Targeting in Multi-Class Screening Under Error and Error Free Measurement Systems

Targeting in Multi-Class Screening Under Error and Error Free

Measurement Systems

Targeting in Multi-Class Screening Under Error and Error Free

Measurement Systems

Thesis Defense Presentationon

byATIQ WALIULLAH SIDDIQUI

PRESENTATION OVERVIEWPRESENTATION OVERVIEW

Introduction

Literature Review

Objectives of the Thesis

Multi-Class Screening Model Presented in Min Koo Lee and Joon Soon Jang (1997)

Model Extensions

Sensitivity Analysis

Model Comparisons

Conclusion

Introduction

Literature Review

Objectives of the Thesis

Multi-Class Screening Model Presented in Min Koo Lee and Joon Soon Jang (1997)

Model Extensions

Sensitivity Analysis

Model Comparisons

Conclusion

Fitness for Use

Conformance to Specifications

Penalties Associated with Product Deviations Off Target (within specification limits)

Fitness for Use

Conformance to Specifications

Penalties Associated with Product Deviations Off Target (within specification limits)

WHAT IS QUALITY?WHAT IS QUALITY?

LL UL Quality characteristic

LL UL Quality characteristic

CustomerTolerance

Cos

t of

re

pair

, $

Cos

t of

re

pair

, $

CustomerTolerance

Goal Post Syndrome Taguchi Loss Function Goal Post Syndrome Taguchi Loss Function

BASIC TARGETING MODELBASIC TARGETING MODEL

Model Formulation:

Per unit profit function:

Per unit Expected profit:

Where:

Model Formulation:



Where:

LYycr

LYycayP 1)(

LYycr

LYycayP 1)(

cra1EPM 211 )()( cra1EPM 211 )()(

y

11

L

y

11

L

y

22

L

y

22

L

LITERATURE REVIEWLITERATURE REVIEW

C. Springer (1951): The problem was to find the mean for a canning process in order to minimize the cost. The price for producing under/over filled cans are assumed to be different.

W. Hunter & C. Kartha (1977): Above problem with under filled item sold in a secondary market. Objective is to maximize of profit.

D. Golhar (1987): Under filled cans are to be emptied and refilled at the expense of a fixed reprocessing cost.

C. Springer (1951): The problem was to find the mean for a canning process in order to minimize the cost. The price for producing under/over filled cans are assumed to be different.

W. Hunter & C. Kartha (1977): Above problem with under filled item sold in a secondary market. Objective is to maximize of profit.

D. Golhar (1987): Under filled cans are to be emptied and refilled at the expense of a fixed reprocessing cost.

Cont…


M. A. Rahim and P. K. Banerjee (1988): considered the process where the system has a linear drift (e.g., tool wear etc).

O. Carlsson (1989): determined, for the case of two variable characteristics, the optimum process mean under acceptance variable sampling.

R. Schmidt & P. Pfeifer (1989): investigated the effects on cost savings from variance reduction.

K. S. Al-Sultan (1994): addressed the problem of two machines in series.

M. A. Rahim and P. K. Banerjee (1988): considered the process where the system has a linear drift (e.g., tool wear etc).

O. Carlsson (1989): determined, for the case of two variable characteristics, the optimum process mean under acceptance variable sampling.

R. Schmidt & P. Pfeifer (1989): investigated the effects on cost savings from variance reduction.

K. S. Al-Sultan (1994): addressed the problem of two machines in series.

Cont…


Liu, Tang and Chun (1995) considered the case of a filling process with limited capacity constraint.

F. J. Arcelus (1996) introduced the consistency criteria in the targeting problem.

J. Roan, L. Gong & K. Tang (1997) considered production decisions such as production setup and raw material procurement policies.

Min Koo Lee & Joon Soon Jang (1997) developed the model for multi-class screening case.

Sung Hoon Hong & E. A. Elsayed (1999) studied the effect of measurement error for targeting problem.

