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Process Capability Assessment. Process Capability vs. Process Control. Evaluating Process Performance Ability of process to produce parts that conform to engineering specifications (CONFORMANCE) Ability of process to maintain a state of statistical control; i.e., be within control limits - PowerPoint PPT Presentation
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Process Capability AssessmentProcess Capability Assessment
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Process Capability vs. Process ControlProcess Capability vs. Process Control
Evaluating Process Performance– Ability of process to produce parts that
conform to engineering specifications
(CONFORMANCE)(CONFORMANCE)– Ability of process to maintain a state of
statistical control; i.e., be within control limits
(CONTROL)(CONTROL)
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– Process Capability…dealing with individual measurements, e.g., X
(LSL, USL)(LSL, USL)
Linkages Between Process Control & Linkages Between Process Control & Process CapabilityProcess Capability
Process must be in statistical control before assessing process capability. Why?
Statistical aspects– Process Control…use summary statistics from a
sample (subgroup); dealing with sampling distributions, e.g., and R
(LCL, UCL)(LCL, UCL)X
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Process may be in statistical control, but not capable (of meeting specifications)
– Process is off-center from nominal (bias)
– Process variability is too large relative to specifications (variation)
– Process is both off-center and has large variation.
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Relationship Between Process Relationship Between Process Variability and Product SpecificationVariability and Product Specification
(a) Process variation is small relative to the specifications so that the process mean can shift about without causing the process to degrade its capability. This will reduce the defects per million (DPM), reduce the cost of quality (COQ), and hence increase profitability.
Upper Specification
Lower Specification
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Relationship Between Process Relationship Between Process Variability and Product SpecificationVariability and Product Specification
(b) Process variation is large relative to the specifications such that the process must remain well centered for the process capability to be maintained at a tolerable level. Variation, however, must be reduced. This will increase process capability, reduce DPMs, reduce COQ, and increase profitability.
Upper Specification
Lower Specification
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Relationship Between Process Relationship Between Process Variability and Product SpecificationVariability and Product Specification
(c) Process variation is large relative to the specifications so that the process cannot be considered capable regardless of the process centering. Hence we have a severe and urgent problem. Process variation must be reduced drastically.
Upper Specification
Lower Specification
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Statistical Assessment of Process Statistical Assessment of Process CapabilityCapability
Get Process in Statistical Control
Statistical Assessment (Minitab or Excel)– Construct histogram of individual measurements
– Compute probability of exceeding specifications P(•)• Empirically (observed)
• Mathematically: … assume N(,) and compute (expected)
• Convert to defects per million (DPM) and sigma capability
• Compute process capability indices … Cp, Cpk
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Alternatives for Improving Process Alternatives for Improving Process CapabilityCapability
If bias– Recenter and recompute P(•), dpm, and sigma
capability If too much variation
– Sort by 100% inspection – Widen tolerance – Use a more precise process (e.g., better or new
technology) to reduce variation – Use statistical methods to identify variation reduction
opportunities for existing process
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Summary: Comparison of Specification Summary: Comparison of Specification Limits and Control LimitsLimits and Control Limits
Spec limits or tolerances for product quality
characteristics are:– Characteristic of the part/item (product) in question
– Based on functional design considerations
– Related to/compared with an individual part
measurement
– Used to establish a part’s conformance to design intent
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Control limitsControl limits on a control chart on a control chart are:are:
Characteristic of the process in question
Based on the process mean and variation
Dependent on sampling parameters, viz., sample
size and -risk (Type I error)
Used to identify presence/absence of special-cause
variation in the process
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Some Common Indices of Process CapabilitySome Common Indices of Process Capability
Cp Formula
x
p 6
LSLUSLC
Specification Range
Variation of Distribution of Individual Product
USL ~ LSLUSL ~ LSL
N(N(XX, , XX))
rejectrejectrejectreject
-3X(1) +3X(1)
(1)
TTXX
-3X(2)+3X(2)
(2)Cp(1) < Cp(2)
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Relationship Between CRelationship Between Cpp, DPM, and , DPM, and
Sigma of Process (Assumes No Bias)Sigma of Process (Assumes No Bias)Cp = (USL-LSL) / 6 *Sigma of
Process = 3Cp
DPM2-sided spec limits
DPM1-sided spec limits
Remarks
0 0 Very high Very High Worst Case: Itemsproduced haveenormous variation
0.5 1.50 133,615 66,8080.75 2.25 24,449 12,2251.00 3.00 2,700 1,350 Minimally
Acceptable Case1.25 3.75 177 891.33 4.00 63 31.51.50 4.50 7 3.51.75 5.25 0.2 0.12.00 6.00 0.0 0.0 Motorola 0.0 0.0 Ideal Case: Each item
produced right ontarget (cloned)
*DPM = (Probability of Exceeding Specs) * 106
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CCpkpk
Purpose: To promote adherence of process mean to target (nominal) value of spec.
