graduate school of industrial engineering & management, national yunlin university of science...

Graduate school of Industrial Engineering & Management,National Yunlin University of Science & Technology

System Reliability Lab. http://campusweb.yuntech.edu.tw/~qre/index.htm

Economical Statistical Design of Combined Double Sampling and

Variable Sampling Interval Joint X-bar and S Control Charts

Student: Yi-Chun ChenAdvisor: Chau-Chen Torng

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Contents

Introduction1

Literature Review2

Design of control charts3

Expect result4

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1.Introduction

1

Background&

Motivation

2

Purposes

Introduction Literature Review

Design of control charts Expect result

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Background & Motivation(1/5)Introduction Literature

ReviewDesign of

control charts Expect result

Manufacturers

Product

Customers

QuantitySupply < Demand

Before

QuantitySupply > Demand

New

Qualityvs

Cost

Qualityvs

Cost

Source: Business Weekly , vol. 974

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Background & Motivation(2/5)

Statistical Process Control (SPC)



Shewhart (1924) development

X-bar control chart

Control charts

Source: Montgomery, 2006

μ-3σ

μ+3σ

μ

Out of control

Out of control

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Traditional Control chart Advantage Easy to operate,

Detection large shift is sensitive

Disadvantage Detection small & medium shift

is not sensitive



Control chartS

X-bar &

S

X-bar &

R

Control chart

Control chart

Control chart

adaptive

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Background & Motivation(4/5)Introduction Literature

ReviewDesign of


Variable Sampling

Size

DoubleSampling

Variable SamplingIntervals

Controlchart

Variable Sampling

Size & Variable

Sampling Intervals

VariableParameters

Double Sampling

& Variable

Sampling Interval

Adaptive control chart

Daudin(1992) development DS control chart

Costa(1994) development VSS control chart

Reyonlds (1988)development VSI control chart Costa (1999)development

VP control chart

Costa (1997)development VSSI control chart

Carot (2002)development DSVSI control chart better than single DS and single VSI control chart

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On the other hand, cost is very important for Manufacturers.Duncan(1956)construct economic design of control chart, that model purpose is minimize the average unit cost.

The other some researcher find that single economic design of control chart model would made higher false alarm and lower power. Saniga(1989) economical statistical design of control chart.



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Purposes

1) Seek appropriate economic model and statistical model used on design of control charts.

2) By the analysis tool to find the best parameters of Economical statistical design of DSVSI X-bar & S control chart.

3) By sensitivity analysis to find the relationship between cost parameters and statistical performance.



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2.Literature Review

1

JointX-bar & S

control charts

2

Adaptive control chart

3

Design of control chart model



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2.1 Joint X-bar & S control charts

X-bar control chartS control chart

Why do we want combination of X-bar and S control chart?



(a) (c)(b)

μ0 μ0 μ0UCLLCL UCLLCL UCLLCL

Source: Montgomery,2008

σ0 σ0σ1

μ1

μ-3σ

μ+3σ

μ

Out of control

Out of control

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2.2 Adaptive control chart

A. Double Sampling(DS) control chart.

B. Variable Sampling Interval(VSI) control chart.

C. Double Sampling and Variable Sampling Interval(DSVSI) control chart.



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DS X-bar control chart

Dudin(1992) development DS X-bar control chart.



Action Region

Warning Region

Central Region

Warning Region

Action Region

Action Region

Central Region

Action Region

First stage DS X-bar control chart

Second stage DS X-bar control chart

Change sampling size

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VSI X-bar control chart

Reyonlds et al. (1988) development VSI X-bar control chart.



Action Region

Warning Region

Central Region

Warning Region

Action Region

VSI X-bar control chart

Change sampling interval

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DSVSI X-bar & S control chart(1/2)

Lee(2012)development DSVSI X-bar&S control chart.

Design background



Lee et al.

Joint S

and DSVSI

DSVSI S

Torng et al.

Showed DSVSI X-bar batter than

CUSUMnon-normal

He et al.

Combing DS X-bar

and DS S

DS X-bar&S

Carot et al.

Combing DS X-bar

and VSI X-bar

DSVSI X-bar

2002 2006 2010 2012

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DSVSI X-bar & S control chart(2/2)Introduction Literature

ReviewDesign of


AR

WR II

WR I

CR

WR I

WR II

AR

AR

CR

AR

AR

WR II

WR I

CR

AR

CR

First stage X-bar chart Second stage X-bar chart

First stage S chart Second stage S chart

5S

n

h

n

h

n

h

X 5

X 1

X 2

X 2

X 3

X 3

X 5

X 4

X 6

X 6

3S

2S

1S

6S

4S

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2.3 Design of control chart model

A. Economic design of control chart

B. Statistical design of control chart



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Economic design of control chartIntroduction Literature

ReviewDesign of


τD

gn

h/(1-β)

1/λ

In control Out of control

Cycle start

Assignable cause

Assignable cause

removed

Last sampling

before Assignable

cause

First sampling

after Assignable

cause

Lack of control

detected

Assignable cause

located

Ν Ν

Duncan(1956)

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Statistical design of control chart

Woodall(1986) find economic design of control chart will have higher false alarm and lower power.

