Download - Impact of Statistical Process Control (SPC) on the Performance of Production Systems M. Colledani, T. Tolio Dipartimento di Meccanica Sezione Tecnologie.

Impact of Statistical Process Impact of Statistical Process Control (SPC) on the Control (SPC) on the

Performance of Production Performance of Production SystemsSystems

M. Colledani, T. TolioM. Colledani, T. Tolio

Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione

Dipartimento di Meccanica

Sezione Tecnologie Meccaniche e Produzione< Impact of SPC on System Performance >< Impact of SPC on System Performance >

Outline of the presentationOutline of the presentation

1- Literature review2- Problem definition

3- Isolated machine with local monitoring4- Two machines one buffer with local monitoring5- Two machines one buffer with remote monitoring 6- Long lines with local monitoring

7- Numerical results8- Conclusion and future research



1- Literature Review1- Literature Review- Montgomery,D.C, Introduction to Statistical Process Control, John Wiley and Sons, Inc, 1991. - Ho C., Case K., Economic Design of Control Charts: A Literature Review for 1981-1991, Journal of Quality Technology, 26,: 39-53,1994.- Raz T.,” A Survey of Models for Allocating Inspection Effort in Multistage Production Systems”, Journal of Quality Technology, 18-239-246, 1986. - Dallery, Y.,Gershwin, S.B., “Manufacturing Flow Line Systems: A Review of Models an Analytical Results, Queueing Systems Theory and Applications, Special Issue on Queueing Models of Manufacturing Systems, 12(1-2). 1992. - Gershwin S.B., Matta A. and Tolio T., Analisys of Two Machine Lines with Multiple Failure Modes, IIE Transaction, 34(1) : 51 - 62, 2002.

- Gershwin S.B., Kim J.,Integrated Quality and Quantity Modeling of a Production Line”, OR Spectrum, 2005. - Colosimo B.M., Semeraro Q. and Tolio T. “Designing X bar Control Charts in Multistage Serial Manufacturing System, CIRP Journal of Manufacturing Systems, 31-6, 2002 - Tempelmeier H., Burger M, Performance Evaluation of Unbalanced Flow Lines with General Distributed Processing Times, Failures and Imperfect Production, IIE Transactions,33,293-302, 2000.- Helber S. Performance Anaiysis of Flow Lines with Non-Linear Flow of Material, Springer, 1999.

- Inman R., Blumenfeld D., Huang N. and Li J..,” Designing Productivity Systems for Quality: research opportunities from an automotive industry perspective “, IJPR, 41(9), 2003.



t )IID(0,~ 2

tY

Example

2- Process in Statistical control

PROCESS IN CONTROLPROCESS IN CONTROL each quality measure is in a statistical control state.

STATISTICAL CONTROL STATESTATISTICAL CONTROL STATE is a state where all the variations within the observed data can be related to a set of causes not identifiable which do not change over time (i.e. the distribution is stable)



2- Specifications and bad parts

t

Lower Specification Limit - LSL

Upper Specification Limit - USL

Even if the process is in control it can produce bad parts.

However, if the process goes out of control the number of bad parts produced changes (in general, infinite out of control modes are possible).

LSL

USL

t



2- Detecting out of control states

In order to understand if the process is in control or out of control, we can sample the produced parts (in the extreme case we can have 100% inspection). Then we measure the parts in the sample and we perform a statistical test with the following hypotheses.H0: the process is in control

H1: the process is out control

The outcome of the test is subject to two types of errors:

error: the process is in control but the test detects an out of control (false alarm)

error: the process is out control but the test does not detect it (out of control not detected)



If the process is out of control

2- Detecting out of control states

1

ARL0 1

1ARL1

If we repeat the test many times and each test has the same and errors than we can evaluate the average number of samples we have to take in order to have an alarm (ARL = Average Run Length).

t t

If the process is in control

4.370ARL0.0027 0 For example:



2- Detecting out of control statesIf we consider a single machine in isolation and a control chart attached to it then how many parts does the machine produce before getting an alarm?

