chapter 7 - statistical process control

To Accompany Krajewski & Ritzman Operations Management: Strategy and Analysis, Sixth Edition © 2002 Prentice Hall, Inc. All rights reserved.

Chapter 7 -Chapter 7 -

StatisticalStatisticalProcessProcessControlControl


Common CausesCommon Causes



Figure 7.1



Figure 7.1

x =xi

i=1

n

∑n

σ =xi − x ( )∑

2

n−1


x =xi

i=1

n

∑n

σ =xi − x ( )∑

2

n−1


425 GramsFigure 7.1


Assignable CausesAssignable Causes



(a) LocationGrams

Average

Figure 7.2



(b) SpreadGrams

Average

Figure 7.2



(c) ShapeGrams

Average

Figure 7.2


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)


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)


Sample Means and theSample Means and theProcess DistributionProcess Distribution

Figure 7.4 425 Grams

Mean

Processdistribution


Sample Means and theSample Means and theProcess DistributionProcess Distribution

Processdistribution

Figure 7.4 425 Grams

MeanDistribution ofsample means


The NormalThe NormalDistributionDistribution

Figure 7.5



Mean

σ = Standard deviation

Figure 7.5



Mean

68.26%


Figure 7.5

–1σ+1σ



Mean

68.26%95.44%


Figure 7.5

–2σ–1σ+1σ+2σ



–3σ–2σ–1σ+1σ+2σ+3σMean

68.26%95.44%99.74%


Figure 7.5


Control ChartsControl Charts



UCL

Nominal

LCL

Figure 7.6



UCL

Nominal

LCL

1 2 3SamplesFigure 7.6



UCL

Nominal

LCL

Assignable causes likely

1 2 3SamplesFigure 7.6


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


Control Chart ExamplesControl Chart Examples

Nominal

UCL

LCL

Sample number

Var

iati

on

s

Figure 7.7 (a)



Nominal

UCL

LCL

Sample number

Var

iati

on

s

Figure 7.7 (b)



Nominal

UCL

LCL

Sample number

Var

iati

on

s

Figure 7.7 (c)



Nominal

UCL

LCL

Sample number

Var

iati

on

s

Figure 7.7 (d)



Nominal

UCL

LCL

Sample number

Var

iati

on

s

Figure 7.7 (e)


Control Limits and ErrorsControl Limits and Errors



LCL

Processaverage

UCL

Figure 7.8 (a) Three-sigma limits



LCL

Processaverage

UCL

Three-sigma limits

Type I error:Probability of searching for a cause when none exists

Figure 7.8 (a)



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)



UCL

LCL

Processaverage

Figure 7.8 (b) Two-sigma limits


Control Limits and ErrorsControl Limits and ErrorsType I error:Probability of searching for a cause when none exists


UCL

LCL

Processaverage


Control Limits and ErrorsControl Limits and ErrorsType I error:Probability of searching for a cause when none exists


Type II error:Probability of concludingthat nothing has changed

Shift in process average

UCL

LCL

Processaverage


Control ChartsControl Chartsfor Variablesfor Variables

West Allis IndustriesWest Allis Industries

Example 7.1



Sample Sample

Number 1 2 3 4

1

2

3

4

5

Special Metal Screw

Example 7.1



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



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

_



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

_



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

_



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

_



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

_



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

=

_


Example 7.1


Control Charts – Special Metal Screw

R-Charts R = 0.0021

UCLR = D4RLCLR = D3R


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


Example 7.1


Control Charts - Special Metal Screw

R - Charts R = 0.0020 D4 = 2.2080

Control Chart FactorsControl Chart Factors


((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


Example 7.1


Control Charts—Special Metal Screw

R-Charts R = 0.0021 D4 = 2.282D3 = 0



Example 7.1



R-Charts R = 0.0021 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0021) = 0.00479 in.



Example 7.1



R-Charts R = 0.0021 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.



Range Chart - Range Chart - Special Metal ScrewSpecial Metal Screw

Figure 7.9


Example 7.1



X-Charts

UCLx = x + A2RLCLx = x - A2R

==

R = 0.0021x = 0.5027=


Example 7.1


Control Charts - Special Metal Screw

R = 0.0020x = 0.5025

x - Charts


Control Chart FactorsControl Chart Factors


((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


Example 7.1



x- Charts


==

R = 0.0021 A2 = 0.729x = 0.5027=


Example 7.1



x-Charts

UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.


==

R = 0.0021 A2 = 0.729x = 0.5027=


Example 7.1



x-Charts

UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.


