1. (20 pts) A publisher sells books to Barnes & Noble at S16 each. The unit prod
ID: 334964 • Letter: 1
Question
1. (20 pts) A publisher sells books to Barnes & Noble at S16 each. The unit production cost for the publisher is $4 per book. Barnes & Noble prices the book to its customers at S30 and expects demand over the next two months to be normally distributed, with a mean of 16,000 and a standard deviation of 5,000. Barnes & Noble places a single order with the publisher for delivery at the beginning of the two-month period. Currently, Barnes & Noble discounts any unsold books at the end of two months down to S5, and any books that did not sell at full price sell at this price a) How many books should Barnes & Noble order? What is its expected profit? How many books does it b) What is the profit that the publisher makes given Barnes & Noble's actions above? Use the following information to solve parts c) and d). expect to sell at a discount? A plan under discussion is for the publisher to refund Barnes & Noble S4 per book that does not sell during the two-month period. Because Barnes & Noble shares the point-of-sales data with the publisher, the publisher does not require Barnes & Noble to return all unsold books for verification purpose. As before, Barnes & Noble will discount them to S5 and sell any that remain ts & Noble shares all unsold c) Under this plan, how many books will Barnes & Noble order? What is the expected profit for Barnes & Noble? How many books are expected to be unsold during the two-month period? [Hint: Since Barnes & Noble gets a refund of S4 from the publisher for each unsold book as well as a S5 of clearing each unsold book after the two-month period, Co c-b - Sr. Moreover, SM 0] d) What is the expected profit for the publisher under the plan and given Barnes & Noble's actions in part c)?Explanation / Answer
a) Underage cost, Cu = lost profit due to shortage of a unit compared to demand = selling price - cost = 30-16 = 14
Overage cost, Co = net cost incurred due to one excess unit leftover = Cost - Salvage = 16 - 5 = 11
Critical ratio = Cu/(Cu+Co) = 14/(14+11) = 0.56
z-stat = NORMSINV(0.56) = 0.151
Number of books Barnes & Noble should order, Q = m + z*s = 16000 + 0.151*5000 = 16,755
For z=0.151, Standard loss variate, L(z) = 0.328 (taken from standard normal loss table)
Expected lost sales L(Q) = s*L(z) = 5000*0.328 = 1640
Expected sales, S(Q) = m - L(Q) = 16000 - 1640 = 14360
Expected leftover inventory V(Q) = Q - S(Q) = 16755 - 14360 = 2395 (this is the number of books to sell at discount)
Expected number of books to sell at discount = 2,395 books
Expected profit =S(Q)*Cu - V(Q)*Co = 14360*14 - 2395*11 = $ 174,695
b) Profit earned by publisher = Q*(selling price to Barnes & Nobles - production cost) = 16755*(16-4) = $ 201,060
c) In this scenario, Cu is same as before, Cu = 14
Co = 16 - 5 - 4 = 7
Critical ratio = Cu/(Cu+Co) = 14/(14+7) = 0.667
z-stat = NORMSINV(0.667) = 0.4308
Number of books Barnes & Noble should order, Q = m + z*s = 16000 + 0.4308*5000 = 18,154
For z=0.4308, Standard loss variate, L(z) = 0.220 (taken from standard normal loss table)
Expected lost sales L(Q) = s*L(z) = 5000*0.220 = 1100
Expected sales, S(Q) = m - L(Q) = 16000 - 1100 = 14900
Expected leftover inventory, V(Q) = Q - S(Q) = 18154 - 14900 = 3254 (this is the number of unsold books)
Expected profit for Barnes and Nobles =S(Q)*Cu - V(Q)*Co = 14900*14 - 3254*7 = $ 185,822
Expected number of books unsold = 3,254 books
d) Expected profit for the publisher = 18154*(16-4) - 3254*4 = $ 204,832
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