advanced tecniques spc

Advanced Statistical Quality Control and Monitoring

Prof. Fugee TsungHong Kong University

of Science and Technology

Quality Lab: http://qlab.ielm.ust.hk

Dr. Fugee Tsung, HKUST 2

Outline

Module A: Overview of SPC; Autocorrelated SPCModule B: Multivariate SPC; Profile MonitoringModule C: Integrating EPC and SPC; Run-to Run ControlModule D: Multistage SPC; Multi-ChartsModule E: Categorical SPC; Nonparametric SPC


Ignore

DesensitizeRobust D

esign

Compensate EPC

Reduce/Eliminate

SPC

How to Handle Variation


Statistical Process Control (SPC): Role in Six Sigma

SPC is emphasized as the major tool used to control and sustain a Six Sigma process

Define

Improve Analyze

MeasureControl


SPC: Basic Idea

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1.5

1.0

0.5

0.0

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-1.0

-1.5

Sample NumberS

ampl

e M

ean

Mean=0

UCL=1.342

LCL=-1.342

? y

In most SPC applications it is assumed that the quality of a process or product can be adequately represented by the distribution of a univariate quality characteristic or by the general multivariate distribution of a vector consisting of several quality characteristics (Woodall et al. 2004).

),(~

),(~222

211

σμ

σμ

Ny

Ny

i

i

.,...,1 ;,...,2,1 nifor

ifor+=

=τ

τ

SPC for Profiles


SPC: A Different Idea

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ean

Mean=0

UCL=1.342

LCL=-1.342

? yx

In many practical situations, however, the quality of a process or product is better characterized and summarized by a relationship between a response variable and one or more explanatory variables (Woodall et al. 2004).

)( ii xfy =


A Profile Example


Another Profile Example

SPC for engineering controlled processes


A Engineering Controlled Process

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ampl

e M

ean

Mean=0

UCL=1.342

LCL=-1.342

? yxProcess

EPC Controller

iDiI

iPi

iD

i

jiIiPi

yBkyB

kykx

ykykykx

)1(1

1

−+−

+=

∇++= ∑=


Example: CMP Run-to-Run Control

Drift in removal rate during processingVariation in incoming thicknessHead-to-head variation in processingUnmeasurable tool characteristics affect product outputIncomplete product measurementsSignificant measurement delay

4

51

2

3

CMPProcess

EPCController

+xt

dt

yt

yt

SPC for Multistage processes


A Multistage Process

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1.5

1.0

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Sample NumberS

ampl

e M

ean

Mean=0

UCL=1.342

LCL=-1.342

? ykyk-1

kkkk

kkkkkk

vyCYwByAy

+=++= −− μ11

Stage 1 Stage NStage kStage k-1

),0(~),,0(~ 22wkvk NwNv σσ


Example: Printed Circuit Board Manufacturing

Exposure

Black Oxide

Lay-up

Hot Press

Cutting

Drilling

Inspection

…….

From Ding, Shi, and Ceglarek (2002)

Module A: Overview of Statistical Process Control (SPC)/ Statistical Quality Control (SQC)

Based on Montgomery, Introduction to Statistical Quality Control, 4th edition, John Wiley, 2001.


Outline

I. What is SPC

II. Common and special causes of quality

variation

III. Statistical basis of control charts

IV. Introduction of popular control charts

I. What is SPC


Statistical process control (SPC)Control charts, plus other problem-solving toolsUseful in monitoring processes, reducing variability through elimination of special causesOn-line technique


The Tools of SPCThe primary tool of SPC is the Control ChartThe Control Chart looks similar to a run chart, however the addition of control limits plays an important role.

II. Common and special causes of quality variation


Purpose for Using Control Charts

The Control Limits probabilistically separate common cause and special cause variability.


Common and Special Causes of Quality Variation

A process that is operating with only common cause/chance causes of variation present is said to be in statistical control.A process that is operating in the presence of special/assignable causes is said to be out of control.The eventual goal of SPC is reduction or elimination of variability in the process by identification of assignable causes.


Common and Special Causes of Quality Variation

III. Statistical basis of control charts


Statistical Basis of the Control Chart

Basic PrinciplesA typical control chart has control limits set at values such that if the process is in control, nearly all points will lie between the upper control limit (UCL) and the lower control limit (LCL).


Statistical Basis of the Control ChartBasic Principles


The Out-of-Control SignalThe control limits define a probabilistic level of occurrence of an extreme reading.



Out-of-Control SituationsIf at least one point plots beyond the control limits, the process is out of controlIf the points behave in a systematic or nonrandom manner, then the process could be out of control.


