Download Unit IX

STATISTICAL PROCESS CONTROL AND
QUALITY MANAGEMENT
1
Quality
• Fitness for use, acceptable standard
• Based on needs, expectations and customer
requests
Types of Quality
• Quality of design
• Quality of conformance
• Quality of performance
2
Quality of Design
• Differences in quality due to design differences,
intentional differences
Quality of Conformance
• Degree to which product meets or exceeds
standards
Quality of Performance
• Long term consistent functioning of the product,
reliability, safety, serviceability, maintainability
3
Quality and Productivity
• Improved quality leads to lower costs and
increased profits
4
Statistics and Quality
Management
• Statistical analysis is used to assist with product
design, monitor the production process, and
check quality of the finished product
5
Checking Finished Product Quality
• Random samples selected from batches of
finished product can be used to check for
product quality
6
Assisting With Production Design
• A variety of experimental design techniques are
available for improving the production process
7
Monitoring the Process
• Control charts is used to monitor the process as
it unfolds
• Sampling from the production line to see if
variation in product quality is consistent with
expectations
8
The Control Chart
• A special type of sequence plot which is used to
monitor a process
• Throughout the process measurements are
taken and plotted in a sequence plot
• Plot also contains upper and lower control limits
indicating the expected range of the process
when it is behaving properly
9
Examples
PROCESS CONTROL CHART
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51
50
49
48
97
93
89
85
81
77
73
69
65
61
57
53
49
45
41
37
33
29
25
21
17
13
9
5
47
1
BATCH MEAN
52
BATCH NUMBER
MEAN
TRUE MEAN
LOWER
UPPER
10
Variation
• There is no two natural items in any
category are the same.
• Variation may be quite large or very small.
• If variation very small, it may appear that
items are identical, but precision
instruments will show differences.
3 Categories of variation
• Within-piece variation
– One portion of surface is rougher than another
portion.
• Apiece-to-piece variation
– Variation among pieces produced at the same time.
• Time-to-time variation
– Service given early would be different from that given
later in the day.
Source of variation
• Equipment
– Tool wear, machine vibration, …
• Material
– Raw material quality
• Environment
– Temperature, pressure, humadity
• Operator
– Operator performs- physical & emotional
Control Chart Viewpoint

Variation due to
Common or chance causes
 Assignable causes


Control chart may be used to discover
“assignable causes”
Control chart functions
• Control charts are powerful aids to
understanding the performance of a process
over time.
Input
PROCESS
What’s causing variability?
Output
Control charts identify variation
• Chance causes - “common cause”
– inherent to the process or random and not
controllable
– if only common cause present, the process is
considered stable or “in control”
• Assignable causes - “special cause”
– variation due to outside influences
– if present, the process is “out of control”
Control charts help us learn more about
processes
• Separate common and special causes of
variation
• Determine whether a process is in a state of
statistical control or out-of-control
• Estimate the process parameters (mean,
variation) and assess the performance of a
process or its capability
Control charts to monitor processes
• To monitor output, we use a control chart
– we check things like the mean, range, standard
deviation
• To monitor a process, we typically use two
control charts
– mean (or some other central tendency measure)
– variation (typically using range or standard
deviation)
Types of Data
• Variable data
– Product characteristic that can be measured
• Length, size, weight, height, time, velocity
• Attribute data
Product characteristic evaluated with a discrete choice
• Good/bad, yes/no
Types of Control Charts
• Control Chart For The Mean(X chart)
• Control Chart For The Range(R chart)
• Control Chart For A Proportion(p chart)
• Control Chart For Attribute Measures(c chart)
20
Control Chart For The Sample Mean
• Assuming that the sample mean is
approximately normal with mean  and
standard deviation , a control chart for the
mean usually consists of three horizontal lines
• The vertical axis is used to plot the magnitude
of observed sample means while the horizontal
axis represents time or the order of the
sequence of observed means
21
Control Chart For The Sample Mean
• The center line is at the mean,  and upper and lower
control limits are at
• +3 / n
and
-3
/ n
• Since  (standard deviation)is usually unknown the
• term  / n is usually replaced by an estimator
based on the sample range
22
Control Chart For The Sample Mean
The formula is given by
x  A2 R
where R = average value of the range
k
=
 Ri / k
i 1
k = number of samples
The values of A2 depend on the sample size
23
Control Chart For The Sample Mean
• LCL =
x  R A2
• UCL =
x  R A2
24
Control Chart For The Range
• Designed to monitor variability in the product
• Range easier to determine than standard
deviation
• Distribution of sample range assumes product
measurement is normally distributed
25
Control Chart For The Range
• Upper and lower control limits and center line
obtained from R and the values of D3, D4
• according to the formulae
• LCL =
D3 R
• UCL =
D4 R
• The values of D3, D4 are based on sample size
26
Control Chart For The Sample
Proportion
• Population proportion 
• The sample mean is now a mean proportion
given by p
• Where p = total number of objects in sample
with characteristic divided by total sample size
27
Control Chart For The Sample
Proportion
• Using the central limit theorem the control
limits are given by:
 (1   )
p3
n
28
Control Chart For The Sample
Proportion
• Since the true proportion  is usually unknown
we replace it by the average proportion
p3
p (1  p )
n
• If the sample size varies then the upper and
lower limits will vary and the equations
become:
p 3
p (1  p )
ni
• where ni = sample size in sample i
29
Control Chart For Attribute Measures
• Alternative method of counting good and bad items.
