PROCESS CONTROL

1
S
TATISTICAL
PROCESS CONTROL
CUSTOMER COMPETITIVE INTELLIGENCE FOR PRODUCT,
PROCESS, SYSTEMS ENTERPRISE EXCELLENCE
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT
2
Quality Management Statistical Process
Control
3
Statistical Process Control

• Statistical Process Control (SPC) can be thought
  of as the application of statistical methods for
  the purposes of quality control and improvement.
• Quality Improvement is perhaps foremost among
  all areas in business for application of
  statistical methods.

4
Data Driven Decision Making

• In God we trust. ... all others must bring
  data. --- The Statisticians Creed
• SPC is one method that assists in enabling
  data-driven decision making.
• SPC is a key quantitative aid to quality
  improvement efforts.

5
Control ChartsRecognizing Sources of Variation

• Why Use a Control Chart?
• To monitor, control, and improve process
  performance over time by studying variation and
  its source.
• What Does a Control Chart Do?
• Focuses attention on detecting and monitoring
  process variation over time
• Distinguishes special from common causes of
  variation, as a guide to local or management
  action
• Serves as a tool for ongoing control of a
  process
• Helps improve a process to perform consistently
  and predictably for higher quality, lower cost,
  and higher effective capacity
• Provides a common language for discussing process
  performance.

6
Control ChartsRecognizing Sources of Variation

• How Do I Use Control Charts?
• There are many types of control charts. The
  control charts that you or your team decides to
  use should be determined by the type of data that
  you have.
• Use the following tree diagram to determine which
  chart will best fit your situation. Only the most
  common types of charts are addressed.

7
Control Chart Selection Variable DataMeasured
Plotted on a Continuous Scale such as Time,
Temperature, Cost, Figures.
n is small 3
n is large n 10
2
n 1
X Rm
X R
X R
X S
8
Control Chart Selection Attribute Data Counted
or Plotted as Discrete Events Such as Shipping
Errors, Waste or Absenteeism.
Defect or Nonconformity Data
Defective Data
Constant Variable
Constant Variable sample size
sample size n 50
n 50
c chart u chart p or
np chart p chart
9
Control Chart Construction

• Select the process to be charted
• Determine sampling method and plan
• How large a sample needs to be selected? Balance
  the time and cost to collect a sample with the
  amount of information you will gather.
• As much as possible, obtain the samples under the
  same technical conditions the same machine,
  operator, lot, and so on.
• Frequency of sampling will depend on whether you
  are able to discern patterns in the data.
  Consider hourly, daily, shifts, monthly,
  annually, lots, and so on. Once the process is
  in control, you might consider reducing the
  frequency with which you sample.
• Generally, collect 20-25 groups of samples before
  calculating the statistics and control limits.
• Consider using historical data to establish a
  performance baseline.

10
Control Chart Construction

• Initiate data collection
• Run the process untouched, and gather sampled
  data.
• Record data on an appropriate Control Chart sheet
  or other graph paper. Include any unusual events
  that occur.
• Calculate the appropriate statistics and control
  limits
• Use the appropriate formulas.
• Construct the control chart(s) and plot the data.

11
Control Chart Interpretation Time, Production
Spatial Analysis Still-Life Photography

• An event taken in isolation or a group of items
  each selected from a process during the same
  (brief) time span can generally provide
  information about process performance ONLY during
  that brief span.
• Unless process performance is static through time
  this will be true.
• Dynamic processes vary through time.

12
Control Chart Interpretation Time, Production
Spatial Analysis The Video Generation

• If a process varies through time, it is often
  useful to know how the process varies so that it
  can be controlled or guided in its behavior.
• This requires monitoring through time, similar to
  videotaping the process - in some sense, the
  process has a life of its own and we want to
  nurture that life.

13
Control Chart Interpretation Persistence
Through Time

• A process can be characterized by
• Examining its behavior during a sufficiently
  brief interlude of time
• Examining its behavior across a greater expanse
  of time.
• Stable process one which performs with a high
  degree of consistency at an essentially constant
  level for an extended period of time
• In-control
• A process that is not stable is referred to as
  being in an out-of-control state

14
Data Plot with PAT Zones
36.10 (A)
36 34 32 30 28 26
34.36 (B)
32.62 (C)
30.88
29.14 (C)
27.40 (B)
25.66 (A)
Item
15
Control Chart InterpretationPattern Analysis
Tests (PATs)

