2.852 lecture 25: an integrated quality/quantity model of a transfer line jongyoon kim stanley b....

2.852 Lecture 25:

An Integrated Quality/Quantity Model of a Transfer Line

Jongyoon Kim

Stanley B. Gershwin

Copyright 2003 Jongyoon Kim, Stanley B. GershwinAll right reserved.

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Copyright 2003 Jongyoon Kim, Stanley B. GershwinAll rights reserved

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AGENDAAGENDA

1. Motivation

2. Research Objectives

3. Research Directions

4. Model Descriptions

5. Preliminary Results

6. Future Research Plan


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MOTIVATIONMOTIVATION

“Effective implementation of quality program leads to significant wealth creation.” (38% to 46% of higher stock value)*

“At 3 sigma level, quality cost accounts for 25 – 40% of Revenue.”**

“A dissatisfied customer will tell 9 to 10 people about an unhappy experience, even if the problem is not serious.”***

*Hendricks and Singhal, Management Science**Harry and et al., “6 Sigma”***Pande, Holpp, “What is 6 Sigma?”

Goal: Perfect Quality!

Quality


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System Effect on Quality

Design and control of production system have significant impact on product quality”*

*Inman et al., General Motors R&D center** Smith, Wards Auto World, July 2001*** Monden, Toyota Production System

• Jaguar**After Ford acquired Jaguar, Jaguar’s quality improved rapidly. Product design wasn’t changed, but production system changed.

• Toyota***Numerous study show that Toyota’s particular attention to the production system’s impact on quality



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MOTIVATIONMOTIVATIONQuality, Quantity and

Production System Design

*Womack, et al. (1990)Machine that changed the world

Is there any relationships among quality, productivity, and production system design?If then, can we quantify them?


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MOTIVATIONMOTIVATION Toyota Production System

Does reduction of inventory lead to higher quality?

Does JIDOKA (stopping the lines whenever abnormalities occur) improve quality and productivity in every case?

Scientific conceptual and computational model is needed.

Existing arguments about this are based on anecdotal experience or qualitative reasoning that lack sound scientific foundation.


c

Gain in-depth understanding to investigate how manufacturing system design and operations simultaneously influence quality and productivity.

RESEARCH RESEARCH OBJECTIVEOBJECTIVE

Mission


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MOTIVATIONMOTIVATIONSystem Yield

System yield is the fraction of production that is of acceptable quality.

System yield depends on•Individual operation yields,•Inspection strategies,•Operation policies,•Environmental conditions

in a very complex way.•Buffer sizes, and

Comprehensive approaches are needed to manage system yield effectively.


c

M1 B1 M2 B8 M9 B9 M10

For typical mfg. operations, Cp =1.3, which means yield = 99%

•For one operation – 99%•For 10 operations – 90.4%•For 100 operations – 36.6%

Probability of non-defective output is

Limit of PartialApproachMOTIVATIONMOTIVATION

Focusing on the yield of individual operations gives limited influence on the system yield.


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• Lead time :

3 weeks

• Chronic Scrap: 5 - 6 %(50,000 ppm)

• Machine capability:

Cp = 1.67 (233 ppm)

• Inventory :

300,000 units

WHEELABRATOR SHOT-PEEN (7)

INCOMING PINION FORGINGS

KASPER CNC SPINDLE LATHES (12)

GLEASON #116 ROUGHERS (57)

GLEASON #960 (12)GLEASON #116FINISHERS (64)

HEAT TREAT

ANNEAL CELL

PRATT & WHITNEY GRINDERS (14)

INCOMING RING FORGINGS

KASPER BORING LATHES (8)

KASPER TURNING LATHES (8)

BARNES DRILLS (4) SNYDER DRILL STANDARD DRILL

GLEASON 606/607GEAR CUTTERS (43)

HEAT TREAT

ID HONING MACHINES (6)GEAR SET MATCHINGGLEASON #514GLEASON #506

GLEASON #19

OERLIKON

24LAPPERS

36LAPPERS

36LAPPERS

GLEASON 17A ROLL TESTERS (21)

Assembly

Limit of Robust Processes

Hypoid gear set factory in Michigan



c

MOTIVATIONMOTIVATIONInfluence of Material

Flow Control

A material control scheme can affect the performance of a factory dramatically.*

Various practices are used to control material flow.MRP/ERPKanbanConwip

*Bonvik Couch and Gershwin, “A comparison of production line control mechanism”, International Journal of Production Research, 1997


c

Managerial Practices Statistical Quality Control (SQC) Poka-Yoke -> Early detection of defective parts Total Quality Management (TQM) Six Sigma-> Root cause elimination

Academic Approaches Inspection Location Problem-> Only heuristic solutions using simulation have been

proposed. Optimal Quality Control Chart Design

What has been done?- Quality Control MOTIVATIONMOTIVATION


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Managerial Practices Kanban CONWIP Base Stock MRP ERP

Academic Approaches Control Point Policy Hybrid Control Generalized Kanban Extended Kanban

What has been done?- Material Flow Control MOTIVATIONMOTIVATION


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Practices in Quality Control have focused on one operation; each machine is treated separately.

But quality is a system-wide problem.

Practices in material flow control assume that each machine produces non-defective parts.

But capacity is wasted if machines are working on already defective parts.

Quality control and material flow control are interrelated and need to be treated together.

