Matching Supply withDemand Q16.9 Each year the admissions committee at a top bus
ID: 408074 • Letter: M
Question
Matching Supply withDemand Q16.9 Each year the admissions committee at a top business school receives a large number of applications 6.9 (MBA Admissions) Each year the admissions committee at a top business school receives a large number of applications for admission to the MBA program and they have to decide on the number of offers to make. Since some of the admitted students may decide to pursue other opportunities, the committee typically admits more students than the ideal class size of 720 students. You were asked to help the admissions committee estimate the appropri- ate number of people who should be offered admission. It is estimated that in the coming year the number of people who will not accept the admission offer is normally distributed with mean 50 and standard deviation 21. Suppose for now that the school does not main- tain a waiting list, that is, all students are accepted or rejected. a. Suppose 750 students are admitted. What is the probability that the class size will be at least 720 students? b. It is hard to associate a monetary value with admitting too many students or admitting too few. However, there is a mutual agreement that it is about two times more expen- sive to have a student in excess of the ideal 720 than to have fewer students in the class. What is the appropriate number of students to admit? c. A waiting list mitigates the problem of having too few students since at the very last moment there is an opportunity to admit some students from the waiting list. Hence, the admissions committee revises its estimate: It claims that it is five times more expensiveExplanation / Answer
Using the matching framework of the Newsvendor model, the random quantity D is the number of people who will not accept the admission offer while the matching quantity Q is the overbooking quantity. (Overbooking quantity Q = number of students admitted – 720. If D is not random, we can simply set Q = D.) The program incurs overage costs when Q is larger than D and it incurs underage costs otherwise.
a. The class size will be at least 720 if there are 30 or fewer students who decline the offer. Because D has a normal distribution with mean 50 and standard deviation 21, the corresponding safety factor z = (30 – 50)/21 = -0.9524. From Excel, Prob (D < 30) = normsdist (-0.9524) = 0.1704 (i.e. or from Z-table, p = 0.1704). Hence the probability for the class size to be at least 720 is about 17%.
b. The cost of admitting too many (overage cost) is twice the cost of admitting too few (underage cost), so the critical ratio is Cu / (Co +Cu) = Cu / (2 × Cu + Cu) = 0.3333. The required safety factor z = normsinv (0.3333) = -0.4308 (or from Z-table, z = -0.4308). So overbooking quantity is 50 – 0.4308*21 = 41. Admit 720 + 41 = 761 students
c. The cost of admitting too many (overage cost) is five times the cost of admitting too few (underage cost), so the critical ratio is u Cu / (Co +Cu) = Cu / (5 × Cu + Cu) = 0.1667. The required safety factor z = normsinv (0.1667) = -0.9673 (or from Z-table, z = -0.9673). So overbooking quantity is 50 – 0.9673*21 = 29.7. Admit 720 + 30 = 750 students.