Chapter 13, Problem 1PA

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.

1. 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]
2. 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]
3. c. What is the probability that demand for this book will be within 20 percent of the mean forecast? [LO13-1]
4. d. What order quantity maximizes Dan’s expected profit? [LO13-1]
5. 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]
6. f. Suppose Dan orders 300 copies of the book. What is Dan’s expected leftover inventory? [LO13-2]
7. g. Suppose Dan orders 300 copies of the book. What are Dan’s expected sales? [LO13-2]
8. h. Suppose Dan orders 300 copies of the book. What is Dan’s expected profit? [LO13-2]
9. i. How many books should Dan order if he wants to achieve a 95 percent in-stock probability? [LO13-3]

Summary Introduction

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:

$\begin{array}{c}\text{Z}=\frac{T-M}{SD}\\ =\frac{400-200}{80}\\ =\frac{200}{80}\\ =2.5\end{array}$

Using the Excel =NORMSDIST (2.5) function, the probability value is 0.99379.

Calculation of probability of being a blockbuster:

$\begin{array}{c}\text{Probability}=1-\text{F}\\ =1-0.99379\\ =0.00621\end{array}$

The probability that the book will be a blockbuster is 0.0062.

Summary Introduction

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:

$\begin{array}{c}\text{Z}=\frac{T-M}{SD}\\ =\frac{100-200}{80}\\ =\frac{-100}{80}\\ =-1.25\end{array}$

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.

Summary Introduction

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:

$\begin{array}{c}\text{Z}=\frac{\left(M-\left(M\times \frac{20}{100}\right)\right)-M}{SD}\\ =\frac{\left(200-\left(200\times \frac{20}{100}\right)\right)-200}{80}\\ =\frac{160-200}{80}\\ =-0.5\end{array}$

When sales is 20% more:

$\begin{array}{c}\text{Z}=\frac{\left(M+\left(M\times \frac{20}{100}\right)\right)-M}{SD}\\ =\frac{\left(200+\left(200\times \frac{20}{100}\right)\right)-200}{80}\\ =\frac{240-200}{80}\\ =0.5\end{array}$

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:

$\begin{array}{c}\text{Probability}=0.69146-0.30854\\ =0.38292\end{array}$

The probability that the demand of the book will be within 20% of the mean forecast is 0.3830.

Summary Introduction

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:

$\begin{array}{c}\text{Critical ratio}=\frac{RP-WP}{\left(RP-WP\right)+SC}\\ =\frac{20-12}{\left(20-12\right)+SC}\\ =\frac{8}{8+4}\\ =\frac{8}{12}\\ =0.6667\end{array}$

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:

$\begin{array}{c}\text{Order quantity}=M+\left(Z\times SD\right)\\ =200+\left(0.5\times 80\right)\\ =200+40\\ =240\text{ units}\end{array}$

The order quantity that maximizes the profit is 240 units.

Summary Introduction

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:

$\begin{array}{c}\text{Probability}=100-\text{In-stock probability}\\ =100-95\%\\ =5\%\end{array}$

The probability that some customers won’t be able to purchase a copy of the book is 5%.

Summary Introduction

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:

$\begin{array}{c}\text{Z}=\frac{T-M}{SD}\\ =\frac{300-200}{80}\\ =\frac{100}{80}\\ =1.25\end{array}$

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:

$\begin{array}{c}\text{Expected leftover inventory}=SD\times I\\ =80\times 1.3455\\ =107.64\end{array}$

The expected leftover inventory is 107.64 units.

Summary Introduction

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:

$\begin{array}{c}\text{Expected sales}=\text{Target sales}-\text{Expected leftover inventory}\\ =300-107.64\\ =192.36\end{array}$

The expected sales is 192.36 units.

Summary Introduction

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:

$\begin{array}{c}\text{Expected profit}=\left(RP\times \text{Expected sales}\right)+\left[\left(RP-WP\right)\times \text{Leftover inventory}\right]-\left(WP\times T\right)\\ =\left(20\times 192.36\right)+\left[\left(20-12\right)\times 107.64\right]-\left(12\times 300\right)\\ =3,847.2+861.12-3,600\\ =\$1,108.32\end{array}$

The expected profit is $108.32.

Summary Introduction

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.

$\begin{array}{c}\text{Order quantity}=M+\left(Z\times SD\right)\\ =200+\left(1.7\times 80\right)\\ =200+136\\ =336\text{ units}\end{array}$

The number of books that must be ordered to achieve an in-stock probability of 95% is $336 books.