Let us define:m the sample size.h the number of parts produced between two samples.

)(1

mh

1

1m) (h

β

If the process is out of controlIf the process is in control



2- Inspection stationsTesting allows to draw information on the process but also on the inspected parts.

•Inspected parts which are within the specification limits may proceed downstream (if testing does not destroy the parts).•Inspected parts which are outside the specification limits may be either scrapped or reworked

Therefore the logic at an inspection station decides two things:

•control charts which send the alarms related to the out of control conditions•scrap/rework policies which decide the final destination of the inspected parts

qMqiC ,

scrap

rework

good(If m=1 and h=0 than 100% inspection is performed). qi

Out of control



2- Some Assumptions of the Model2- Some Assumptions of the Model-The flow of material in the system is considered as discrete.-Each machine is characterized by the same processing time, scaled to time unit.-Buffers have finite capacity.-Machines can be of three types: manufacturing machines, inspection machines or integrated machines.-Inspection machines are perfectly accurate.-Failures and shifts to out of control are Operation Dependent. -Once an out of control has been detected, the time to repair it is geometrically distributed.-Machines can fail in different modes. We identify two classes of failures:

-f type local failures: are those for which the repairing intervention also set the machine to the in control state; type local failures: are those for which the repairing intervention reset the machine to conditions it had before the failure occurred.



3- Modelling a single machine in isolation Wi: operative in control

Di,fi: f type local down stateDi,i: type local down stateAi

1: out of control detected but not realAi

2: out of control detected and real Oi: out of control non

detected

1iA)( ,qi

falsei Cp

ifiD , iO

2iA

)( ,qii Cp

qualityip

falseir

qualityirifip .

fiir .

iW

i

i

WiD ,

fiip .

iir .

iip .

iir .

iip .

i

i

OiD ,

)]()()[(

1

)(

1)(

,,,0,,

qiqiqiqiiqi

false

CmChCARLCMTTFACp

i

Quality link equations:

)]()()[(

1

)(

1)(

,,,1,,

qiqiqiqiiqii CmChCARLCMTTD

Cp



3- The Isolated Machine Case3- The Isolated Machine CaseOnce the Markov chain has been solved and all the state probabilities have been calculated, the performance measures for the single machine case can be derive as follows:Total average production rate: )()( ii

TOT

i OWE

Effective average production rate: )1)(()1)(( ii Oii

Wii

EFF

i OWE

System yield (fraction of conforming parts produced):

)()(

)1)(()1)((

ii

Oii

Wii

TOT

i

EFF

ii OW

OW

E

EY

ii

Average production rate of parts to be scrapped:

Average production rate of parts to be reworked:

SOsii

SWii

S ii OWE ,,

, )()(

RWOii

RWWii

RW ii OWE ,, )()(



qM jMiB KMqiC , jjC ,

iM qBjB

RWjjB ,

RWqiB ,

scrap

rework

rework

scrap

4- The General Case4- The General Case

Quality has an impact on production system performance:

- Control charts allow to identify out of control states. The search for a cause for the out of control reduces the up time of the machine.- Scrap/rework policies allow to identify defective parts and to decide whether to scrap or to rework them.

The system architecture impacts on the quality system performance:- The presence of buffers causes a delay in the transmission

of the quality signal.

Total throughput Yield

Total throughput

Yield



CONVERGENCE

4- Two machine one buffer with local 4- Two machine one buffer with local monitoringmonitoring

C2,2

M2 B

1,1p 1,1r2,1p

2,1r

1,1 Fp1,1 Fr

1,2p 1,2r2,2p

2,2r

2,2 Fp2,2 Fr

M1

C1,1

This system is formed by two machines M1 and M2 locally controlled by C1,1 and C2,2.

