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TOTAL QUALITY MANAGEMENT
Part VI.
Statistical process control
CHE 460
BASIC CONCEPTS
Remember: 3 ways of eliminating variation:
 Removing
the defected product Quality
control
 Eliminating
the cause of variation 
Quality assurance, problem solving process,
Statistical process control
 Robust
product/process (eliminating the
effects of noises) Parameter design
CHE 460
BASIC CONCEPTS
 Measure
of central tendency: Where
the data are located?
1 n
   yi
n i 1
• Mean ( X’ for sample, µ for population)
• Median (middle measurement)
• Mode (most repeated measurement)
 Measure
of variation: How the data are
distributed?
• Range (R=Xmax-Xmin)
• Standard deviation (s for sample, σ for
population)
1 n
2


 
y


 i
n  1 i 1
2
CHE 460
BASIC CONCEPTS

Symmetric

Approaches to infinity
in both side

The area under the
curve within ± σ is
68.27 %
CHE 460
FREQUENCY
BASIC CONCEPTS
Event
(example: type of
defects)
CHE 460
BASIC CONCEPTS
Common
cause of variation
• Comes from the nature of the system
• Totally random, can not be explained by a specific cause
• Gives normal distribution curve
• Example: Variation among the heights of 20 years old men in
Istanbul
Special
(assignable) cause of variation
• Comes from a special cause
• Not random, it can be explained
• Deviates from normal distribution
• Example: mixing of some women or children to above group
CHE 460
BASIC CONCEPTS

Common cause of variation
• Can not be predicted, does not form any trend.
• Can be removed only by fundamental improvement of the system
• Treating common cause as special cause is “tampering”

Special cause of variation
• Can be explained and predicted.
• Can be removed by eliminating the special cause
• Treating special cause as common cause is waste of time and
resource
CHE 460
BASIC CONCEPTS
Is
process in control or not?
• In control: only common cause of variation exists
• Out of control: there are some special cause of variation

Is process capable or not?
• Capable: Process can produce product characteristics
within the desired specification limits.
• Incapable: Process can not produce product
characteristics within the desired specification limits.
CHE 460
BASIC CONCEPTS
In Improve the system
control
DESIRED
STATUS
Achieve control
Out of
control status then improve
system
Achieve control
status
Control
Status
Incapable
Capable
Capability status
CHE 460
BASIC CONCEPTS
•A process with only common causes of
variation present is in statistical control.
•A process operating in the presence of
assignable causes is out of control.
•The goal of SPC is to identify and elimination
of variability taking appropriate action for both
type of variations.
CHE 460
BASIC CONCEPTS
A typical control chart has control limits set at values
such that if the process is in control, nearly all points
will lie within the upper control limit (UCL) and the
lower control limit (LCL).
Figure 15-1 A typical control chart.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
BASIC CONCEPTS
Figure 15-2 Process
improvement using the
control chart.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
BASIC CONCEPTS
where
k = distance of control limit from the center line
w = mean of some sample statistic, W.
w = standard deviation of some statistic, W.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
BASIC CONCEPTS
Important uses of the control chart
•
Most processes do not operate in a state of statistical control
•
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 eliminate causes.
•
Control charts are a proven technique for improving productivity.
•
Control charts are effective in defect prevention.
•
Control charts prevent unnecessary process adjustment.
•
Control charts provide diagnostic information.
•
Control charts provide information about process capability.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
BASIC CONCEPTS
Two groups of control charts are used depending on data type:

Continuous data
• Single sample: x-R charts (moving range)
• Small sample size: X’-R
• Large sample size: X’-s

Attribute Data
• Fraction of defective : p chart
• Defective per sample: np chart (for constant sample size)
• Fraction of defect per item: u chart
• Defect per item : c chart (for constant sample size)
CHE 460
BASIC CONCEPTS
Central Limit Theorem
Even the distribution of a population is
not normal, the means of samples from
this population result a normal
distribution if the number of sample is
large.
CHE 460
BASIC CONCEPTS

Since distribution of sample means is normal, the mean of any
sample should be on the normal distribution curve if
everything is O.K. (variation is due to common cause)

If the mean of a sample is not on the curve, then there is
some problem (variation is due to special cause)

