Question

Dan McClure owns a thriving independent bookstore in artsy New Hope, Pennsylvania. He must decide how many copies to order of a new book, Power and Self-Destruction, an exposé on a famous politician’s lurid affairs. Interest in the book will be intense at first and then fizzle quickly as attention turns to other celebrities. The book’s retail price is $20, and the wholesale price is $12. The publisher will buy back the retailer’s leftover copies at a full refund, but McClure Books incurs $4 in shipping and handling costs for each book returned to the publisher. Dan believes his demand forecast can be represented by a normal distribution with a mean of 200 and a standard deviation of 80.

A. If Dan orders the quantity needed to achieve a 95 percent in-stock probability, what is the probability that some customer won’t be able to purchase a copy of the book?

B. Suppose Dan orders 300 copies of the book. What is Dan’s expected leftover inventory?

C. Suppose Dan orders 300 copies of the book. What are Dan’s expected sales?

D. Suppose Dan orders 300 copies of the book. What is Dan’s expected profit?

E.How many books should Dan order if he wants to achieve a 95 percent in-stock probability?

Answer 1

Cu = cost of underage = 20 - 12 = 8
Co = cost of overgae = 4

Critical factor = Cu / (Co+Cu) = 4 / (8+4) = 0.3333

(A)

In-stock probability = 95%

So, the probability that some customer won’t be able to purchase a copy = stockout probability = 5%

(B)

Order qty, Q = 300 copies

z = (300 - 200) / 80 = 1.25

Corresponding normal loss function value, L(z) = 0.05

So, expected left-over inventory, V(Q) = L(z) * Stdev + z * Stdev = 0.05*80 + 1.25*80 = 104 units

(C)

Order qty, Q = 300 copies

z = (300 - 200) / 80 = 1.25

Corresponding normal loss function value, L(z) = 0.05

Expected sales, S(Q) = Mean demand - L(z) * Stdev = 200 - 0.05*80 = 196 units

(D)

Expected profit = Cu * S(Q) - Co * V(Q) = 8*196 - 4*104 = $1,152

(E)

F(z) = 0.95

Corresponding z = normsinv(0.95) = 1.645

So, order quantity, Q = Mean demand + z * Stdev = 200 + 1.645*80 = 332 units