MU(1) MD(1)

)1(1,1

Up )1(1,1

Ur

)( 1,1)1( CpU

false )( 1,1)1( CrU

false

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

)1( CrU

)1(,1 1

UFp )1(

,1 1

UFr

)1(1,2

Dp )1(1,2

Dr

)( 2,2)1( Cp D

false )( 2,2)1( Cr D

false

)( 2,2)1( Cr D)( 2,2

)1( Cp D

)1(,2 2

DFp )1(

,2 2

DFr

B

Bfr2,2

)( 1,11 Cp

Bfr2,2qualityp1

falser1

qualityr11.1 fp

1.1 fr

)( 1,11 Cp false 11A

Bfp2,2 2

1A

1,1 fD 1O

1W

1

2,2W

fB

Bfp2,21.1 fp

1

2,2O

fB

)( 1,1)1( CpU

falseBfr2,2

1.1 fp

1.1 fr

)1(,1 UA

)1(,2 UA

)1(UW

)1(2,2

UfB

)1(1,1

UfD

)( 1,1)1( CpU

)( 1,1)1( CrU

false

)( 1,1)1( CrU

Bfp2,2

Markov chain solution

Stationary state

probability distribution

New failure probabilities calculation

Upstream pseudo-machine MU(1)

Building Block evaluation“Gershwin, Matta, Tolio 2002”

New blocking and

starvation probabilitiesUpstream machine

M1

Downstream machine M2



)1()()()()()1( 221121 EOWOWEEE TOTTOTTOT

Total average throughput:

System yield: TOTEYYY 21)1(

Average buffer level: )1(nn

Bfr2,2

)( 1,11 Cp

Bfr2,2qualityp1

falser1

qualityr11.1 fp

1.1 fr


Bfp2,2 2

1A

1,1 fD 1O

1W

1

2,2W

fB

Bfp2,21.1 fp

1

2,2O

fB)1(

,...,1 ))1((

))1((

,,2

,2

,

,2

2

22

Dfj

Bf

djB

f

fjBf

j

j

rr

FfrE

Bp

Blocking (starvation) probabilities

equations

Monitored

machine model

)( 1,1)1( CpU

falseBfr2,2

1.1 fp

1.1 fr

)1(,1 UA

)1(,2 UA

)1(UW

)1(2,2

UfB

)1(1,1

UfD

)( 1,1)1( CpU

)( 1,1)1( CrU

false

)( 1,1)1( CrU

Bfp2,2

)()())1(( 11 OWW U

)())1(( 11

,1 AA U

)())1(( 21

,2 AA U

)()())1((

)())1((

1

2

1

22,2

11,1

,2,2

,1

Of

Wf

U

fU

BBB

DD

f

f

Stationary state probability distribution

)()()(

)()(

))1((

))1(()( 1,1

)1(

11

11

1,1)1(

,1

1,1)1( Cr

OW

ACr

W

ACp UU

U

U

Ufalsefalsefalse

)()()(

)()(

))1((

))1(()( 1,1

)1(

11

21

1,1)1(

,2

1,1)1( Cr

OW

ACr

W

ACp UU

U

U

U

falseU rCr false 11,1)1( )(

Transition probabilitiesFalse alarm state:

qualityU rCr 11,1)1( )(

Detected out of control state:

Pseudo-machine model

4- Two machine one buffer with local 4- Two machine one buffer with local monitoringmonitoring



C 1,2

M 2

B

1,1p

1,1r

2,1p

2,1r

1,1 Fp

1,1 Fr

1,2p

1,2r

2,2p

2,2r

2,2 Fp

2,2 Fr

M 1

Monitored machine MiControl chart Ci,q (i<q)

p1,2delay represents the delay of the quality information due to the

presence of the finite capacity buffer B. It can be calculated by using the following equation:

1E

ndelay En

E

delaypdelay

12,1

5- Two machine one buffer with remote 5- Two machine one buffer with remote monitoringmonitoring

A12: out of control correctly

detected by the control chart (i<q)

