Question

A daily newspaper is stocked by a coffee shop so its patrons can purchase and read it while they drink coffee. The newspaper costs $ 1.13 per unit and sells for $ 1.75 per unit. If units are unsold at the end of the day, the supplier takes them back at a rebate of $ 1 per unit. Assume that daily demand is approximately normally distributed with mu=150 and sigma=30 .

(a) What is your recommended daily order quantity for the coffee shop?
(b) What is the probability that the coffee shop will sell all the units it orders?
(c) In problems such as these, why would the supplier offer a rebate as high as $ 1 ? For example, why not offer a nominal rebate? Find the recommended order quantity at 25 $ per unit.

Answer 1

The recommended daily order quantity for the coffee shop is 156 newspapers, and there is a 58.3% probability of selling all units on any given day. The supplier offers a high rebate to encourage retailers to order more units and avoid stockouts, which can lead to lost sales and unhappy customers.

Step-by-step explanation:

Calculating the Recommended Daily Order Quantity

To determine the recommended daily order quantity for the coffee shop, we need to calculate the newspaper's profitability and balance the costs of underordering and overordering. The coffee shop buys each newspaper for $1.13, sells at $1.75, and receives a rebate of $1 for unsold units. Critically point, where the cost of overordering is equal to the cost of underordering, is given by:

Z = (Cost per unit - Rebate) / (Selling price - Cost per unit)

Z = ($1.13 - $1.00) / ($1.75 - $1.13)

Z = $0.13 / $0.62 = 0.2097

Using a standard normal distribution table, a Z-score of 0.2097 corresponds to a cumulative probability of approximately 58.3%. This means that the coffee shop should stock enough newspapers to meet the demand of 58.3% of the days. Translating this into the actual quantity and using the daily demand distribution (mu=150 and sigma=30), we calculate the newspaper order quantity:

Order Quantity = mu + Z * sigma

Order Quantity = 150 + 0.2097 * 30

Order Quantity = 150 + 6.291 = 156.291

Therefore, the recommended daily order quantity is 156 newspapers.

The Probability of Selling All Units Ordered

The probability of selling all units ordered is equivalent to the probability that demand will be at least as high as the order quantity. Since we are ordering 156 units and a normal distribution of demand with a mean of 150 and a standard deviation of 30, the Z-score for selling all units is calculated as:

Z = (Order Quantity - mu) / sigma

Z = (156 - 150) / 30

Z = 6 / 30 = 0.2

The cumulative probability for a Z-score of 0.2 is about 58.3%, which indicates that there is a 58.3% chance of selling out on any given day.

Reason for a High Rebate Offer by the Supplier

Suppliers might offer a high rebate to encourage retailers to order more units, reducing the likelihood of stockouts, which can result in lost sales and dissatisfied customers. A nominal rebate might not be sufficient to offset the risk of unsold inventory, thereby discouraging retailers from maintaining an adequate stock.

If we consider a rebate of $25 per unit, practically it does not make sense because it is much higher than the cost of the newspaper, leading to the supplier incurring huge losses. Therefore, such a rebate would not be sustainable for the supplier.