1 © the mcgraw-hill companies, inc., 2004 technical note 7 process capability and statistical...
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©The McGraw-Hill Companies, Inc., 2004
Technical Note 7
Process Capability and
Statistical Quality Control
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©The McGraw-Hill Companies, Inc., 2004
• What is quality?
• Process Variation
• Process Capability
• Process Control Procedures— Variable data— Attribute data
• Acceptance Sampling— Operating Characteristic Curve— Standard table of sampling plans
OBJECTIVES
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• Quality defined— Durability, reliability, long warrantee— Fitness for use, degree of conformance— Maintainability
• Measures of quality— Grade—measurable characteristics, finish— Consistency—good or bad, predictability— Conformance—degree product meets
specifications Consistency versus conformance
What Is Quality?
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Basic Forms of Variation
Assignable variation (special) is caused by factors that can be clearly identified and possibly managed
Common variation (chance or random) is inherent in the production process
Example: A poorly trained employee that creates variation in finished product output.
Example: A poorly trained employee that creates variation in finished product output.
Example: A molding process that always leaves “burrs” or flaws on a molded item.
Example: A molding process that always leaves “burrs” or flaws on a molded item.
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Taguchi’s View of Variation
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Traditional View
IncrementalCost of Variability
High
Zero
LowerSpec
TargetSpec
UpperSpec
Taguchi’s View
Exhibits TN7.1 & TN7.2
Exhibits TN7.1 & TN7.2
Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.
Traditional view is that quality within the LS and US is good and that the cost of quality outside this range is constant, where Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs.
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Process Capability
• Process (control) limits— Calculated from data gathered from the process— It is natural tolerance limits— Defined by +/- 3 standard deviation— Used to determine if process is in statistical control
• Tolerance limits— Often determined externally, e.g., by customer— Process may be in control but not within
specification
• How do the limits relate to one another?
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Process Capability
• Case 1: Cp > 1— USL-LSL > 6 sigma— Situation desired— Defacto standard is 1.33+
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LSLUSLCp
6
LNTL UNTL
LSL USL
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Process Capability
• Case 1: Cp = 1— USL-LSL = 6 sigma— Approximately 0.27% defectives will be made— Process is unstable
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LSLUSLC p
6
LNTL UNTL
LSL USL
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Process Capability
• Case 1: Cp < 1— USL-LSL < 6 sigma— Situation undesirable— Process is yield sensitive— Could produce large number of defectives
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LSLUSLCp
6
LNTL UNTL
LSL USL
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©The McGraw-Hill Companies, Inc., 2004
Process Capability Index,
• Most widely used capability measure
• Measures design versus specification relative to the nominal value
• Based on worst case situation
• Defacto value is 1 and processes with this score is capable
• Scores > 1 indicates 6-sigma subsumed by the inspection limits
• Scores less than 1 will result in an incapable process
pkC
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Process Capability Index, Cpk
3
X-UTLor
3
LTLXmin=C pk
Shifts in Process Mean
Capability Index shows how well parts being produced fit into design limit specifications.
Capability Index shows how well parts being produced fit into design limit specifications.
As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.
As a production process produces items small shifts in equipment or systems can cause differences in production performance from differing samples.
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Types of Statistical Sampling
• Attribute (Go or no-go information)— Defectives refers to the acceptability of product
across a range of characteristics.— Defects refers to the number of defects per unit
which may be higher than the number of defectives.
— p-chart application
• Variable (Continuous)— Usually measured by the mean and the
standard deviation.— X-bar and R chart applications
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UCL
LCL
Samples over time
1 2 3 4 5 6
UCL
LCL
Samples over time
1 2 3 4 5 6
UCL
LCL
Samples over time
1 2 3 4 5 6
Normal BehaviorNormal Behavior
Possible problem, investigatePossible problem, investigate
Possible problem, investigatePossible problem, investigate
Statistical Process Control (SPC) Charts
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Control Limits are based on the Normal Curve
x
0 1 2 3-3 -2 -1z
mean
Standard deviation units or “z” units.
Standard deviation units or “z” units.
