Design and Robustness of Some Design and Robustness of Some Statistical Quality Control ToolsStatistical Quality Control Tools

Dr. Maria Calzada

Loyola University New Orleans

Statistical Quality ControlStatistical Quality Control

Statistical quality control (SQC) is the use of modern statistical methods towards the end of assessing and improving a process.

The father of SQC is Walter A. Shewhart. The most widely used tool in quality control is named after him: the Shewhart chart.

Walter A. ShewhartWalter A. ShewhartDeveloped concepts in quality control based

on his work at Bell Laboratories in the 1920’s.Basic idea: NO TWO THINGS ARE

ALIKE.Two items could be different solely because

of “chance causes.”But also, two items could be different because

of “assignable causes.”

Chance CausesChance Causes

CONTROLLED VARIATION: A stable consistent pattern of variation over time. One is able to predict what will happen in the future based on past data.

This type of variation accepts the fact that even when two items are “created equal” there will be differences between the items.

Visualizing Chance CausesVisualizing Chance Causes

Target

At a fixed point in time

Time Target

Over time

Think of a manufacturing process producing distinct parts with measurable characteristics. These measurements vary because of materials, machines, operators, etc. These sources make up chance causes of variation.

Assignable CausesAssignable Causes

UNCONTROLLED VARIATION: A pattern of variation that changes over time. One is not able to predict what will happen in the future based on past data.

This type of variation tells you that two items were not “created equal.”

Visualizing Assignable CausesVisualizing Assignable Causes

TargetTime

Here not only the process not “on-target” but its variance is increasing. Possible reasons for this: machines need adjustment, materials are slightly different, workers need training, etc.

Statistical Quality ControlStatistical Quality Control

The use of statistical methods to “catch” assignable causes of variation as soon as possible to do something about them.

The most widely used tool in statistical quality control is the Shewhart chart.

Shewhart ChartShewhart ChartWe take measurements at different points in

time, average them, and plot them together with 3 control limits.

One should also have a Shewhart chart for monitoring the variation. However, for the moment we will disregard this.

An ExampleAn ExampleFrom Montgomery, page 237.

The fill volume of soft drinks is an important characteristic. This can be measured by placing a gauge over the crown and comparing the height of the liquid in the neck of the bottle against a coded scale. A reading of zero corresponds to the correct fill height. Samples of size 10 are taken at regular times

1 0.52 0.453 -0.14 -0.575 06 07 0.058 -0.159 0.2

10 -0.1511 0.312 013 -0.5514 -0.1515 0.15

Sample Number Average

The target is at zero and the variation is three standard deviations from the target.

This process is in-control.

-1.5

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0

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This shows the process leaving the control limits at sample 19.However, we can “see” that there may have been a change in the process earlier than that. The problem should be corrected so that the process returns to statistical control.

Average Run LengthAverage Run LengthThe Average Run Length (ARL) is the

average number of points that most be plotted before a point indicates an out-of-control condition.

For the Shewhart chart this can be calculated as

ARL=1/p where p is the probability that any one point

exceeds the control limit.

We would like the ARL to be long when the process is in-control and short when the process is out-of-control.

For a 3 Shewhart chart and an underlying normally distributed process we have

ARL=1/(.0027)=370.37

when the process is in-controlFor many practical applications this chart

has proven very effective and robust. It should be used more.

The EWMA Control ChartThe EWMA Control Chart

Given individual readings x1, x2, x3,…, the Exponentially Weighted Moving Average Control Chart is defined by.

were 0<1 is a design parameter. An out-of-control signal is given when |Zi|>h, where h is another design parameter.

Z i x i 1 Z i 1

930

940

950

960

970

980

990

0 5 10 15 20 25

EWMA Chart for some data

When to use EWMAWhen to use EWMASamples of size 1 (it can be applied to other

sizes)

We need to quickly recognize a small change in the mean.

For large changes in the mean the Shewhart chart has been shown to be superior.

ARL for EWMAARL for EWMA

If L(u) is the ARL given a start at u, then

which is an Fredholm Integral Equation of the second kind.

Lu 1 1

h

h

Lx f x 1 u dx

Robustness of EWMARobustness of EWMA

Borror et al (1999) showed that a properly designed EWMA chart is robust to non-normality.

Stoumbos et all (2000) is a study of robustness to non-normality and autocorrelation with recommendations.

New tools are being developedNew tools are being developed

The Synthetic Control chart was introduced in 2000 by Wu and Trevor. Their paper shows that the synthetic control chart detects moderate shifts in the mean faster than the Shewhart chart and the EWMA chart.

The Synthetic Control Chart

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Synthetic RobustnessSynthetic Robustness

Calzada and Scariano (2001) is a study of the robustness of the Synthetic control chart to non-normality. It is robust to moderate departures from the normal distribution.

There is still much work to be done here.

The MaxEWMA ChartThe MaxEWMA Chart

Introduced in 2001 by Chen et al.

Designed to jointly monitor changes in the mean and variance.

For normal processes this chart shows all the advantages of the EWMA chart.

The ARL for the MaxEWMA chart can be written as

Chen, Scariano,and Calzada are working on optimizing the MaxEWMA chart for users.

Lu,v 1 1

2 h

h h

hLs, tf s 1 u

g t 1 v dsdt.

Robustness of MaxEWMARobustness of MaxEWMA

Calzada and Scariano have shown that while the MaxEWMA chart is promising for normally distributed processes, it is not robust to heavy tails or skewness. This chart must be used with caution.

There is still much work to be done here.

ConclusionConclusionIndustrial applications, like statistical

quality control, can be very rewarding areas of research for people with interests in mathematics.

Jobs in this area include academic, industrial, and consulting jobs.

Strong mathematics, statistics and computations background is important.

ReferencesReferences

Wheeler and Chambers, Understanding Statistical Process Control

Montgomery, Introduction to Statistical Quality Control, Fourth Edition

Borrow et al, “Robustness of the EWMA chart to non-normality”, Journal of Quality Technology, Vol. 31, 1999.

Wu and Trevor, “A synthetic control chart for detecting small shifts in the process mean”, Journal of Quality Technology, Vol. 32, 2000.

Chen et al, Monitoring process mean and variability with one EWMA chart, Journal of Quality Technology, Vol. 33, 2001

Calzada and Scariano, “The robustness of the synthetic control chart to non-normality, Vol. 30, 2001.