O11: out of control, not

detected stateO2

1: out of control, detected if the machine was locally monitored (i=q)

Additional states:

The approach remains the same as in the previous case, only the Markov chain is more complicated:

Bfr2,2

)( 1,11 CpB

fr2,2

qualityp1

falser1

qualityr11.1 fp

1.1 fr


Bfp2,2 2

1A

1,1 fD 11O

1W

1

2,2W

fB

Bfp2,2

1.1 fp 11

2,2O

fB

21

2,2O

fB

21O

Bfp2,2

Bfr2,22,1

delayp



6- Long lines with local monitoring – 6- Long lines with local monitoring – the approachthe approach KKC ,1,1 iiCiiC ,2,2C1,1C

1iM1B 2B iB 1iBiM KM1M 2M

By solving the two locally monitored machines systems with the presented method and by using decomposition equations the

performance of the original line can be estimated.

As for the two locally monitored machines system, the approach follows a two level decomposition, since alternately the Markov chain representing each machine (machine level) and each building block (buffer level) are studied .



)1(DM)1(UM )1(B )2(B )2(DM)2(UM

)1(1,1

Up )1(1,1

Ur)1(

,1 1

UFp )1(

,1 1

UFr

)( 1,1)1( CpU

false)( 1,1

)1( CrUfalse

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

)1( CrU

)( 2,2)1( CpU

false

)( 2,2)1( CpU

)1(1,3

Dp )1(1,3

Dr)1(

,3 3

DFp )1(

,3 3

UFr

)( 3,3)1( Cp D

false)( 3,3

)1( Cr Dfalse

)( 3,3)1( Cp D )( 3,3

)1( Cr D

)1(1,2

Dp )1(1,2

Dr)1(

,2 2

DFp )1(

,2 2

DFr

)( 2,2)1( Cp D

false)( 2,2

)1( Cr Dfalse

)( 2,2)1( Cp D )( 2,2

)1( Cr D

)1(1,3

Dp )1(1,3

Dr)1(

,3 3

DFp )1(

,3 3

DFr

)1(2,3 3

DFp

)1(2,3 3

DFr

)1(1,2

Up )1(1,2

Ur)1(

,2 2

UFp )1(

,2 2

UFr

)( 2,2)1( CrU

false

)( 2,2)1( CrU

)1(1,1

Up )1(1,1

Ur)1(

,1 1

UFp )1(

,1 1

UFr

)1(2,1 1

UFp

)1(2,1 1

UFr

Local failure probabilities: are simply equal to those of the correspondent machine in the original line.

Quality linked failure probabilities: can be evaluated by using quality link equations provided.Remote failure probabilities: can be evaluated by using the following decomposition equations. They acts exactly in the same way as type local failure modes for the pseudo-machines.

jj fjU

fj rir ,, )( 2,...,1,1,...,1 )()1(

)1()(

,

,

,

jj

UfjUfj Ffijir

iE

iPsip

jfj

j

j

jj fjD

fj rir ,, )( 2,...,1;,...1 )()1(

)1()(

,

,,

Kk

DfkDfk FfKikir

iE

iPbip

kfk

k

k

Upstream pseudo-machinesDownstream pseudo-machines

6- Long lines with local monitoring - 6- Long lines with local monitoring - failuresfailures



KKC ,1,1 iiCiiC ,2,2C1,1C

1iM1B 2B iB 1iBiM KM1M 2M

An iterative algorithm, inspired to the DDX, has been used to efficiently solve the decomposition equations. It behaves as follows, after the initialization phase:

- Visiting all the upstream pseudo-machines for i=2,..,K-1

- Unknown failures probabilities are calculated by using decomposition equations;- The performance of the building block l(i) are evaluated by using the two level approach used for the two monitored machines system;- The same steps are performed visiting all the downstream pseudo-

machines for i=K-2,..,1At the convergence, the system yield can be evaluated as:

)1(...)2()1( KYYYYsystem )()()( iYiYiY DU

K

ik

kD

i

j

jU YiYYiY

11

)( )(

totsystemsystem

eff EYE

The total and the effective average production rates:

)1(...)2()1( KEEEE totsystem

6- Long lines with local monitoring – 6- Long lines with local monitoring – the algorithmthe algorithm



7- Numerical Results – two locally monitored 7- Numerical Results – two locally monitored machinesmachinesMore than one hundred test cases with random parameters have been carried out and compared with simulation. Some of those cases randomly selected are reported:

CASE 1 2 3 4 5 6 7 8 9 10

CT 1.6780 0.955 2.4 2.4138 1.9848 5.1083 1.9835 3.023 18.429 1.4289

SIM 1.6648 0.939 2.386 2.399 1.9847 5.0487 1.9844 0.034 18.632 1.4244

n (1

)

e % 0.3295 0.199 0.343 0.35 0.0023 0.4969 0.0232 0.1813 0.9195 0.03

CT 0.6562 0.1532 0.3557 0.3563 0.4067 0.4301 0.4316 0.113 0.2804 0.5368

SIM 0.6575 0.1527 0.3559 0.3567 0.4088 0.4314 0.4434 0.114 0.2809 0.537

E_e

ff

e % 0.1864 0.336 0.049 0.134 0.5158 0.3047 0.649 0.8546 0.152 0.0275

CT 0.9115 0.2211 0.3755 0.3772 0.4259 0.6070 0.4612 0.4333 0.5735 0.5567

SIM 0.9132 0.2205 0.3753 0.3777 0.4280 0.6087 0.4642 0.4367 0.5743 0.5569 E

_tot

e % 0.1904 0.287 0.047 0.139 0.5057 0.2687 0.6618 0.7868 0.1372 0.0287

CT 0.7199 0.693 0.9472 0.9444 0.9550 0.7085 0.9359 0.261 0.489 0.9642

SIM 0.7199 0.6927 0.9473 0.9444 0.9551 0.7087 0.9358 0.2611 0.4891 0.9642

Yie

ld

e % 0.0038 0.048 0.0014 0.005 0.0100 0.0361 0.011 0.0689 0.0163 0.0005

System Yield Total E Effective E Buffer level Average 0.05 0.432 0.516 0.388

<1% 98 97 95 98 <2% 100 100 100 100



7- Numerical Results – one remotely 7- Numerical Results – one remotely monitored machine monitored machine

CASE 1 2 3 4 5 6 7 8

CT 6.5074 2.1908 2.2900 8.0193 1.2400 2.4120 2.2066 1.3486

SIM 6.5550 2.1900 2.2679 7.8694 1.2362 2.4526 2.2119 1.3583

n (1

) e % 0.3968 0.0205 0.5525 1.2493 0.0549 0.4058 0.1317 0.1210

CT 0.3384 0.9019 0.8995 0.9018 0.6497 0.4209 0.5395 0.2955

SIM 0.3344 0.9017 0.9008 0.9046 0.6421 0.4171 0.5386 0.2950

E_e

ff

e % 1.2171 0.0210 0.1444 0.3098 1.1703 0.9278 0.1622 0.1932

CT 0.6059 0.9124 0.9139 0.9236 0.6734 0.6225 0.5725 0.3311

SIM 0.6060 0.9121 0.9137 0.9235 0.6635 0.6260 0.5735 0.3299 E

_tot

e % 0.0191 0.0406 0.0216 0.0165 1.4851 0.5612 0.1701 0.3509

CT 0.5585 0.9884 0.9842 0.9763 0.9647 0.6762 0.9423 0.8927

SIM 0.5517 0.9886 0.9858 0.9795 0.9677 0.6662 0.9391 0.8941

Yie

ld

e % 1.2347 0.0203 0.1659 0.3264 0.3101 1.4974 0.3327 0.1565


<1% 90 88 85 92 <2% 100 100 100 100



7- Numerical Results – K locally 7- Numerical Results – K locally monitored machinesmonitored machinesCASES E tot E eff Yield N1 N2

