Performance Evaluation Of Flow Lines With Multiple Products POLITECNICO DI MILANO Dipartimento di Meccanica Colledani M., Matta A. and Tolio T

Performance Evaluation Performance Evaluation Of Flow Lines With Of Flow Lines With Multiple ProductsMultiple Products

POLITECNICO DI MILANODipartimento di Meccanica

Colledani M., Matta A. and

Tolio T.

Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione

Outline

System: description and assumptions

Analytical model: building block

Analytical model: decomposition

Numerical results

Conclusions and future developments


System description

1M ...M iM KM...M1,1B

,...1B

zB ,1

1,1iB

,...1iB

ziB ,1

1,iB

,...iB

ziB ,

1,1KB

,...1KB

zKB ,1

K machines, i=1,…,K z products, q=1,…,z Homogeneous buffers Discrete material/discrete time Deterministic and equal processing times Saturated system


System description

Machines can fail with multiple failures, j=1,…,Fi Machines can be failed in only one mode at the same time period MTTF and MTTR are geometrically distributed The production rule at machine is local and stochastic (production

parameters i,q) Buffers have finite capacity, Ni,q Blocking Before Service

1M ...M iM KM...M1,1B

,...1B

zB ,1

1,1iB

,...1iB

ziB ,1

1,iB

,...iB

ziB ,

1,1KB

,...1KB

zKB ,1

1,1q,1z,1

1...,q...,z...,

1,iqi,zi,

1...,q...,z...,

1,KqK ,zK ,


System description

Production parameters of machine Mi at time t are adjusted depending on the state of the immediately upstream and downstream buffers.

iM1,1iB

,...1iB

ziB ,1

1,iB

,...iB

ziB ,

1,iqi,zi,

Ki

zqt

ririri Ntnortnqrri

qi

qi ,,1

,,1

1)(

,,,1 )(0)(:,

,*,

(Nemec, 1999) and (Syrowicz, 1999) deal with lines with two products and priority rules.(Colledani et al., 2003 and 2005) deals with lines with two products and same assumptions.


Two-machine line with z products

1,u

uM dM

zB

1B

qB

zqt

zqt

tnqr

ur

dqd

q

Ntnqr

ur

uqu

q

r

rr

,,11

)(

,,11

)(

0)(:

*

)(:

*

qu ,

Zu ,

1,d

qd ,

Zd ,

Production parameters may changebecause of the emptying and/or filling of buffers.

Production parameters are adjusted as follows:

uuF

uuF

uu

uu

rp

rp

rp

11

ddF

ddF

dd

dd

rp

rp

rp

11


ZP2M: two-machine line with z products

1,u

uM dM

zB

1B

qBqu ,

Zu ,

1,d

qd ,

Zd ,

The state of the system is represented by: (n1,…, nz,xu,xd).

The total number of possible states is:

z

qq

du NFFTNS1

111

The corresponding Markov Chain is too complex to be solved numerically with traditional techniques.


The aggregation technique

uMdM

zB

1B

qB

The behaviour of z-1 products can be modelled in an approximate way by considering an equivalent aggregate product.

uM dM

aB

1B

a aAggregate product



1,u

uM dM

zB

1B

qBqu ,

Zu ,

1,d

qd ,

Zd ,

)(ZM u

)1(uM

)(qM u

)1(dM

)(qM d

)(ZM d

)(quq

)(qua

)1(1u

)(qdq

)(zuz

)(zua

)1(ua

)(qda

)1(1d

)1(da

)(zdz

)(zda

)1(1B

)1(aB

)(zBz

)(zBa

…

…

…

…

The system with z products is represented by a set of equivalent z systems, each one crossed by 2 products: product q and the corresponding aggregate product.

(Baynat and Dallery, 1995) first proposes this technique for analyzing multiclass queuing systems.


The aggregation techniqueThe production probabilities in the original system are adjusted as follows:

zqt

zqt

tnqr

ur

dqd

q

Ntnqr

ur

uqu

q

r

rr

,,11

)(

,,11

)(

0)(:

*

)(:

*

However, the aggregation of all products except q does not allow to recognize the buffer levels of single aggregated products in the analysis of the two-machine two-product system, thus we are forced to find new values for the production parameters.



)(qM d)(qM u

)(quq

)(qua

)(qdq

)(qda

)(1 qB

)(qBa

For each system it is necessary to calculate the parameters: Buffer capacities: Nq(q) and Na(q) Upstream production parameters: u

q(q) and u

a(q) Downstream production parameters: d

q(q) and d

a(q)

There are totally 6z unknowns.