Formulas:
,USL
ZX
XUSL
X
XLSL
LSLZ
]Z,Z[MINZ LSLUSLmin
3/ZC minpk
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ExampleExampleUSLUSLLSLLSL
100100 190 x190 xN(130, 10)N(130, 10) T = 145T = 145
((XX-T) = bias-T) = bias
50.1)10(6
100190Cp
00.13/ZC
3)]3(,6[MINZ
310
130100Z
610
130190Z
)Bias(:C
minpk
min
LSL
USL
pk
50.13/ZC
5.4)]5.4(,5.4[MINZ
5.410
145100Z
5.410
145190Z
)Centering:Unbias(:C
minpk
min
LSL
USL
pk
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Questions?Questions?
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PCB Exercise (USL = +8, LSL = -8)PCB Exercise (USL = +8, LSL = -8)
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Microeconomics of Quality: Loss Microeconomics of Quality: Loss Due to Variation (Taguchi)Due to Variation (Taguchi)
Linking Cost of Quality Due to Bias and Variation to DPM and Process Capability in PCB Manufacture
Variation is Related to Functional Form (Distribution) of Process Output
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Loss Function Representation of Loss Function Representation of Quality for PCBsQuality for PCBs
LSLLSL(-8 microns)(-8 microns)
Target (T)Target (T)(Normal)(Normal)
USLUSL(+8 microns)(+8 microns)
Quality LossQuality Loss
Reject Reject (Scrap)(Scrap)
Reject Reject (Scrap)(Scrap)
•Failure Cost @ Failure Cost @ USL = $2.40/unitUSL = $2.40/unit•Failure Rate Failure Rate (Probability) = (Probability) = 100%100%
Probability Distribution of Quality Characteristic Produced by Process
Quadratic Loss Function
Poor
FairGood
BestGood
Fair
Poor
X (Microns)X (Microns)
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Loss Function ApproachLoss Function Approach Measured loss Function: LM(X)
LM(X) = k (x - T)2
where k is an unknown constantx is a value of the quality characteristicT is the target
Determining the Constant kLM(x) @ USL = k (USL - T)2
where LM(x) @ USL is a known measured cost of scrap(=Cost of a failure @ USL * failure rate (probability))
USL & T are knownk = (2.40)(1.0)/(8)2
=0.0375
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Total Loss Function (Measured + Hidden): LT(x)
LT(x) = ak (x - T)2
where a is the hidden “cost of quality” multiplier
(6 < a < 50)
If we assume a = 28, then ak = 28 * 0.0375 = 1.05
Evaluation of Expected Total Process loss: ET{L(x)}
ET{L(x)} = ak {x2 + (x - T2)}
Where x2 is process variance and
(x - T) is process bias (mean from target)
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Effect of Bias & Variation (Variance) of Process Variable on Cost of Quality in
Millions of Dollars
(ak = 1.05; produce 10 million units/year)
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Questions?Questions?
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(x -T) = 0
x = 6
ET {L(x)} = 1.05 {62 + 02} * 107 = $378 million
Figure 2. Evaluation of Quality Loss Function (N(0,62))
TT
Loss ($)Loss ($)
LSLLSL USLUSL x (microns)
Probability Probability DistributionDistribution
Loss Loss FunctionFunction
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Analysis (Figure 2.) Analysis (Figure 2.)
Process Capability
No. Standard Deviation from Mean (z)
z = 3Cp = 3 (0.444) = 1.332
This is a 1.332 Process.
444.0)6(6
16
6
LSLUSLCp
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Defects Per Million (dpm): ExcelDefects Per Million (dpm): Excel
dpm = (1-NORMSDIST(1.332)) * 2 * 10 ^ 6 = 182,423 dpm
MMLSLLSL-1.332-1.332
RR RR
USLUSL+1.332+1.332
AA
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(x -T) = 0
x = 2
ET {L(x)} = 1.05 {22 + 02} * 107 = $42 million
Figure 3. Evaluation of Quality Loss Function (N(0,22))
TT
Loss ($)Loss ($)
LSLLSL USLUSL x (microns)
Probability Probability DistributionDistribution
Loss Loss FunctionFunction
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Analysis (Figure 3)Analysis (Figure 3)
Process Capability
No. Standard Deviation from Mean (z)
z = 3Cp = 3 (1.333) = 4.000
This is a 4 Process.
333.1)2(6
16
6
LSLUSLCp
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Defects Per Million (dmp): Excel
dpm = (1-NORMSDIST(4.000)) * 2 * 106
= 63 dpm
Expected Cost Change (ECC)
9
1
36
4
378
42
)}7({
)}2({
LE
LEECC
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ET {L(x)} = 1.05 {22 + 22} * 107 = $84 million
Figure 4. Quality Loss Consequences of Shifting the Process Mean Toward the Upper Specification
TTLSLLSL USLUSLx (microns)(x -T) = 2
x = 2
(x -T) = 2
(x -T) = 2
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Analysis (Figure 4)Analysis (Figure 4)
Process Capability
333.1)2(6
16
6
LSLUSLCp
1
}67.1;1{Value Abs Min
)2(3
)2(8;
)2(3
28Value Abs Min
3;
3Value Abs Min
LSLUSLCpk
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Defects Per Million (dpm): Excel
dpm = (1-NORMSDIST(3))+ NORMSDIST (-5.01)) * 10^6
= 1349.97 + .27 = 1350.24
Expected Cost Change (ECC)
9
2
36
8
378
84
)}1({
)}3({
LE
LEECC
No. Standard Deviation from Mean (z)
ZUSL = 3 (1) = 3
ZLSL = 3(-1.67) = -5.01
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Questions?Questions?
