Degan routinely drove the 400-mile round trip to his favorite grocery store to stock up on bread, which cost $1.50 per loaf. Running out would be disastrous - an unplanned trip to this store would mean a custom order that would run $7.75 per loaf. Overstocking wasn't a big issue, he could sell it to his teaching partner for $0.10 per loaf. With a demand that is normally distributed with a mean of 125 and standard deviation of 15, what is Degan's optimal purchase quantity on his next trip to the store?
A)139
B)173
C)115
D)128

Question

Quizplus · Accepted Answer

The optimal purchase quantity can be determined using the Newsvendor model, which calculates the optimal inventory level to minimize the costs associated with overstocking and understocking. The formula for the critical ratio (CR) in the Newsvendor model is:$$ CR = \frac{Cu}{Cu + Co} $$where $Cu$ is the cost of understocking (the cost of a stockout) and $Co$ is the cost of overstocking (the cost of having excess inventory).Given:- Cost of understocking ($Cu$) = $7.75 (cost of a custom order) - $1.50 (regular cost) = $6.25- Cost of overstocking ($Co$) = $1.50 (regular cost) - $0.10 (resale value) = $1.40$$ CR = \frac{6.25}{6.25 + 1.40} = \frac{6.25}{7.65} \approx 0.817 $$The critical ratio corresponds to the service level, which can be translated into a Z-score in the context of a normal distribution. Looking up the Z-score for a CR of approximately 0.817 gives us a Z-score of about 0.91.The optimal order quantity (Q*) can be found using the formula:$$ Q* = \mu + Z \cdot \sigma $$where:- $\mu$ is the mean demand (125 loaves),- $Z$ is the Z-score corresponding to the desired service level (0.91),- $\sigma$ is the standard deviation of demand (15 loaves).$$ Q* = 125 + 0.91 \cdot 15 \approx 125 + 13.65 = 138.65 $$Rounding to the nearest whole number, the optimal purchase quantity is 139 loaves.

Degan Routinely Drove the 400-Mile Round Trip to His Favorite