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 (22 points total for a-d) a) On what % of deliveries do you expect to receive a bonus? b) On what % of deliveries do you expect to pay a penalty? c) In a year how many deliveries will be on time 6.5-9 hours)? d) What is your estimate for the total yearly profit from this customer? Assume 3 deliveries per day and 300 business days in a year 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

Note : Allowed to solve only one question with 4 sub part. Hence question one solved.

We are given
mean delivery time = 8 hours
std. dev for delivery time = 1 hour

total deliveries in the year = 300

Profit for deliveries within 6.5 – 9 hour = 5
Profit for deliveries less 6.5 hours = 6
Profit for deliveries more than 9 hours = 3

a) On what % of deliveries do you expect to receive a bonus?

b) On what % of deliveries do you expect to pay a penalty?

c) In a year how many deliveries will be on time 6.5-9 hours)?
Deliveries will be on time 6.5-9 hours = Expected percentage * average number of deliver in the year = 0.7745*300 = 232.35

d) What is your estimate for the total yearly profit from this customer?
We need to find the percentage deliver done less than 6.5 hours

The total yearly profit is worked out in the table below
We find the expected number of delivery for each time limit of delivery and multiply it with the profit.