Hypothesis Testing & Control Charting


Statistical Basis of the Control ChartRelationship between hypothesis testing andcontrol charts

We have a process that we assume the true process mean is μ = 74 and the process standard deviation is σ = 0.01. Samples of size 5 are taken giving a standard deviation of the sample average, , as x

0045.0501.0

nx ==σ

=σ


Statistical Basis of the Control ChartRelationship between hypothesis testing and control charts

Control limits can be set at 3 standard deviations from the mean.This results in “3-Sigma Control Limits”UCL = 74 + 3(0.0045) = 74.0135

CL= 74LCL = 74 - 3(0.0045) = 73.9865


Statistical Basis of the Control ChartRelationship between hypothesis testing and control charts

Choosing the control limits is equivalent to setting up the critical region for testing hypothesisH0: μ = 75H1: μ ≠ 75


Statistical Basis of the Control ChartRelationship between the process and the control chart


Statistical Basis of the Control ChartImportant uses of the control chart

Most processes do not operate in a state of statistical control.Consequently, the routine and attentive use of control charts will identify assignable causes. If these causes can be eliminated from the process, variability will be reduced and the process will be improved.The control chart only detects assignable causes. Management, operator, and engineering action will be necessary to eliminate the assignable causes.Out-of-control action plans (OCAPs) are an important aspect of successful control chart usage.



Types the control chartVariables Control Charts

These charts are applied to data that follow a continuous distribution (measurement data).

Attributes Control ChartsThese charts are applied to data that follow a discrete distribution.


Statistical Basis of the Control ChartType of Process Variability

Stationary behavior, uncorrelated dataStationary behavior, autocorrelated dataNonstationary behavior



Type of VariabilityMost conventional control charts are effective when the in-control process data is stationary and uncorrelated.


Statistical Basis of the Control ChartPopularity of control charts

1) Control charts are a proven technique for improving productivity.

2) Control charts are effective in defect prevention.3) Control charts prevent unnecessary process

adjustment.4) Control charts provide diagnostic information.5) Control charts provide information about process

capability.


Choice of Control LimitsGeneral model of a control chart

where L = distance of the control limit from the center line

= mean of the sample statistic, w.= standard deviation of the statistic, w.

WW

W

WW

LLCLLineCenter

LUCL

σ−μ=μ=σ+μ=

Wμ

Wσ


Choice of Control Limits“99.7% of the Data”

If approximately 99.7% of the data lies within 3σof the mean (i.e., 99.7% of the data should lie within the control limits), then 1 - 0.997 = 0.003 or 0.3% of the data can fall outside 3σ (or 0.3% of the data lies outside the control limits). (Actually, we should use the more exact value 0.0027)0.0027 is the probability of a Type I error or a false alarm in this situation.


Choice of Control Limits

Three-Sigma Limits

The use of 3-sigma limits generally gives good results in practice. If the distribution of the quality characteristic is reasonably well approximated by the normal distribution, then the use of 3-sigma limits is applicable.These limits are often referred to as action limits.


Sample Size and Sampling Frequency

In designing a control chart, both the sample size to be selected and the frequency of selection must be specified.Larger samples make it easier to detect small shifts in the process.Current practice tends to favor smaller, more frequent samples.



Average Run LengthThe average run length (ARL) is a very important way of determining the appropriate sample size and sampling frequency.Let p = probability that any point exceeds the control limits. Then,

p1ARL =



IllustrationConsider a problem with control limits set at 3standard deviations from the mean. The probability that a point plots beyond the control limits is again, 0.0027 (i.e., p = 0.0027). Then the average run length is

3700027.01ARL ==



What does the ARL tell us?The average run length gives us the length of time (or number of samples) that should plot in control before a point plots outside the control limits.For our problem, even if the process remains in control, an out-of-control signal will be generated every 370 samples, on average.



Average Time to SignalSometimes it is more appropriate to express the performance of the control chart in terms of the average time to signal (ATS). Say that samples are taken at fixed intervals, h hours apart.

)h(ARLATS =


Rational Subgroups

Subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized, while the chance for differences due to these assignable causes within a subgroup will be minimized.


Analysis of Patterns on Control ChartsNonrandom patterns can indicate out-of-control

conditionsPatterns such as cycles, trends, are often of considerable diagnostic value.Look for “runs” - this is a sequence of observations of the same type (all above the center line, or all below the center line)Runs of say 8 observations or more could indicate an out-of-control situation.