==

R = 0.0021 A2 = 0.729x = 0.5027=


xx-Chart—-Chart—Special Metal ScrewSpecial Metal Screw

Figure 7.10


xx-Chart— -Chart— Special Metal ScrewSpecial Metal Screw

Figure 7.10


xx-Chart—-Chart—Special Metal ScrewSpecial Metal Screw

Figure 7.10

Measure the process Find the assignable cause Eliminate the problem Repeat the cycle



for Variables Using for Variables Using σσ

Example 7.2

UCLUCLxx = = xx + + zzσσxx

LCLLCLxx = = xx – – zzσσxx

σx = σ/n

==

==




Example 7.2



σx = σ/n

==

==

Sunny Dale Bank

x = 5.0 minutesσ = 1.5 minutesn = 6 customersz = 1.96

=




UCLx = 5.0 + 1.96(1.5)/ 6 = 6.20 min

UCLx = 5.0 – 1.96(1.5)/ 6 = 3.80 minExample 7.2



σx = σ/n

==

==

Sunny Dale Bank

x = 5.0 minutesσ = 1.5 minutesn = 6 customersz = 1.96

=


Control ChartsControl Chartsfor Attributesfor Attributes

HOMETOWN BANK

Hometown BankHometown Bank




UCLUCLpp = = pp + + zzσσpp

LCLLCLpp = = pp – – zzσσpp

σσpp = = pp(1 – (1 – pp))//nn

Example 7.3


Example 7.3




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


Example 7.3





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


Example 7.3





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


Example 7.3





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






σσpp = = pp(1 – (1 – pp))//nn

n = 2500 p = 0.0049

Example 7.3






σσpp = 0.0049(1 – 0.0049)/2500 = 0.0049(1 – 0.0049)/2500

n = 2500 p = 0.0049

Example 7.3






σσpp = 0.0014 = 0.0014

n = 2500 p = 0.0049

Example 7.3




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)


UCLUCLpp = 0.0091 = 0.0091

LCLLCLpp = 0.0007 = 0.0007



σσpp = 0.0014 = 0.0014

n = 2500 p = 0.0049

Example 7.3


p-ChartWrong Account Numbers

Figure 7.11


p-ChartWrong Account Numbers

Figure 7.11

Measure the process Find the assignable cause Eliminate the problem Repeat the cycle



WoodlandWoodlandPaper Paper CompanyCompany



c = 20 z = 2

UCLc = c + z c

LCLc = c – z cExample 7.4

WoodlandWoodlandPaperPaperCompanyCompany



Example 7.4

c = 20 z = 2

UCLc = 20 + 2 20

LCLc = 20 – 2 20




Example 7.4

c = 20 z = 2

UCLc = 28.94

LCLc = 11.06




Example 7.4





Example 7.4

Measure the process Find the assignable cause Incorporate the problem Repeat the cycle


Process CapabilityProcess Capability


Process CapabilityProcess CapabilityNominal

value

800 1000 1200 Hours

Upperspecification

Lowerspecification

Process distribution

(a) Process is capableFigure 7.13


Process CapabilityProcess CapabilityNominal

value

Hours

Upperspecification

Lowerspecification

Process distribution

(b) Process is not capableFigure 7.13

800 1000 1200



Lowerspecification

Mean

Upperspecification

Two sigma

Nominal value

Figure 7.14



Lowerspecification

Mean

Upperspecification

Four sigma

Two sigma

Nominal value

Figure 7.14



Lowerspecification

Mean

Upperspecification

Six sigma

Four sigma

Two sigma

Nominal value

Figure 7.14


Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours σ = 48 hours

Example 7.5


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



Cp = 1200 – 800

6(48)




Cp = 1.39




Cp = 1.39

Example 7.5



Cp = 1.39

Cpk = Minimum of

ProcessProcessCapabilityCapabilityIndexIndex

Example 7.5

Upper specification – x

3σ

x – Lower specification

3σ,

=

=



1200 – 900

3(48)

900 – 800

3(48)


,

Example 7.5

Cpk = Minimum of

Cp = 1.39



Cpk = Minimum of [ 0.69, 2.08 ]


Example 7.5

Cp = 1.39



Cpk = 0.69


Example 7.5

Cp = 1.39



Cp = 1.39Cpk = 0.69


ProcessProcessCapabilityCapabilityRatioRatio

Example 7.5


Taguchi's Quality Taguchi's Quality Loss FunctionLoss Function


Taguchi's Quality Taguchi's Quality Loss FunctionLoss Function

Figure 7.15

Lo

ss (

do

llar

s)

Lower Nominal Upperspecification value specification

chapter 7 - statistical process control

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