Missing LinksMOTIVATIONMOTIVATION


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Ideal Inspections MOTIVATIONMOTIVATION

Production

M 1

Inspection

M1 B1 M2 B8 M9 B9 M10

Ideally, inspection is ubiquitous. Bad parts are caught and scrapped or reworked immediately.

No downstream capacity is wasted on parts that will be scrapped. Problems are identified and corrected immediately.


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Actual Inspections MOTIVATIONMOTIVATION

M1 B1 M2 B8 M9 B9 M10

However, ubiquitous inspection is expensive. Inspection is often done at inspection stations.

Question: What is the best distribution of inspection stations?


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Find the best way to achieve high quality with low cost by bringing quality control and material flow control together.

Develop a conceptual and computational tool which avoids conflict between productivity and quality. The tool will efficiently assess quality and throughput simultaneously.

RESEARCH RESEARCH OBJECTIVEOBJECTIVE

Mission


c

We choose a continuous material analytical model because

Considerably less computation required time than simulation -> Larger searchable design space Same inputs always give same outputs ->Easy to evaluate reliable direction for improvement Continuous optimization is much easier than discrete optimization.

But…

Tough to develop. Approximate.

Research DirectionResearch DirectionAnalytic or Simulation?


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Research PlanResearch PlanBig picture

1. Immediate Research Plan Characterization 2M 1B Long line decomposition

2. Long term possible research tasks Optimization Case study for PSA

3. Ultimate Goal Redesign factory layouts. Improving inspection policies Revamping material flow controls SAVE BIG MONEY


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Characterization

Simulator Construction – For a comparison purpose

Development of a new decomposition method for long line

2M-1B case

Zero buffer/ Infinite buffer case

Transition Equations

Finding probability density function

Boundary conditions

Performance measures evaluations

Research PlanResearch PlanImmediate Research Plan


c

Model DescriptionModel DescriptionQuality

Assignable Variation: variation due to a specific, identifiable cause which changes the process mean.Random Variation: variation that is inherent in the design of the process and cannot be removed.

Two sources of process variation

d

Meand

Tool Change

Assignable Variation

Random Variation

In this research, we focus on Assignable Variation.


c

REVIEWREVIEWQuality Failures

Common Cause Variation: random variation that is inherent in the design of the process and cannot be removed.Assignable Cause Variation: variation due to a specific, identifiable cause.

Two kinds of process variation

Two types of quality failuresBernoulli-type: quality failure due to common cause of variation-> quality of each part is independent of others.Markovian-type: quality failure due to assignable cause of variation.-> Once a bad part is produced, all the subsequent parts will be bad until the operation is repaired.


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Objective – Capture important features but try to be simple!

1 -1

0

0’

pp

r

rgf

r

01 -1

p

r

g f

•Each machine can produce ‘good’ or ‘bad’ parts•Each machine has inspection

• How many states do we need? -> as few as possible!

Characterization Model DescriptionModel Description

State of One Machine

0’’


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1 -1 0

p

g f

r

Each machine has 3 states

•State 1:Machine is producing non-defective parts.

•State –1: Machine is producing defective parts but the operator doesn’t know it.

•State 0: Machine is not operating

State of One Machine

Model DescriptionModel DescriptionCharacterization


c

= a speed at which a machine processes material while it is operating and not constrained by the other

machine or buffer. p= probability rate that machine fails (=1/MTTF)

r= probability rate that machine is repaired (= 1/MTTR)

f = rate of transition from state –1 to state 0 (=1/Mean Time to Detect and Stop)

-> more inspection leads smaller MTDS & larger f

g = rate of transition from state 1 to state –1 (=1/Mean Time to Quality Failure)

-> more stable operation leads larger MTQF & smaller g

REVIEWREVIEWSingle Machine Analysis


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State of 2M1B

M 1 M 2B 1

),,,( 21 yx

State Definition

x : total amount of material at B1

y : amount of defective material at B1

1 : status of machine 1 (1,-1,0)


•Blockage of machine 1 and starvation of machine 2 are dependent on x and independent of y.

•Inspection at Machine 2 can trigger state change at machine 1.

Model DescriptionModel DescriptionCharacterization


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Buffer Size

Prod

ucti

on R

ate

Zero Buffer

Infinite Buffer

Finite Buffer

2M1B Special Cases

Develop formulas with special cases: zero buffer and infinite buffer cases.

Results are very good.