• Defects are measured by merely counting the no. of
defects.
• Where c
= total number of defects in a sample
30
Control Chart For Attribute Measures
• Process average or central line
c=
• UCL: c  3
• LCL: c  3
c
n
p
c
c
31
Example: Control Charts for Variable Data
Sample
1
2
3
4
5
6
7
8
9
10
Slip Ring Diameter (cm)
1
2
3
4
5.02 5.01 4.94 4.99
5.01 5.03 5.07 4.95
4.99 5.00 4.93 4.92
5.03 4.91 5.01 4.98
4.95 4.92 5.03 5.05
4.97 5.06 5.06 4.96
5.05 5.01 5.10 4.96
5.09 5.10 5.00 4.99
5.14 5.10 4.99 5.08
5.01 4.98 5.08 5.07
5
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99
X
R
Example: Control Charts for Variable Data
Sample
1
2
3
4
5
6
7
8
9
10
Slip Ring Diameter (cm)
1
2
3
4
5.02 5.01 4.94 4.99
5.01 5.03 5.07 4.95
4.99 5.00 4.93 4.92
5.03 4.91 5.01 4.98
4.95 4.92 5.03 5.05
4.97 5.06 5.06 4.96
5.05 5.01 5.10 4.96
5.09 5.10 5.00 4.99
5.14 5.10 4.99 5.08
5.01 4.98 5.08 5.07
5
4.96
4.96
4.99
4.89
5.01
5.03
4.99
5.08
5.09
4.99
X
4.98
5.00
4.97
4.96
4.99
5.01
5.02
5.05
5.08
5.03
50.09
R
0.08
0.12
0.08
0.14
0.13
0.10
0.14
0.11
0.15
0.10
1.15
Calculation
From Table above:
• Sigma X-bar = 50.09
• Sigma R = 1.15
• m = 10
Thus;
• X-Double bar = 50.09/10 = 5.009 cm
• R-bar = 1.15/10 = 0.115 cm
Note: The control limits are only preliminary with 10 samples.
It is desirable to have at least 25 samples.
Trial control limit
• UCLx-bar = X-D bar + A2 R-bar
= 5.009 + (0.577)(0.115) = 5.075 cm
• LCLx-bar = X-D bar - A2 R-bar
= 5.009 - (0.577)(0.115) = 4.943 cm
n=5
• UCLR = D4R-bar
= (2.114)(0.115) = 0.243 cm
• LCLR = D3R-bar
= (0)(0.115) = 0 cm
For A2, D3, D4: see Table B, Appendix
3-Sigma Control Chart Factors
Sample size
n
2
3
4
5
6
7
8
X-chart
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
R-chart
D3
0
0
0
0
0
0.08
0.14
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
X-bar Chart
R Chart
Subgroup
18
2
1
12.45
12.39
2
12.55
3
3
4
5
12.40
12.37
12.40
12.38
12.36
12.38
12.44
12.46
12.44
12.30
12.39
12.36
4
12.38
12.39
12.37
12.55
12.37
5
12.37
12.44
12.44
12.37
12.38
6
12.45
12.37
12.36
12.41
12.39
7
12.46
12.38
12.51
12.44
12.55
8
12.44
12.39
12.38
12.39
12.37
9
12.44
12.55
12.41
12.44
12.39
10
12.35
12.38
12.37
12.44
12.38
11
12.36
12.40
12.41
12.35
12.44
12
12.51
12.36
12.41
12.36
12.39
13
12.38
12.30
12.45
12.37
12.44
14
12.41
12.37
12.45
12.45
12.37
15
12.37
12.44
12.45
12.46
12.38
39
Sample No.
