Question

York University bookstore purchases T-shirts with different school names and logos from a vendor. The lead time required to receive an order of T-shirts from the vendor is 4 weeks. The weekly demand for York T-shirts is normally distributed with mean of 20 pieces and with a variance of 10 pieces per week. York bookstore wants to limit the probability of stockout to 5%.

Q1. Using the periodic-review inventory system, the bookstore checks the store's inventory every 6 weeks. If today is the time to review the inventory and 70 T-shirts are left, calculate the appropriate order quantity

Answer 1

To limit the stockout probability to 5%, the York University bookstore should order 147 T-shirts during this inventory review cycle, taking into account the 10-week time horizon of lead time plus review period and the demand distribution.

Step-by-step explanation:

To calculate the appropriate order quantity for the York University bookstore using a periodic-review inventory system, we need to take into account the lead time, the review period, the demand distribution, and the target stockout probability.

To begin with, we know that the lead time is 4 weeks, and the review period is 6 weeks, so the time horizon we must cover with our order is 10 weeks in total. The weekly demand for T-shirts follows a normal distribution with a mean (\(\mu\)) of 20 pieces and a variance (\(\sigma^2\)) of 10 pieces, which gives us a standard deviation (\(\sigma\)) of \(\sqrt{10}\), or approximately 3.16 pieces.

Next, we need to determine the safety stock that will limit the stockout probability to 5%. We use the z-score corresponding to a 95% service level, which is approximately 1.645. Then, we calculate the safety stock as follows:
Safety Stock = z-score * \(\sigma\) * \(\sqrt{Lead Time + Review Period}\)

Plugging in the numbers: Safety Stock = 1.645 * 3.16 * \(\sqrt{10}\) ≈ 16.45 pieces. Since we cannot order a fraction of a T-shirt, we round this up to 17 pieces.

We need to ensure we have enough stock to cover the expected demand during the 10-week horizon, plus the safety stock. The expected demand over this time period is 20 shirts/week * 10 weeks = 200 shirts. Therefore, the target inventory level is 200 shirts + 17 shirts = 217 shirts.

Finally, since there are currently 70 T-shirts in stock, the order quantity would be the target inventory level minus the current inventory: Order Quantity = 217 shirts - 70 shirts = 147 shirts.

Therefore, the York University bookstore should order 147 T-shirts in this inventory review cycle.