# Statistical Process Control

## Introduction

Statistical Process Control (SPC) is a self-control technique designed to guarantee production conformity. The birth of SPC dates back to Shewhart's famous work in the late 1920s, when he proposed to identify the causes of product non-quality using statistical tests in graphical form; the control chart was born. In the decades that followed, M.S.P. was enriched with modern tools to meet the new expectations of manufacturers.

These tools, however different they are, are based on two essential concepts of SPC :

- Capability analysis
- Monitoring with control charts

Although the concepts of capability and control chart were not introduced at the same time, they are very closely related. A capability study defines whether the manufacturing process is capable of delivering a product with the required level of quality. Capability indices compare the quality of a production run over a given period of time, against a given target. Control cards, on the other hand, are used to monitor a process in order to maintain and improve its capability.

Control chart and capability analysis with SOSstat

## The capability

The capability of a means of production is defined as a quantification of the actual performance of the process in relation to the desired performance. Translating this definition into mathematical language is still the subject of much debate. Two questions arise, namely: What is the translation of the desired performance?

*How to measure the actual performance ?*

Two generations of capability indicators have emerged:

- The first generation of indicators based on a calculation of the dispersion of the measurements.
- Second generation indicators, which are based on the loss function introduced by Taguchi.

### Pp and Ppk indicators

The capability indicators *Pp* and *Ppk* are widely used in industry to define
the quality of a product delivered to the customer. Let's recall the formulas
for these indicators. Xbar and s are the population mean and standard deviation,
IT is the tolerance interval, TS and TI are the Upper and Lower Tolerances
respectively.

These capability indicators are inseparable if one wishes to properly formalize
the quality of a production. The *Ppk* alone reflects the proportion of
non-compliance, the gap between the *Pp* and *Ppk* indicates a misalignment of
the production.

We consider that there are two families of variability:

- The short-term variability due primarily to the machine or process.
- The long-term variability due to the remaining 4M (Man, Material, Methods and Environment), causing the process instabilities.

To quantify the short-term phenomena indicators *Cp* and *Cpk* are used while
indicators *Pp* and *Ppk* are used for long-term phenomena.

Since the aim of quality is to reduce the influence of the causes of dispersion, we naturally target our action on the 5Ms of the process. However, it is often difficult to act on machine dispersion, unless major investments are made.

This residual dispersion, also known as “natural dispersion”, determines the maximum achievable capability: this is “process capability”.

Capability Study performed with SOSstat

**SOSstat** Conducts capability studies according to several standards. It can
perform calculations on one or more characteristics simultaneously (all the
characteristics of a control plan). **SOSstat** also has the ability to make the
capability calculation with non-normal populations to provide an accurate
estimate of the proportion of non-conforming.

Example of implementation of a process capability analysis with SOSstat

## Control charts

Control charts are essential tools for rational process control. Rigorous application of this method can significantly improve process capability for two reasons: The use of statistical decision criteria reduces errors due to inappropriate settings or lack of settings. The result is an increase in stability performance. In addition, the application of a target-value policy improves feature centering.

To reduce the overall dispersion of the process, and thus for the process performance tends to the process capability, it is necessary to improve the stability of the machine and the control. The problem that arises is to differentiate deviations of the process that are natural, from those who will cause an adjustment, ie keep the process under control.

A detailed analysis of the dispersion of a process allows the identification of two essential causes of dispersion. These are the common causes that are related to random phenomena, and special causes of dispersion that are identifiable. Unlike common cause, special causes require action on the process.

Control charts have been developed to detect the appearance of special causes and to dissociate them from common causes which do not require intervention in the process. To achieve this, two statistical tests are carried out: the first to ensure that the machine is not out of adjustment, and the second to check that the natural dispersion has not changed.

### The Shewhart control chart family

The control chart introduced by Shewhart consists of two hypothesis tests, to verify that the process centering has not changed, on the one hand, and that the process dispersion is stable, on the other hand.

Monitoring is done graphically with two *charts* :

- The average or the median are used to monitor the centering
- The range or the standard deviation are used to monitor the variability

**SOSstat** can easily build control charts, automatically or from information of
the observation phase provided by the user.

Shewhart's control chart in SOSstat

### Other control charts

There are many control charts in the literature that meet specific industrial constraints. Some are adaptations of Shewhart's charts :

- Individual value control chart
- Short run and moving average control chart
- Control charts with expanded limites

Carte moyenne mobile avec SOSstat

Other charts offer an original approach to enhance detection :

- CUSUM control chart
- EWMA control chart

EWMA control chart in SOSstat

## Bibliography

Pillet, M. - Appliquer la maîtrise statistique des processus MSP/SPC , Editions d'Organisation, 552 pages, 2005, EAN13 : 9782708133495

Duclos, E - La Maîtrise Statistique des Procédés MSP/SPC , LULU , 2016 , ISBN13 : 580-0-1151067-2-8

Duclos, E - L'ABC de la MSP en BD , Comics - Lulu - 40 pages - French edition

Duclos, E - The ABC of SPC in comic form, Bande dessinée - Lulu - 40 pages - English version

Montgomery D.C. - Statistical Quality Control, Wiley, 768 pages , 2012 , ISBN-13: 978-1118146811 GoogleBooks

Donald J. Wheeler, David Smith ChambersUnderstanding Statistical Process Control, SPC Press, 406 pages, ISBN : 9780945320692, 0945320698