Number of defects
1
10
2
9
3
8
4
11
5
7
6
12
7
7
8
10
9
13
10
12
11
13
12
14
From a lot of 100
40
Sample No.
Number of defects
Proportion
1
10
0.10
2
9
0.9
3
8
0.8
4
11
0.11
5
7
0.7
6
12
0.12
7
7
0.7
8
10
0.10
9
13
0.13
10
12
0.12
11
13
0.13
12
14
0.14
From a lot of 100
41
The Normal
Distribution
 = Standard deviation
Mean
-3 -2 -1
+1 +2 +3
68.26%
95.44%
USL 99.74%
LSL
-3
CL
+3
•
•
•
•
•
34.13% of data lie between  and 1 above the mean ().
34.13% between  and 1 below the mean.
Approximately two-thirds (68.28 %) within 1 of the mean.
13.59% of the data lie between one and two standard deviations
Finally, almost all of the data (99.74%) are within 3 of the mean.
Normal Distribution Review

Define the 3-sigma limits for sample means as follows:
3
3(0.05)
Upper Limit   
 5.01 
 5.077
n
5
3
3(0.05)
Lower Limit   
 5.01 
 4.943
n
5


What is the probability that the sample means will lie outside
3-sigma limits?
Note that the 3-sigma limits for sample means are different
from natural tolerances which are at
  3
Common Causes
Process Out of Control
• The term out of control is a change in the
process due to an assignable cause.
• When a point (subgroup value) falls outside its
control limits, the process is out of control.
Assignable Causes
(a) Mean
Average
Grams
Assignable Causes
Average
(b) Spread
Grams
Assignable Causes
Average
Grams
(c) Shape
Assignable
causes likely
Control Charts
UCL
Nominal
LCL
1
2
Samples
3
Control Chart Examples
Variations
UCL
Nominal
LCL
Sample number
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(a) Three-sigma limits
UCL
Process
average
LCL
Control Limits and Errors
Type I error:
Probability of searching for
a cause when none exists
(b) Two-sigma limits
UCL
Process
average
LCL
Control Limits and Errors
(a) Three-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
Process
average
average
LCL
Control Limits and Errors
(b) Two-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
Process
average
average
LCL
Achieve the purpose
Our goal is to decrease the variation inherent
in a process over time.
As we improve the process, the spread of the
data will continue to decrease.
Quality improves!!
Improvement
Examine the process
• A process is considered to be stable and in
a state of control, or under control, when
the performance of the process falls
within the statistically calculated control
limits and exhibits only chance, or
common causes.
Consequences of misinterpreting
the process
• Blaming people for problems that they cannot control
• Spending time and money looking for problems that do
not exist
• Spending time and money on unnecessary process
adjustments
• Taking action where no action is warranted
• Asking for worker-related improvements when process
improvements are needed first
Process variation
• When a system is subject to only chance
causes of variation, 99.74% of the
measurements will fall within 6 standard
deviations
– If 1000 subgroups are measured, 997
will fall within the six sigma limits.
Mean
-3 -2 -1
+1 +2 +3
68.26%
95.44%
99.74%
Chart zones
• Based on our knowledge of the normal curve, a
control chart exhibits a state of control when:
♥ Two thirds of all points are near the center value.
♥ The points appear to float back and forth across
the centerline.
♥ The points are balanced on both sides of the
centerline.
♥ No points beyond the control limits.
♥ No patterns or trends.
Acceptance Sampling
Acceptance sampling is a form of testing
that involves taking random samples of
“lots,” or batches, of finished products and
measuring them against predetermined
standards.
• A “lot,” or batch, of items can be inspected
in several ways, including the use of single,
double, or sequential sampling.
Single Sampling
• Two numbers specify a single sampling plan:
They are the number of items to be sampled
(n) and a pre specified acceptable number of
defects (c). If there are fewer or equal defects
in the lot than the acceptance number, c, then
the whole batch will be accepted. If there are
more than c defects, the whole lot will be
rejected or subjected to 100% screening.