Infinite Buffer Case

Zero Buffer Case

REVIEWREVIEW

Case # Pe(A) Pe(S) % Difference1 0.657 0.662 -0.73%

2 0.620 0.627 -1.15%

3 0.614 0.621 -1.03%

4 0.529 0.534 -0.99%

5 0.480 0.484 -0.77%

6 0.647 0.651 -0.57%

7 0.706 0.712 -0.91%

8 1.377 1.526 -9.79%

9 0.706 0.711 -0.77%

10 1.377 1.380 -0.22%

Case # Pe(A) Pe(S) % Difference1 0.762 0.761 0.17%

2 0.708 0.708 0.00%

3 0.657 0.657 -0.06%

4 0.577 0.580 -0.50%

5 0.527 0.530 -0.42%

6 0.745 0.745 0.01%

7 0.762 0.760 0.30%

8 1.524 1.522 0.14%

9 0.762 0.762 0.01%

10 1.524 1.526 -0.13%


c

9 transition equations are derived),,1,0(),,0,1(),,1,1()(

),,1,1(),,1,1()(

),,1,1(122211

212 yxfryxfryxfgpgp

y

yxf

x

y

x

yxf

t

yxf

),,0,0(),,1,1(),,0,1()(),,0,1(

),,1,1(),,0,1(

1221112 yxfryxffyxfrgpx

yxfyxfp

t

yxf

),,1,0(),,1,1()(),,1,1(),,1,1(

)(),,1,1(),,1,1(

12112

122 yxfryxffgpy

yxf

x

y

x

yxfyxfg

t

yxf

),,1,1(),,0,0(),,1,0()(),,1,0(),,1,0(

),,1,1(),,1,0(

122212

21 yxffyxfryxfgpry

yxf

x

y

x

yxfyxfp

t

yxf

),,0,1(),,1,0(),,0,0()(),,1,0(),,0,1(),,0,0(

122121 yxffyxffyxfrryxfpyxfpt

yxf

),,1,1(),,1,0(),,1,0(

),,1,0()(),,1,0(),,1,1(),,1,0(

12

22121 yxffyxx

y

x

yxfyxffryxfgyxfp

t

yxf

),,0,1(),,1,1(

)(),,1,1(

)(),,1,1()(),,1,1(),,1,1(

212

121221 yxfry

yxf

x

y

x

yxfyxffgpyxfg

t

yxf

),,1,1(),,1,1(),,0,1()(),,0,1(),,0,1(

),,0,1(),,0,1(

2212111 yxffyxfpyxffry

yxf

x

yxfyxfg

t

yxf

),,1,1()(),,1,1(

)(),,1,1(

)(),,1,1(),,1,1(),,1,1(

2112

1221 yxfffy

yxf

x

y

x

yxfyxfgyxfg

t

yxf

Model DescriptionModel Description2M1B Internal Transition

Equations


c

The transition equations are linear partial differential equations in t,x,y with coefficients that are nonlinear functions or x and y.

),,( 21 x

Unlikely to be solved

Facts• Starvation and Blockage of machines are independent of y.• Average value of y can be calculated from other formulations.

Redefine the 2M1B state as


Equations


c

1 2( , , )x State Definition

x : total amount of material at B



2M1B

No part is scrapped: defective parts are marked and reworked later.

REVIEWREVIEW

M 1 M 2B


c

2M1B Finite Buffer Case

Finite buffer case:

Approach: Develop Markov process model;

write and solve transition equations and boundary equations.

When buffer B is neither empty nor full the behavior is described by differential equations with probability density function f(x,1,2).

When buffer B is either empty or full, the behavior is described by boundary equations with probability mass functions P(0,1,2), P(N,1,2).

REVIEWREVIEW


c

9 transition equations for f(x,1,2) are derived

2 1 1 1 2 2 2 1

( ,1,1) ( ,1,1)( ) ( ) ( ,1,1) ( ,1,0) ( ,0,1)

f x f xp g p g f x r f x r f x

t x

2 1 1 1 2 2 1

( ,1,0) ( ,1,0)( ,1,1) ( ) ( ,1,0) ( ,1, 1) ( ,0,0)

f x f xp f x p g r f x f f x r f x

t x

2 2 1 1 1 2 1

( ,1, 1) ( ,1, 1)( ,1,1) ( ) ( ) ( ,1, 1) ( ,0, 1)

f x f xg f x p g f f x r f x

t x

1 2 1 2 2 2 1

( ,0,1) ( ,0,1)( ,1,1) ( ) ( ,0,1) ( ,0,0) ( , 1,1)

f x f xp f x r p g f x r f x f f x

t x

1 2 1 2 2 1

( ,0,0)( ,1,0) ( ,0,1) ( ) ( ,0,0) ( ,0, 1) ( , 1,0)

f xp f x p f x r r f x f f x f f x

t

1 2 1 2 2 1

( ,0, 1) ( ,0, 1)( ,1, 1) ( ,0,1) ( ) ( ,0, 1) ( , 1, 1)

f x f xp f x g f x r f f x f f x

t x

1 2 2 1 2 1 2

( , 1,1) ( , 1,1)( ,1,1) ( ) ( , 1,1) ( ) ( , 1,0)

f x f xg f x p g f f x r f x

t x

1 1 2 1 2 2

( , 1,0) ( , 1,0)( ,1,0) ( ) ( , 1,0) ( , 1,1) ( , 1, 1)

f x f xg f x r f f x p f x f f x

t x

1 2 2 1 1 2

( , 1, 1) ( , 1, 1)( ,1, 1) ( , 1,1) ( ) ( ) ( , 1, 1)

f x f xg f x g f x f f f x

t x

REVIEWREVIEW2M1B Internal



c

)()(),,( 221121 GGexf xAssume a solution in a form of .

After much mathematical manipulation, the equations are

simplified into 2 equations and 2 unknowns.

It turns out that there are multiple roots

depending on machine parameters. (3 to 7 roots)

A general solution to the transition equations is a linear

combination of the roots.