Saniga(1989)development economical statistical design of control chart.



AATSAATS

AATSAATS

OOCOOC

ICICts

CEMin

*

*..

)(

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3.Design of control charts

1

Symbol definition

&description

2

Model assumptions

3

Modeling



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Sampling size(small) Warning limit to change sampling interval for X-bar

Sampling size(big) Warning limit to change sampling interval for S

Sampling interval(long) control limit to the first stage X-bar

Sampling interval(short) control limit to the first stage S

Warning limit to change sampling size for X-bar

control limit to the second stage X-bar

Warning limit to change sampling size for S

control limit to the second stage S

Fixed costs False alarm costs

Variable costs Expected cost of the production defective products, in control

Search assignable cause costs Expected cost of the production defective products, out of control

Symbol definition & description(1/2)Introduction Literature

ReviewDesign of


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Expected sample size of control chart

Expected number of assignable cause occur per unit time

Expected sample size of process in control

The sampling interval h between jth and j+1th sampling, expected time of assignable cause occur

Expected sample size of process out of control

The sampling interval h1 between jth andJ+1th sampling, expected time of assignable cause occur

Expected sampling interval of control chart

The sampling interval h2 between thf andJ+1th sampling, expected time of assignable cause occur

Expected sampling interval of process in control

Expected time of assignable cause occur between jth and j+1th sampling

Expected sampling interval of process out of control

The average sampling and inspection time is a multiple of the expected size of the number of samples

Expected cycle time of control chart Searched assignable cause to removed time, constant

Expected cost of cycle time per unit Expected cost of control chart cycle time per unit

Symbol definition & description(2/2)Introduction Literature

ReviewDesign of


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3.2 Model assumptions

The initial state of a process is in control and the mean and standard deviation are μ= μ0 and σ= σ0.

When an assignable cause occurs, μ1= μ0+δμ0 or σ1= γσ0.

Quality characteristics X is subject to the normal distribution.

The process just have only one assignable cause.

The relationship between assignable cause and process variation time is exponential distribution, the expected value of 1/λ.

Process variation is instantaneous.

The process does not have the ability to repair itself.



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3.3 Modeling

A. Cycle Time Computation

B. Cost Model Development

C. Statistical Constraints Formulation



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Cycle Time Computation(1/2)

E(nІδ,γ) =n1+n2×P(change sampling sizeІδ,γ)

P(change sampling sizeІδ,γ)

A

ISIX

ISIX

s

s

XP

XP

511

311

3

3

ISIX SX

P35

11



hhAATS QIT5.0)(~

1

Second Warring Region

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Cycle Time Computation(2/2)

1

2

0

)1(

)1(1)(

1

1

1

1

1

1 1)1(

1

1

1

eeh

e

ehh

h

hh

dx

hh

dxx

j

j

x

j

j

x

)1(

)1(1)(

2

2

2

2

2

2 2)1(

1

2

2

eeh

e

ehh

h

hh

dx

hh

dxx

j

j

x

j

j

x

2211

~~

hphphp

2112

111 )1(~

hphphp

2112

212 )1(

)1(~

j j+1h1



h2j j+1

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Cycle Time Computation

ζD

GE(n)

AATS

1/λ

In control Out of control

Cycle start

Assignable cause

Assignable cause

removed

Last sampling

before Assignable

cause

First sampling

after Assignable

cause

Lack of control

detected

Assignable cause

located

Ν Ν

1/λ

In control

AATS

GE(n) D

E(T)=1/λ+AATS+GE(n)+D

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E(C)

sampling and inspection

haaTE

nE1

21

)()1,0(

haaTE

nE2

21

)()1,0(

Cost Model Development

4

3

2

1

defective products cost C0 ×1/λ+C1 ×AATS

False alarm timesα ×sFalse alarm costsa3’× α ×s

Search assignable cause costsa3 sAATSCE aCC

'

310

1)(

haaTE

nE1

21

)()1,0(

haaTE

nE2

21

)()1,0(

)(

)()'(

TE

CECE

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Statistical Constraints Formulation

Mins.t.

)'(CE

AATSAATS ICIC

*

AATSAATS OOCOOC

*

hh 120

Nnn 21

0

kww XXX nt 1

0

kk XX 12

0

kww ssnst 10

kk ss 120 S control chart

X-bar control chart

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4.Expect result

The economical statistical design of Combined DSVSI Joint X-bar and S Control Charts optimal combination of parameters obtained by solving tool.

Use sensitivity analysis to find relationship between cost parameters and cost, relationship between cost and statistical performance.Lorenzen(1986) ±30%



a1 a2 a3 a3’ C0 C1 λ G D

0 4.22 1086 977.4 114.24 949.2 0.02 5/60 5/60

Graduate school of Industrial Engineering & Management,National Yunlin University of Science & Technology

System Reliability Lab. http://campusweb.yuntech.edu.tw/~qre/index.htm

graduate school of industrial engineering & management, national yunlin university of science...

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