Liu, Tang and Chun (1995) considered the case of a filling process with limited capacity constraint.

F. J. Arcelus (1996) introduced the consistency criteria in the targeting problem.

J. Roan, L. Gong & K. Tang (1997) considered production decisions such as production setup and raw material procurement policies.

Min Koo Lee & Joon Soon Jang (1997) developed the model for multi-class screening case.

Sung Hoon Hong & E. A. Elsayed (1999) studied the effect of measurement error for targeting problem.

OBJECTIVES OF THE THESISOBJECTIVES OF THE THESIS

1. Extend the Min et al. (1997) targeting model for multi class screening incorporating the effects of measurement error.

2. Develop a targeting model for multi class screening incorporating the product uniformity under error free measurement system.

3. Extend the model resulting in objective 2 for the case of measurement systems with error.

4. Study the effects of error in the measurement on the models that will be developed in objective 2 and 3 above

1. Extend the Min et al. (1997) targeting model for multi class screening incorporating the effects of measurement error.

2. Develop a targeting model for multi class screening incorporating the product uniformity under error free measurement system.

3. Extend the model resulting in objective 2 for the case of measurement systems with error.

4. Study the effects of error in the measurement on the models that will be developed in objective 2 and 3 above

MIN et al. (1997) MODEL MIN et al. (1997) MODEL

Model Assumptions:

A single item is to be sold in two different markets with different cost/profit structures.

The quality characteristic ‘Y’ is assumed normally distributed with unknown process mean and known variance .

The production cost per item is

Model Assumptions:


The quality characteristic ‘Y’ is assumed normally distributed with unknown process mean and known variance .

The production cost per item is

raa 21 raa 21

Cont…

ycc0 ycc0

MIN et al. (1997) Model MIN et al. (1997) Model

Specification limits on different grades are:

100% inspection is considered

Specification limits on different grades are:

100% inspection is considered

Specification limits on ‘Y’ for grade 1 areSpecification limits on ‘Y’ for grade 1 are

Specification limits on ‘Y’ for grade 2 areSpecification limits on ‘Y’ for grade 2 are

Specification limitSpecification limits on ‘Y’ for scrap ares on ‘Y’ for scrap are

Y L1,

L2Y< L1,

Y< L2,

Y L1,

L2Y< L1,

Y< L2,

MIN et al. (1997) Model MIN et al. (1997) Model

Model Formulation:



Where:

Model Formulation:



Where:

2i0

12i02

1i01

LYccycr

LYLccyca

LYccyca

yP

)(

)(

)(

)(

2i0

12i02

1i01

LYccycr

LYLccyca

LYccyca

yP

)(

)(

)(

)(

y11i0221211 Lcccraa1EPM )()()()( y11i0221211 Lcccraa1EPM )()()()(

y

11

L

y

11

L

y

22

L

y

22

L

OBJECTIVE 1:(Measurement error)OBJECTIVE 1:(Measurement error)

Model Assumptions:


The quality characteristic ‘Y’ is assumed normally distributed with unknown process mean and known variance.

The inspection process is error prone.

The measurement ‘X’ is assumed to be unbiased and distributed normally across the true value.

The inspection is based on ‘X’ (observed) as opposed to ‘Y’ (actual) as in Min et al. (1997).

Model Assumptions:





The inspection is based on ‘X’ (observed) as opposed to ‘Y’ (actual) as in Min et al. (1997).


