^- + s₂^- + s₃^- + s₄^- + s₅^-

s.t.

x₁ + x₂ + x₃ + s₁^- - s₁⁺ = 1,200 -----------Goal-1

(27 - 25) x₁ + (26 - 25) x₂ + (23 - 25) x₃ + s₂^- - s₂⁺ = 0 -----------Goal-2
or, 2 x₁ + 1 x₂ - 2 x₃ + s₂^- - s₂⁺ = 0

x₃ - 0.10 (x₁ + x₂ + x₃) + s₃^- - s₃⁺ = 0 -----------Goal-3
or, - 0.10 x₁ - 0.01 x₂ + 0.90 x₃ + s₃^- - s₃⁺ = 0

(1 x₁ + 2 x₂ + x₃) - (3/4) (1 x₁ + 3 x₂ + 8 x₃)  + s₄^- - s₄⁺ = 0 -----------Goal-4
or, 0.75 x₁ - 0.25 x₂ - 5 x₃ + s₄^- - s₄⁺ = 0

x₂ - 0.20 (x₁ + x₂ + x₃) + s₅^- - s₅⁺ = 0 -----------Goal-5
or, - 0.20 x₁ + 0.80 x₂ - 0.20 x₃ + s₅^- - s₅⁺ = 0

x₁, x₂, x₃, s_i^-, s_i⁺ >= 0 for i=1,2,...,5