Quality Control

The systematic approach to quality control was developed in industrial manufacturing in the interwar years. With the impact of mass production required during World War II, it became necessary to introduce a more rigorous form of quality control. Some of the initial work is credited to Walter Shewhart of Bell Labs, starting with his famous one-page memorandum of 1924.

Control charts are often employed to detect changes in a process mean over time. In the traditional approach, a sample is drawn, and the sample mean (

) is calculated and plotted on a Shewhart chart having control limits, which depict the extremes of pure chance fluctuations. A point outside the limits suggests that the process is off target. While a Shewhart chart is relatively easy to use and interpret, according to W H. Woodall, a cumulative sum (CUSUM) chart is more capable of detecting small changes in the process mean, as well as pinpointing the exact time when the process goes “out of control.” Faster detection of significant changes means tighter control, which is necessary if corrective action is to be taken promptly.

[Page 1335]Like Shewhart and CUSUM control schemes, an exponentially weighted moving average (EWMA), as discussed by J. M. Lucas and M. S. Saccucci, control scheme is easy to implement and interpret. The ability of the EWMA chart to detect small shifts in the process mean is on a par with the CUSUM chart and superior to the Shewhart

chart. Lucas and Saccucci argued that the EWMA chart is simpler to explain to the lay user than the CUSUM chart, by noting its similarity to the classical Shewhart chart. Both the CUSUM and EWMA charts are more suitable for single sampling schemes.

A control chart procedure has been proposed for which the Shewhart

chart, the cumulative sum chart, and the exponentially weighted moving average chart are special cases. The procedure for constructing these charts has been described by C. W Champ and colleagues.

A typical statistical test examines the validity of a null hypothesis, H0 (the process is on target), against an alternative, HA (the process has changed). A Type I error is said to occur if H0 is rejected when it is true and occurs with a probability α. That is, the process is on target, but, by chance, lies outside the control limits. A Type II error is said to occur if H0 is accepted when HA is true and occurs with a probability β. That is, the process has deviated, but lies within the control limits. The power of a test is defined to be the probability of correctly rejecting a false null hypothesis, and is equal to 1-β. For a given α, one test is more powerful than another if 1-β for the former is greater than for the latter for all possible changes in the process mean.

Referring to the process under consideration, assume a batch has a true mean μ. Let r be an acceptable target value for μ; therefore a batch is acceptable if μ=τ. In this case H0 is μ=τ and HA is μ∗τ. If a batch is rejected when, in fact, the mean is μ=τ, this is unfair to the producer. This is called the producer s risk or Type I error, and occurs with a probability α. Conversely, if a batch is accepted when, in fact, the mean is μ≠τ, this is unfair to the consumer. This is called the consumer s risk or Type II error, and occurs with a probability β.

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