Relationship between ‘X’ and ‘Y’:

The ‘X’ is the observed value of ‘Y’ i.e.,

Where ‘’ is the error in measurement:

The expected value of the observed value ‘X’:


The ‘X’ is the observed value of ‘Y’ i.e.,

Where ‘’ is the error in measurement:

The expected value of the observed value ‘X’:

Cont…

Y X Y X

),0(N~ 2 ),0(N~ 2

)x(E )x(E

The variance of the observed value ‘X’:

The joint distribution of ‘X’ and ‘Y’:

or

where

The variance of the observed value ‘X’:

The joint distribution of ‘X’ and ‘Y’:

or

where

),y(Cov2)(Var)y(Var)x(Var ),y(Cov2)(Var)y(Var)x(Var

22y

2x 22

y2x

)Y(Var)x(Var )Y(Var)x(Var


yx

2

y

2

x2

yx2

yx

12

1

2xy

e12

1xy ),(

yx

2

y

2

x2

yx2

yx

12

1

2xy

e12

1xy ),(

uv2vu2

1 22

e2

1)u,v(

uv2vu2

1 22

e2

1)u,v(

22y

2

2x

2y

2x

2

1

22y

2

2x

2y

2x

2

1


Cut off Points:

Instead of the specification limits, inspection is based on ‘Cut off Points’

Why ‘Cut off Points’?

Cut off Points:



Grade 2

Grade 1

Scrape

L2 L1

w2 w1

Targeting with Measurement Error. wi show the ‘Cut off

Points’ for the Inspection

Targeting with Measurement Error. wi show the ‘Cut off Points’ for the Inspection


Penalties Associated with Misclassifications:Penalties Associated with Misclassifications:

OBSERVEDOBSERVEDACTUALACTUAL

Grade 1Grade 1 Grade 2Grade 2 ScrapeScrape

Grade1Grade1 -- bb2121 bbs1s1

Grade 2Grade 2ImplicitImplicit

aa11-a-a22

-- bbs2s2

ScrapScrap

ImplicitImplicit

aa11-a-ass

ImplicitImplicit

aa22-a-ass --

Model Formulation:


Model Formulation:



122i0

12i0

22i0

2122si02

112i02

1212i02

211si01

12121i01

11i01

LYLwXccycr

LYwXccycr

LYwXccycr

LYwXwbccyca

LYwXwccyca

LYLwXwccyca

LYwXbccyca

LYLwXbccyca

LYwXccyca

yP

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

122i0

12i0

22i0

2122si02

112i02

1212i02

211si01

12121i01

11i01

LYLwXccycr

LYwXccycr

LYwXccycr

LYwXwbccyca

LYwXwccyca

LYLwXwccyca

LYwXbccyca

LYLwXbccyca

LYwXccyca

yP

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

Cont…

y11i0

2s1s21

221211

Lccc

dvdu)u,v(bdvdu)u,v(bdvdu)u,v(b

)(r)()(a)(a2EPM1

2

2

1

2

1

1

2

y11i0

2s1s21

221211

Lccc

dvdu)u,v(bdvdu)u,v(bdvdu)u,v(b

)(r)()(a)(a2EPM1

2

2

1

2

1

1

2

y

11

L

y

11

L

y

22

L

y

22

L

2e

2y

11

w

2e

2y

11

w

Model Formulation:

Expected Per unit profit:

Where

Model Formulation:


Where


2e

2y

22

w

2e

2y

22

w

Sensitivity Analysis: Effect of Error on Expected Profit

Sensitivity Analysis: Effect of Error on Expected Profit


E(p) versus measurment error (rho) at a2-r/ a1-a2 = 4 sigma = 1.25

-3.00-2.00-1.000.001.002.003.004.005.006.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

E(P

)

0.01 0.15 0.3c/ a2-a1

90 problems:

= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

Cont…

Mean versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma= 1.25

42.50

43.00

43.50

44.00

44.50

45.00

45.50

46.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Mea

n

0.01 0.15 0.3c/ a2-a1

Sensitivity Analysis: Effect of Error on Optimal Mean

Sensitivity Analysis: Effect of Error on Optimal Mean


90 problems:

= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

Cont…

w1, w2 versus measurment error (rho) at a2-r/ a1-a2 = 4 sigma = 1.25

37.5038.0038.5039.0039.5040.0040.5041.0041.5042.0042.50

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Cut

off

poi

nts

0.01 0.01 0.15 0.15 0.3 0.3

c/ a2-a1

Sensitivity Analysis: Effect of Error on Cut off PointsSensitivity Analysis: Effect of Error on Cut off Points


90 problems:

= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

Cont…

L1

L2

w1

w2

OBJECTIVE 2:(Uniformity Penalty)OBJECTIVE 2:(Uniformity Penalty)

Model Assumptions:



The inspection process is error free.