Double Sampling
• Often a lot of items is so good or so bad that we can
reach a conclusion about its quality by taking a
smaller sample than would have been used in a
single sampling plan. If the number of defects in this
smaller sample (of size n1) is less than or equal to
some lower limit (c1), the lot can be accepted. If the
number of defects exceeds an upper limit (c2), the
whole lot can be rejected. But if the number of
defects in the n1 sample is between c1 and c2, a
second sample (of size n2) is drawn. The cumulative
results determine whether to accept or reject the lot.
The concept is called double sampling.
Sequential Sampling
• Multiple sampling is an extension of double
sampling, with smaller samples used
sequentially until a clear decision can be
made. When units are randomly selected from
a lot and tested one by one, with the
cumulative number of inspected pieces and
defects recorded, the process is called
sequential sampling.
OPERATING CHARACTERISTIC (OC) CURVES
• The operating characteristic (OC) curve
describes how well an acceptance plan
discriminates between good and bad
lots. A curve pertains to a specific plan,
that is, a combination of n (sample size)
and c (acceptance level). It is intended to
show the probability that the plan will
accept lots of various quality levels.
• Figure shows a perfect discrimination plan for
a company that wants to reject all lots with
more than 2 ½ % defectives and accept all
lots with less than 2 ½ % defectives.
OC Curves for Two Different Acceptable Levels of Defects
(c = 1, c = 4) for the Same Sample Size (n = 100).
• So one way to increase the probability of accepting only
good lots and rejecting only bad lots with random
sampling is to set very tight acceptance levels.
• OC Curves for Two Different Sample Sizes (n = 25,
n = 100) but Same Acceptance Percentages (4%).
Larger sample size shows better discrimination.
Sampling Terms
• Acceptance quality level (AQL): the
percentage of defects at which consumers are
willing to accept lots as “good”
• Lot tolerance percent defective (LTPD): the
upper limit on the percentage of defects that a
consumer is willing to accept
• Consumer’s risk: the probability that a lot
contained defectives exceeding the LTPD will be
accepted
• Producer’s risk: the probability that a lot
containing the acceptable quality level will be
rejected
THE OPERATING-CHARACTERISTIC (OC)
CURVE
• For a given a sampling plan and a specified true fraction
defective p, we can calculate
– Pa -- Probability of accepting lot
• If lot is truly good, 1 - Pa = a (Producers' Risk)
• If lot is truly bad, Pa = b (Consumer ‘s Risk)
• A plot of Pa as a function of p is called the OC curve for a
given sampling plan
The OC curve shows the features of a particular
sampling plan, including the risks of making
a wrong decision.
Construction of OC curve
• In attribute sampling, where products are
determined to be either good or bad, a binomial
distribution is usually employed to build the OC
curve. The binomial equation is
where
n = number of items sampled (called trials)
p = probability that an x (defect) will occur on any one trial
P(x) = probability of exactly x results in n trials
In a Poisson approximation of the binomial distribution,
the mean of the binomial, which is np, is used as the
mean of the Poisson, which is λ; that is,
λ = np
Example
• Probability of acceptance A shipment of 2,000
portable battery units for microcomputers is about
to be inspected by a Malaysian importer. The
Korean manufacturer and the importer have set
up a sampling plan in which the risk is limited to
5% at an acceptable quality level (AQL) of 2%
defective, and the risk is set to 10% at Lot
Tolerance Percent Defective (LTPD) = 7% defective.
We want to construct the OC curve for the plan of
n = 120 sample size and an acceptance level of c ≤
3 defectives. Both firms want to know if this plan
will satisfy their quality and risk requirements.
Example
N=1000
n = 100
AQL=1%
LTPD=5%
ß=10%
α = 5%
C<=2
Does the plan meet the producer’s and
consumer’s requirement.
AVERAGE OUTGOING QUALITY
In most sampling plans, when a lot is rejected, the
entire lot is inspected and all of the defective items
are replaced. Use of this replacement technique
improves the average outgoing quality in terms of
percent defective. In fact, given (1) any sampling plan
that replaces all defective items encountered and (2)
the true incoming percent defective for the lot, it is
possible to determine the average outgoing quality
(AOQ) in percent defective.
AVERAGE OUTGOING QUALITY
The equation for AOQ is
Example
Selected Values of
% Defective
Mean of Poisson,
λ = np
P (acceptance)
.01
1
0.920
.02
2
0.677
.03
3
0.423
.04
4
0.238
.05
5
0.125
.06
6
0.062
Example
• The percent defective from an incoming lot in
is 3%. An OC curve showed the probability of
acceptance to be .515. Given a lot size of
2,000 and a sample of 120, what is the
average outgoing quality in percent defective?