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Control Limits
We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations from some x-bar or mean value. Based on this we can expect 99.7% of our sample observations to fall within these limits.
xLCL UCL
99.7%
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Example of Constructing a p-Chart: Required Data
1 100 42 100 23 100 54 100 35 100 66 100 47 100 38 100 79 100 1
10 100 211 100 312 100 213 100 214 100 815 100 3
Sample No. m
No. in eachSample, n
No. of defects found in each sample, D
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Statistical Process Control Formulas:Attribute Measurements (p-Chart)
p =Total Number of Defectives
Total Number of Observationsp =
Total Number of Defectives
Total Number of Observations
ns
)p-(1 p = p n
s)p-(1 p
= p
p
p
z - p = LCL
z + p = UCL
s
s
p
p
z - p = LCL
z + p = UCL
s
s
Given:
Compute control limits:
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1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample
1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample
Sample n Defectives p1 100 4 0.042 100 2 0.023 100 5 0.054 100 3 0.035 100 6 0.066 100 4 0.047 100 3 0.038 100 7 0.079 100 1 0.01
10 100 2 0.0211 100 3 0.0312 100 2 0.0213 100 2 0.0214 100 8 0.0815 100 3 0.03
Example of Constructing a p-chart: Step 1
i
ii nDp
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2. Calculate the average of the sample proportions2. Calculate the average of the sample proportions
0.036=1500
55 = p 0.036=1500
55 = p
3. Calculate the standard deviation of the sample proportion
3. Calculate the standard deviation of the sample proportion
.0188= 100
.036)-.036(1=
)p-(1 p = p n
s .0188= 100
.036)-.036(1=
)p-(1 p = p n
s
Example of Constructing a p-chart: Steps 2&3
mp
nD
p i
i
i
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4. Calculate the control limits4. Calculate the control limits
3(.0188) .036 3(.0188) .036
UCL = 0.0924LCL = -0.0204 (or 0)
UCL = 0.0924LCL = -0.0204 (or 0)
p
p
z - p = LCL
z + p = UCL
s
s
p
p
z - p = LCL
z + p = UCL
s
s
Example of Constructing a p-chart: Step 4
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Example of Constructing a p-Chart: Step 5
5. Plot the individual sample proportions, the average of the proportions, and the control limits
5. Plot the individual sample proportions, the average of the proportions, and the control limits
Example of Constructing a p -Chart
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Observation
p
p
UCL
LCL
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Example of x-bar and R Charts: Required Data
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 51 10.682 10.689 10.776 10.798 10.7142 10.787 10.86 10.601 10.746 10.7793 10.78 10.667 10.838 10.785 10.7234 10.591 10.727 10.812 10.775 10.735 10.693 10.708 10.79 10.758 10.6716 10.749 10.714 10.738 10.719 10.6067 10.791 10.713 10.689 10.877 10.6038 10.744 10.779 10.11 10.737 10.759 10.769 10.773 10.641 10.644 10.72510 10.718 10.671 10.708 10.85 10.71211 10.787 10.821 10.764 10.658 10.70812 10.622 10.802 10.818 10.872 10.72713 10.657 10.822 10.893 10.544 10.7514 10.806 10.749 10.859 10.801 10.70115 10.66 10.681 10.644 10.747 10.728
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Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.
Sample Obs 1 Obs 2 Obs 3 Obs 4 Obs 5 Avg Range1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.86 10.601 10.746 10.779 10.755 0.2593 10.78 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.73 10.727 0.2215 10.693 10.708 10.79 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 10.11 10.737 10.75 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.132
10 10.718 10.671 10.708 10.85 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 10.75 10.733 0.34914 10.806 10.749 10.859 10.801 10.701 10.783 0.15815 10.66 10.681 10.644 10.747 10.728 10.692 0.103
Averages 10.728 0.220400
mx
x mR
R = 160.926 = 3.306
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Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and
Necessary Tabled Values
x Chart Control Limits
UCL = x + A R
LCL = x - A R
2
2
x Chart Control Limits
UCL = x + A R
LCL = x - A R
2
2
R Chart Control Limits
UCL = D R
LCL = D R
4
3
R Chart Control Limits
UCL = D R
LCL = D R
4
3
From Exhibit TN7.7From Exhibit TN7.7
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 1.82
10 0.31 0.22 1.7811 0.29 0.26 1.74
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Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values
10.601
10.856
=).58(0.2204-10.728RA - x = LCL
=).58(0.2204-10.728RA + x = UCL
2
2
10.601
10.856
=).58(0.2204-10.728RA - x = LCL
=).58(0.2204-10.728RA + x = UCL
2
2
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
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Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot Values
0
0.46504
)2204.0)(0(R D= LCL
)2204.0)(11.2(R D= UCL
3
4
0
0.46504
)2204.0)(0(R D= LCL
)2204.0)(11.2(R D= UCL
3
4
0.000
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
RUCL
LCL
R
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Basic Forms of Statistical Sampling for Quality Control
• Acceptance Sampling is sampling to accept or reject the immediate lot of product at hand— Does not necessarily determine quality level— Results subject to sampling error
• Statistical Process Control is sampling to determine if the process is within acceptable limits— Takes steps to increase quality
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Acceptance Sampling
• Purposes— Make decision about (sentence) a product— Ensure quality is within predetermined level?