The inspection is based on ‘y’.

Model Assumptions:



The inspection process is error free.

The inspection is based on ‘y’.

Cont…


Model Assumptions:

A quadratic penalty, similar to that of ‘Taguchi Quadratic Loss Function’ is used as a penalty for the item being off target.

Model Assumptions:


Cont…

Grade 2

Grade 1

Scrape

L2 L1

Targeting with Uniformity Penalty ‘t’ shows the Target Value

Targeting with Uniformity Penalty ‘t’ shows the Target Value

penalty $

Target

‘t’

Model Formulation:



Where:

Model Formulation:



Where:


2i0

122

i02

12

i01

LY)ccyc(r

LYLtyK)ccyc(a

LYtyK)ccyc(a

)y(P

2i0

122

i02

12

i01

LY)ccyc(r

LYLtyK)ccyc(a

LYtyK)ccyc(a

)y(P

)()()()( 2222 1tdzzzt2dzzzK

1EPM3EPM

22


1EPM3EPM

22

y

22

L

y

22

L

Special Cases:

Special case I: (target set at mean)

Special case II: (target set at L1)

Special Cases:




2

dzzzK1EPM3EPM 22I )(

2


)()()()( 22

1122

II

1LdzzzL2dzzzK

1EPM3EPM

22

)()()()( 22

1122

II

1LdzzzL2dzzzK

1EPM3EPM

22

Sensitivity Analysis: Effect on Expected ProfitSensitivity Analysis: Effect on Expected Profit


Special Case I252 problems:

= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Cont…

E(p) versus a2-r/ a1-a2 at sigma = 1.5

-3.000-2.000-1.0000.0001.0002.0003.0004.0005.0006.000

3.8 4.3 4.8 5.3 5.8 6.3 6.8 7.3 7.8

(a2-r)/ (a1-a2)

E(P

)

0.01 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

E(p) versus c/ a1-a2 at sigma = 1.5

-3.000

-2.000

-1.000

0.000

1.000

2.000

3.000

4.000

5.000

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

E(P

)

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)

Optimal Mean versus a2-r/ a1-a2 at sigma = 1.5

42.500

43.000

43.500

44.000

44.500

45.000

45.500

46.000

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

(a2-r)/ (a1-a2)

Mu

0.01 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

Sensitivity Analysis: Effect on Optimal MeanSensitivity Analysis: Effect on Optimal Mean



= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Cont…

Optimal Mean versus c/ a1-a2 at sigma = 1.5

42.500

43.000

43.500

44.000

44.500

45.000

45.500

46.000

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

Mu

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)



Special Case II252 problems:

= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Cont…

E(p) versus a2-r/ a1-a2 at sigma = 1.5

-3.000

-1.000

1.000

3.000

5.000

3.8 4.3 4.8 5.3 5.8 6.3 6.8 7.3 7.8

(a2-r)/ (a1-a2)

E(P

)

0.01 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

E(p) versus c/ a1-a2 at sigma = 1.5

-3.000-2.000-1.0000.0001.0002.0003.0004.0005.000

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

E(P

)

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)

Optimal Mean versus a2-r/ a1-a2 at sigma = 1.5

42.400

42.500

42.600

42.700

42.800

42.900

43.000

43.100

43.200

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

(a2-r)/ (a1-a2)

Mea

n

0.01 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)




= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Cont…

Optimal Mean versus c/ a1-a2 at sigma = 1.5

42.400

42.500

42.600

42.700

42.800

42.900

43.000

43.100

43.200

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

Mea

n

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)

OBJECTIVE 3:(Integrated Model)OBJECTIVE 3:(Integrated Model)

Cont…

Model Assumptions:





The inspection is based on ‘X’ (observed) as opposed to ‘Y’ (actual) in last Min et al. (1997).

