performance evaluation of flow lines with multiple products
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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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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 ,
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
du NFFTNS1
111
The corresponding Markov Chain is too complex to be solved numerically with traditional techniques.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
)(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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
1
1
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)(
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
…
…
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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:
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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)(
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
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 ,
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
)(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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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.
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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
Dipartimento di MeccanicaSezione Tecnologie Meccaniche e Produzione
The aggregation technique
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
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