• Advantages– Economy– Less handling damage– Fewer inspectors– Upgrading of the inspection job– Applicability to destructive testing– Entire lot rejection (motivation for improvement)
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Acceptance Sampling (Continued)
• Disadvantages— Risks of accepting “bad” lots and
rejecting “good” lots— Added planning and documentation— Sample provides less information than
100-percent inspection
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Acceptance Sampling: Single Sampling Plan
A simple goal
Determine:
(1) how many units, n, to sample from a lot, and
(2) the maximum number of defective items, c, that can be found in the sample before the lot is rejected
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Risk
• Acceptable Quality Level (AQL)— Max. acceptable percentage of defectives defined
by producer
• The alpha (Producer’s risk)— The probability of rejecting a good lot
• Lot Tolerance Percent Defective (LTPD)— Percentage of defectives that defines consumer’s
rejection point
• The (Consumer’s risk)— The probability of accepting a bad lot
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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
Percent defective
Pro
bab
ility
of
acce
pta
nce
=.10(consumer’s risk)
a = .05 (producer’s risk)
The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together
The OCC brings the concepts of producer’s risk, consumer’s risk, sample size, and maximum defects allowed together
The shape or slope of the curve is dependent on a particular combination of the four parameters
The shape or slope of the curve is dependent on a particular combination of the four parameters
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Example: Acceptance Sampling Problem
Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.
Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.
Zypercom, a manufacturer of video interfaces, purchases printed wiring boards from an outside vender, Procard. Procard has set an acceptable quality level of 1% and accepts a 5% risk of rejecting lots at or below this level. Zypercom considers lots with 3% defectives to be unacceptable and will assume a 10% risk of accepting a defective lot.
Develop a sampling plan for Zypercom and determine a rule to be followed by the receiving inspection personnel.
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Example: Step 1. What is given and
what is not?
In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.
In this problem, AQL is given to be 0.01 and LTDP is given to be 0.03. We are also given an alpha of 0.05 and a beta of 0.10.
What you need to determine is your sampling plan is “c” and “n.”
What you need to determine is your sampling plan is “c” and “n.”
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Example: Step 2. Determine “c”
First divide LTPD by AQL.First divide LTPD by AQL.LTPD
AQL =
.03
.01 = 3
LTPD
AQL =
.03
.01 = 3
Then find the value for “c” by selecting the value in the TN7.10 “n(AQL)”column that is equal to or just greater than the ratio above.
Then find the value for “c” by selecting the value in the TN7.10 “n(AQL)”column that is equal to or just greater than the ratio above.
Exhibit TN 7.10Exhibit TN 7.10
c LTPD/AQL n AQL c LTPD/AQL n AQL0 44.890 0.052 5 3.549 2.6131 10.946 0.355 6 3.206 3.2862 6.509 0.818 7 2.957 3.9813 4.890 1.366 8 2.768 4.6954 4.057 1.970 9 2.618 5.426
So, c = 6.So, c = 6.
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Example: Step 3. Determine Sample Size
c = 6, from Tablen (AQL) = 3.286, from TableAQL = .01, given in problem
c = 6, from Tablen (AQL) = 3.286, from TableAQL = .01, given in problem
Sampling Plan:Take a random sample of 329 units from a lot. Reject the lot if more than 6 units are defective.
Sampling Plan:Take a random sample of 329 units from a lot. Reject the lot if more than 6 units are defective.
Now given the information below, compute the sample size in units to generate your sampling plan
Now given the information below, compute the sample size in units to generate your sampling plan
n(AQL/AQL) = 3.286/.01 = 328.6, or 329 (always round up)n(AQL/AQL) = 3.286/.01 = 328.6, or 329 (always round up)
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Standard Table of Sampling Plans
• MIL-STD-105D— For attribute sampling plans— Needs to know:
The lot size N The inspection level (I, II, III) The AQL Type of sampling (single, double,
multiple) Type of inspection (normal, tightened,
reduced)
• Find a code letter then read plan from Table
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Standard Table of Sampling Plans:Single Sampling Plan
• Example: If N=2000 and AQL=0.65% find the normal, tightened, and reduced single sampling plan using inspection level II.