May/June 2010 ## What Do We Mean by “Zero Defects”?## Part 2 of 3: Non-Conformities When Sampling a Continuum
## Q: How do we handle zero defects or non-conformities when sampling a continuum?A. In our first article on this topic, we found that while “zero defects” is a standard in quality excellence, the occurrence of In this article we develop a similar upper confidence bound for a rate of occurrence when sampling a continuum, and we observe zero defects or non-conformities within the inspection region. Before this is taken up, we briefly revisit the binary data scenario and consider how an inspection error probability may be accounted for in the upper confidence bound. By inspection error we mean the error of misclassifying non-conforming items as conforming. This is also called “consumer risk,” or the risk to a customer. We also assume that the second type of error, the “producer’s risk,” ## The Poisson DistributionIn a sample When sampling a continuum we call the events that are found either defects or non-conformities. By a continuum, we mean any type of inspection region defined by space, time, area, volume or other similar metric. An event within the inspection region is the occurrence of a defect or non-conformity. Thus, for this type of sampling, we are not classifying objects, rather we are counting events. We further assume that the inspection region being observed is homogeneous and that events occur randomly throughout the region. This means that the rate of event occurrence remains constant throughout the inspection region. Some examples include: a) paper rolling off a mill, where events are defined as any abnormality on the paper’s surface, e.g., the rate might be defined as abnormalities per square yard; b) surface pits or other abnormality per square foot for sheet metal stock; or c) blemishes on a painted surface per square foot. There are numerous applications for this type of inspection process. Wherever there is an event of interest within some defined interval we can use this method. The statistical model that governs this type of sampling is the Poisson distribution. In this distribution there is a single parameter, λ, interpreted as the unknown rate of occurrence for the event in question. Sometimes there is an auxiliary parameter, In the Poisson distribution with rate λ, the probability of observing Using an argument similar to that in part 1, when we observe zero events we require that this probability be at least some small value such as 0.05 pr 0.01. This is cast as the quantity 1 - Then solving the inequality for the rate λ gives the upper confidence bound when We can also take (3) and solve for either The interpretation of (4) is that the confidence is at least ## Illustrations1) Suppose the inspection region is 20 board feet of processed timber used in high end furniture manufacturing. The event is defined as a certain size of abnormality in wood products. At confidence Use (3) with 2) Suppose in illustration 1 that there is a potential for an inspection error of approximately 10 percent. This means that the inspector may miss a real event about 10 percent of the time. Use (5) with 3) A business unit manager of a large manufacturing cell is interested in tracking the number of returns or “turnbacks” of cell product per thousand units produced. Last week, a typical week for this operation, the cell produced 17,500 units with 0 turnbacks observed. Under similar controlled conditions, what rate per thousand units is being demonstrated with 90 percent confidence? Assume that there is a risk of approximately 3 percent of missing a real turnback. Use (5) with
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