Normal distribution curve goes to infinite in both direction in
theory. Accept some practical limits for the curve (for
example 3 sigma: 99.97% of area)
CHE 460
DESIGN OF CONTROL CHARTS
Suppose we have a process that we assume
the true process mean is  = 74 and the
process standard deviation is  = 0.01.
Samples of size 5 are taken giving a
standard deviation of the sample average,
is
 0.01
x 

 0.0045
n
5
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
DESIGN OF CONTROL CHARTS
• Control limits can be set at 3 standard
deviations from the mean in both
directions.
• “3-Sigma Control Limits”
UCL = 74 + 3(0.0045) = 74.0135
CL= 74
LCL = 74 - 3(0.0045) = 73.9865
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
DESIGN OF CONTROL CHARTS
Figure 15-3
X-bar control chart for piston ring diameter.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
DESIGN OF CONTROL CHARTS
• Choosing the control limits is equivalent to
setting up the critical region for
hypothesis testing
H0:  = 74
H1:   74
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
DESIGN OF CONTROL CHARTS
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.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
DESIGN OF CONTROL CHARTS
Constructing Rational Subgroups
•
•
Select consecutive units of production.
–
Provides a “snapshot” of the process.
–
Good at detecting process shifts.
Select a random sample over the entire sampling
interval.
–
Good at detecting if a mean has shifted
–
out-of-control and then back in-control.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
X-BAR & R CHARTS
 Sample
size n is small (if n>10-12 or n is variable)
 Examples:
•
Measurement of a specific characteristics of n products every
hour, day....
•
Taking n different sample from a chemical mixture
•
Measuring temperature in n different location in a reactor
•
Taking n samples from a raw material from a suppliers.
•
......
CHE 460
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
X-BAR & R CHARTS
3-sigma control limits:
The grand mean:
CHE 460
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
X-BAR & R CHARTS
x Control Chart (from R):
R Chart:
CHE 460
Montgomery, Introduction to Statistical Process Control, Wiley, 2009
X-BAR & R CHARTS
Example: We measure a characteristics (viscosity, concentration.... of a
product stream by taking 3 sample every hour, or we measure the
temperature of a tank in 3 different location every hour)
CHE 460
X-BAR & R CHARTS
X'
52.0
51.0
50.0
49.0
48.0
47.0
46.0
45.0
44.0
43.0
42.0
UCL(X)
X''
LCL(X)
1
2
3
4
5
6
7
8
9
10
R
HOUR
8
7
6
5
4
3
2
1
0
UCL(R)
R'
1
2
3
4
5
6
HOUR
CHE 460
7
8
9
• The second point
is out of control,
there is a special
cause for this,
• Variation in other
points is due to
common cause
• All the points are
in control,
variation is due to
commn cause
10
PROCESS IS OUT OF CONTROL
Montgomery,
Introduction to
Statistical Process
Control, Wiley,
2009
CHE 460
CHE 460
Montgomery, Introduction to Statistical Process Control, Wiley, 2009
CHE 460
Montgomery, Introduction to Statistical Process Control, Wiley, 2009
X-BAR & S CHARTS
 Sample
size (n) is large (if n>10-12)
(for same type of systems as x’ and R chart)
_
n
s 
2
2
(
x

x
)
 i
i 1
n 1




UCL( x)  x  A3 s
LCL( x)  x  A3 s

UCL( s )  B4 s

LCL( s )  B3 s
CHE 460
X-BAR & S CHARTS
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2001
CHE 460
X-BAR & S CHARTS
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2001
CHE 460
X-BAR & S CHARTS
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2001
CHE 460
x-R CHARTS (MOVING RANGE)
 There
is a single sample for some reason:
• There is only one measurement (for example well mixed
mixture)
• Sampling is very difficult or long
• Sampling is very expensive (like destruction test)
 Use
moving average for X’ and R
• Decide n (have many successive data points will be taken)
• All the other calculations are the same as X’-R
CHE 460
x-R CHARTS (MOVING RANGE)
Example: We measure a characteristics (viscosity, concentration.... of a
product stream by taking only one sample every hour, we decided to use
n=2 for moving range
CHE 460
X'
x-R CHARTS (MOVING RANGE)
53.0
52.0
51.0
50.0
49.0
48.0
47.0
46.0
45.0
44.0
43.0
42.0
UCL(X)
X''
LCL(X)
1
2
3
4
5
6
7
8
9
10
• Both charts
are in control
R
HOUR
8
7
6
5
4
3
2
1
0
UCL(R)
R'
1
2
3
4
5
6
HOUR
CHE 460
7
8
9
10
p- CHARTS:Fraction nonconforming
Di
pi 
n
Di: Defective in batch
n: sample size
n
_
p
p
m: number of samples
i
i 1
m
_
_
_
_
p (1  p )
UCL  p  3
n
_
p (1  p )
LCL  p  3
n
_
CHE 460
CHE 460
Montgomery, Introduction to Statistical Process Control, Wiley, 2009
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
np- CHARTS: number of nonconforming
_
_
_
UCL  np  3 np(1  p )
_
_
_
LCL  np  3 np(1  p)
CHE 460
np: number of defective per batch
n: sample size
m: number of samples
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
u-CHART Fraction of defect per item
x
u
n