REVIEWREVIEWSolution to Internal


1 2 1 1 2 2( , , ) ( ) ( )i x i iif x e G G

1 2 1 1 2 21

( , , ) ( ) ( )i

RNx i i

ii

f x c e G G


c

)()(),,( 221121 GGexf xAssume that Then, the transition equations in steady state become

0)1()0()0()1()}1()1()(){( 21121221221112 GGrGGrGGgpgp

0)0()0()1()1()1()1()0()1()}({ 211212212212111 GGrGGfGGpGGrgp

0)1()0()1()1()1()1()}(){( 2112122121112 GGrGGgGGfgp

0)1()1()0()0()1()1()1()0()}({ 211212211212212 GGfGGrGGpGGgpr

0)0()1()1()0()0()0()()1()0()0()1( 2112122121212211 GGfGGfGGrrGGpGGp

0)1()1()1()0()1()1()1()0()}({ 21121221121212 GGfGGgGGpGGfr

0)0()1()1()1()1()1()}(){( 2122112112212 GGrGGgGGfgp

0)1()1()1()1()0()1()0()1()}({ 21221221121121 GGfGGpGGgGGfr

0)1()1()1()1()1()1()}(){( 212211212112 GGgGGgGGff


Equations


c

021

212 121

2121 )( 2121 )(

121 212

1221 )( 2121 )(

)0(

)1(

)0(

)1(

i

iii

i

iii G

Gfr

G

Gp

)1(

)0(

i

iiiii G

Grgp

)1(

)1(

i

iiii G

Ggf

If we define

Then, 9 transitionequations become

Model DescriptionModel Description2M1B Transition

Equations (5)


c

021 1 1 1 1 1 2 2 2 2 2( )p Y r f Z p Y r f Z M

1 1 2 21 1 1 1 2 2 2 2

1 1 2 2,r Y r Yp g f g p g f gY Z Y Z

1 21 1 2 2

1 11 1(1 ) (1 ) NY Z Y Z

In addition to that, we already have


Equations

And introduce two parameters: M, N

(1) ( 1),

(0) (0)i i

i ii i

G GY Z

G G

Define


c

After multiple mathematical manipulation we get

21 1 1 1 1 1 1 1 1 1 1

11 1 1 1 1 1

22 2 2 2 2 2 2 2 2 2 2

22 2 2 2 2 2

{( )( 1) } {( ) ( 1)}{( )( 1) }0 ( )

( )( 1) ( )( 1)

{( )( 1) } {( ) ( 1)}{( )( 1) }0 ( )

( )( 1) ( )( 1)

M r N f p g f r N M r N fr red

f p N f p N

M r N f p g f r N M r N fr blue

f p N f p N

It turns out that this formulation is easy to be solved because• Curves only have asymptotes perpendicular to M or N• Locations of gaps at red curves and blue curves are easily calculated • Locations and number of roots are easily estimated


Equations


c


Equations


c

From the characterization of red curves and blue curveswhat we found are:

•There are 3 roots at the top•There are no roots at the bottom•There are maximum 4 roots at RHS or LHS•If there are any roots at RHS, then there are no roots at LHS and vice versa

Root finding

Based on the characterization of the curves, a special algorithm to find all the roots is developed.

Model DescriptionModel Description2M1B Transition

Equations (10)


c

2M1B Boundary Equations

1 1 2 2 1( ) (0,1,1) (0,0,1) 0b bp g p g P r P

2 1 1 2 1(0,1,1) ( ) (0,1, 1) (0,0, 1) 0b bg P p g f P r P

1 1 2 1 2(0,1,1) (0,0,1) (0,0,1) (0, 1,1) (0,0,0) 0p P r P f f P r P

1 1 2 1(0,1, 1) (0,0, 1) (0,0, 1) (0, 1, 1) 0p P r P f f P

1 1 2 2(0,1,1) ( ) (0, 1,1) 0b bg P f p g P

1 2 1 2(0,1, 1) (0, 1,1) ( ) (0, 1, 1) 0b bg P g P f f P

How to determine coefficients in probability density functionsand probability masses ?

ic

1 2 1 2(0, , ), ( , , )P P N

22 Boundary equations are derived for case and solved.1 2

…

REVIEWREVIEW


c

After finding all of probability densities and masses,

Total production rate is calculated from

2

1

11 2 2 2 2

1,0,1 0

2 1 11,0,1

[ { ( , 1, ) ( ,1, )} (0,1, ) (0, 1, )]

{ ( , ,1) ( , , 1)}

N

T TP P f x f x dx P P

P N P N

REVIEWREVIEW2M1B

Performance Measures

Average inventory is expressed as

1 2

1 2 1 21,0,1 1,0,1 0

( , , ) ( , , )N

x xf x dx NP N


c

1 2E E

E TT T

P PP PP P

where2

11 2 2 2

1,0,1 0

[ ( ,1, ) (0,1, )] { ( ,1, 1) ( ,1,1)}N

EP f x dx P P N P N

1

22 1 1 1

1,0,1 0

[ ( , ,1) ( , ,1)] { (0, 1,1) (0,1,1)}N

EP f x dx P N P P

REVIEWREVIEW2M1B

Performance Measures

Effective production rate is calculated from

After finding all of probability densities and masses,


c

Preliminary ResultPreliminary Result2M1B Validation

Solution for 2M1B with equal production rate case is found and validated through comparison with simulation.

Case # PR(Analytic) PR(Sim) %Difference Inv(Analytic) Inv(Sim) %Difference1 0.806 0.808 -0.25% 2.500 2.619 -4.53%2 0.855 0.858 -0.37% 25.000 24.883 0.47%3 0.936 0.938 -0.23% 4.709 4.989 -5.60%4 0.944 0.946 -0.22% 12.654 12.757 -0.81%5 0.909 0.911 -0.19% 2.781 2.832 -1.81%6 0.922 0.924 -0.24% 9.213 9.318 -1.13%7 0.909 0.910 -0.07% 2.220 2.321 -4.39%8 0.925 0.926 -0.18% 7.242 7.080 2.30%9 0.840 0.843 -0.38% 20.020 20.149 -0.64%

10 0.763 0.767 -0.49% 4.983 5.110 -2.48%


c

2M1B Validation (1)

Analytic solution for 2M1B with case is found and validated through comparison with simulation.