Sim. 0.64898 0.35825 0.55202 2.75398 2.2599 An. 0.64605 0.35652 0.551846 2.75398 2.2599 1

Err % 0.45194 0.48318 0.03152 1.3770 0.7390 Sim. 0.47543 0.24270 0.51049 3.4071 1.2432 An. 0.47651 0.24265 0.509222 3.41109 1.2308 2

Err % 0.22737 0.02060 0.24839 0.0998 0.3110 Sim. 0.5748 0.5567 0.9686 19.302 11.706 An. 0.5753 0.5574 0.9687 19.71 11.782 3

Err % 0.086 0.125 0.01 1.63 0.304 Sim. 0.0273 0.0269 0.9866 3.443 0.344 An. 0.0271 0.0268 0.9868 3.414 0.313 4

Err % 0.732 0.37 0.02 0.116 0.124

3 machine cases

CASES E tot E eff Yield N1 N2 N3 N4 N5 N6 N7 N8 N9 Sim. 0.5522 0.0849 0.1538 2.892 2.604 2.381 2.184 1.997 1.809 1.616 1.393 1.106 An. 0.5559 0.0854 0.1538 2.9 2.618 2.317 2.179 2 1.82 1.682 1.381 1.099 1

Err% 0.67 0.58 0 0.2 0.35 1.6 0.125 0.075 0.275 1.65 0.3 0.175 Sim. 0.8025 0.4566 0.569 4.035 3.650 3.401 3.188 2.990 2.785 2.584 2.336 1.951 An. 0.8062 0.4588 0.5692 4.048 3.648 3.357 3.189 2.999 2.81 2.642 2.351 1.951 2

Err% 0.46 0.481 0.035 0.21 0.033 0.73 0.016 0.15 0.416 0.96 0.25 0

10 machine cases


<1% 98 93 92 90 <2% 100 100 100 100

Summary of results



7- System Behavior7- System BehaviorTotal Throughput and System yield vs. N

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

3 5 7 9 11 13 15 17 19 21 23 25 27 29

N

Total throughput

System Yield

Effective throughput vs. N

0.485

0.49

0.495

0.5

0.505

0.51

0.515

0.52

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

N

EE

FF

h=3 Effective throughput vs. N

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

0.505

0.51

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

N

EE

FF

h=5


0.515

0.52

0.525

0.53

0.535

0.54

0.545

0.55

0.555

0.56

0.565

3 5 7 9 11 13 15 17 19 21 23 25 27 29

N

EE

FF

h=0 - 100% insp



8- Conclusion and Future Research8- Conclusion and Future Research

- Quality issues and productivity aspects must be jointly considered in the design phase of production systems, since their correlation is evident.

-The proposed method paves the way to the integrated analysis and solution of other system design problem such as:

- Optimal design of control chart parameters;

- Optimal allocation of inspection devices;

- Optimal allocation of buffer space.

- New improvement of the method will be the integration of various scrap and rework policies in order to identify the optimal scrap/rework parameters.

- The proposed method, dealing with the interaction between SPC theory principles and production system design issues, provides accurate results in the performance analysis of such systems.



Thank you for your attention.



10- System Behavior10- System Behavior


0.515

0.52

0.525

0.53

0.535

0.54

0.545

0.55

0.555

0.56

0.565

3 5 7 9 11 13 15 17 19 21 23 25 27 29

N

EE

FF

Total Throughput and System yield vs. N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3 5 7 9 11 13 15 17 19 21 23 25 27 29

N

Total throughput

System Yield

piquality ri

qua lity rifalse pi ri iWi iOi hi(Ci,i) mi(Ci,i) i(Ci,i) i(Ci,i)

0.08 0.5 0.94 0.1 0.5 0.03 0.9 0 1 0.0027 0.6 0.1 0.47

The behavior of systems in which the first machine is monitored by the second one have been observed.

