mba 8452 systems and operations management
DESCRIPTION
MBA 8452 Systems and Operations Management. Statistical Quality Control. Objective: Quality Analysis. Process Variation Be able to explain Taguchi’s View of the cost of variation. Statistical Process Control Charts and Process Capability Be able to develop and interpret SPC charts. - PowerPoint PPT PresentationTRANSCRIPT
MBA 8452 Systems and Operations ManagementMBA 8452 Systems and Operations Management
Statistical Quality Statistical Quality ControlControl
2
Objective: Quality Analysis Process Variation
Be able to explain Taguchi’s View of the cost of variation.
Statistical Process Control Charts and Process Capability 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
size
3
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
4
Line graph shows plot of dataand variation from the average
1 2 3 4 5 6 7 8 9 10
Sample number
Processtarget or average
5
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
6
Types of Variation
Natural (common) Assignable (special) Exogenous to process Not random Controllable Preventable Examples
tool wear human factors (fatigue) poor maintenance
Inherent to process Random Cannot be controlled Cannot be prevented Examples
weather accuracy of measurements capability of machine
7
Cost of Variation: Traditional vs. Taguchi’s View
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Traditional View
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Taguchi’s View
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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
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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
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Statistical Process Control Control Charts
Charts for attributes p-chart (for proportions) c-chart (for counts)
Charts for variables R-chart (for ranges) -chart (for means) X
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Control Chart General Structure
1 2 3 4 5 6 7 8 9 10
Sample number
Uppercontrollimit (UCL)
Processtarget or average
Lowercontrollimit (LCL)
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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
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Common Out-of-control Signs
One observation outside the limits
Sample observations consistently below or above the average
Sample observations consistently decrease or increase
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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
99.74 %
95 %
1 +2 -1-2-3
If we establish control limits at +/- 3 standard deviations (), then we would expect 99.74% of observations (X) to fall within these limits.
X
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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)
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Control Charts for Attributes p-charts
p
p
z- p = LCL
z + p = UCL
s
s
ns
)p-(1 p = p
nsObservatio ofNumber Total
Defectives ofNumber Total=
sample in the defectives of proportion Average =p
size sample n
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p-Chart
Example 20 Samples of 100 pairs of jeans each were randomly selected from the Western Jean
Company’s production line.
.03= 100
.10)-.10(1=
)p-(1 p = p n
s
0.01 = 3(.03) - .10 =z - p = LCL
0.19 = 3(.03) .10 =z + p = UCL
p
p
s
s
total defectives total sample observations
200 20 (100)
p =
= = 0.10
n=100 jeans in each sample
Proportion Sample Defect Defective
1 6 .06 2 12 .12 3 4 .04
. . .
. . .
. . .
20 18 .18Total 200
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p-Chart Example
. .
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 2 4 6 8
10 12 14 16 18 20
Pro
port
ion
defe
ctiv
e
Sample number
UCL
LCL
p
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Control Charts For Variables
X-bar charts and R-charts
Where X = average of sample means = Xi / mR = average of sample ranges = Ri / mXi = mean of sample i, i = 1,2,…,mRi = range of sample i, i = 1,2,…,mm = total number of samples
A2, D3, and D4 are constants from Exhibit TN7.7
RA - x = LCL
RA + x = UCL
2
2
Limits ControlChart x R Chart Control Limits
UCL = D R
LCL = D R
4
3
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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.
X-bar and R Charts Example
Sample SampleSample 1 2 3 4 5 Mean Range
1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.860 10.601 10.746 10.779 10.755 0.2593 10.780 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.730 10.727 0.2215 10.693 10.708 10.790 10.758 10.671 10.724 0.1196 10.749 10.714 10.738 10.719 10.606 10.705 0.1437 10.791 10.713 10.689 10.877 10.603 10.735 0.2748 10.744 10.779 10.110 10.737 10.750 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.13210 10.718 10.671 10.708 10.850 10.712 10.732 0.17911 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.25013 10.657 10.822 10.893 10.544 10.750 10.733 0.34914 10.806 10.749 10.859 10.801 10.701 10.783 0.15815 10.660 10.681 10.644 10.747 10.728 10.692 0.103
10.728 0.220
Observation
Overall Averages
(Xi) (Ri)
X R
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X-bar and R Charts
Example
n A2 D3 D42 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.8210 0.31 0.22 1.7811 0.29 0.26 1.74
Exhibit TN7.7Exhibit TN7.7
Since n=5, from Exhibit TN7.7 (also right table), we find
A2=0.58
D3=0
D4=2.11
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X-bar and R Charts: ExampleR chart
R
0
0.464
)220.0)(0(RD = LCL
)220.0)(11.2(RD = UCL
3
4
UCL
LCL0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sample
R
25
X-bar and R Charts
Example
10.600
10.856
=.58(0.220)-10.728RA - x = LCL
=.58(0.220)10.728RA + x = UCL
2
2
X-bar chart
10.550
10.600
10.650
10.700
10.750
10.800
10.850
10.900
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sample
Mea
ns
UCL
LCL
X
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Process Capability
The ability of a process to meet product design/technical specifications Design specifications for products
(Tolerances) upper and lower specification limits (USL, LSL)
Process variability in production process natural variation in process (3 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
specification specification
specification specification
natural variation natural variation
(a) (b)
natural variation natural variation
(c) (d)
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Process Capability
Further Illustrations
Capable process
LSL USLTarget USLTargetLSL
Process variation
Highly capable process
Process not capable Process not capable
Tolerance variation
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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
IllustrationsUSL
LSLLCL
UCL
(1) In control and within specifications
USL
LCL
UCL
LSL
(2) In control but exceeds specifications
USL
LSL
UCL
LCL
(3) Out-of-control and within specifications
31
Process Capability Index:Cp -- Measure of Potential Capability
6 variationprocess
variationprocess LSLUSL
actual
allowableC p
Cp =
1
Cp <
1
Cp >
1
LSL USL
32
Process Capability Index:Cpk -- Measure of Actual Capability
3,
3min
XUSLLSLXC pk
is the standard deviation of the production process
Cpk considers both process variation () and process location (X)
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Process Capability Index
ExampleA 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.
56.0]56.0,11.1min[
)03.0(3
205.2,
)03.0(3
90.12min
3,
3min
XUSLLSLX
C pk
83.0)03.0(6
90.105.2
6
LSLUSL
C p
Therefore, the process is not capable (the variation is too much and the process mean is not on the target)
34
Impact of Process Location on Process Capability
= 2
5044 56 6238
5044 56 6238
5044 56 62 38
5044 56 6238
LSL USL
53
Cp = 2.0Cpk = 2.0
Cp = 2.0Cpk = 1.5
Cp = 2.0Cpk = 1.0
Cp = 2.0Cpk = 0
35
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
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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
37
Producer’s & Consumer’s Risk Producer’s Risks
reject a good lot (TYPE I ERROR) = producer’s risk = P(reject good lot) 5% is common
Consumer’s Risks accept a bad lot (TYPE II ERROR) = consumer’s risk = P(accept bad lot) 10% is typical
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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
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Operating Characteristic Curve
n = 99c = 4
AQL LTPD
00.10.20.30.40.50.60.70.80.9
1
1 2 3 4 5 6 7 8 9 10 11 12
Pro
bab
ilit
y of
acc
epta
nce
=.10(consumer’s risk)
= .05 (producer’s risk)
Percent defective in a lot