Transcript TN7

MBA 8452 Systems and Operations Management
Statistical Quality
Control
Objective: Quality Analysis

Process Variation


Statistical Process Control Charts and
Process Capability




Be able to explain Taguchi’s View of the cost of
variation.
Be able to develop and interpret SPC charts.
Be able to calculate and interpret Cp and Cpk
Be able to explain the difference between
process control and process capability
Sample Size

Be able to explain the importance of sample size2
Statistical Quality Control
Approaches

Statistical Process Control (SPC)


Sampling to determine if the process is
within acceptable limits (under control)
Acceptance Sampling

Inspects a random sample of a product
to determine if the lot is acceptable
3
Line graph shows plot of data
and variation from the average
Process
target or
average
1
2
3
4
5
6
7
8
9
10
Sample number
4
Why Statistical Quality Control?

Variations in Manufacturing/Service Processes


Any process has some variations: common and/or
special
Variations are causes for quality problems



If a process is stable (no special variation), it is able to
produce product/service consistently
As variation is reduced, quality is improved
Statistics is the only science that is dedicated to
dealing with variations.

Measure, monitor, and reduce variations in the process
5
Types of Variation
Assignable (special)
Natural (common)
Inherent to process
 Random
 Cannot be controlled
 Cannot be prevented
 Examples

 weather
 accuracy of measurements
 capability of machine





Exogenous to process
Not random
Controllable
Preventable
Examples
 tool wear
 human factors (fatigue)
 poor maintenance
6
Cost of Variation:
Traditional vs. Taguchi’s View
High
High
Incremental
Cost of
Variability
Incremental
Cost of
Variability
Zero
Zero
Lower Target Upper
Spec Spec Spec
Traditional View
Lower Target Upper
Spec Spec Spec
Taguchi’s View
7
Statistical Process Control



On-line quality control tool used when the
product/service is being produced
Purpose: prevent systematic quality problems
Procedure

Take periodic random samples from a process

Plot the sample statistics on control chart(s)

Determine if the process is under control


If the process is under control, do nothing
If the process is out of control, investigate and fix the
cause
8
Statistical Process Control
Types Of Data

Attribute data (discrete values)

Quality characteristic evaluated about
whether it meets the required specifications


Good/bad, yes/no
Variable data (continuous values)

Quality characteristic that can be measured
 Length, size, weight, height, time, velocity
9
Statistical Process Control
Control Charts

Charts for attributes



p-chart (for proportions)
c-chart (for counts)
Charts for variables


R-chart (for ranges)
X -chart (for means)
10
Control Chart
General Structure
Upper
control
limit (UCL)
Process
target or
average
Lower
control
limit (LCL)
1
2
3
4
5
6
7
8
9
10
Sample number
11
A Process Is In Control If ...

No sample points outside control limits

Most points near the process average


About an equal # points above & below
the centerline
Points appear randomly distributed
12
Common Out-of-control Signs
One observation
outside the limits
Sample observations
consistently below or
above the average
Sample observations
consistently decrease
or increase
13
Issues In Building Control Charts

Number of samples: around 25

Size of each sample: large (100) for
attributes and small (25) for variables

Frequency of sampling: depends

Control limits: typically 3-sigma away from
the process mean
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Control Limits:
The Normal Distribution
95 %
99.74 %
m-3s m-2s m-1s m m+1s m+2s m+3s
X
If we establish control limits at +/- 3 standard deviations (s),
then we would expect 99.74% of observations (X) to fall
within these limits.
15
Control Limits: General Formulas
UCL = mean + z (stand dev)
LCL = mean – z (stand dev)



