Problem 1
Pony Express Creations Inc. (www.pony-ex.com) is a manufacturer of party hats, primarily for the Halloween season. 80% of their annual sales occur over a six-week period. One of their popular products is Elvis wig, complete with sideburns and metallic glasses. The Elvis wig is produced in China. So Pony Express must make a single order well in advance of the upcoming season. They expect the demand to be normally distributed with a mean of 25000 and a standard deviation of 7000. The Elvis wig retails for $25 but Pony Express charges a wholesale price of $12 from a retailer. Their production cost is $6 and the leftover inventory can be sold to discounters for $2.50.
a. Suppose Pony Express manufactures 40,000 Elvis wigs. What is the chance that they have to liquidate 10,000 or more wigs to a discounter?
b. What is the chance that the demand is within 20% of the expected value?
c. What order quantity maximizes Pony Express' expected profit?
d. If Pony Express wants to have 90% service level (or in-sock probability), how many Elvis wigs should be ordered?
Problem 2
Each year the admissions committee at the business school receives a large number of applications for admissions to the MBA program and they have to decide 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 appropriate number of students who should be offered admission. Based on the historical data, the committee has estimated that the number of students who will not accept the admissions offer is normally distributed with mean 50 and standard deviation 21. Suppose now that the school does not maintain a waiting list, i.e., an applicant is either accepted or rejected.
a. Suppose 750 students are admitted. What is the chance that the class size will be at least 720 students?
b. It is hard to associate a monetary value with admitting too many or too few students. However, there is an agreement in the business school that it is about two times more expensive 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 expensive to have a student in excess of 720 than to have fewer students accept among the initial group of admitted students. What is your revised suggestion?
Problem 3
a. Dan's Independent Book Store is trying to decide on how many copies of a book to purchase at the start of the upcoming selling season. The book retails at $28.00. The publisher sells the book to Dan at $20.00. Dan will dispose of all of the unsold copies of the book at 50% off the retail price, at the end of the season. Dan estimates that demand for this book during the season is Normal with a mean of 1000 and a standard deviation of 250. What is the quantity that Dan should order, to maximize his expected profits?
b. The publisher is thinking of offering the following scheme to Dan. At the end of the season, they will buy back unsold copies at a pre-determined price of $17.00. However, Dan would have to bear the costs of shipping unsold copies back to the publisher at $1.00 per copy. What is the quantity that Dan should order, to maximize his expected profits now?