After the admissions decisions have been made for the graduate engineering school, it is the Scholarship Committee's job to award financial aid. There is never enough money to offer as much scholarship aid as each needy applicant requires. Each admitted applicant's financial need is determined by comparing an estimate of his sources of revenue with a reasonable school and personal expense budget for the normal academic year. An admit tee's need, if any, is the difference between the standard budget and the expected contribution from him and his family. Scholarship offers provide an amount of aid equal to some fraction of each applicant's need. In cases where need is not met in full, the school is able to supply low-cost loans to cover the difference between scholarships and need. Besides receiving funds from the university, a needy admit tee might receive a scholarship from nonuniversity funds. In this case the admit tee, if he decides to matriculate, is expected to accept the outside funds. His total scholarship award is then the greater of the university offer or the outside offer, because the university supplements any outside offer up to the level awarded by the scholarship committee. Prior to the deadline for determining a school scholarship-offer policy, the committee has a good estimate of the amount of outside aid that each needy admit tee will be offered. The most important function of the scholarship policy is to enroll the highest-quality needy admit tees possible. The admissions committee's rank list of all needy admit tees is used as the measurement of quality for the potential students. In using this list, the top 100α% of the needy group ordered by quality is expected to yield at least βT enrollees, where T is the total desired number of enrollees from the needy group. In addition to satisfying the above criteria, the dean wants to minimize the total expected cost of the scholarship program to the university.
As a last point, Pi, the probability that needy admittee i enrolls, is an increasing function of yi, the fraction of the standard budget B covered by the total scholarship offer. An estimate of this function is given in Fig. E13.3.Here. xi B is the dollar amount of aid offered admittee f and ni B is the dollar amount of need for admittee i.
a) Formulate a nonlinear-programming model that will have an expected number T of enrolling needy admit tees, and minimize the scholarship budget. (Assume that Pi can be approximated by a linear function.)
b) Suggest two different ways of solving the model formulated in (a). [Hint. What special form does the objective function have?]
c) Reformulate the model so as to maximize the expected number of enrolling needy students from the top 100α% of the need group, subject to a fixed total scholarship budget. Comment on how to solve this variation of the model.
d) Suppose that, in the formulation proposed in (a), the probability Pi that admittee i enrolls is approximated by Pi= ai + biyi1/2. Comment on how to solve this variation of the model.=