u
_
m
i 1
x: number of defect in sample
u
n: sample size
m
m: number of samples
_
_
u
UCL  u  3
n
_
_
u
LCL  np  3
n
CHE 460
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
u-CHART Fraction of defect per item
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
CHE 460
INTERPRETING CONTROL CHARTS

X’ and R charts should be used together and interpreted depending
on the process and the parameter x.

All data points should be within the control limits for “in control”
status (one data point outside the limits is sufficient to show that
the process is out of control)

There should be no special pattern for in control status:
• Totally random variation
• Approximately equal number of data points in both side
• Most data points are close to the center line

Only the first condition is exact. The second condition can be used
with experimental observation. For example: 8 consecutive data
points in one side of mean or all increasing/decreasing are
considered as out of control status.
CHE 460
INTERPRETING CONTROL CHARTS
•
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 increasing
–
Run down: a series of observations are decreasing
50
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
INTERPRETING CONTROL CHARTS
Figure 15-4
X-bar 51
control chart.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
INTERPRETING CONTROL CHARTS
Figure 15-5 An X-bar chart with a cyclic pattern.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
INTERPRETING CONTROL CHARTS
Figure 15-6 (a) Variability with the cyclic pattern. (b) Variability with the cyclic pattern
eliminated.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
INTERPRETING CONTROL CHARTS
Western Electric Handbook Rules
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.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and
Runger.
INTERPRETING CONTROL CHARTS
Figure 15-7 The Western Electric zone rules.
© John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger.
PROCESS CAPABILITY

Control limits show what the processes can do and determined
by statistics NOT BY THE USER

The user can determine the specification limits (what the user
wants from the process)

Process may or may not be capable to deliver what the user
wants. (even if it is in control)

If the process is capable, the specification limits should be
outside of control limits in X’ chart.

Out of control status is from special cause that should be
fixed, incapability is due to the common cause and it requires
system level improvement

We make the process in control first and then improve
capability
CHE 460
PROCESS CAPABILITY
56.0
54.0
USL
UCL(X)
X'
52.0
50.0
48.0
46.0
X''
CAPABLE
LCL(X)
44.0
42.0
40.0
LSL
1
2
3
4
5
6
7
8
9
10
X'
HOUR
56.0
54.0
USL
52.0
UCL(X)
50.0
48.0
X''
46.0
LCL(X)
44.0
42.0
LSL
40.0
1
2
3
4
5
6
HOUR
CHE 460
7
8
9
10
NOT CAPABLE
PROCESS CAPABILITY
Capability index:
Cplower
X ' ' LSL

3
  R' / d 2
USL  X ' '
Cpupper 
3
If Cplower, and Cpupper 1.3, the process is considered as capable
CHE 460
PROCESS CAPABILITY
Example: Suppose USL=50, LSL=45 in moving range
chart. Is this process capable?
σ=2.3/1.128= 2.04
Cplower = (46.7-45)/3*2.04=0.27
Cpupper = (50-46.7)/3*2.04=0.54
The process is not capable
CHE 460
EXAMPLE
CHE 460
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
EXAMPLE
CHE 460
Montgomery,
Introduction to
Statistical
Process Control,
Wiley, 2009
Guidelines
to use
control
charts
Guidelines to use control charts
Montgomery, Introduction to Statistical Process Control, Wiley, 2009
Guidelines
to use
control
charts
Montgomery, Introduction to Statistical Process Control, Wiley, 2009