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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
- a. Dan will consider this book to be a blockbuster for him if it sells more than 400 units. What is the probability that Power and Self-Destruction will be a blockbuster? [LO13-1]
- b. Dan considers a book a “dog” if it sells less than 50 percent of his mean forecast. What is the probability this exposé is a “dog”? [LO13-1]
- c. What is the probability that demand for this book will be within 20 percent of the mean forecast? [LO13-1]
- d. What order quantity maximizes Dan’s expected profit? [LO13-1]
- e. 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? [LO13-2]
- f. Suppose Dan orders 300 copies of the book. What is Dan’s expected leftover inventory? [LO13-2]
- g. Suppose Dan orders 300 copies of the book. What are Dan’s expected sales? [LO13-2]
- h. Suppose Dan orders 300 copies of the book. What is Dan’s expected profit? [LO13-2]
- i. How many books should Dan order if he wants to achieve a 95 percent in-stock probability? [LO13-3]
a)
To determine: The probability that the book will be a blockbuster.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 400 units
Calculation of Z – value:
Using the Excel =NORMSDIST (2.5) function, the probability value is 0.99379.
Calculation of probability of being a blockbuster:
The probability that the book will be a blockbuster is 0.0062.
b)
To determine: The probability that the book will be a dog.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 50% of mean
= 100 units
Calculation of Z – value:
Using the Excel =NORMSDIST (-1.25) function, the probability value is 0.10565.
The probability that the book will be a dog is 0.1057.
c)
To determine: The probability that the demand of the book will be within 20% of the mean forecast.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Expected demand = within 20% of the mean forecast
Calculation of probability:
The expected demand value is said to be within 20% of the mean forecast. It means, the demand will be 20 % less or 20% more than the mean.
When sales is 20% less:
When sales is 20% more:
Using the Excel =NORMSDIST (-0.50) function, the probability value is 0.30854.
Using the Excel =NORMSDIST (0.50) function, the probability value is 0.69146.
The two probabilities are subtracted to identify the demand probability as shown below:
The probability that the demand of the book will be within 20% of the mean forecast is 0.3830.
d)
To determine: The order quantity that maximizes the profit.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Calculation of critical ratio:
Using the Excel =NORMSINV (0.6667) function, the value of Z using the round up rule is 0.5.
Calculation of order quantity that maximizes the expected profit:
The order quantity that maximizes the profit is 240 units.
e)
To determine: The probability that some customers won’t be able to purchase a copy of the book.
Explanation of Solution
Given information:
In-stock probability = 95%
Calculation of probability of some customers not able to purchase the book:
The in-stock probability is 95% which means 95% percent of customers are able to purchase the book. Therefore, the probability that the customers will not be able to purchase the book will be:
The probability that some customers won’t be able to purchase a copy of the book is 5%.
f)
To determine: The expected leftover inventory.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 300 units
Calculation of Z – value:
From the standard normal distribution table, using the roundup rule, the value of expected inventory distribution (I) for a Z-value of 1.3 is 1.3455.
Calculation of expected leftover inventory:
The expected leftover inventory is 107.64 units.
g)
To determine: The expected sales.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 300 units
Calculation of expected sales:
The expected sales is 192.36 units.
h)
To determine: The expected profit.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 300 units
Calculation of expected profit:
The expected profit is $108.32.
i)
To determine: The number of books that must be ordered to achieve an in-stock probability of 95%.
Explanation of Solution
Given information:
Retail price (RP) = $20
Wholesale price (WP) = $12
Shipping and handling cost (SC) = $4
Mean (M) = 200
Standard deviation (SD) = 80
Target sales (T) = 300 units
Calculation of order quantity:
The in-stock probability of 95% and the roundup rule corresponds to z- value of 1.70 from the standard distribution table.
The number of books that must be ordered to achieve an in-stock probability of 95% is $336 books.
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