• PAT 1 One point plots beyond zone A on either
  side of the mean
• PAT 2 Nine points in a row plot on the same
  side of the mean
• PAT 3 Six consecutive points are strictly
  increasing or strictly decreasing
• PAT 4 Fourteen consecutive points which
  alternate up and down

16
Control Chart InterpretationPattern Analysis
Tests

• PAT 5 Two out of three consecutive points plot
  in zone A or beyond, and all three points plot on
  the same side of the mean
• PAT 6 Four out of five consecutive points plot
  in zone B or beyond, and all five points plot on
  the same side of the mean

17
Control Chart InterpretationPattern Analysis
Tests

• PAT 7 Fifteen consecutive points plot in zones
  C, spanning both sides of the mean
• PAT 8 Eight consecutive points plot at more
  than one standard deviation away from the mean
  with some smaller than the mean and some larger
  than the mean

18
Control Chart InterpretationMonitoring
Improving Processes

• The performance of every process will be composed
  of two primary components
• Controlled or guided performance which is
  predictable in both an instantaneous and
  long-term sense
• Uncontrolled variation
• Special or assignable causes
• Common causes

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Control Chart InterpretationMonitoring
Improving Processes

• True process improvement is typically a result of
  either
• Breakthrough thinking
• Efforts to identify and reduce or eliminate
  common causes of variation methodical
  quantitatively oriented tools which monitor a
  process over time --- the approach taken
  generally by control charts.

20
Control Chart Interpretation

• The vertical axis coordinate of a point plotted
  on the chart corresponding to the value of an
  appropriate PPM and the horizontal axis
  coordinate of a point plotted on the chart
  corresponding to the time in sequence at which
  the observation was made with the time between
  observations divided into equal increments.

21
Control Charts Colors Used
UCL
A B C C B A

U2SWL


U1SL



CL







L1SL


L2SWL
LCL
22
P Charts for the Process Proportion
Based on m preliminary samples from the process.
While the number of items, n, may vary from
sample to sample, it is customary for each of the
samples in a given application to include the
same number of items, n. For the ith of these m
samples, let Then the proportion defective for
the ith sample is
Yi number of defective units in the sample
pi Yi / ni
23
Control Chart Interpretation

• Center line (CL) positioned at the estimated
  mean
• Upper and lower one standard deviation lines
  (U1SL and L1SL) positioned one standard
  deviation above and below the mean.
• Upper and lower two standard deviation warning
  lines (U2SWL and L2SWL) positioned at two
  standard deviations above and below the mean.
• Upper and lower control lines (UCL and LCL)
  positioned at three standard deviations above and
  below the mean.

24
P Charts for the Proportion
An estimate of the overall process proportion
defective is
p (Y1Y2... Ym) / (n1n2... nm)
(total defectives) / (total items) When
all samples have n items each then p (p1 p2
... pm)/m The estimated standard deviation of
the process proportion defective is
Sp v p (1-p)/ ni
25
P Chart Control Lines Limits
The coordinates for the seven lines on the P
chart are positioned at CL p U1SL p
Sp L1SL p - Sp U2SWL p 2Sp L2SWL p -
2Sp UCL p 3Sp LCL p - 3Sp
26
South of the Borders, Inc.
Custom Wallpapers Borders
Free Estimates [phone]
27
South of the Borders, Inc.
South of the Borders, Inc. is a custom wallpapers
and borders manufacturer. While their products
vary in visual design, the manufacturing process
for each of the products is similar. Each day a
sample of 100 rolls of wallpaper border is
sampled and the number of defective rolls in the
sample is noted. The number of defective rolls
in samples from 25 consecutive production days
follows. Determine all coordinates construct
interpret the p chart. PATs 1, 2, 3 and 4
apply to p charts.
28
South of the Borders, Inc.
Day
Defective Rolls
Day Defective Rolls
1 2 3 4 5 6 7 8 9 10 11 12 13
13 4 7 11 8 10 2 9 12 6 4 7 9
14 15 16 17 18 19 20 21 22 23 24 25
8 9 3 5 14 10 11 6 6 9 3 10
29
South of the Borders, Inc.
Total of items sampled 2500 Total of
defective items 196 p 196/2500 .0784 Sp
v .0784(.9216)/100 .02688
30
South of the Borders, Inc.
CL .0784 UCL .0784 3(.0269) .1590 LCL
.0784 - .0806 -.0022 (na) U2SWL .0784
2(.0269) .1322 L2SWL .0784 - .0538
.0246 U1SL .0784 .0269 .1053 L1SL
.0784 - .0269 .0515
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(No Transcript)
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South of the Borders, Inc. P Chart Interpretation

• No violations of PATs one through four are
  apparent. This implies that the process is in a
  state of statistical control.
• It does not indicate that we are satisfied with
  the performance of the process.
• It does, however, indicate that the process is
  stable enough in its performance that we may
  seriously engage in PDCA for the purpose of
  long-term process improvement.

