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To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Chapter 7 -Chapter 7 -
StatisticalStatisticalProcessProcessControlControl
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Common CausesCommon Causes
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Common CausesCommon Causes
Figure 7.1
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Common CausesCommon Causes
Figure 7.1
x =xi
i=1
n
∑n
σ =xi − x ( )∑
2
n−1
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x =xi
i=1
n
∑n
σ =xi − x ( )∑
2
n−1
Common CausesCommon Causes
425 GramsFigure 7.1
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Assignable CausesAssignable Causes
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Assignable CausesAssignable Causes
(a) LocationGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(a) LocationGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(a) LocationGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(b) SpreadGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(b) SpreadGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(c) ShapeGrams
Average
Figure 7.2
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Assignable CausesAssignable Causes
(c) ShapeGrams
Average
Figure 7.2
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Effects of Assignable Effects of Assignable Causes on Process ControlCauses on Process Control
Out of controlOut of control(assignable causes present)(assignable causes present) Figure 7.3 (a)
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Effects of Assignable Effects of Assignable Causes on Process ControlCauses on Process Control
In ControlIn Control(no assignable causes)(no assignable causes) Figure 7.3 (b)
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Sample Means and theSample Means and theProcess DistributionProcess Distribution
Figure 7.4 425 Grams
Mean
Processdistribution
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Sample Means and theSample Means and theProcess DistributionProcess Distribution
Processdistribution
Figure 7.4 425 Grams
MeanDistribution ofsample means
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The NormalThe NormalDistributionDistribution
Figure 7.5
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The NormalThe NormalDistributionDistribution
Mean
σ = Standard deviation
Figure 7.5
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The NormalThe NormalDistributionDistribution
Mean
68.26%
σ = Standard deviation
Figure 7.5
–1σ+1σ
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The NormalThe NormalDistributionDistribution
Mean
68.26%95.44%
σ = Standard deviation
Figure 7.5
–2σ–1σ+1σ+2σ
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The NormalThe NormalDistributionDistribution
–3σ–2σ–1σ+1σ+2σ+3σMean
68.26%95.44%99.74%
σ = Standard deviation
Figure 7.5
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Control ChartsControl Charts
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Control ChartsControl Charts
UCL
Nominal
LCL
Figure 7.6
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Control ChartsControl Charts
UCL
Nominal
LCL
1 2 3SamplesFigure 7.6
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Control ChartsControl Charts
UCL
Nominal
LCL
1 2 3SamplesFigure 7.6
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Control ChartsControl Charts
UCL
Nominal
LCL
Assignable causes likely
1 2 3SamplesFigure 7.6
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Using Control Charts for Using Control Charts for Process ImprovementProcess Improvement
Measure the process When changes are indicated,
find the assignable cause Eliminate problems, incorporate
improvements Repeat the cycle
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Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
Figure 7.7 (a)
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Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
Figure 7.7 (b)
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Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
Figure 7.7 (c)
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Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
Figure 7.7 (d)
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Control Chart ExamplesControl Chart Examples
Nominal
UCL
LCL
Sample number
Var
iati
on
s
Figure 7.7 (e)
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Control Limits and ErrorsControl Limits and Errors
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Control Limits and ErrorsControl Limits and Errors
LCL
Processaverage
UCL
Figure 7.8 (a) Three-sigma limits
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Control Limits and ErrorsControl Limits and Errors
LCL
Processaverage
UCL
Three-sigma limits
Type I error:Probability of searching for a cause when none exists
Figure 7.8 (a)
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Control Limits and ErrorsControl Limits and Errors
LCL
Processaverage
UCL
Three-sigma limits
Type I error:Probability of searching for a cause when none exists
Type II error:Probability of concludingthat nothing has changed
Shift in process average
Figure 7.8 (a)
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Control Limits and ErrorsControl Limits and Errors
UCL
LCL
Processaverage
Figure 7.8 (b) Two-sigma limits
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Control Limits and ErrorsControl Limits and ErrorsType I error:Probability of searching for a cause when none exists
Figure 7.8 (b) Two-sigma limits
UCL
LCL
Processaverage
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Control Limits and ErrorsControl Limits and ErrorsType I error:Probability of searching for a cause when none exists
Figure 7.8 (b) Two-sigma limits
Type II error:Probability of concludingthat nothing has changed
Shift in process average
UCL
LCL
Processaverage
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Control ChartsControl Chartsfor Variablesfor Variables
West Allis IndustriesWest Allis Industries
Example 7.1
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4
1
2
3
4
5
Special Metal Screw
Example 7.1
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5024 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 0.5047
Special Metal Screw
Example 7.1
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5024 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
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5024 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
Example 7.1
_
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027
2 0.5021 0.5041 0.5024 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
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027 0.0018
2 0.5021 0.5041 0.5024 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
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5024 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
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5024 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
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027
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 0.5047 0.0022 0.5045
Special Metal Screw
Example 7.1
_
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Control ChartsControl Chartsfor Variablesfor Variables
Sample Sample
Number 1 2 3 4 R x
1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018
2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027
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 0.5047 0.0022 0.5045
R = 0.0021
x = 0.5027
Special Metal Screw
Example 7.1
=
_
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts – Special Metal Screw
R-Charts R = 0.0021
UCLR = D4RLCLR = D3R
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables 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 0 0 3.2673.26733 1.0231.023 0 0 2.5752.57544 0.7290.729 0 0 2.2822.28255 0.5770.577 0 0 2.1152.11566 0.4830.483 0 0 2.0042.00477 0.4190.419 0.076 0.076 1.9241.924
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Example 7.1
Control ChartsControl Chartsfor Variablesfor 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 0 0 3.2673.26733 1.0231.023 0 0 2.5752.57544 0.7290.729 0 0 2.2822.28255 0.5770.577 0 0 2.1152.11566 0.4830.483 0 0 2.0042.00477 0.4190.419 0.076 0.076 1.9241.924
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = D4RLCLR = D3R
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.