Model Assumptions:





The inspection is based on ‘X’ (observed) as opposed to ‘Y’ (actual) in last Min et al. (1997).

Model Assumptions:


Model Assumptions:


Grade 2

Grade 1

Scrape

L2 L1

Targeting with Measurement Error & Uniformity Penalty ‘t’ shows the Target Value & wi represent the Cut off Points

Targeting with Measurement Error & Uniformity Penalty ‘t’ shows the Target Value & wi represent the Cut off Points

penalty $

Target

‘t’w2 w1




The ‘X’ is the observed value of ‘Y’ and the joint distribution is:

or


The ‘X’ is the observed value of ‘Y’ and the joint distribution is:

or

yx

2

y

2

x2

yx2

yx

12

1

2ey

e12

1xy ),(

yx

2

y

2

x2

yx2

yx

12

1

2ey

e12

1xy ),(

uv2vu2

1 22

e2

1)u,v(

uv2vu2

1 22

e2

1)u,v(

Cut off Points:



Cut off Points:




Penalties for misclassification:Penalties for misclassification:

OBSERVEDOBSERVEDACTUALACTUAL

Grade 1Grade 1 Grade 2Grade 2 ScrapeScrape

Grade1Grade1 -- bb2121 bbs1s1

Grade 2Grade 2ImplicitImplicit

aa11-a-a22

-- bbs2s2

ScrapScrap

ImplicitImplicit

aa11-a-ass

ImplicitImplicit

aa22-a-ass --


Model Formulation:


Model Formulation:


22

122i0

12i0

22i0

2122si02

112i02

1212i02

211si01

12121i01

11i01

LYXtyK

LYLwXccycr

LYwXccycr

LYwXccycr

LYwXwbccyca

LYwXwccyca

LYLwXwccyca

LYwXbccyca

LYLwXbccyca

LYwXccyca

yP

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(

22

122i0

12i0

22i0

2122si02

112i02

1212i02

211si01

12121i01

11i01

LYXtyK

LYLwXccycr

LYwXccycr

LYwXccycr

LYwXwbccyca

LYwXwccyca

LYLwXwccyca

LYwXbccyca

LYLwXbccyca

LYwXccyca

yP

)(

)(

)(

)(

)(

)(

)(

)(

)(

)(


y

22

L

y

22

L

Model Formulation:


Where

Model Formulation:


Where


2EPM4EPM

22


2EPM4EPM

22


Special Cases:



Special Cases:



2


2


)()()()( 22

1122

II

1LdzzzL2dzzzK

2EPM4EPM

22

)()()()( 22

1122

II

1LdzzzL2dzzzK

2EPM4EPM

22


OBJECTIVE 3 :(Integrated Model)OBJECTIVE 3 :(Integrated Model)

Cont…

E(p) versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma = 1.25

-4.00

-2.00

0.00

2.00

4.00

6.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

E(P

)

0.01 0.15 0.3c/ a2-a1


= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Mu versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma = 1.25

42.50

43.00

43.50

44.00

44.50

45.00

45.50

46.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Mea

n

0.01 0.15 0.3c/ a2-a1


Cont…


= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


Sensitivity Analysis: Effect on Cut off PointsSensitivity Analysis: Effect on Cut off Points

Cont…


= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05

w1, w2 versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma = 1.25

37.5038.0038.5039.0039.5040.0040.5041.0041.5042.0042.50

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Cut

off

poi

nts

0.01 0.01 0.15 0.15 0.3 0.3

c/ a2-a1


w1

w2

L1

L2


Cont…


= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05

E(p) versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma = 1.25

-4.00-3.00-2.00-1.000.001.002.003.004.005.006.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