33
C and U Charts for Nonconformities

• When data originates from a Poisson process, it
  is customary to monitor output from the process
  with a defects or C chart
• Recall the Poisson Distribution with mean c
  and standard deviation ?c
• P(y) cye-c/y!

34
C U Charts for Nonconformities

• C represents the average number of defects
  (nonconformities) per measured unit with all
  units assumed to be of the same size and all
  samples are assumed to have the same number of
  units
• m 20 to 40 initial samples
• C (number of defects in the m samples) / m
• Estimated standard deviation v C

35
C Control Chart Coordinates

• CL C
• UCL C3 C and LCL C-3 C
• U2SWL C2 C and L2SWL C- 2 C
• U1SL C C and L1SL C- C

36
Scientific Technical Materials, Inc.
37
Scientific Technical Materials, Inc.

• Scientific Technical Materials, Inc. produces
  material for use as gaskets in scientific,
  medical, and engineering equipment. Scarred
  material can adversely affect the ability of the
  material to fulfill its intended use.
• A sample of 40 pieces of material, taken at a
  rate of 1 per each 25 pieces of material produced
  gave the results on the following slide. Use
  this information to construct and interpret a C
  chart.

38
Scientific Technical Materials, Inc.
Piece Scars Piece Scars Piece Scars Piece Scars
1 4 11 1 21 2 31 2
2 4 12 1 22 1 32 1
3 2 13 2 23 0 33 1
4 3 14 3 24 3 34 3
5 1 15 0 25 5 35 2
6 2 16 4 26 4 36 0
7 0 17 3 27 2 37 1
8 2 18 2 28 1 38 5
9 3 19 2 29 4 39 9
10 1 20 1 30 2 40 1
39
Scientific Technical Materials, Inc.

• C 90/40 2.25 CL, Sc 2.25 1.5
• UCL 2.25 3(1.5) 6.75
• LCL 2.25- 4.5 -2.25 (NA)
• U2SWL 2.25 2(1.5) 5.25
• L2SWL 2.25- 3 -0.75 (NA)
• U1SL 2.25 1.5 3.75
• L1SL 2.25- 1.5 0.75

40
Scientific Technical Materials, Inc. C Chart
for Gasket Material Data
UCL
U2SWL
U1SL
CL
L1SL
41
Scientific Technical Materials, Inc. C
Chart Interpretation

• Application of PATs one through four indicates a
  violation of PAT 1 at sample number 39 where 9
  scars appear on the surface of the sampled
  material.
• Corrective measures would be identified and
  implemented.
• After process stability was (re) assured, we
  would move into PDCA mode.

42
Variation of the C chart where Sample size may
vary

• CL U
• UCL U 3 U/ni, LCL U-3 U/ni
• U2SWL U 2 U/ni, L2SWL U- 2 U/ni
• U1Sl U U/ni, L1SL U- U/ni

U Chart
43
Control Charts for theProcess Mean and
Dispersion

• X bar Chart
• Typically used to monitor process centrality (or
  location)
• Limits depend on the measure is used to monitor
  process dispersion
• (R or S may be used).
• S or Standard Deviation Chart
• Used to monitor process dispersion
• R or Range Chart
• Also used to monitor process dispersion

44
Sample Summary Information

• m 20 to 40 initial samples of n observations
  each.
• Xi mean of ith sample
• Si standard deviation of ith sample
• Ri range of ith sample

X (X1 X2 ... Xm) / m
R (R1 R2 ... Rm)/m S (S1 S2 ...
Sm)/m ? R/d2 where d2 depends only on n
45
Coordinates for the X-bar Control Chart R

• CL X,
• UCL X A2R,
• UCL X- A2R
• U2SWL X 2A2R/3
• L2SWL X- 2A2R/3
• U1SL X A2R/3
• L1SL X- A2R/3

A2 is a constant that depends only on n.
46
Coordinates for anR Control Chart

• CL R
• UCL D4R
• LCL D3R
• U2SWL R 2(D4-1)R/3
• L2SWL R- 2(D4-1)R/3
• U1SL R (D4-1)R/3
• L1SL R- (D4-1)R/3
• where D3 and D4 depend only on n