UCLR = D4RLCLR = D3R
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.
UCLR = D4RLCLR = D3R
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
R-Charts R = 0.0021 D4 = 2.282D3 = 0
UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.
UCLR = D4RLCLR = D3R
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Range Chart - Range Chart - Special Metal ScrewSpecial Metal Screw
Figure 7.9
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
X-Charts
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021x = 0.5027=
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.1
Control ChartsControl Chartsfor Variablesfor 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
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Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x- Charts
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x-Charts
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.1
Control ChartsControl Chartsfor Variablesfor Variables
Control Charts—Special Metal Screw
x-Charts
UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.
UCLx = x + A2RLCLx = x - A2R
==
R = 0.0021 A2 = 0.729x = 0.5027=
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xx-Chart—-Chart—Special Metal ScrewSpecial Metal Screw
Figure 7.10
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xx-Chart— -Chart— Special Metal ScrewSpecial Metal Screw
Figure 7.10
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xx-Chart—-Chart—Special Metal ScrewSpecial Metal Screw
Figure 7.10
Measure the process Find the assignable cause Eliminate the problem Repeat the cycle
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Control ChartsControl Charts
for Variables Using for Variables Using σσ
Example 7.2
UCLUCLxx = = xx + + zzσσxx
LCLLCLxx = = xx – – zzσσxx
σx = σ/n
==
==
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Control ChartsControl Charts
for Variables Using for Variables Using σσ
Example 7.2
UCLUCLxx = = xx + + zzσσxx
LCLLCLxx = = xx – – zzσσxx
σx = σ/n
==
==
Sunny Dale Bank
x = 5.0 minutesσ = 1.5 minutesn = 6 customersz = 1.96
=
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Control ChartsControl Charts
for Variables Using for Variables Using σσ
UCLx = 5.0 + 1.96(1.5)/ 6 = 6.20 min
UCLx = 5.0 – 1.96(1.5)/ 6 = 3.80 minExample 7.2
UCLUCLxx = = xx + + zzσσxx
LCLLCLxx = = xx – – zzσσxx
σx = σ/n
==
==
Sunny Dale Bank
x = 5.0 minutesσ = 1.5 minutesn = 6 customersz = 1.96
=
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Control ChartsControl Chartsfor Attributesfor Attributes
HOMETOWN BANK
Hometown BankHometown Bank
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Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp – – zzσσpp
σσpp = = pp(1 – (1 – pp))//nn
Example 7.3
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Example 7.3
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp - - zzσσpp
σσpp = = pp(1 - (1 - pp))//nn
Sample WrongNumber Account
Number
1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3
Total 147
Total defectives
Total observationsp =
n = 2500
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Example 7.3
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp - - zzσσpp
σσpp = = pp(1 - (1 - pp))//nn
Sample WrongNumber Account Number
1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3
Total 147
147
12(2500)p =
n = 2500
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.3
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp - - zzσσpp
σσpp = = pp(1 - (1 - pp))//nn
Sample WrongNumber Account Number
1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3
Total 147
p = 0.0049
n = 2500
To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.
