Quality Control and Genetic Algorithms - Quality Control

Quality Control

Alternative quality control (QC) procedures can be applied on a process to test statistically the null hypothesis, that the process conforms to the quality requirements, therefore that the process is in control, against the alternative, that the process is out of control. When a true null hypothesis is rejected, a statistical type I error is committed. We have then a false rejection of a run of the process. The probability of a type I error is called probability of false rejection. When a false null hypothesis is accepted, a statistical type II error is committed. We fail then to detect a significant change in the process. The probability of rejection of a false null hypothesis equals the probability of detection of the nonconformity of the process to the quality requirements.

The QC procedure to be designed or optimized can be formulated as:

Q₁(n₁,X₁)# Q₂(n₂,X₂) #...# Q_q(n_q,X_q) (1)

where Q_i(n_i,X_i) denotes a statistical decision rule, n_i denotes the size of the sample S_i, that is the number of the samples the rule is applied upon, and X_i denotes the vector of the rule specific parameters, including the decision limits. Each symbol # denotes either the Boolean operator AND or the operator OR. Obviously, for # denoting AND, and for n₁ < n₂ <...< n_q, that is for S₁ S₂ .... S_q, the (1) denotes a q-sampling QC procedure.

Each statistical decision rule is evaluated by calculating the respective statistic of a monitored variable of samples taken from the process. Then, if the statistic is out of the interval between the decision limits, the decision rule is considered to be true. Many statistics can be used, including the following: a single value of the variable of a sample, the range, the mean, and the standard deviation of the values of the variable of the samples, the cumulative sum, the smoothed mean, and the smoothed standard deviation. Finally, the QC procedure is evaluated as a Boolean proposition. If it is true, then the null hypothesis is considered to be false, the process is considered to be out of control, and the run is rejected.

A quality control procedure is considered to be optimum when it minimizes (or maximizes) a context specific objective function. The objective function depends on the probabilities of detection of the nonconformity of the process and of false rejection. These probabilities depend on the parameters of the quality control procedure (1) and on the probability density functions (see probability density function) of the monitored variables of the process.