Question

Wholemark is an Internet order business that sells one popular New Year 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 $0.25 per card. The company receives $4.00 per card sold. Since the cards have the current year printed on them, unsold cards have no salvage value. Their 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 mean 5,000 and standard deviation 290. (Assume these four are independent.)

What is the optimal production quantity for the card? (Use Excel's NORMSINV() function to find the correct critical value for the given α-level. Do not round intermediate calculations. Round your answer to the nearest whole number.)

Optimal production quantity

Answer 1

Selling price(SP) = $4.00

Cost price (CP) = paper cost + printing cost $0.35+$0.25 = 0.60

Salvage value(V) = $0

Average demand (d) = 5000 cards

Standard deviation of daily demand (d) = 290 cards

Overage cost(Co) = CP - V = $0.60-$0 = $0.60

Underage cost(Cu) = SP-CP = $4.00-$0.60 = $3.40

Service level = Cu/(Co + Cu) = 3.40/(0.60+3.40) = 3.40/4 = 0.85 = 85%

So the optimal service level is 85%

At 85% service level value of Z = 1.04

Optimal production quantity = d + Z x d

= 5000 + 1.04 x 290

= 5000 + 301.6

= 5301.6 or rounded up to 5302 cards

Answer 2

To find the optimal production quantity for the card, we need to use the Newsvendor model, which helps determine the order quantity that maximizes expected profit when facing uncertain demand.

Given data: Cost of paper per card (Cp) = $0.35 Cost of printing per card (Cv) = $0.25 Selling price per card (Sp) = $4.00 Mean number of customers (μ) = 5,000 Standard deviation of customers (σ) = 290

Let's calculate the optimal production quantity (Q_optimal):

Step 1: Calculate the critical ratio (CR): The critical ratio represents the ratio of the opportunity cost of overstocking to the opportunity cost of understocking. It is calculated as (Selling price - Cost of paper) / (Selling price - Cost of paper + Cost of printing).

CR = (Sp - Cp) / (Sp - Cp + Cv)

CR = (4.00 - 0.35) / (4.00 - 0.35 + 0.25) CR ≈ 0.93457944

Step 2: Calculate the service level (SL): The service level represents the probability of meeting demand. It is the inverse of the standard normal cumulative distribution function (CDF) evaluated at the critical ratio (CR).

SL = NORMSINV(CR)

Using Excel's NORMSINV() function with CR = 0.93457944, we get:

SL ≈ 1.884852232

Step 3: Calculate the optimal production quantity (Q_optimal): The optimal production quantity is given by:

Q_optimal = μ + SL * σ

Q_optimal = 5,000 + 1.884852232 * 290

Q_optimal ≈ 5,546.21

Since the production quantity must be a whole number, the optimal production quantity is approximately 5,546 cards.