E(P

)

0.01 0.15 0.3c/ a2-a1




= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05

Cont…

Mu versus measurment error (rho) at a2-r/ a1-a2 = 4, sigma = 1.25

42.40

42.60

42.80

43.00

43.20

43.40

43.60

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Mea

n

0.01 0.15 0.3c/ a2-a1


w1, w2 versus measurment error (rho) at, a2-r/ a1-a2 = 4, sigma = 1.25

38.5039.0039.5040.0040.5041.0041.5042.0042.50

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

Cut

off

poi

nts

0.01 0.01 0.15 0.15 0.3 0.3

c/ a2-a1

Sensitivity Analysis: Effect on Cut off PointsSensitivity Analysis: Effect on Cut off Points

Cont…


= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


w1

w2

L1

L2

% Gain in expected profit versus measurment error (rho) at a2-r/ a1-a2 = 8, sigma = 1.75

0.00

5.00

10.00

15.00

20.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

% G

ain

0.01 0.15 0.3c/ a2-a1

MODEL COMPARISONMODEL COMPARISON

EMP1 vs. EMP290 problems:

= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

Numerical Comparison: % Gain in ProfitNumerical Comparison: % Gain in Profit


Analytic Comparison:

To show that some models are special cases of the others

Analytic Comparison:

To show that some models are special cases of the others

EPM1 vs. EPM2:

It is clear that as

Also,

Since it was assumed that i.e., it is clear that

EPM1 vs. EPM2:

It is clear that as

Also,

Since it was assumed that i.e., it is clear that

2x

2

1

2x

2

1

02 02 1 1

22y

2x 22

y2x 2

y2x 2

y2x

),0(N~ 2 ),0(N~ 2 0E )( 0E )(

YX YX


Since, , & at , it can be shown that:

Also by using L’hospital rule it can be shown that the penalty terms containing the bivariate normal distribution will vanish at

Since, , & at , it can be shown that:

Also by using L’hospital rule it can be shown that the penalty terms containing the bivariate normal distribution will vanish at

y

Yv

y

Yv

y

Xu

y

Xu 02 02 1 1

2211 & 2211 &

1 1


y11i0

2s1s21

221211

Lccc

dvduuvbdvduuvbdvduuvb

raa2EPM1

2

2

1

2

1

1

2

),(),(),(

)()()()(

y11i0

2s1s21

221211

Lccc

dvduuvbdvduuvbdvduuvb

raa2EPM1

2

2

1

2

1

1

2

),(),(),(

)()()()(

Therefore at the relationship for EPM2 i.e.,

can be written as:

EMP2 EMP1 as

Therefore at the relationship for EPM2 i.e.,

can be written as:

EMP2 EMP1 as

1 1

y11i0221211 Lcccaraa2EPM )()()( y11i0221211 Lcccaraa2EPM )()()(

1 1


Analytic comparison:Analytic comparison:

EPM1 vs. EPM3:

At K = 0

EMP3 EMP1 as

EPM1 vs. EPM3:

At K = 0

EMP3 EMP1 as


1EPM3EPM

22


1EPM3EPM

22

1EPM3EPM 1EPM3EPM

0K 0K



EPM1 vs. EPM3: (Special cases)

Care must be required in approximating EPM3II as this special case will converge slowly to EPM1 as compared to EPM3I as k 0

EPM1 vs. EPM3: (Special cases)

Care must be required in approximating EPM3II as this special case will converge slowly to EPM1 as compared to EPM3I as k 0

2


2


)()()()( 22

1122

II

1LdzzzL2dzzzK

1EPM3EPM

22

)()()()( 22

1122

II

1LdzzzL2dzzzK

1EPM3EPM

22


Numerical Comparison: % Gain in ProfitNumerical Comparison: % Gain in Profit% Gain in expected profit versus c/ (a1-a2)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

% G

ain

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)


= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05




= 1.5, 1.75, 2, 2.25

a2-r/a1-a2 = 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8

c/a1-a2 = 0.01, 0.05 0.1, 0.15, 0.2, 0.25, 0.3

L1 = 41.5, L2 = 40

K = 0.05

% Gain in expected profit versus c/ (a1-a2)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0 0.05 0.1 0.15 0.2 0.25 0.3

c/ (a1-a2)

% G

ain

4 5 5.5 6 6.5 7 7.5 8

(a2-r)/ (a1-a2)



EPM2 vs. EPM4:

At K = 0

EMP4 EMP2 as

EPM2 vs. EPM4:

At K = 0

EMP4 EMP2 as

2EPM4EPM 2EPM4EPM

0K 0K




= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

% G

ain

0.01 0.15 0.3c/ a2-a1




= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


0.05.0

10.015.020.025.030.035.040.0

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

% G

ain

0.01 0.15 0.3c/ a2-a1



EPM4 vs. EPM3:

The uniformity penalty term in both the models are same. The only difference is of EMP2 in EPM4 as compared to EPM1 in EPM3

As it is already shown that EMP2 EMP1 as error goes to zero

Therefore the proof of EMP4 EMP3 as is trivial

EPM4 vs. EPM3:

The uniformity penalty term in both the models are same. The only difference is of EMP2 in EPM4 as compared to EPM1 in EPM3

As it is already shown that EMP2 EMP1 as error goes to zero

Therefore the proof of EMP4 EMP3 as is trivial

1 1




= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


0.005.00

10.0015.0020.0025.0030.0035.0040.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

% G

ain

0.01 0.15 0.3c/ a2-a1




= 0.85, 0.98 ,0.93,

0.97, 0.995

= 1.25, 1.75

a2-r/a1-a2 = 4, 6, 8

c/a1-a2 = 0.01, 0.15, 0.3

L1 = 41.5, L2 = 40

K = 0.05


0.00

5.00

10.00

15.00

20.00

25.00

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1

rho

% G

ain

0.01 0.15 0.3c/ a2-a1

CONCLUSIONSCONCLUSIONS

Three models were developed for various multi-class screening targeting problems

EPM2 (measurement error) performed well in minimizing the effect of error

This effect is relatively less at higher cost

The effect of cost is more pronounced as compared to selling prices

In EPM3, the effect of uniformity penalty, if the target is set other than Mean, is high even at very low values of K

Three models were developed for various multi-class screening targeting problems

EPM2 (measurement error) performed well in minimizing the effect of error

This effect is relatively less at higher cost

The effect of cost is more pronounced as compared to selling prices

In EPM3, the effect of uniformity penalty, if the target is set other than Mean, is high even at very low values of K

CONCLUSIONSCONCLUSIONS

In EPM4 the effect of measurement error and uniformity penalty were integrated

The numerical comparison show significant gain in profit if more realistic model is used

Gain in the case of error is higher with the higher error

Gains are also significant, even at low values of K, if the Target is set farther from the mean

In EPM4 the effect of measurement error and uniformity penalty were integrated

The numerical comparison show significant gain in profit if more realistic model is used

Gain in the case of error is higher with the higher error

Gains are also significant, even at low values of K, if the Target is set farther from the mean

FUTURE RESEARCHFUTURE RESEARCH

Process Targeting is an active area for research.

It has the potential to be appreciably beneficial for the industry.

The few of the direction from this work are

Process Targeting is an active area for research.

It has the potential to be appreciably beneficial for the industry.

The few of the direction from this work are

Generalization for the n-class screening situation

Other sampling plans, instead of 100% inspection

Processes with drift

Machines in series

Integration of other production decisions

Generalization for the n-class screening situation

Other sampling plans, instead of 100% inspection

Processes with drift

Machines in series

Integration of other production decisions

THANK YOUTHANK YOU

Documents

Targeting in Multi-Class Screening Under Error and Error Free Measurement Systems