﻿ An Introduction to Design and Optimization of Statistical Quality Control

# An Introduction to Design and Optimization of Statistical Quality Control

Statistical quality control (QC) procedures are essential for testing whether a process conforms to quality specifications (null hypothesis) or is out of control (alternative hypothesis). In this context, a type I error occurs when a true null hypothesis is rejected, resulting in a false rejection of the process run. The probability of a type I error is known as the probability of false rejection. A type II error occurs when a false null hypothesis is accepted, leading to a failure in detecting significant changes in the probability density function of a process's quality characteristic. The probability of rejecting a false null hypothesis is equivalent to the probability of detecting nonconformity in the process.

The statistical QC procedure to be designed or optimized can be represented as:

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

Here, Qi(ni, Xi) denotes a statistical decision rule, ni refers to the sample size Si the rule is applied to, and Xi denotes the vector of rule specific parameters, including the decision limits. The symbol # represents either the Boolean operator AND or the operator OR. For # denoting AND, and for n1 < n 2 < ... < n q, that is for S1 S2 ⊂ ... ⊂ Sq , the procedure (1) denotes a q‑sampling statistical QC procedure.

Various statistics, such as a single value, range, mean, standard deviation, cumulative sum, smoothed mean, and smoothed standard deviation, are used to evaluate each statistical decision rule by calculating the respective statistic of the measured quality characteristic of the sample. If the statistic falls outside the interval between the decision limits, the decision rule is considered true. The statistical QC procedure is then evaluated as a Boolean proposition, and if true, the null hypothesis is considered false, the process is deemed out of control, and the run is rejected.

An optimum statistical QC procedure minimizes (or maximizes) a context-specific objective function, which depends on the probabilities of detecting nonconformity and false rejection. These probabilities rely on the parameters of the procedure (1) and the probability density function of the measured quality characteristic of the process.

The probabilities of detecting nonconformity and false rejection can be estimated using simulation1 (see Other publications by HCSL Fellows) or calculated, using numerical methods2 (see HCSL publications on statistical QC design).

Generally, algebraic methods are not suitable for optimizing statistical QC procedures. Enumerative methods can be cumbersome, particularly with multi-rule procedures, since the number of points in the parameter space to be searched increases exponentially with the number of parameters to be optimized. Genetic algorithms (GAs) provide an attractive alternative, as they are robust search algorithms that do not require knowledge of the objective function and can efficiently search through large spaces. GAs have been inspired 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 employed to address a wide range of complex optimization problems. Additionally, the design process for novel QC procedures is inherently more complex than optimizing predefined ones. Classifier systems and the genetic programming paradigm have demonstrated that GAs can be applied to tasks as intricate as program induction.

Since 1993 we have successfully used the GAs to optimize and to design novel statistical QC procedures3 (see HCSL publications on the GAs based design and optimization of statistical QC).

## References

1. Hatjimihail AT. A tool for the design and evaluation of alternative quality control procedures. Clin Chem 1992;38:204-10.
2. Hatjimihail AT. Tool for quality control design and evaluation. Wolfram Demonstrations Project, Champaign: Wolfram Research, Inc., 2010.
3. Hatjimihail AT. Genetic algorithms based design and optimization of statistical quality control procedures. Clin Chem 1993;39:1972-8.