Average absolute error = 0.73%

Effective production rate estimation Average inventory estimation

Average absolute error = 2.75%

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Case Number

% e

rror

of

PE

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Case Number

% e

rror

of

In

v

REVIEWREVIEW

1 2


c

Boundary Equations

26 Boundary equations are derived and solved.

22M1B with M1B with 1 2

1 2 2(0,1,0) (0,1,1) (0,1, 1)b bf p P f P

1 2 2(0, 1,0) (0, 1,1) (0, 1, 1)b bf p P f P

2 ( ,0,1) 0f N

2 ( ,0, 1) 0f N

2 1 2( ) ( ,1,1) ( ,1,0)f N r P N

2 1 2( ) ( , 1,1) ( , 1,0)f N r P N

1 1 2 2 2 1 1( ) (0,1,1) ( ) (0,1,1) (0,0,1) 0b bp g p g P f r P

…


c

22M1B with M1B with 1 2 2M1B Validation (2)

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 4 7 10

13

16

19

22

25

28

31

34

37

40

43

46

49

Case Number

% e

rror

of

P E

Analytic solution for 2M1B with case is found and validated through comparison with simulation.

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Case Number

% e

rror

of

In

v

Effective production rate estimation

1 2

Average inventory estimation

Average absolute error = 0.68% Average absolute error = 3.41%


c

Preliminary ResultPreliminary ResultQuality Information

Feedback

M 1 M 2

Inspection @ M2 notify M1 isproducing bad parts

Inspection @ M1 notify M1 isproducing bad parts

Downstream machines can detect defective parts made by an upstream machine and notify the operator at the machine.

Quality Information Feedback


c

Preliminary ResultPreliminary ResultQuality Information

Feedback

If there is quality information feedback, the yield of the system depends on the time gap between making a defect and identifying the defect.

System yield is a function of buffer size: A smaller buffer increases system yield since lower inventory level leads to a smaller the time gap.


c

Preliminary ResultsPreliminary ResultsQuality Information

Feedback

0 5 10 15 20 250.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.8

Buffer Size

Pro

duct

ion

Rat

e

Pr w/o feedbackPr w/ feedback

0 5 10 15 20 250.65

0.66

0.67

0.68

0.69

0.7

0.71

0.72

0.73

0.74

Buffer Size

Effe

ctiv

e P

rodu

ctio

n R

ate

ePr w/o feedbackePr w/ feedback

*The effective production rate is the rate of production of good parts.

Quality feedback slightly decrease total production rate but increases *effective production rate significantly.


c

Preliminary ResultsPreliminary ResultsHow to Increase

Quality

0 50 100 150 200 250 300 350 400 450 5000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mean Time To Defect

Eff

ectiv

e P

rodu

ctio

n R

ate

0 20 40 60 80 100 120 140 160 180Mean Time To Identify

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Eff

ectiv

e P

rodu

ctio

n R

ate

Effect of maintenance (MTTD) on quality monotonically decreases.Effect of inspection (MTTI) on quality is monotonically decreases

There is a combination of maintenance and inspection policies that achieve target quality with minimum cost.


c

Preliminary ResultsPreliminary ResultsHow to increase

productivity

In some situations, increasing inspection reliability is more effective than increasing buffer size to boost productivity.

0 5 10 15 20 25 30 35 400.63

0.635

0.64

0.645

0.65

0.655

0.66

Buffer Size

Eff

ectiv

e P

rodu

ctio

n R

ate

0 5 10 15 20 25 30 35 400.885

0.89

0.895

0.9

0.905

0.91

0.915

0.92

0.925

0.93

Buffer Size

Eff

ectiv

e P

rodu

ctio

n R

ate


c

Preliminary ResultPreliminary ResultHow to increase productivity?

0 5 10 15 20 25 30 35 400.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

Buffer Size

Eff

ectiv

e P

rodu

ctio

n R

ate

0 5 10 15 20 25 30 35 400.24

0.25

0.26

0.27

0.28

0.29

0.3

0.31

0.32

0.33

Buffer Size

Eff

ectiv

e P

rodu

ctio

n R

ate

In some situations, increasing machine stability is more effective than increasing buffer size to enhance productivity.


c


Downstream machines can detect defective parts made by an upstream machine and notify the operator at the machine.

REVIEWREVIEW

M 1 M 2

Inspection @ M2

Inspection @ M1

System yield is a function of buffer size: A smaller buffer increases system yield since lower inventory level leads to a early detection of quality failures.

Quality information feedback is captured by modifying f.


c

0 5 10 15 20 250.65

0.7

0.75

0.8

Buffer Size

Effe

ctiv

e P

rodu

ctio

n R

ate Without feedback

With feedback

0 5 10 15 20 250.65

0.7

0.75

0.8

Without feedbackWith feedback

0 5 10 15 20 250.65

0.7

0.75

0.8


0 5 10 15 20 250.65

0.7

0.75

0.8

Buffer Size

Tot

al p

rodu

ctio

n R

ate


0 5 10 15 20 250.65

0.7

0.75

0.8


0 5 10 15 20 250.65

0.7

0.75

0.8



Quality feedback results in more effective production rate and less total production rate.

Increase of buffer size is beneficial contrary to TPS.

