lect 4 spc

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Stat is t ic a l Process Cont ro l

Statistical Process Control (SP C) can be defined as: a collection of methods for controlling the quality of a product by

collecting and interpreting data to determine the capability and currentperformance of a process.

SPC methods make a distinction between what is called commoncause variat ion and special cause variation.

Common cause variation (sometimes called inherentvariation) is always present. It normally arises from severalsources, each of which usually makes a relatively smallcontribution. Common cause is typically quantified using measuressuch as the sample standard deviation s or the range R.

Reduction of common cause variation requires fundamental

changes in an operation, requiring management authorization (i.e.fine tuning will have little effect).

SPC - Termino logy

Processes exhibiting only common cause variation are said to be instatistical control, even if they may not be meetingspecifications. Such processes are stable, and hence predictable(within appropriate limits identified by confidence intervals). Themagnitude of common cause variation determines the systemcapability.

In contrast, special cause variation is sporadic, often upsetting aprocess when it occurs. Special cause variation can bedistinguished from common cause variation by the size or patternof change that occurs in process behaviour. Detection of special

cause variation is often subjective, with guidance from objectivetechniques. Special cause variation is abnormal variation, and itmay be harmful or beneficial.

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SPC - Process Im provement

Detecting and acting upon special cause variation is a responsibilityof everyone in an organization, from operators to management.Systems exhibiting special cause variation which is not acted uponare not in statistical control. System capability has no meaningfor such systems (i.e. it is important to ensure that a process isstable before evaluating its capability).

Process Improvement comes about through theidentification of special cause variation and then itsdeliberate elimination or persistence.

Is the process

operating

normally?

SPC (Shewhart) Chart

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1 3 5 7 911

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ontrolled

Variable


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ControlledV

ariable

Normal Operating Region

Atypical Operating Region

Atypical Operating Region

Features of Cont ro l Chart sSPC (Shewhart) Chart

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1 3 5 7 911

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Upper Confidence Limit

Lower Confidence Limit

Target

Variable Being

Monitored


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ontrolled

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Suggests something

has changed .Acts as a warning

for production

personnel

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Shew har t Char t on Ind iv idual Measurements

Plot each new individual measurement x i on a chart with center lineequal to the target or long term mean and control limits at 3x Example: In manufacturing a polymer the viscosity (xi) of a sample of

polymer collected from the process is measured in the quality controllab every hour.

How to determine x? x is the variance of the common cause disturbances in the process

It should be estimated using data from the process when it is deemed to beoperating in an acceptable manner with only common cause variationpresent.

Why put limits at 3x ? Corresponds to = .0027 (99.73% confidence interval)

Sta t i s t i c a l Process Con t ro l Shewhar t Charts

Shewhart at Bell Labs in 1920s

Suggested plotting quality data on control charts

3 sigma control limits based on common cause variability (ie.when process is working normally)

Any violation is a special event to be investigated

By detecting & correcting process long term improvement

o00

0 0

0 00

0

0 0

00

00

x

3

orX=

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Shewhar t Charts on Averages

Charts

Often parts are made very rapidly e.g. injection molded plasticparts

it is not possible to measure the quality (e.g. a dimension) on everypart

Take a random sample of n parts from every hours production

Plot Shewhart Chart on the average with limits 3sx/n centeredabout the grand average

sx is estimated from the pooled variation of the replicates at each hour

Plot a chart on the standard deviation sx estimated from the replicates

Provides a means to monitor both changes in the process mean andalso changes in the variance chart

X

X

xsX

X

Char ts t o Moni tor Process Var iab i l i ty

Var y=0.47 Var y=2.2382

An Exam ple - Mould Level Cont ro l a t #1 Caster

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S Char ts - Cont ro l L imi ts

The control limits can be obtained using the 100 (1-) %confidence intervals for x2

Even though s is not normally distributed, both the lower and upper

control limits are sometimes set as

3 (standard deviation of s ) = sc

c13s

4

24

2

2/1,

2

2

2/,

2

,

ss

where c4 is chosen to correspond to an approx. 100 (1-) % limit (tables)

s2 can be calculated over time windows or on various rational

subgoups (e.g. random sample of parts produced every hour

on each machine).

Range Charts

Often the Range R = (xmax-xmin) from each sample of n parts is usedinstead of sx to monitor the process variation

This is due to computers not being available years ago and hascontinued because of tradition

Tables are available for setting the upper and lower control limits for Rcharts

charts are common combinations in parts manufacturingRX _

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X-bar Chart(Shew har t Char t us ing means of samples)

To build a control chart is to determine values for the three lineson the chart

Centre line

Upper control limit

Lower control limit

The centre line value for an X-bar chart may be

the target value for the performance characteristic of interest

or

the overall sample mean of values from recent samples

of the measured characteristic, where

X

samplesofnumber

sampleslalforvaluesXofTotal

X =

X-bar Char ts - Cont ro l L imi t s

Upper and lower control limits for an X-bar chart are determinedfrom the pdf of the individual sample means , which is N(,2/n),

where denotes the population standard deviation of

individual X measurements undercommon cause variation

and n denotes the size of each sample

Essentially, the limits represent 100(1-)% confidence interval forthe mean. The UCL and LCL are determined from the following.

nzX

zX

x2/

x2/

=

n

xx

=

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X-bar Char ts - Cont ro l L imi t s

Of course, we generally dont know the true value of the varianceand we have to use an estimate in its place. In this case, theconfidence limits (and consequently, the control limits) aredetermined using the t distribution instead of the normaldistribution.