47
Championship Card Company
Championship
48
Championship Card Company
Championship Card Company (CCC) produces
collectible sports cards of college and
professional athletes. CCCs card-front design
uses a picture of the athlete, bordered all-the-wa
y-around with one-eighth inch gold foil.
However, the process used to center an athletes
picture does not function perfectly. Five cards
are randomly selected from each 1000 cards
produced and measured to determine the degree of
off-centeredness of each cards picture. The
measurement taken represents percentage of total
margin (.25) that is on the left edge of a card.
Data from 30 consecutive samples is included
with your materials, and summarized on the
following slides.
49
Championship Card Company
Sample X-bar R Sample X-bar R
Sample X-bar R 1 55.6 22
11 51.2 15 21
50.0 11 2 61.0 23
12 49.4 14 22 47.0
14 3 45.2 20 13
44.0 32 23 50.6 15
4 46.2 11 14 51.6
14 24 48.8 16 5
46.8 18 15 53.2 12
25 44.6 22 6 49.8
23 16 52.4 23 26
46.8 16 7 46.8 18
17 50.6 8 27
49.2 8 8 44.2 20
18 56.0 18 28 45.6
19 9 50.8 32 19
50.2 19 29 57.6 40
10 48.4 16 20 44.0
23 30 51.4 17
50
Championship Card CompanySummary Information
51
Championship Card Company X-bar and R Control
Chart Limits
R
UCL U2SWL U1SL CL L1SL L2SWL LCL
60.38 56.80 53.22 49.63 46.05 42.47 38.89
39.40 32.48 25.55 18.63 11.71 4.79 ------
52
Championship Card Company
Limits Based on R
53
Championship Card Company
54
Championship Card CompanyX-bar R Chart
Interpretation

• Application of all eight PATs to the X-bar chart
  indicated a violation of PAT 1 (one point
  plotting above the UCL) at sample 2. Apparently,
  a successful process adjustment was made, as
  suggested by examination of the remainder of the
  chart.
• Application of PATs one through four to the R
  chart indicated a violation of PAT 1 at sample
  29. Measures would be investigated to reduce
  process variation at that point. The violation
  was a close call and was out of character with
  the remainder of the data.
• We are close to being able to apply PDCA to the
  process for the purpose of achieving lasting
  process improvements.

55
Coordinates for the X bar Control Chart S

• CL X
• UCL X A3S
• LCL X- A3S
• U2SWL X 2A3S/3
• L2SWL X- 2A3S/3
• U1SL X A3S/3
• L1SL X- A3S/3
• where A3 depends only on n

56
Coordinates on an S Control Chart

• CL S
• UCL B4S
• LCL B3S
• U2SWL S 2(B4-1)S/3
• L2SWL S- 2(B4-1)S/3
• U1SL S (B4-1)S/3
• L1SL S- (B4-1)S/3
• where B3 and B4 depend only on n

57
Championship Card Company
Sample X-bar S Sample X-bar S
Sample X-bar S 1 55.6 9.63
11 51.2 6.83 21
50.0 5.15 2 61.0 8.63
12 49.4 5.46 22 47.0
5.15 3 45.2 7.40 13
44.0 14.35 23 50.6 5.55
4 46.2 4.09 14 51.6
5.18 24 48.8 6.50 5
46.8 7.22 15 53.2 5.36
25 44.6 8.96 6 49.8
8.76 16 52.4 9.48 26
46.8 6.50 7 46.8 6.72
17 50.6 3.44 27 49.2
3.19 8 44.2 8.53 18
56.0 7.00 28 45.6 7.96
9 50.8 11.95 19 50.2
7.60 29 57.6 14.38 10
48.4 6.19 20 44.0 8.46
30 51.4 6.80
58
Championship Card Company X-bar and S Chart
Limits
S
UCL U2SWL U1SL CL L1SL L2SWL LCL
60.22 56.69 53.16 49.63 46.11 42.58 39.05
15.49 12.80 10.11 7.42 4.72 2.03 ------
59
Championship Card Company
Limits Based on S
60
Championship Card Company
61
Championship Card CompanyX-bar S Chart
Interpretation

• Application of all eight PATs to the X-bar chart
  indicates a violation of PAT 1 (one pt. above the
  UCL) at sample 2. Judging from the remainder of
  the chart, the process was successfully adjusted.
• Application of the first four PATs to the S chart
  indicates no violations.
• In summary, the process appears to have been
  temporarily out-of-control w.r.t. its mean at
  sample 2. The process was successfully adjusted
  and may now be subjected to PDCA for permanent
  improvement purposes.

