Students planning classes. Second year students who are planning their junior year program in civil and environmental engineering are choosing from among 10 elective courses in thaw fields. Professors have set upper limits for enrollment in the courses that we number as 1 to 10. These upper limits are b,. bj. Sixty-four students arc each seeking to get their three best choices of classes for their junior year pro. gram. Accordingly. they each make out cards that rate their preferenees. The associate chairperson. who does the actual assignments. has consulted a faculty member in the civil systems program, and she has come up with a plan that definitely will improve on the technique of shuffling cards-that the students liken to throwing the cards down a set of stain-to obtain rankings and assignments. She plans to optimize the course assignments in which each student will be assigned to three of the ten courses.
Each student i is asked to rank each of the courses with a number from Ito 10, with 10 denoting the first choice or greatest preference, 9 the second greatest preference. and so on. The student can give only one course a 10. only one course a 9, and so on. We will assume that each student follows this rule, because the chairperson has announced that those students who violate the ranking method will be assigned last-after all others have been assigned-to classes that still have room. Thus, in general, student i gives course j a preference of pη.
The optimization process will attempt to maximize the total preference score achievement of all students.
(a) Structure the student-class assignment model that the associate chairperson needs to solve. Use summation notation.
(b) Explain conceptually how you could modify the model so that no student is as-signed to any count below their fifth choice-if that is feasible to achieve. A.