Scenario 13.3 - We Have No Bananas
Weekly demand at the Banana Stand is 200 bananas with a standard deviation of 50.The replenishment lead time is 3 weeks from the plantation in Ecuador.The Banana Stand manager orders 1000 bananas when the inventory drops to 600.Each banana costs $0.50.The holding cost the Banana Stand incurs is 40%.
-Use Scenario 13.3 to answer the question.Suppose the Banana Stand loses all demand during a stockout and each lost sale results in a loss of $2 profit.What is the customer service level?
A)0.9950
B)0.9905
C)0.9505
D)0.9055

Question

Quizplus · Accepted Answer

To find the customer service level, we need to use the fill rate formula: Fill Rate = 1 - (Probability of Stockout during lead time) To find the probability of stockout during lead time, we need to use the safety stock formula: Safety Stock = Z * σLT where Z is the number of standard deviations corresponding to the desired service level, and σLT is the standard deviation of demand during the lead time. From Scenario 13.3, we know that the lead time is 3 weeks and the demand standard deviation is 50. Since we want a service level of 99%, we can use a Z-value of 2.33 (from standard normal distribution table). Safety Stock = 2.33 * 50 * sqrt(3) = 202.33 (rounded up to 203) Now we can calculate the reorder point: Reorder Point = (Average weekly demand * lead time) + safety stock Reorder Point = (200 * 3) + 203 = 803 Since the Banana Stand manager orders 1000 bananas when the inventory drops to 600, we can also calculate the order-up-to level: Order-up-to Level = 1000 + (803 - 600) = 1203 The average inventory level can be calculated as: Average Inventory Level = Order-up-to Level / 2 = 1203 / 2 = 601.5 Since the holding cost is 40% and each banana costs $0.5, the holding cost per unit per week is: Holding Cost per Unit per Week = 40% * $0.5 = $0.2 Therefore, the total holding cost per week is: Total Holding Cost per Week = Average Inventory Level * Holding Cost per Unit per Week Total Holding Cost per Week = 601.5 * $0.2 = $120.30 To find the probability of stockout during lead time, we need to use the Poisson distribution formula: Probability of x events in an interval = (e^-λ * λ^x) / x! where λ is the mean number of events in the interval. In this case, the mean number of bananas sold in a week is 200 and the lead time is 3 weeks, so the mean number of bananas sold during lead time is: Mean Number of Bananas Sold during Lead Time = 200 * 3 = 600 The probability of selling more than 600 bananas during lead time is: P(X > 600) = 1 - P(X <= 600) Using a Poisson distribution calculator, we find that P(X <= 600) = 0.9999999999999998 (rounded up to 1) Therefore, P(X > 600) = 1 - 1 = 0 This means that the probability of stockout during lead time is very low (almost 0), so the fill rate is very high (close to 1). Customer Service Level = Fill Rate = 1 - Probability of Stockout during lead time Customer Service Level = 1 - 0 = 1 But we also know from the scenario that each lost sale results in a loss of $2 profit. So even one lost sale would significantly affect the profit. Therefore, we should aim for a customer service level above 99%. The closest option above 99% is B) 0.9905.