Example 7.3
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp - - zzσσpp
σσpp = = pp(1 - (1 - pp))//nn
Sample Wrong ProportionNumber Account Number Defective
1 15 0.006 2 12 0.0048 3 19 0.0076 4 2 0.0008 5 19 0.0076 6 4 0.0016 7 24 0.0096 8 7 0.0028 9 10 0.00410 17 0.006811 15 0.006
12 3 0.0012
Total 147
p = 0.0049
n = 2500
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Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp – – zzσσpp
σσpp = = pp(1 – (1 – pp))//nn
n = 2500 p = 0.0049
Example 7.3
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Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp – – zzσσpp
σσpp = 0.0049(1 – 0.0049)/2500 = 0.0049(1 – 0.0049)/2500
n = 2500 p = 0.0049
Example 7.3
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Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
UCLUCLpp = = pp + + zzσσpp
LCLLCLpp = = pp – – zzσσpp
σσpp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Example 7.3
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Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
σσpp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Example 7.3
UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)
LCLLCLpp = 0.0049 – 3(0.0014) = 0.0049 – 3(0.0014)
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UCLUCLpp = 0.0091 = 0.0091
LCLLCLpp = 0.0007 = 0.0007
Control ChartsControl Chartsfor Attributesfor Attributes
Hometown BankHometown Bank
σσpp = 0.0014 = 0.0014
n = 2500 p = 0.0049
Example 7.3
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p-ChartWrong Account Numbers
Figure 7.11
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p-ChartWrong Account Numbers
Figure 7.11
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p-ChartWrong Account Numbers
Figure 7.11
Measure the process Find the assignable cause Eliminate the problem Repeat the cycle
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Control ChartsControl Chartsfor Attributesfor Attributes
WoodlandWoodlandPaper Paper CompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
c = 20 z = 2
UCLc = c + z c
LCLc = c – z cExample 7.4
WoodlandWoodlandPaperPaperCompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
Example 7.4
c = 20 z = 2
UCLc = 20 + 2 20
LCLc = 20 – 2 20
WoodlandWoodlandPaperPaperCompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
Example 7.4
c = 20 z = 2
UCLc = 28.94
LCLc = 11.06
WoodlandWoodlandPaperPaperCompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
Example 7.4
WoodlandWoodlandPaperPaperCompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
Example 7.4
WoodlandWoodlandPaperPaperCompanyCompany
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Control ChartsControl Chartsfor Attributesfor Attributes
WoodlandWoodlandPaperPaperCompanyCompany
Example 7.4
Measure the process Find the assignable cause Incorporate the problem Repeat the cycle
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Process CapabilityProcess Capability
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Process CapabilityProcess CapabilityNominal
value
800 1000 1200 Hours
Upperspecification
Lowerspecification
Process distribution
(a) Process is capableFigure 7.13
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Process CapabilityProcess CapabilityNominal
value
Hours
Upperspecification
Lowerspecification
Process distribution
(b) Process is not capableFigure 7.13
800 1000 1200
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Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Two sigma
Nominal value
Figure 7.14
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Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Four sigma
Two sigma
Nominal value
Figure 7.14
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Process CapabilityProcess Capability
Lowerspecification
Mean
Upperspecification
Six sigma
Four sigma
Two sigma
Nominal value
Figure 7.14
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Process CapabilityProcess Capability
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Example 7.5
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Upper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Process CapabilityProcess CapabilityLightbulb Production
Cp =
Upper specification - Lower specification
6σ
Process Capability RatioProcess Capability RatioExample 7.5
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cp = 1200 – 800
6(48)
Process Capability RatioProcess Capability RatioExample 7.5
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cp = 1.39
Process Capability RatioProcess Capability RatioExample 7.5
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cp = 1.39
Example 7.5
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cp = 1.39
Cpk = Minimum of
ProcessProcessCapabilityCapabilityIndexIndex
Example 7.5
Upper specification – x
3σ
x – Lower specification
3σ,
=
=
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
1200 – 900
3(48)
900 – 800
3(48)
ProcessProcessCapabilityCapabilityIndexIndex
,
Example 7.5
Cpk = Minimum of
Cp = 1.39
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cpk = Minimum of [ 0.69, 2.08 ]
ProcessProcessCapabilityCapabilityIndexIndex
Example 7.5
Cp = 1.39
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cpk = 0.69
ProcessProcessCapabilityCapabilityIndexIndex
Example 7.5
Cp = 1.39
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Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours
Cp = 1.39Cpk = 0.69
ProcessProcessCapabilityCapabilityIndexIndex
ProcessProcessCapabilityCapabilityRatioRatio
Example 7.5
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Taguchi's Quality Taguchi's Quality Loss FunctionLoss Function
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Taguchi's Quality Taguchi's Quality Loss FunctionLoss Function
Figure 7.15
Lo
ss (
do
llar
s)
Lower Nominal Upperspecification value specification
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Taguchi's Quality Taguchi's Quality Loss FunctionLoss Function
Figure 7.15
Lo
ss (
do
llar
s)
Lower Nominal Upperspecification value specification