Students planning classes, Second year students who are planning their junior year program...

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 those fields. Professors have set upper limits for enrollment in the courses that we number as 1 to 10. These upper limits are bl. b2 ... b10. Sixty-four students are each seeking together three best choices of classes for their junior year program. Accordingly they each make out cards that rate their preferences. 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 stairs-to obtain rankings and assignments. She plans to optimize the course assignments in which each student will be assigned to three of the ten

Each student i is asked to rank each of the courses with a number from I to 10. with 10 de noting the first choice or greatest preference to the second greatest preference, and so on. The students give only one course a 10, only one course a 9, and so on. We will assume that each student follow this rule, because the chairperson has announced that those students who violet the ranking method will be assigned last after all others have been assigned-to class that still have room. Thus in general student i give course j a preference of pij.

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 assigned to any course below their fifth choice-if that is feasible to achieve