We can estimate the variance of the mean values as follows.

DOF==k(n-1) Then we compute the control limits as:

samplesofNumber

samplesindividualallforvaluessofTotalss X ==

n

stX

stX

x2/.

x2/.

=

Western Elect r ic Rules

A set of supplementary guidelines to improve the sensitivity of X-barcharts. The following is an incomplete set of the rules:

2 out of 3 consecutive values of X-bar on the same side of the centre line andmore than 2 standard deviations from the centre line

4 out of 5 consecutive values of X-bar on the same side of the centre line andmore than 1 standard deviation from the centre line

8 consecutive points are the same side of the centre line

7 or more consecutive values is a consistently rising or falling pattern

a recurring cyclic pattern

abnormal clustering close to the centre line (signals a decrease in variation inthe process)

clustering of values close to both control limits (suggests X-bar is following twodistributions instead of one).

Note: These rules need to be applied with caution they are really onlyreasonable if the observations are uncorrelated over time (this usually isnot the case in the process industries - more common in parts manufacture

Control limits need to be reviewed periodically!

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CUSUM Chart

Cumulative Sum (CUSUM) Chart

A Cumulative Sum Chart, developed by E. S. Page, monitors Si, thecumulative sum (cusum) of departures of sample mean values of

measurements x, up to and including sample i , from their target value.

=

=i

1j

ji )etargTX(S

This definition of Si includes the case of samples of size 1. Note that each

value Si includes all of the data collected up to that point, in contrast to the

previous control charts in which only the current sample value is used as a

basis for decision.

A change in mean of the process will show up as a change in slope of the CUSUM

chart (i.e. Si continues to integrate (sum) the change as long as the change persists)

Time Ser ies Plot of M i leage

Mileage

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48

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Day

Mileage

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CUSUM Plot

Mileage

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-1

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CUSUM

Summary

SPC charts are used to monitor a process and detect when specialevents arise.

Chart to monitor the mean value of quality variables

Shewhart Chart on individual observations or on the mean value over someperiod of operation

CUSUM chart (Other more specialized charts also exist)

Chart to monitor process variability s or R chart

The idea is to have operators/engineers follow through on alarms todetermine an assignable cause, and then try to modify procedures,etc. so that these do not occur again.

This will lead to continuous improvement over time Spectacular successes Japanese auto and electronics industries

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First five defects all related to soldering

Montgomery D.C & Runger G.C. (2007), p.693, fig 16-24

Paret o Charts (order ing faul ts by f requenc y)

Useful to summarize process

Shows potential causes of defects in products and interrelationships


Fishbone (Cause & Effect) Charts

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Process Capabi l i t y

In SPC, we specify control limits based on the performance of theprocess. In Process capability, the Specification Limits areestablished by the customer

Process Capability Indexes

We want Cpk to be greater than 1. Most industies want Cpk to begreater than 1.33 (basis for choosing supplier, process is said to be

capable). A value of 2.0 corresponds to 6

Another variation for when the limits are symmetrical about the meanis

= 3

LSLX,

3

XUSLMinCpk

6

LSLUSLCp

=

Process natural tolerance limits lie inside specifications

Very few defective units produced

Montgomery D.C & Runger G.C. (2007), p.671, fig 16-14a

Proce ss Capabi l i t y Rat i o (PCR): PCR>1Proce ss Capabi l i t y Rat i o (PCR): PCR>1

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Small number of nonconforming units

0.27% for normal process

Montgomery D.C & Runger G.C. (2007), p.671, fig 16-14b

Proce ss Capabi l i t y Rat i o (PCR): PCR=1Proce ss Capabi l i t y Rat i o (PCR): PCR=1

Process very yield sensitive

Large number of nonconforming units produced

Montgomery D.C & Runger G.C. (2007), p.671, fig 16-14c

Proce ss Capabi l i t y Rat i o (PCR): PCR

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Six-Sigma Stat is t ic a l concept

Idea in parts manufacturing is to aim at achieving 6-sigmacapability

I.e. Achieve a process with Cpk= 2.0

In that way if the process were at target there would be 0.0018PPM defective.

But process is rarely exactly at target. Allowing for a deviation inthe process mean of 1.5 from target, then still have 4.5 margin.This corresponds to 3.4 PPM defective (a goal).

Mean of six-sigma process shifts by 1.5 standard deviations

Still 4.5 from the Spec limit 3.4 defects per million parts


Si xSi x --S igma ProcessSigma Process

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Six Sigma Qual i t y Improvem ent Programs

The Six Sigma term has been adopted as a title for currentmanagement quality improvement programs

Allied Chemical (Honeywell) and Motorola started it, GE popularized it.

Original concept is good

Project driven

Define and carry out quality improvement projects

Use any data analysis tools appropriate (training in basic methods)

Successful completion: Green belt Black belt

As with all corporate management programs, it has become jargonfilled and the approaches more rigorously defined:

Define: Objectives & critical to quality (CTQ) variables

Measure: Measurements of CTQ variables, DOEs in CTQs

Analyze: Analysis of the data

Improve: Optimize/improve the process using knowledge gained

Control: Maintain gains through control charts, etc.

lect 4 spc

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