(Colledani et al., 2003 and 2005) is used to calculate the performance.



1

1

qq

qqda

dq

ua

uq

The sum of production parameters of each machine must be equal to one:

zq ,,1

The buffer capacity of the product q corresponds to that in the original system:

The buffer capacity of the aggregate product is the sum of the buffer capacities in the original system of the single aggregated products:

zqNqN qq ,...,1)(

zqNqNqrqa ,...,1)(



zqqP

qP

q

uq

uq

ss Nns

us

uq

uq ,,1

,

1,

:

zqPqPqr

rr ,,1,

zqNqss sszqquq ,,1,: moreor one :),,...,( ,111

We define uq as the set containing all the combinations, in

the two-machine line original system, of buffers full and not full obtained without considering buffer Bq:

The new value of the upstream production probability of product q is calculated as a weighted combination of the adjusted u

q values overall the possible combinations belonging to the set u

q :

probability associated to the occurrence of combination u

q


The aggregation techniqueWe define d

q as the set containing all the combinations, in the original system, of buffers empty and not empty obtained without considering buffer Bq:

The new value of the upstream production probability of product q is calculated as a weighted combination of the adjusted d

q values overall the possible combinations belonging to the set d

q :

probability associated to the occurrence of combination d

q

zqqP

qP

q

dq

dq

ss

ds

dq

dq ,,1

,

1,

0:

zqPqPqr

rr ,,1,

zqqss szqqdq ,,10,: moreor one :),,...,( ,111



0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

5 7 9 11 13 15 17 19 21 23 25N1

E

E tot

E1

E2

E3

0

2

4

6

8

10

12

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

n1

n2,n3

Pb1

Ps1

n a

vera

ge

N1

Pb

, P

s


Long lines with Z products

)(il

)(il

)1(l

)2(UM

)(iM U

)1( iM U

)1(DM

)2(DM

)(iM D

)1(UM

)1( iM D

…

…


Long lines with Z products

uq

uq

s

iP

iP

i ssi

qi

uq

1,

11,

0:)()(,

,

1,,1;,,11,

11,

0:)()(,1

,1

KizqiP

iP

i

dq

dq

sssi

qi

dq

)(il

)(iM u )(iM d

uuF

uuF

uu

uu

rp

rp

rp

11

ddF

ddF

dd

dd

rp

rp

rp

11

Local failures Local failures

Remote failures(starvation)

Remote failures(blocking)

urem

urem rp d

remdrem rp

(Tolio and Matta, 1998)

The new production parameters are calculated as follows:


Numerical results: 3P3M

CASE 1 (buffer 4-4-4/4-4-4) 2 (buffer 4-4-4/4-4-4) 3 (buffer 6-6-6/6-6-6) 4 (buffer 4-4-4/4-4-4)