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Normal Distribution: N(0, 2.672)
x (microns)
Los
s ($)
LSL USL(x -T) = 0x = 2.67
(Note: 3x = 8, thus x = 8/3 = 2.67)
ET{L(x)} = 1.05 {2.672 + 02} * 107 = $74.85 million
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Uniform Distribution: U(0, 4.622)
x (microns)
Los
s ($)
LSL USL(x -T) = 0x = 4.62
(Note: x2 = (b-a)2/12 = 21.33; x = 4.62)
ET{L(x)} = 1.05 {4.622 + 02} * 107 = $224.12 million
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Analysis (Figure 5)Analysis (Figure 5)
Process Capability
Normal Distribution
000.1)67.2(6
16
6
LSLUSLCp
Uniform Distribution (Inspection; Adjustment)
577.0)62.4(6
16
6
LSLUSLCp
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Defects Per Million (dmp): Excel
Normal Distribution
dpm = (1-NORMSDIST(3.000)* 2 * 106 = 2700
Uniform Distribution
Theoretically, there are no units exceeding the specification limits; however, there are many more units further away from the target value (T) than with a normal distribution. This accounts for the higher variance (4.622 vs 2.672) and, as we shall see, higher cost of quality.
No. Standard Deviation from Mean (z)
Normal Distribution Z = 3 Cp= 3 (1.000) = 3.000 This is a 3 Process.
Uniform Distribution Z = 3 Cp= 3 (0.577) = 1.731 Process
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Cost of QualityCost of QualityNormal (N) Distribution
E{L(x)} = ak {2 + ( - T)2} = ak * 107 {2.672 + 02} = ak * 107 * 7.13
Uniform (U) DistributionE{L(x)} = ak * 107 { 4.622 + 02}
= ak * 107 * 21.34
Expected Cost Change (ECC): Normal vs UniformExpected Cost Change (ECC): Normal vs Uniform
334.034.21
13.7
)}U(L{E
)}N(L{EECC
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SummarySummary
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Comparison of Process (1 vs 2)Comparison of Process (1 vs 2)
(. , .) ( - T), bias, variation
AA(0, 6)(0, 6)
BB(0, 2)(0, 2)Process
T = 0
Max
Min
Max
Min
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Comparison of Process (1 vs 3)Comparison of Process (1 vs 3)
AA(0, 6)(0, 6)
CC(2, 2)(2, 2)Process
T = 0
Max
Min
Max
Min
(. , .) ( - T), bias, variation
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Questions?Questions?
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Understanding the DifferencesUnderstanding the Differences
3 Capability Historical Standard
4 Capability Current Standard
6 Capability New Standard
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Understanding the DifferencesUnderstanding the Differences
3Floor space of a small hardware store
1.5 misspelled words per page in a book
$2.7 million indebtedness per $1 billion in assets
3 1/2 months per century
Coast-to-coast trip
4Floor space of a typical living room
1 misspelled word per 30 pages in a book
$63,000 indebtedness per $1 billion in assets
2 1/2 days per century
45 minutes of freeway driving (in any direction)
5Size of the bottom of your telephone
1 misspelled word in a set of encyclopedias
$570 indebtedness per $1 billion in assets
30 minutes per century
A trip to the local gas station
Sigma Area Spelling Money Time Distance
6Size of a typical diamond
1 misspelled word in all of the books contained in a small library
$2 indebtedness per $1 billion assets
6 seconds per century
4 steps in any direction
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Understanding the DifferenceUnderstanding the Difference
Suppose a process produced 294,118 units of product. If the process capability was 4then the defects produced could be represented by the matrix of dots given below. If the capability was 6, only one dot would appear in the entire matrix.
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Understanding the DifferenceUnderstanding the Difference• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
4 Capability: Defect Dots = 1849
6 Capability: Defect Dots = 1
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The Inspection ExerciseThe Inspection ExerciseTask: Count the number of times the 6th letter of the alphabet appears in the following text.
The Necessity of Training Farm Hands for First Class Farms in the Fatherly Handling of Farm Live Stock is Foremost in the Eyes of Farm Owners. Since the Forefathers of the Farm Owners Trained the Farm Hands for First Class Farms in the Fatherly Handling of Farms Live Stock, the Farm Owners Feel they should Carry on with the Family Tradition of Training Farm Hands of First Class Farmers in the Fatherly Handling of Farm Live Stock Because They Believe it is the basis of Good Fundamental Farm Management.
Questions?Questions?