z is the # of standard deviations
z = 3.00 is the most commonly used value with
99.7% confidence level
Other z values can be used (e.g. z=2 for 95%
confidence and z=2.58 for 99% confidence)
16
Control Charts for Attributes
p-charts
UCL = p + zs p
LCL = p - zs p
p = Average proportion of defectives in the sample
Total Number of Defectives
=
Total Number of Observatio ns
sp =
p (1 - p)
n
n  sample size
17
p-Chart
Example
20 Samples of 100 pairs of jeans each were randomly
selected from the Western Jean Company’s production line.
Sample Defect
1
6
2
12
3
4
..
.
20
Total
..
.
18
200
Proportion
Defective
.06
.12
.04
..
.
.18
n=100 jeans in each sample
total defectives
total sample observations
200
= 0.10
=
20 (100)
p =
sp =
p (1 - p)
=
n
.10(1 - .10)
= .03
100
UCL = p + zsp = .10 + 3(.03) = 0.19
LCL = p - zsp = .10 - 3(.03) = 0.01
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p-Chart
Example
0.2
UCL
0.14
0.12
p
0.1
0.08
0.06
0.04
0.02
0
20
18
16
14
12
..
10
8
6
4
2
LCL
0
Proportion defective
0.18
0.16
Sample number
19
Control Charts For Variables
X-bar charts and R-charts
x Chart Control Limits
R Chart Control Limits
UCL = x + A 2 R
UCL = D 4 R
LCL = x - A 2 R
LCL = D 3 R
Where X = average of sample means = Xi / m
R = average of sample ranges = Ri / m
Xi = mean of sample i, i = 1,2,…,m
Ri = range of sample i, i = 1,2,…,m
m = total number of samples
A2, D3, and D4 are constants from Exhibit TN7.7
20
Example

If a company makes jeans, there are a
specifications that must be met.

The back pockets of the jeans can’t be too
small or too large.


The control chart can be established to monitor
the measurements of the back pocket
Given 15 samples with 5 observations each, we
can determine the Upper and Lower control
limits for the range and x-bar charts.
21
X-bar and R Charts
Example
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
10.682
10.787
10.780
10.591
10.693
10.749
10.791
10.744
10.769
10.718
10.787
10.622
10.657
10.806
10.660
2
10.689
10.860
10.667
10.727
10.708
10.714
10.713
10.779
10.773
10.671
10.821
10.802
10.822
10.749
10.681
Observation
3
10.776
10.601
10.838
10.812
10.790
10.738
10.689
10.110
10.641
10.708
10.764
10.818
10.893
10.859
10.644
4
5
10.798
10.714
10.746
10.779
10.785
10.723
10.775
10.730
10.758
10.671
10.719
10.606
10.877
10.603
10.737
10.750
10.644
10.725
10.850
10.712
10.658
10.708
10.872
10.727
10.544
10.750
10.801
10.701
10.747
10.728
Overall Averages
Sample
Sample
Mean (Xi) Range (Ri)
10.732
0.116
10.755
0.259
10.759
0.171
10.727
0.221
10.724
0.119
10.705
0.143
10.735
0.274
10.624
0.669
10.710
0.132
10.732
0.179
10.748
0.163
10.768
0.250
10.733
0.349
10.783
0.158
10.692
0.103
10.728
0.220
X
R
X-bar and R Charts
Example
Exhibit TN7.7
Since n=5, from Exhibit
TN7.7 (also right table),
we find
A2=0.58
D3=0
D4=2.11
n
2
3
4
5
6
7
8
9
10
11
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
D3
0
0
0
0
0
0.08
0.14
0.18
0.22
0.26
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
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X-bar and R Charts: Example
R chart
UCL = D 4 R  ( 2.11)(0.220)  0.464
LCL = D 3 R  (0)( 0.220)  0
0 .8 0 0
0 .7 0 0
0 .6 0 0
0 .5 0 0
R
UCL
0 .4 0 0
0 .3 0 0
R
0 .2 0 0
0 .1 0 0
0 .0 0 0
LCL
1
2
3
4
5
6
7
8
S a m p le
9
10
11
12
13
14
15
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X-bar and R Charts
Example
X-bar chart
UCL = x + A 2 R  10.728 + .58(0.220) = 10.856
LCL = x - A 2 R  10.728 - .58(0.220) = 10.600
1 0 .9 0 0
UCL
1 0 .8 5 0
M eans
1 0 .8 0 0
1 0 .7 5 0
X
1 0 .7 0 0
1 0 .6 5 0
1 0 .6 0 0
LCL
1 0 .5 5 0
1
2
3
4
5
6
7
8
S a m p le
9
10
11
12
13
14
15
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Process Capability

