Statistical Process Control A. A. Elimam A. A. Elimam.
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Transcript of Statistical Process Control A. A. Elimam A. A. Elimam.
Statistical ProcessStatistical ProcessControlControl
A. A. ElimamA. A. Elimam
Two Primary Topics in Two Primary Topics in Statistical Quality ControlStatistical Quality Control
Statistical process control (SPC) is a statistical method using control charts to check a production process - prevent poor quality. In TQM all workers are trained in SPC methods.
Two Primary Topics in Two Primary Topics in Statistical Quality ControlStatistical Quality Control
Acceptance Sampling involves inspecting a sample of product. If sample fails reject the entire product - identifies the products to throw away or rework. Contradicts the philosophy of TQM. Why ?
InspectionInspection Traditional Role: at the beginning and end
of the production process Relieves Operator from the responsibility of
detecting defectives & quality problems It was the inspection's job
In TQM, inspection is part of the process & it is the operator’s job
Customers may require independent inspections
How Much to Inspect?How Much to Inspect?
Complete or 100 % Inspection.• Viable for products that can cause safety
problems• Does not guarantee catching all defectives• Too expensive for most cases
Inspection by Sampling• Sample size : representative• A must in destructive testing (e.g...
Tasting food)
Where To Inspect ?Where To Inspect ?
In TQM , inspection occurs throughout the production process
IN TQM, the operator is the inspector Locate inspection where it has the most
effect (e.g.... prior to costly or irreversible operation)
Early detection avoids waste of more resources
Quality Testing Quality Testing Destructive Testing
• Product cannot be used after testing (e.g.. taste or breaking item)
• Sample testing
• Could be costly Non-Destructive Testing
• Product is usable after testing
• 100% or sampling
Quality Measures:AttributesQuality Measures:Attributes
• Attribute is a qualitative measure
• Product characteristics such as color, taste, smell or surface texture
• Simple and can be evaluated with a discrete response (good/bad, yes/no)
• Large sample size (100’s)
Quality Measures:VariablesQuality Measures:Variables
• A quantitative measure of a product characteristic such as weight, length, etc.
• Small sample size (2-20)
• Requires skilled workers
Variation & Process Control ChartsVariation & Process Control Charts
Variation always exists Two Types of Variation
• Causal: can be attributed to a cause. If we know the cause we can eliminate it.
• Random: Cannot be explained by a cause. An act of nature - need to accept it.
Process control charts are designed to detect causal variations
Control Charts: Definition & TypesControl Charts: Definition & Types
A control chart is a graph that builds the control limits of a process
Control limits are the upper and lower bands of a control chart
Types of Charts:• Measurement by Variables: X-bar and R
charts• Measurement by Attributes: p and c
Process Process ControlControl Chart Chart & Control Criteria& Control Criteria
1. No sample points outside control limits.
2. Most points near the process average.
3. Approximately equal No. of points above
& below center.
4. Points appear to be randomly distributed
around the center line.
5. No extreme jumps.
6. Cannot detect trend.
Basis of Control ChartsBasis of Control Charts
Specification Control Charts• Target Specification: Process Average
• Tolerances define the specified upper and lower control limits
• Used for new products (historical measurements are not available)
Historical Data Control Charts • Process Average, upper & lower control limits:
based on historical measurements• Often used in well established processes
Common CausesCommon Causes
xx
n
ii
n
1
x x
ni
2
1
425 Grams
Assignable CausesAssignable Causes
(a) LocationGrams
Average
Assignable CausesAssignable Causes
(b) SpreadGrams
Average
Assignable CausesAssignable Causes
(b) SpreadGrams
Average
Assignable CausesAssignable Causes
(c) ShapeGrams
Average
Effects of Assignable Causes on Effects of Assignable Causes on Process ControlProcess Control
Assignable Assignable causes presentcauses present
Effects of Assignable Effects of Assignable Causes on Process ControlCauses on Process Control
No No assignable causesassignable causes
Sample Means and theSample Means and theProcess DistributionProcess Distribution
425 Grams
Mean
Processdistribution
Distribution ofsample means