Run up: a series of observations are increasingRun down: a series of observations are decreasing


Analysis of Patterns on Control ChartsWestern Electric Handbook Rules (Should be used carefully because of the increased risk of false alarms)

A process is considered out of control if any of the following occur:1) One point plots outside the 3-sigma control limits.2) Two out of three consecutive points plot beyond the 2-

sigma warning limits.3) Four out of five consecutive points plot at a distance of 1-

sigma or beyond from the center line.4) Eight consecutive points plot on one side of the center line.

IV. Introduction of popular control charts

1. The Shewhart chart (X-bar & R chart)


Notation for variables control chartsn - size of the sample (sometimes called a subgroup) chosen at a point in timem - number of samples selected

= average of the observations in the ith sample (where i = 1, 2, ..., m)

= grand average or “average of the averages (this value is used as the center line of the control chart)

ix

x


Notation and valuesRi = range of the values in the ith sample

Ri = xmax - xmin

= average range for all m samplesμ is the true process meanσ is the true process standard deviation

R


Statistical Basis of the ChartsAssume the quality characteristic of interest is normally distributed with mean μ, and standard deviation, σ.If x1, x2, …, xn is a sample of size n, then the average of this sample is

is normally distributed with mean, μ, and standard deviation,

nxxxx n21 +++

=L

x

n/x σ=σ


Statistical Basis of the ChartsThe probability is 1 - α that any sample mean will fall between

The above can be used as upper and lower control limits on a control chart for sample means, if the process parameters are known.

nZZ

andn

ZZ

2/x2/

2/x2/

σ−μ=σ−μ

σ+μ=σ+μ

αα

αα


The Operating Characteristic Function

How well the and R charts can detect process shifts is described by operating characteristic (OC) curves.Consider a process whose mean has shifted from an in-control value by k standard deviations. If the next sample after the shift plots in-control, then you will not detect the shift in the mean. The probability of this occurring is called the α-risk.

x


The Operating Characteristic FunctionThe probability of not detecting a shift in the process mean on the first sample is

L= multiple of standard error in the control limitsk = shift in process mean (#of standard deviations).

)nkL()nkL( −−Φ−−Φ=β

2. The Cumulative Sum (CUSUM) Charts


Basic Principles: The Cusum Control Chart for Monitoring the Process Mean

The cusum chart incorporates all information in the sequence of sample values by plotting the cumulative sums of the deviations of the sample values from a target value.If μ0 is the target for the process mean, is the average of the jth sample, then the cumulative sum control chart is formed by plotting the quantity

∑ μ−==

i

1j0ji )x(C

jx


The Tabular or Algorithmic Cusum for Monitoring the Process Mean

Let xi be the ith observation on the processIf the process is in control thenAssume σ is known or can be estimated.Accumulate derivations from the target μ0 above the target with one statistic, C+

Accumulate derivations from the target μ0 below the target with another statistic, C-

C+ and C- are one-sided upper and lower cusums, respectively.

),(N~x 0i σμ


The statistics are computed as follows:The Tabular Cusum

starting values areK is the reference value (or allowance or slack value)If either statistic exceed a decision interval H, the process is considered to be out of control. Often taken as a H = 5σ

[ ][ ]−

−−

+−

+

+−−μ=

++μ−=

1ii0i

1i0ii

Cx)k(,0maxC

C)k(x,0maxC

0CC 00 == −+


Selecting the reference value, KK is often chosen halfway between the target μ0 and the out-of-control value of the mean μ1 that we are interested in detecting quickly.Shift is expressed in standard deviation units as μ1= μ0+δσ, then K is

22K 01 μ−μ

=σδ

=

3. The Exponentially Weighted Moving Average (EWMA) Charts


3. The EWMA chart


Other ChartsCuscore Charts (Box and Ramirez, 1992) Reference-Free Cuscore Charts (Han and Tsung, JASA, 2006) General EWMA Charts (Han and Tsung, Annals of Stat, 2004)GLRT Charts


Related referencesHan, D., and Tsung, F.(2004), " A Generalized EWMA Control Chart and its

Comparison with the Optimal EWMA, CUSUM and GLR Schemes," The Annals of

Statistics, 32.

Han, D. and Tsung, F. (2005), " Comparison of the Cuscore, GLRT and CUSUM

Control Charts for Detecting a Dynamic Mean Change," The Annals of the Institute of

Statistical Mathematics, 57.

Han, D. and Tsung, F.(2006) " A Reference-Free Cuscore Chart for Dynamic Mean

Change Detection and a Unified Framework for Charting Performance Comparison ,"

Journal of the American Statistical Association, 101, 368-386.

Ou, Y. J., Wu, Z. and Tsung, F. (2012), " Comparison Study of Effectiveness and

Robustness of Control Charts for Monitoring Process Mean," International Journal of

Production Economics, 135, 479-490.

advanced tecniques spc

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