An Introduction to the Design and Optimization of Statistical Quality Control Using Genetic Algorithms

Alternative quality control (QC) procedures can be applied on a process to test statistically the null hypothesis, 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 for 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 distribution of error in the process. The probability for rejection of a false null hypothesis is called probability for error detection.

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

Q1(n1,X1) # Q2(n2,X2) #...# Qq(nq,Xq)  (1)

where Qi(ni,Xi) denotes a statistical decision rule, ni denotes the size of the sample Si, that is the number of the measurements the rule is applied upon, and Xi 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 <...< nq, that is for S1 subset of S2 subset of ....subset of Sq, the (1) denotes a q-sampling QC procedure.

Each statistical decision rule is evaluated by calculating the respective statistic of the sample of the measurements. 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 sample, the range of the sample, the mean of the sample, the standard deviation of the sample, 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 QC procedure is considered to be optimum when it minimizes (or maximizes) a context specific objective function. The objective function depends on the probabilities for error detection and for false rejection. The probabilities for error detection and for false rejection depend on the parameters of the QC procedure (1) and on the probability density function of the error in the process.

In general, we can not use algebraic methods to optimize the QC procedures. Usage of enumerative methods would be very tedious, especially with multi-rule procedures, as the number of the points of the parameter space to be searched grows exponentially with the number of the parameters to be optimized. Optimization methods based on the genetic algorithms (GAs) offer an appealing alternative as they are robust search algorithms, that do not require knowledge of the objective function and search through large spaces quickly. GAs have been derived from the processes of the molecular biology of the gene and the evolution of life. Their operators, cross-over, mutation, and reproduction, are isomorphic with the synonymous biological processes. GAs have been used to solve a variety of complex optimization problems. Furthermore, the complexity of the design process of novel QC procedures is obviously greater than the complexity of the optimization of predefined ones. The classifier systems and the genetic programming paradigm have shown us that GAs can be used for tasks as complex as the program induction.

In fact, since 1993, we have successfully used the GAs to optimize and to design novel QC procedures, as it is described in the HCSL publications on the GAs based QC.

--Aristides T. Hatjimihail, MD, PhD, ath@hcsl.com