MFG. SYSTEM MFG. SYSTEM BEHAVIORBEHAVIOR


c


The effective production rate may decrease as the buffer size increases when•M1 is faster than M2 and quality feedback exists.•M1 produces bad parts frequently.•Inspection at M1 is poor and inspection at M2 is good.

This is a case when inventory reduction is good as TPS advocates.

Harmful Buffer Case

0 5 10 15 20 25 30 35 400.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Buffer Size

Eff

ectiv

e Pr

oduc

tion

Rat

e



c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f

Eff

ectiv

e Pr

oduc

tion

Rat

e

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f

Eff

ectiv

e Pr

oduc

tion

Rat

e

0.398

0.560

M1 B1 M2 M1 B1 M2 B2 M3 B3 M4

Effect of inspection (MTDS) on effective production rate

decreases as f increases. increases as a line gets longer.


Effectiveness of Inspection


c


Effectiveness of Operation Stabilization

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

MTQF

Eff

ecti

ve P

rodu

ctio

n R

ate

0 50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

MTQF

Eff

ecti

ve P

rodu

ctio

n R

ate

0.609

0.701

Effect of machine stabilization (MTQF) on effective production rate

decreases as MTQF increases. increases as a line gets longer.

M1 B1 M2 M1 B1 M2 B2 M3 B3 M4


c

How to increase productivity

In some situations, increasing inspection reliability is more effective than increasing buffer size to boost productivity.

0 5 10 15 20 25 30 35 400.5

0.55

0.6

0.65

0.7

0.75

0.8

Buffer Size

Eff

ect

ive

Pro

du

ctio

n R

ate

MTDS = 20MTDS = 10MTDS = 2



c

How to increase productivity

In some situations, increasing machine stability is more effective than increasing buffer size to enhance productivity.

0 5 10 15 20 25 30 35 400.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Buffer Size

Eff

ect

ive

Pro

du

ctio

n R

ate

MTQF = 20MTQF = 100MTQF = 500



c

LONG LINELONG LINEExtension

Longer line increases the dimension of partial differential equations used at internal transitions equations.

No good exact method known for longer lines.

It is reasonable to use approximation methods to obtain solutions for transfer lines with more than two machines.

Decomposition techniques have been successfully used for various kind of long line analysis.

Tandem long line Assembly/disassembly line Closed loop


c

Decomposition TechniqueLONG LINELONG LINE

M1 B1 M2 M3 M4B2 B3

1 1 1

1 1

, ,

,

r p

g f

2 2 2

2 2

, ,

,

r p

g f

3 3 3

3 3

, ,

,

r p

g f

4 4 4

4 4

, ,

,

r p

g f

L

(1), (1), (1)

(1), (1)u u u

u u

r p

g f

Mu (1) B(1) L(1)Md (1)

1N

B(2) L(2)Mu (2) Md (2)

B(3) L(3)Mu (3) Md (3)

(1), (1), (1)

(1), (1)d d d

d d

r p

g f

Decomposition

(2), (2), (2)

(2), (2)u u u

u u

r p

g f

2N

(2), (2), (2)

(2), (2)d d d

d d

r p

g f

(3), (3), (3)

(3), (3)u u u

u u

r p

g f

3N (3), (3), (3)

(3), (3)d d d

d d

r p

g f

1N 2N3N


c

Decomposition TechniqueLONG LINELONG LINE

Decomposition TechniqueDecompose the line (L) into a set of two-machine lines L(i) in a way that performance measures of the L(i)s are close to those of the original line L.

Pseudo-machine Mu(i) models the part of the line upstream of Bi and Md(i) models the part of line downstream from Bi.

Decomposition techniques work well even though no mathematical proof is available.

Procedures

Develop equations for 10(k-1) pseudo-machine parameters for k-machine line.

Develop algorithm to solve the equations efficiently.


c

Ubiquitous InspectionCaseLONG LINELONG LINE

Each machine has both of operational failures and quality failures Each operation works on different features. Inspection at machine Mi can detect defective features made by Mi

not others.

For each decomposed line L(i), incoming parts from upstream are viewed as non-defective ones. -> gi is independent of other machine parameters: gu(i) = gi , gd(i) = gi+1

Outgoing defective parts from L(i) are not checked from inspection downstream -> fi is independent of others: fu(i) = fi , fd(i) = fi+1

M1 B1 M2 M3 M4B2 B3


c

3-state-machine / 2-state-machine

Stopped Operating

1 -1 0

r

p

g f

1' 0

p'

r

LONG LINELONG LINE

How to determine and ? Strategy: approximate 3-state machine with 2-state

machine.

The strategy will work well if the transition time from up states (1 or -1) to down state closely follow to the exponential distribution.

0 100 200 300 400 500 600 70010

-5

10-4

10-3

10-2

10-1

100

Freq

uenc

y di

stri

butio

n

Transition time from 1 to 0

( ), ( ), ( ), ( ), ( )u u u d di r i p i i r i ( )dp i


c

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

Case Number

% e

rror

of

PE

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97

Case Number

% e

rror

of

In

v


A 2-state-machine with parameter adjustments closely approximates the corresponding 3-state-machine.

LONG LINELONG LINE3-state-machine / 2-state-machine


c

Long Line ValidationLONG LINELONG LINE

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Case Number

% e

rror

of

PE

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Case Number

% e

rror

of

Inv

1

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Case Number

% e

rror

of

Inv

2

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Case Number

% e

rror

of

Inv

3

M1 B1 M2 M3 M4B2 B3




c

UPCOMING TASKUPCOMING TASKEffective of Jidoka

Does JIDOKA (stopping the lines whenever abnormalities occur) improve quality and productivity in every case?

