Question

You are negotiating a contract with a new customer to deliver one of the components for their product. You estimate that on average you should be able to deliver the product in 8 hours once the customer places the order. Based on historical data, you believe it is reasonable to assume the standard deviation is 1 hour and that actual delivery times are normally distributed. The customer offers you a bonus for deliveries when you are able to deliver the component in less than 6.5 hours. The customer also wants you to pay a penalty on deliveries that do not arrive within 9 hours. You expect an average of 3 deliveries per day for this customer. Assume 300 business days a year. You have estimated that average profit per on-time (within 6.5 – 9 hour) delivery will be $5. However, if you get a bonus (delivery < 6.5 hours), your profit becomes $6 per delivery. On the other hand, if you have to pay penalty (delivery > 9 hours), your profit goes down to $3 per delivery

From your statistics class, you know that you can achieve this by reducing the -1) average delivery time; or, 2) the standard deviation of delivery time; or 3) both average and standard deviation of delivery time. e) If you are unable to change the standard deviation of the delivery time, what should be the average delivery time so that no more than 5% of the deliveries are late (late = takes more than 9 hours. Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour) f) On the other hand, if you are unable to change the average delivery time, what should be the standard deviation of the delivery process so that no more than 5% of the deliveries are late (late = takes more than 9 hours, Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour)

Based on the calculations above, you consider the 9-hour time limit is too high a risk. Your customer is also adamant that you must deliver within 9 hours, otherwise pay a penalty. To keep the customer happy, you are thinking of making some changes in your delivery process to reduce the chance of late deliveries (> 9 hours). From your statistics class, you know that you can achieve this by reducing the -1) average delivery time; or, 2) the standard deviation of delivery time; or 3) both average and standard deviation of delivery time.

e) If you are unable to change the standard deviation of the delivery time, what should be the average delivery time so that no more than 5% of the deliveries are late (late = takes more than 9 hours. Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour)

f) On the other hand, if you are unable to change the average delivery time, what should be the standard deviation of the delivery process so that no more than 5% of the deliveries are late (late = takes more than 9 hours, Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour)

Answer 1

e) If you are unable to change the standard deviation of the delivery time, what should be the average delivery time so that no more than 5% of the deliveries are late (late = takes more than 9 hours. Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour)

Not more 5% deliveries are late means that 95% of the delivers must be made within 9 hours.

The zscore that corresponds to 95% percentile is 1.64

f) On the other hand, if you are unable to change the average delivery time, what should be the standard deviation of the delivery process so that no more than 5% of the deliveries are late (late = takes more than 9 hours, Current average delivery time = 8 hours, standard deviation of delivery time = 1 hour)

Not more 5% deliveries are late means that 95% of the delivers must be made within 9 hours.

The zscore that corresponds to 95% percentile is 1.64