MACH

M1 M2 M3 M1 M2 M3 M1 M2 M3 M1 M2 M3

p 0.049 0.092 0.009 0.104 0.09 0.064 0.104 0.09 0.064 0.12 0.039 0.105

r 0.62 0.16 0.102 0.62 0.12 0.102 0.7 0.28 0.32 0.42 055 0.37

0.6 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.5 0.5 0.5

0.3 0.3 0.3 0.2 0.2 0.2 0.35 0.35 0.35 0.4 0.4 0.4

0.1 0.1 0.1 0.2 0.2 0.2 0.25 0.25 0.25 0.1 0.1 0.1

PART P1 P2 P3 P1 P2 P3 P1 P2 P3 P1 P2 P3

E sim 0.366 0.191 0.068 0.271 0.110 0.110 0.296 0.261 0.190 0.369 0.301 0.084

E anal 0.362 0.198 0.064 0.265 0.114 0.114 0.296 0.261 0.190 0.368 0.304 0.094

Err -0.59 1.31 -0.70 -1.31 0.69 0.69 -0.03 -0.02 -0.01 -0.16 0.4 1.37

n1 sim 3.38 3.58 3.75 3.43 3.62 3.62 4.98 5.02 5.11 1.69 1.67 1.6

n1 anal 3.06 3.24 3.46 3.11 3.18 3.18 4.26 4.33 4.45 1.60 1.58 1.9

Err -8.03 -8.51 -7.2 -8.0 -10.7 -10.7 -11.9 -11.4 -11 -2.27 -2.05 7.55

n2 sim 0.67 0.5 0.37 1.75 1.71 1.71 1.79 1.76 1.68 2.25 2.26 2.29

n2

anal

0.87 0.71 0.56 1.8 1.86 1.86 2.11 2.26 2.19 2.15 2.21 1.94

Err 5.21 5.22 4.88 1.12 3.76 3.72 5.34 6.69 8.5 -2.45 -1.28 -8.75

,...,zqN

nnnerr

E

EEEerr

q

simulationq

analyticalq

qtotsimulation

simulationq

analyticalq

q 1 ; 100)( ; 100)(


Numerical results: 4P3M

CASE 5 (buffer 4-4-4-4/4-4-4-4) 6 (buffer 4-4-4-4/4-4-4-4) 7 (buffer 4-4-4-4/4-4-4-4) MACH M1 M2 M3 M1 M2 M3 M1 M2 M3 P 0.12 0.039 0.105 0.067 0.029 0.0376 0.001 0.019 0.08 R 0.42 0.55 0.37 0.319 0.109 0.319 0.125 0.217 0.6 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

PART P1 P2 P3 P4 P1 P2 P3 P4 P1 P2 P3 P4

E sim 0.368 0.156 0.156 0.08 0.287 0.221 0.154 0.083 0.343 0.260 0.176 0.091 E anal 0.362 0.159 0.159 0.083 0.279 0.221 0.157 0.086 0.344 0.257 0.174 0.097 Err -0.78 0.39 0.39 0.4 -1.12 -0.06 0.45 0.4 0.11 -0.28 -0.27 0.69

,...,zqN

nnnerr

E

EEEerr

q

simulationq

analyticalq

qtotsimulation

simulationq

analyticalq

q 1 ; 100)( ; 100)(

Test on 50 cases: 0.66 % error on average throughput 6.4 % error on average buffer levels.


Conclusions and future developmentsNew model to estimate the performance of multiple

product flow lines.

Ongoing work Split/merge systems with z different products

To be developed The continuous model with different processing

times Closed systems with z different products Different production policies



System description

Eq is the average production rate of the system related to product q, with q=1,…,z

E is the overall average production rate of the system

z

qqEE

1

1M ...M iM KM...M1,1B

,...1B

zB ,1

1,1iB

,...1iB

ziB ,1

1,iB

,...iB

ziB ,

1,1KB

,...1KB

zKB ,1

1,1q,1z,1

1...,q...,z...,

1,iqi,zi,

1...,q...,z...,

1,KqK ,zK ,



)(qM d)(qM u

)(1 qu

)(qua

)(1 qd

)(qda

)(1 qB

)(qBa

Some relationships:

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

,,1)()(

,,1)()()(

11

1

zqqEqEqEqE

zqrEqE

zqqEqEqE

aa

qrra

a

2P2M: two- product two- machine system



AlgorithmStep 1: Initialization of 2P2M systems. Set the production probabilities of 2P2M systems to some initial value and the buffer capacities.Step 2: Solve 2P2M systems. For q=1,…,Z solve the 2P2M system producing products q and a(q) using the method in (Colledani et al., 2005).Step 3: Calculate alphas. For q=1,…,Z calculate the new values of production probabilities u

q(q), ua(q), d

q(q) and da(q).

Step 4. Check convergence.The algorithm converges if all the production probabilities do not significantly change from one iteration to another, otherwise go back to Step2.



Case p r 1 2 3 N1 N2 N3

Mu 0.1200 0.12 0.5 0.4 0.1 1

Md 0.1200 0.12 0.5 0.4 0.1 4 4 4

Mu 0.0104 0.12 0.5 0.4 0.1 2

Md 0.2800 0.12 0.5 0.4 0.1 4 4 4

Mu 0.0320 0.06 0.5 0.3 0.2 3

Md 0.0400 0.10 0.5 0.3 0.2 4 4 4



0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

5 7 9 11 13 15 17 19 21 23 25N1

E

E tot

E1

E2

E3

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,42

0,46 0,

50,

540,

580,

620,

66 0,7

0,74

0,78

0,82

2

E

E tot

E1

E2

E3

0

2

4

6

8

10

12

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0,5

n1

n2,n3

Pb1

Ps1

n a

vera

ge

N1

Pb

, P

s

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,04

0,06

0,08 0,

10,

120,

140,

160,

18 0,2

0,22

0,24

0,26

0,28 0,

30,

320,

340,

360,

38 0,4

0,42

0,44

rd

E

E tot

E1

E2

E3

Documents

Performance Evaluation Of Flow Lines With Multiple Products POLITECNICO DI MILANO Dipartimento di Meccanica Colledani M., Matta A. and Tolio T