What is the relationship between type I and type II errors and the width of the upper and lower control limits on SPC charts?

Question

Propose a mechanism for determining SPC chart limit widths based upon the financial consequences of type I and type II errors.

Answer 1

Type I and type II errors move in opposite directions, that is, as the chance of a type I error increases, the chance of a type II error decreases. As the width of SPC control limits increases, the chance of a type I error decreases (and the chance of a type II error increases). The width of control limits might be balanced based upon the cost incurred from false positives and false negatives in the course of monitoring the process. The cost associated with a type I error, sometimes called producer's risk, is that a process that is actually in control is stopped and effort is spent troubleshooting the process. The process might be adjusted, suppliers contacted, etc., all in the name of fixing a problem that in actuality, doesn't exist. The direct financial impact is the labor cost of troubleshooting and the lost productive capacity. The cost associated with a type II error, sometimes called a consumer's risk, is that defective product will reach the hands of a consumer and fail, potentially resulting in warranty repair or replacement, negative word of mouth from a dissatisfied customer, and litigation. Depending on the product, these costs could be substantial. A producer might choose to balance the costs with a calculation similar to expected monetary value. A producer might choose to lower the EMV of the type II costs to something they could pay out (or have insurance for).