Hypothesis

The effectiveness of Jidoka on productivity depends on which type of quality failures (Markovian or Bernoulli) is dominant.

The effective production rate may decrease by adopting Jidoka practice when quality failures are mixture of Bernoulli and Markovian


c

FUTURE PLANFUTURE PLANNext Long Line Analysis Task

Inspection

M1 B1 M2 M3 M4B2 B3

Operational Failures + Quality Failures

Operational Failures

M1 undergoes both of quality failures and operational failures.

Other machines (M2, M3 , M4 ) have only operational failures.

Inspection takes place only at the final machine (M4)


c

FUTURE PLANFUTURE PLANNext Long Line Analysis Task

Inspection

M1 B1 M2 M3 M4B2 B3

Operational Failures + Quality Failures

Each machine has both of quality failures and operational failures.

Inspection is only at the final machine (M4) and detect bad parts made by any of upstream machines.


c


c

RESEARCHRESEARCHPROGRESSPROGRESS

Term Research Progress

1999 - Spring 2002 Research on Toyota Production Systems .Fall 2002 Chacterization.

2M1B model formulation. Simulation building. Infinite buffer and zero buffer validations.

Summer 2003 Internship at General Motors Proposal to General Motors for collaboration. Thesis proposal finalization. 2M1B model completion.

1st committee meeting (Dec, 15th) Long line decomposition without quality feedback. Validation of the decomposition technique.

2nd committee meeting (April 7th) Study on Jidoka practice. Long line decomposition with quality feedback. Numerical experimentations and intuition building.

3rd committee meeting (late May/ early June) Possible case study at GM Finalize the research Finish thesis write up Defense (Early September)

Summer 2004

Fall 2003

Spring 2003

Spring 2003


c

SupplementarySupplementaryNotesNotes

Input Parameters for2M1B Special Cases

Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f21 1.0 1.0 0.400 0.100 0.010 0.010 0.001 0.001 0.100 0.5002 1.0 1.0 0.100 0.100 0.010 0.010 0.001 0.001 0.100 0.5003 1.0 1.0 0.100 0.100 0.010 0.005 0.001 0.001 0.100 0.5004 1.0 1.0 0.100 0.100 0.010 0.005 0.005 0.001 0.100 0.5005 1.0 1.0 0.100 0.100 0.010 0.005 0.001 0.001 0.100 0.100

Infinite Buffer Case

Zero Buffer CaseCase # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f2

1 1.0 1.0 0.400 0.100 0.010 0.010 0.001 0.001 0.100 0.1002 1.0 1.0 0.100 0.100 0.010 0.010 0.001 0.001 0.100 0.1003 1.0 1.0 0.100 0.100 0.010 0.000 0.001 0.001 0.100 0.1004 1.0 1.0 0.100 0.100 0.010 0.010 0.010 0.001 0.100 0.1005 1.0 1.0 0.400 0.100 0.010 0.010 0.001 0.001 0.100 0.300


c


Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f2 N FB?1 1.0 1.0 0.100 0.100 0.005 0.005 0.010 0.010 0.100 0.100 5 N2 1.0 1.0 0.100 0.100 0.005 0.005 0.010 0.010 0.100 0.100 50 N3 1.0 1.0 0.300 0.300 0.010 0.010 0.005 0.005 0.500 0.500 10 N4 1.0 1.0 0.300 0.300 0.010 0.010 0.005 0.005 0.500 0.500 25 N5 1.0 1.0 0.300 0.200 0.010 0.010 0.005 0.005 0.500 0.500 5 N6 1.0 1.0 0.300 0.200 0.010 0.010 0.005 0.005 0.500 0.500 15 N7 1.0 1.0 0.200 0.300 0.010 0.010 0.005 0.005 0.500 0.500 5 N8 1.0 1.0 0.200 0.300 0.010 0.010 0.005 0.005 0.500 0.500 20 N9 1.0 1.0 0.100 0.100 0.010 0.010 0.005 0.005 0.200 0.200 40 Y

10 1.0 1.0 0.100 0.100 0.010 0.010 0.001 0.001 0.200 0.200 10 Y

Input Parameters for2M1B Validation

Intermediate Buffer Size Case


c


How to increase productivity (2)

Input Parameters forPreliminary Result

Mu1 r1 p1 f11 0.1 0.01 0.1

Mu2 r2 p2 f21 0.1 0.01 0.1

How to increase productivity (1)

Mu1 r1 p1 g11 0.1 0.005 0.001

Mu2 r2 p2 g21 0.1 0.005 0.001

Mu1 r1 p1 g1 f11 0.1 0.01 0.01 0.1

Mu2 r2 p2 g2 f21 0.1 0.01 0.01 0.9

Quality feedback

Increase Quality

Mu1 r1 p1 g1 f11 0.1 0.01 0.01 0.1

Mu2 r2 p2 g2 f21 0.1 0.01 0.01 0.1


c

21 1 1 1 1 1 1 1 1 1 1

11 1 1 1 1 1

22 2 2 2 2 2 2 2 2 2 2

22 2 2 2 2 2

{( )( 1) } {( ) ( 1)}{( )( 1) }0 ( )

( )( 1) ( )( 1)

{( )( 1) } {( ) ( 1)}{( )( 1) }0 ( )

