Question

4. A manufacturing plant produces memory chips to be used in cardiac pacemakers.The manufacturing process produces a mix of “good” chips and “bad” chips. The lifetime of good chips follows an exponential law with a rate of failure ??, that is:

P[chip still functioning after time ?] = ?−?t

The lifetime of bad chips also follows an exponential law, but with a much faster failure rate of 500*?.

a. Suppose the fraction of bad chips in a typical batch is?.What is the probability that a randomly selected chip is still functioning after time ???

b. Suppose that in an attempt to weed out the bad chips, every chip is tested for some small time ? prior to leaving the factory. The chips that fail during the test time are discarded and the remainder are sent out to customers. What is the value of test- time ? for which 99.9% of chips sent out to customers are good?

c. Suppose the good chips are designed to have a 50% chance of still being functional after 15 years, and that 5% of chips manufactured are bad. Repeating the test in (b), what is the value of test-time ?? for which 99.9% of chips sent out to customers are good?

d. For these particular chips, testing is an expensive process :suppose that the average manufacturing cost of a single chip is $2, but that to test a single chip for one full day costs the company $5. These chips can be sold to pacemaker companies for $200 each, and currently the company is selling 10,000 of these chips each year. The company’s CEO would like to increase profitability by decreasing the testing time before chips are shipped. However, shipping too many bad chips will cause pacemaker companies to lose faith in the product and stop buying the chips – not to mention the cost to patients who may receive a pacemaker with a faulty chip and need additional surgery to repair or replace the malfunctioning device. Your boss has asked you to present a proposal for a reasonable testing time that will maximize profits while still maintaining a quality standard that will satisfy the customer. What do you propose, and what is your reasoning for this proposal?

Answer 1