62
Common Questions for Investigating
anOut-of-Control Process

• Are there differences in the measurement accuracy
  of instruments / methods used?
• Are there differences in the methods used by
  different personnel?
• Is the process affected by the environment, e.g.
  temperature/humidity?
• Has there been a significant change in the
  environment?
• Is the process affected by predictable conditons
  such as tool wear?
• Were any untrained personnel involved in the
  process at the time?
• Has there been a change in the source for input
  to the process such as a new supplier or
  information?
• Is the process affected by employee fatigue?

63
Common Questions for Investigating an
Out-of-Control Process

• Has there been a change in policies or procedures
  such as maintenance procedures?
• Is the process frequently adjusted?
• Did the samples come from different parts of the
  process? Shifts? Individuals?
• Are employees afraid to report bad news?

64
Process Capability The Control Chart Method for
Variables Data

• Construct the control chart and remove all
  special causes.
• NOTE special causes are special only in that
  they come and go, not because their impact is
  either good or bad.
• Estimate the standard deviation. The approach
  used depends on
• whether a R or S chart is used to monitor
  process variability.
• _ _
• ? R / d2 ? S / c4
• Several capability indices are provided on the
  following slide.

65
Process Capability Indices Variables Data
CP
(engineering tolerance)/6? (USL LSL) / 6?
This index is generally used to evaluate
machine capability. tolerance to the
engineering requirements. Assuming that the
process is (approximately) normally distributed
and that the process average is centered between
the specifications, an index value of 1 is
considered to represent a minimally capable
process. HOWEVER allowing for a drift, a
minimum value of 1.33 is ordinarily sought
bigger is better. A true Six Sigma process
that allows for a 1.5 ? shift will have Cp 2.
66
Process Capability Indices Variables Data
CR 1006? /
(Engineering Tolerance) 100 6? /(USL LSL)
This is called the capability ration.
Effectively this is the reciprocal of Cp so that
a value of less than 75 is generally needed and
a Six Sigma process (with a 1.5? shift) will lead
to a CR of 50.
67
Process Capability Indices Variables Data
CM
(engineering tolerance)/8? (USL LSL) / 8?
This index is generally used to evaluate
machine capability. Note this is only
MACHINE capability and NOT the capability of the
full process. Given that there will be additional
sources of variation (tooling, fixtures,
materials, etc.) CM uses an 8? spread, rather
than 6?. For a machine to be used on a Six Sigma
process, a 10? spread would be used.
68
Process Capability Indices Variables Data
ZU
(USL X) / ? ZL (X LSL) / ?
Zmin Minimum (ZL , ZU) Cpk Zmin / 3
This index DOES take into account how well
or how poorly centered a process is. A value of
at least 1 is required with a value of at least
1.33 being preferred. Cp and Cpk are closely
related. In some sense Cpk represents the current
capability of the process whereas Cp represents
the potential gain to be had from perfectly
centering the process between specifications.
69
Process Capability Example
Assume that we have conducted a capability
analysis using X-bar and R charts with subgroups
of size n 5. Also assume the process is in
statistical control with an average of 0.99832
and an average range of 0.02205. A table of d2
values gives d2 2.326 (for n 5). Suppose LSL
0.9800 and USL 1.0200 _ ? R /
d2 0.02205/2.326 0.00948 Cp (1.0200
0.9800) / 6(.00948) 0.703 CR 100(60.00948)
/ (1.0200 0.9800) 142.2 CM (1.0200
0.9800) / (8(0.00948)) 0.527 ZL (.99832 -
.98000)/(.00948) 1.9 ZU (1.02000
.99832)/(.00948) 2.3 so that Zmin 1.9 Cpk
Zmin / 3 1.9 / 3 0.63
70
Process Capability Interpretation
Cp 0.703 since this is less than 1, the
process is not regarded as being capable. CR
142.2 implies that the natural tolerance
consumes 142 of the specifications (not a good
situation at all). CM 0.527 Being less than
1.33, this implies that if we were dealing with
a machine, that it would be incapable of meeting
requirements. ZL 1.9 This should be at least
3 and this value indicates that approximately
2.9 of product will be undersized. ZU 2.3
should be at least 3 and this value indicates
that approximately 1.1 of product will be
oversized. Cpk 0.63 since this is only
slightly less that the value of Cp the indication
is that there is little to be gained by centering
and that the need is to reduce process variation.
71
S
TATISTICAL
PROCESS CONTROL
End of Session
DEPARTMENT OF STATISTICS
REDGEMAN_at_UIDAHO.EDU
OFFICE 1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR CHAIR SIX SIGMA
BLACK BELT