The ability of a process to meet product
design/technical specifications

Design specifications for products (Tolerances)


Process variability in production process


upper and lower specification limits (USL, LSL)
natural variation in process (3s from the mean)
Process may not be capable of meeting
specifications if natural variation in a process
exceeds allowable variation (tolerances)
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Process Capability
Illustrations
(a)
specification
natural variation
(c)
specification
natural variation
(b)
specification
natural variation
(d)
specification
natural variation
27
Process Capability
Further Illustrations
LSL Target
USL
LSL Target
USL
Process
variation
Tolerance
Capable process
variation
Process not capable
Highly capable process
Process not capable
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Specification Limits  Control Limits


Specification limits are pre-established for
products before production
Control limits are used to monitor the actual
production process performance


It is possible that a process is under control, but
not capable to meet specifications
It is also possible that a process that is within
specifications is out-of-control
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Control Limits Vs. Specification Limits
Illustrations
USL
UCL
LCL
LSL
(1) In control and within specifications
UCL
USL
LSL
LCL
(2) In control but exceeds specifications
USL
UCL
LCL
LSL
(3) Out-of-control and within specifications
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Process Capability Index:
Cp -- Measure of Potential Capability
allowable process variation USL  LSL
Cp 

actual process variation
6s
LSL
USL
Cp > 1
Cp = 1
Cp < 1
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Process Capability Index:
Cpk -- Measure of Actual Capability
C pk
s
 X  LSL USL  X 
 min 
,

3s 
 3s
is the standard deviation of the production process
Cpk considers both process variation (s)
and process location (X)
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Process Capability Index
Example
A manufacturing process produces a certain part with a
mean diameter of 2 inches and a standard deviation of
0.03 inches. The lower and upper engineering
specification limits are 1.90 inches and 2.05 inches.
USL  LSL 2.05  1.90
Cp 

 0.83
6s
6(0.03)
C pk
 2  1.90 2.05  2 
 X  LSL USL  X 
 min 
,
 min 
,


3
s
3
s
3
(
0
.
03
)
3
(
0
.
03
)




 min[ 1.11,0.56]  0.56
Therefore, the process is not capable (the variation is
too much and the process mean is not on the target)
33
Impact of Process Location on
Process Capability
LSL
USL
s= 2
38
44
50
Cp = 2.0
Cpk = 2.0
56
62
Cp = 2.0
Cpk = 1.5
38
44
50
53
56
62
Cp = 2.0
Cpk = 1.0
38
44
50
56
62
Cp = 2.0
Cpk = 0
38
44
50
56
62
34
Acceptance Sampling



Determines whether to accept or reject
an entire lot of goods based on sample
results
Measures quality in percent defective
Usually applied to incoming raw
materials or outgoing finished goods
35
Sampling Plan


Guidelines for accepting or rejecting a lot
Single sampling plan





N = lot size
n = sample size
c = max acceptance number of defects
d = number of defective items in sample
If d <= c, accept lot; else reject
Sampling plan is developed based on the tradeoff
between producer’s risk and consumer’s risk
36
Producer’s & Consumer’s Risk

Producer’s Risks




reject a good lot (TYPE I ERROR)
a = producer’s risk = P(reject good lot)
5% is common
Consumer’s Risks



accept a bad lot (TYPE II ERROR)
b = consumer’s risk = P(accept bad lot)
10% is typical
37
Quality Definitions

Acceptable quality level (AQL)



Acceptable proportion of defects on average
“good lot” = the proportion of defects of the
lot is less than or equal to AQL
Lot tolerance percent defective (LTPD)


Maximum proportion of defects in a lot
“bad lot” = the proportion of defects of the
lot is greater than LTPD
38
Probability of acceptance
Operating Characteristic Curve
a = .05 (producer’s risk)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
n = 99
c=4
b =.10
(consumer’s risk)
1
2
AQL
3
4
5
6
7
8
9 10 11 12
LTPD
Percent defective in a lot
39