( )( 1) ( )( 1)

M r N f p g f r N M r N fr red

f p N f p N

M r N f p g f r N M r N fr blue

f p N f p N

After much mathematical manipulation.the 9 equations and 7 unknowns are simplified to

where1 1 2 2

1 1 1 2 2 21 1 2 2

(1) ( 1) (1) ( 1)( )

(0) (0) (0) (0)

G G G Gp r f p r f M

G G G G

1 1 2 21 2

1 1 2 21

1 1 1 1(1 ) (1 )

(1) ( 1) (1) ( 1)(0) (0) (0) (0)

NG G G GG G G G

Simplified Internal Transition Equations SUPPLEMENTARYSUPPLEMENTARY

NOTESNOTES


c

Shape of SITEs SUPPLEMENTARYSUPPLEMENTARYNOTESNOTES


c

SUPPLEMENTARYSUPPLEMENTARYNOTESNOTES

Input Parameters for2M1B Special Cases

Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f21 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.22 1 1 0.3 0.3 0.005 0.005 0.05 0.05 0.5 0.53 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.24 1 1 0.1 0.1 0.05 0.005 0.01 0.01 0.2 0.25 1 1 0.1 0.1 0.01 0.01 0.05 0.005 0.2 0.26 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.5 0.17 2 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.28 3 2 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.29 1 2 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.210 2 3 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2


c

Input Parameters forPreliminary Results

Mu1 r1 p1 g1 f11 0.1 0.01 0.01 0.1

Mu2 r2 p2 g2 f21 0.1 0.01 0.01 0.9

Quality Feedback

Harmful Buffer Case

Mu1 r1 p1 g1 f1 Byp1 Byn1 SAT12 0.3 0.005 0.05 0.05 1 0 1.832

Mu2 r2 p2 g2 f2 Byp2 Byn1 SAT21 0.1 0.01 0.01 0.9 1 0 0.835


Mu1 r1 p1 g1 f1 N1 0.1 0.01 0.01 0.2 30

Mu2 r2 p2 g2 f21 0.1 0.01 0.01 0.2

Effectiveness…


c


Input Parameters forPreliminary Results

Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f2 N Yp1 Yp2 Yn1 Yn2 Case # Mu1 Mu2 r1 r2 p1 p2 g1 g2 f1 f2 N Yp1 Yp2 Yn1 Yn2

1 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 26 1 1 0.1 0.1 0.01 0.1 0.01 0.01 0.2 0.2 30 1 1 0 0

2 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0 27 1 1 0.1 0.1 0.01 0.001 0.01 0.01 0.2 0.2 30 1 1 0 0

3 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0 28 1 1 0.1 0.1 0.01 0.01 0.1 0.01 0.2 0.2 30 1 1 0 0

4 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0 29 1 1 0.1 0.1 0.01 0.01 0 0.01 0.2 0.2 30 1 1 0 0

5 0.5 0.5 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 30 1 1 0.1 0.1 0.01 0.01 0.01 0.1 0.2 0.2 30 1 1 0 0

6 2 2 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 31 1 1 0.1 0.1 0.01 0.01 0.01 0.001 0.2 0.2 30 1 1 0 0

7 3 3 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 32 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.9 0.2 30 1 1 0 0

8 1 1 0.01 0.01 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 33 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.05 0.2 30 1 1 0 0

9 1 1 0.05 0.05 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 34 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.9 30 1 1 0 0

10 1 1 0.5 0.5 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 35 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.05 30 1 1 0 0

11 1 1 0.1 0.1 0.001 0 0.01 0.01 0.2 0.2 30 1 1 0 0 36 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0

12 1 1 0.1 0.1 0.05 0.05 0.01 0.01 0.2 0.2 30 1 1 0 0 37 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0

13 1 1 0.1 0.1 0.1 0.1 0.01 0.01 0.2 0.2 30 1 1 0 0 38 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0

14 1 1 0.1 0.1 0.01 0.01 0.001 0.001 0.2 0.2 30 1 1 0 0 39 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0

15 1 1 0.1 0.1 0.01 0.01 0.05 0.05 0.2 0.2 30 1 1 0 0 40 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0

16 1 1 0.1 0.1 0.01 0.01 0.1 0.1 0.2 0.2 30 1 1 0 0 41 1 1 0.2 0.05 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0

17 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.02 0.02 30 1 1 0 0 42 1 1 0.05 0.2 0.01 0.01 0.01 0.01 0.2 0.2 5 1 1 0 0

18 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.5 0.5 30 1 1 0 0 43 1 1 0.05 0.2 0.01 0.01 0.01 0.01 0.2 0.2 20 1 1 0 0

19 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.95 0.95 30 1 1 0 0 44 1 1 0.05 0.2 0.01 0.01 0.01 0.01 0.2 0.2 50 1 1 0 0

20 1 1 0.5 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 45 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 0.9 0.9 0 0

21 1 1 0.01 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 46 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0.2 0.2

22 1 1 0.1 0.5 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 47 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 0.8 1 0 0

23 1 1 0.1 0.01 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 48 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 0.8 0 0

24 1 1 0.1 0.1 0.1 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 49 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0.2 0

25 1 1 0.1 0.1 0.001 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0 50 1 1 0.1 0.1 0.01 0.01 0.01 0.01 0.2 0.2 30 1 1 0 0.2

Intermediate buffer case: same machine speed

2.852 lecture 25: an integrated quality/quantity model of a transfer line jongyoon kim stanley b....

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