Question

wholemark is an internet order business that sells one popular new year's greeting card once a year. the cost of the paper on which the card is printed is $0.35 per card, and the cost of printing is $0.45 per card. the company receives $5.60 per card sold. because the cards have the current year printed on them, unsold cards have no salvage value. its customers are from the four areas: los angeles, santa monica, hollywood, and pasadena. based on past data, the number of customers from each of the four regions is normally distributed with a mean of 5,300 and a standard deviation of 550. (assume these four are independent.)what is the optimal production quantity for the card?

Answer 1

To maximize the expected profit while minimizing the risk of overproduction, Wholemark should produce about 5887 greeting cards. ​

How to find optimal production quantity?

To determine the optimal production quantity:

Production cost per card (
`\( C_p \)`):

`\[ C_p = \text{Cost of paper per card} + \text{Cost of printing per card} \]`

`\[ C_p = \$0.35 + \$0.45 = \$0.80 \]`

Selling price per card (P)):

`\[ P = \$5.60 \]`

Overage Cost per Card (
`\( C_o \)`):

`\[ C_o = C_p = \$0.80 \]`

(Since unsold cards have no salvage value, the overage cost is the production cost.)

Underage cost per card (
`\( C_u \)`):

`\[ C_u = P - C_p = \$5.60 - \$0.80 = \$4.80 \]`

The critical ratio:

`\[ \text{Critical Ratio} = (C_u)/(C_u + C_o) \]`

`\[ \text{Critical Ratio} = (\$4.80)/(\$4.80 + \$0.80) \]`

`\[ \text{Critical Ratio} \approx 0.857 \]`

Find the z-score that corresponds to a cumulative probability of approximately 0.857 in the standard normal distribution.

`\[ z \approx 1.068 \]`

The optimal production quantity (
`\( Q^* \)`):

`\[ Q^* = \mu + z * \sigma \]`

`\[ Q^* = 5300 + 1.068 * 550 \]`

`\[ Q^* \approx 5887 \]`

Therefore, the optimal production quantity, considering the normal distribution of demand, the costs, and the selling price, is approximately 5887 cards.