You are called in to consult for the Clothing International Company (CIC). The company is a database marketing / mail order firm that takes orders by mail or, with an “800 number,” by phone. CIC has been receiving complaints from several customers about how long it has taken for customers to receive the merchandise they ordered. In the language of CIC (and the entire database marketing industry), this is the time it takes for orders to flow through your database marketing fulfillment process. The fulfillment process includes order taking, order processing, warehousing issues, various aspects of customer service (e.g., ensuring that the customer receives the correct size, color, and fabric of a shirt), and ultimately getting the product to the purchaser. After some studying, you suspect that bottlenecks are occurring at the warehouse stage with respect to the picking of individual items in a multiple-item order, and packing them together. In the past, the time from the order being received at the warehouse to its leaving the warehouse (“order time”) by UPS (United Parcel Service) has had a mean of m = 3.80 days. Order times is are normally distributed with a standard deviation, s = 1.20 days. To study the current picking and packing (P&P) system, you decide to collect data on the time it takes for each of a sample of n = 25 randomly chosen orders to go through the P&P system. Nearly all orders went by UPS; orders to be shipped by other carriers were ignored in the study.

You propose the following decision rule with respect to any action to be taken: a) if the sample average, X-bar, of the n = 25 times exceeds 4.25 days, it will be concluded that there are serious problems with the P&P system, and suggest that CIC redesign the system, adding capacity; b) if this sample average, X-bar, is under 4.25 days, it will be concluded that there are no (strong indications of) problems with the P&P system, and suggest that no action be taken.

a) Suppose that P&P system is functioning as it had in the past (i.e., average time, m = 3.80, with a standard deviation, s = 1.20). Using the above decision rule, what is the probability that you will incorrectly conclude that you have serious problems with the P&P system (“incorrectly,” since, actually, you don’t have problems, since the true m is 3.80)?

b) Suppose that, unfortunately, average P&P system time has actually increased to 4.60 days (i.e., suppose that the true m is 4.6), while individual times are still normally distributed with the same standard deviation of 1.20 days. Using the above decision rule, what is the probability that you will incorrectly conclude that there are no serious problems with the P&P system (”incorrectly,” since you actually do have serious problems, since the true m is too high, 4.6)?

c) Management believes that you should modify the decision rule above (i.e., change the “cutoff” point of 4.25) to have a probability in question b) of (only) .01, what would be the new cutoff point?

d) Using the new cutoff point found in part c), what now is the probability that you will incorrectly conclude that you have serious problems with the P&P system (i.e., the new answer to part a)?

e) Comment on the trade-off involved in using the original cutoff point of 4.25 vs. the new cutoff point found in c) and used in part d).