The NormalThe NormalDistributionDistribution
-3 -2 -1 +1 +2 +3Mean
68.26%95.44%99.97%
= Standard deviation
Control Charts
UCL
Nominal
LCL
Assignable causes likely
1 2 3Samples
Using Control Charts for Using Control Charts for Process ImprovementProcess Improvement
Measure the processMeasure the process When problems are indicated, When problems are indicated,
find the assignable causefind the assignable cause Eliminate problems, incorporate Eliminate problems, incorporate
improvementsimprovements Repeat the cycleRepeat the cycle
Control Chart Examples
Nominal
UCL
LCL
Sample number(a)
Var
iati
on
s
Control Chart Examples
Nominal
UCL
LCL
Sample number(b)
Var
iati
on
s
Control Chart Examples
Nominal
UCL
LCL
Sample number(c)
Var
iati
on
s
Control Chart Examples
Nominal
UCL
LCL
Sample number(d)
Var
iati
on
s
Control Chart Examples
Nominal
UCL
LCL
Sample number(e)
Var
iati
on
s
The Normal DistributionThe Normal DistributionMeasures of Variability:
• Most accurate measure
= Standard Deviation
• Approximate Measure - Simpler to compute
R = Range
• Range is less accurate as the sample size
gets larger
Average = Average R when n = 2
Control Limits and Errors
LCL
Processaverage
UCL
(a) Three-sigma limits
Type I error:Probability of searching for a cause when none exists
Control Limits and Errors
Type I error:Probability of searching for a cause when none exists
UCL
LCL
Processaverage
(b) Two-sigma limits
Type II error:Probability of concludingthat nothing has changed
Control Limits and Errors
UCL
Shift in process average
LCL
Processaverage
(a) Three-sigma limits
Type II error:Probability of concludingthat nothing has changed
Control Limits and Errors
UCL
Shift in process average
LCL
Processaverage
(b) Two-sigma limits
Control ChartsControl Chartsfor Variablesfor Variables
Mandara Mandara IndustriesIndustries
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5032 0.5020
3 0.5018 0.5026 0.5035 0.5023
4 0.5008 0.5034 0.5024 0.5015
5 0.5041 0.5056 0.5034 0.5039
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5032 0.5020
3 0.5018 0.5026 0.5035 0.5023
4 0.5008 0.5034 0.5024 0.5015
5 0.5041 0.5056 0.5034 0.5039
0.5027 - 0.50090.5027 - 0.5009 == 0.00180.0018
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027 0.0018
2 0.5021 0.5041 0.5032 0.5020
3 0.5018 0.5026 0.5035 0.5023
4 0.5008 0.5034 0.5024 0.5015
5 0.5041 0.5056 0.5034 0.5039
0.5027 - 0.50090.5027 - 0.5009 == 0.00180.0018
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5032 0.5020
3 0.5018 0.5026 0.5035 0.5023
4 0.5008 0.5034 0.5024 0.5015
5 0.5041 0.5056 0.5034 0.5039
0.5027 - 0.50090.5027 - 0.5009 == 0.00180.0018(0.5014 + 0.5022 +(0.5014 + 0.5022 + 0.5009 + 0.5027)/40.5009 + 0.5027)/4 == 0.50180.5018
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5032 0.5020
3 0.5018 0.5026 0.5035 0.5023
4 0.5008 0.5034 0.5024 0.5015
5 0.5041 0.5056 0.5034 0.5039
0.5027 - 0.50090.5027 - 0.5009 == 0.00180.0018(0.5014 + 0.5022 +(0.5014 + 0.5022 + 0.5009 + 0.5027)/40.5009 + 0.5027)/4 == 0.50180.5018
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 Range Mean
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5032 0.5020 0.0021 0.5029
3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026
4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020
5 0.5041 0.5056 0.5034 0.5039 0.0022 0.5043
R = 0.0020
x = 0.5025
Special Metal Screw
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R - Charts R = 0.0020
UCLR = D4RLCLR = D3R
Control Charts for VariablesControl Charts for Variables
Control Charts - Special Metal Screw
R - Charts R = 0.0020 D4 = 2.2080
Control Chart FactorsControl Chart Factors
Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts
((nn)) ((AA22)) ((DD33)) ((DD44))
22 1.8801.880 00 3.2673.26733 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R - Charts R = 0.0020 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0020) = 0.00456 in.LCLR = 0 (0.0020) = 0 in.
UCLR = D4RLCLR = D3R
0.005
0.004
0.003
0.002
0.001
0 1 2 3 4 5 6
Ran
ge
(in
.)
Sample number
UCLR = 0.00456
LCLR = 0
R = 0.0020
Range Chart - Special Metal Screw
Control Charts for Variables
Control Charts - Special Metal Screw
R = 0.0020x = 0.5025
x - Charts
UCLx = x + A2RLCLx = x - A2R
Control Chart FactorsControl Chart Factors
Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts
((nn)) ((AA22)) ((DD33)) ((DD44))
22 1.8801.880 00 3.2673.26733 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R = 0.0020 A2 = 0.729x = 0.5025
x - Charts
UCLx = x + A2RLCLx = x - A2R
UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 in.
