Question

Airco ships 1,000,000 hubs a year in lots of 1000 to a foreign buyer. Airco’s current manufacturing process produces hubs whose mean smoothness follows a normal distribution with a mean of 15 Rz and a variance of .01. Hubs are considered defective if their mean smoothness falls below 14.7 or above 15.3.

The foreign buyer is adopting a new acceptance sampling procedure that will increase Airco’s inspection costs. The buyer will receive the lots of 1000 and randomly sample 80 hubs for testing. If any of those 80 hubs is defective, the buyer will reject the entire lot and return it to Airco for complete inspection (all 1000 hubs). This will cost Airco an additional $5000 per rejected lot.

There is a modification for the manufacturing process that reduces the variance from 0.01 to 0.00826. What are the expected savings for Airco if they adopt this new manufacturing process (compared to the old process)? If the new process involves a one-time switching cost of $500,000, is it worth switching?

Answer 1

The mean of the hub is . The probability that a Hubs is considered defective is

The buyer will receive the lots of 1000 and randomly sample 80 hubs for testing. If any of those 80 hubs is defective, the buyer will reject the entire lot and return it to Airco for complete inspection (all 1000 hubs). The probability of rejecting a lot is (the number of hubs out of 80 that are defective follows binomial distribution).

Expected number of lots rejected is

Average cost of rejection is

Under the new  manufacturing process that reduces the variance from

The probability that a Hubs is considered defective is

The probability of rejecting a lot is

Expected number of lots rejected is

Average cost of rejection is .

The total average cost of switching is

Since , we can conclude that it is worth switching.