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts - Special Metal Screw
R = 0.0020 A2 = 0.729x = 0.5025
x - Charts
UCLx = x + A2RLCLx = x - A2R
UCLx = 0.5025 + 0.729 (0.0020) = 0.5040 in.LCLx = 0.5025 - 0.729 (0.0020) = 0.5010 in.
0.5050
0.5040
0.5030
0.5020
0.5010
1 2 3 4 5 6
Ave
rag
e (i
n.)
Sample number
x = 0.5025
UCLx = 0.5040
LCLx = 0.5010
Average Chart - Special Metal Screw
0.5050
0.5040
0.5030
0.5020
0.5010
Ave
rag
e (i
n.)
x = 0.5025
UCLx = 0.5040
LCLx = 0.5010
1 2 3 4 5 6Sample number
Measure the process Find the assignable cause Eliminate the problem Repeat the cycle
Average Chart - Special Metal Screw
Control ChartsControl Chartsfor Attributesfor Attributes
MANDARA BankMANDARA Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp - - zzpp
pp = = pp(1 - (1 - pp))//nn
MANDARA BankMANDARA Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp - - zzpp
pp = = pp(1 - (1 - pp))//nn
Sample Wrong ProportionNumber Account Number Defective
1 15 0.0062 12 0.00483 19 0.00764 2 0.00085 19 0.00766 4 0.00167 24 0.00968 7 0.00289 10 0.004
10 17 0.006811 15 0.00612 3 0.0012
Total 147
p = 0.0049
n = 2500
Control Charts for AttributesControl Charts for Attributes
Control ChartsControl Chartsfor Attributesfor Attributes
MANDARA BankMANDARA Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp - - zzpp
pp = 0.0049(1 - 0.0049)/2500 = 0.0049(1 - 0.0049)/2500
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
MANDARA BankMANDARA Bank
UCLUCLpp = = pp + + zzpp
LCLLCLpp = = pp - - zzpp
pp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
MANDARA BankMANDARA Bank
UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)
LCLLCLpp = 0.0049 - 3(0.0014) = 0.0049 - 3(0.0014)
pp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Control ChartsControl Chartsfor Attributesfor Attributes
MANDARA BankMANDARA Bank
UCLUCLpp = 0.0091 = 0.0091
LCLLCLpp = 0.0007 = 0.0007
pp = 0.0014 = 0.0014
n = 2500 p = 0.0049
1 2 3 4 5 6 7 8 9 10 11 12 13
Sample number
UCL
p
LCL
0.011
0.010
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0Pro
po
rtio
n d
efec
tive
in s
amp
lep-Chart
Wrong Account Numbers
1 2 3 4 5 6 7 8 9 10 11 12 13
Sample number
UCL
p
LCL
0.011
0.010
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0Pro
po
rtio
n d
efec
tive
in s
amp
lep-Chart
Wrong Account Numbers
Measure the process Find the assignable cause Eliminate the problem Repeat the cycle
Process CapabilityProcess Capability
Nominalvalue
80 100 120 Hours
Upperspecification
Lowerspecification
Process distribution
(a) Process is capable
Process CapabilityProcess Capability
Nominalvalue
80 100 120 Hours
Upperspecification
Lowerspecification
Process distribution
(b) Process is not capable
Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Two sigma
Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Four sigma
Two sigma
Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Six sigma
Four sigma
Two sigma
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Process CapabilityProcess CapabilityLight-bulb Production
Cp =
Upper specification - Lower specification
6s
Process Capability RatioProcess Capability Ratio
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 120 - 80
6(4.8)
Process Capability RatioProcess Capability Ratio
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 1.39
Process Capability RatioProcess Capability Ratio
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 1.39
Cpk = Minimum of
Upper specification - x
3s
x - Lower specification
3s
ProcessProcessCapabilityCapabilityIndexIndex
,
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 1.39
Cpk = Minimum of
120 - 90
3(4.8)
90 - 80
3(4.8)
ProcessProcessCapabilityCapabilityIndexIndex
,
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 1.39
Cpk = Minimum of [ 0.69, 2.08 ]
ProcessProcessCapabilityCapabilityIndexIndex
Process CapabilityProcess CapabilityLight-bulb Production
Upper specification = 120 hoursLower specification = 80 hoursAverage life = 90 hours s = 4.8 hours
Cp = 1.39Cpk = 0.69
ProcessProcessCapabilityCapabilityIndexIndex
ProcessProcessCapabilityCapabilityRatioRatio