lecture slides - assembly/disassembly systems
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MIT 2.852Manufacturing Systems Analysis
Lectures 15–16: Assembly/Disassembly Systems
Stanley B. Gershwin
http://web.mit.edu/manuf-sys
Massachusetts Institute of Technology
Spring, 2010
c2.852 Manufacturing Systems Analysis 1/41 Copyright �2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Assembly System
2.852 Manufacturing Systems Analysis 2/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Assembly-Disassembly System with a Loop
2.852 Manufacturing Systems Analysis 3/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems A-D System without Loops
2.852 Manufacturing Systems Analysis 4/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Disruption Propagation in an A-D System without Loops
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Assembly-Disassembly Systems Models and Analysis
An assembly/disassembly system is a generalization of a transfer line:
◮ Each machine may have 0, 1, or more than one buffer upstream.
◮ Each machine may have 0, 1, or more than one buffer downstream.
◮ Each buffer has exactly one machine upstream and one machine downstream.
◮ Discrete material systems: when a machine does an operation, it removes one part from each upstream buffer and inserts one part into each downstream buffer.
◮ Continuous material systems: when machine Mi operates during [t, t + δt], it removes µi δt from each upstream buffer and inserts µi δt into each downstream buffer.
◮ A machine is starved if any of its upstream buffers is empty. It is blocked if any of its downstream buffers is full.
2.852 Manufacturing Systems Analysis 6/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Models and Analysis
◮ A/D systems can be modeled similarly to lines: ◮ discrete material, discrete time, deterministic processing time,
geometric repair and failure times; ◮ discrete material, continuous time, exponential processing, repair, and
failure times; ◮ continuous continuous time, deterministic processing rate, exponential
repair and failure times; ◮ other models not yet discussed in class.
◮ A/D systems without loops can be analyzed similarly to lines by decomposition.
◮ A/D systems with loops can be analyzed by decomposition, but there are additional complexities.
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Assembly-Disassembly Systems Models and Analysis
◮ Systems with loops are not ergodic. That is, the steady-state distribution is a function of the initial conditions.
◮ Example: if the system below has K pallets at time 0, it will have K pallets for all t ≥ 0. Therefore, the probability distribution is a function of K .
Empty Pallet Buffer
Raw Part Input
Finished Part Output
◮ This applies to more general systems with loops, such the example on Slide 3.
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Assembly-Disassembly Systems Models and Analysis
◮ In general,
p(s|s(0)) = lim t→∞
prob { state of the system at time t = s|
state of the system at time 0 = s(0)}.
◮ Consequently, the performance measures depend on the initial state of the system:
◮ The production rate of Machine Mi , in parts per time unit, is
Ei (s(0)) = prob
�
αi = 1 and (nb > 0 ∀ b ∈ U(i)) and
(nb < Nb ∀ b ∈ D(i))
�
�
�
�
s(0)
�
.
◮ The average level of Buffer b is
n̄b(s(0)) = �
s
nb prob (s|s(0)).
2.852 Manufacturing Systems Analysis 9/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Decomposition
Mj
B (j, i)
B (n, i)
Mn
Mi
B (i, m)
Mm
B (i, q)
Mq
•
•
•
•
•
•
Part of Original Network
2.852 Manufacturing Systems Analysis 10/41 Copyright c�2010 Stanley B. Gershwin.
Assembly-Disassembly Systems Decomposition
Mu (j, i) B (j, i) Md (j, i)
Mu (n, i) B (n, i) Md (n, i)
•
• •
Mu (i, m) B (i, m) Md (i, m)
Mu (i, q) B (i, q) Md (i, q) •
• •
Part of Decomposition
2.852 Manufacturing Systems Analysis 11/41 Copyright c�2010 Stanley B. Gershwin.
Numerical examples Eight-Machine Systems
Deterministic processing time model
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Numerical examples Eight-Machine Systems
7.3351
7.3351
5.6516
5.6516
2.6449
7.3351
7.3351
Case 1:
ri = .1, pi = .1, i = 1, ..., 8;
Ni = 10, i = 1, ..., 7.
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Numerical examples Eight-Machine Systems
7.9444
7.9444
7.0529
7.0529
2.0555
7.9444
7.9444
Case 2:
Same as Case 1 except p7 = .2
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Numerical examples Eight-Machine Systems
7.7077
7.7077
4.1089
6.5276
2.2923
4.5640
7.7077
Case 3:
Same as Case 1 except p1 = .2
2.852 Manufacturing Systems Analysis 15/41 Copyright c�2010 Stanley B. Gershwin.
Numerical examples Eight-Machine Systems
7.9017
7.9017
3.4593
6.9609
2.0983
7.9017
7.9017
Case 4:
Same as Case 1 except p3 = .2
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Numerical ExamplesAlternate Assembly Line Designs
A product is made of threesubassemblies (blue, yellow,and red). Each subassemblycan be assembledindependently of the others.We consider four possibleproduction systemstructures.
Machine 6 (the first machineof the yellow process) is thebottleneck — the slowestoperation of all.
2.852 Manufacturing Systems Analysis 17/41 cCopyright �2010 Stanley B. Gershwin.
Numerical Examples Alternate Assembly Line Designs
c2.852 Manufacturing Systems Analysis 18/41 Copyright �2010 Stanley B. Gershwin.
Numerical Examples Alternate Assembly Line Designs
Now the bottleneck is Machine 5, the last operation of the blue process.
2.852 Manufacturing Systems Analysis 19/41 Copyright c�2010 Stanley B. Gershwin.
Numerical Examples Alternate Assembly Line Designs
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Equivalence Simple models
Consider a three-machine transfer line and a three-machine assembly system. Both are perfectly reliable (pi = 0) exponentially processing time systems.
µ2
1N = 2
µ1
µ3
N = 3 2
N = 2 N = 31 2
µ µ µ1 2 3
c2.852 Manufacturing Systems Analysis 21/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Assembly System State Space
µ2
1N = 2
µ1
µ3
N = 3 2
µ3
µ3
µ3
µ
µ
µ
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
00 10
01 11
1202
03 13 23
22
21
20
µ1 µ1
3
3
3
2.852 cManufacturing Systems Analysis 22/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Transfer Line State Space
µ3
µ3
µ3
µ
µ
µ
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
03 13
02 12
1101
00 10 20
21
22
23
µ1 µ1
3N = 2 N = 3
1 2
3µ µ µ
1 2 3
3
c2.852 Manufacturing Systems Analysis 23/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Unlabeled State Space
◮ The transition graphs of the two systemsare the same except for the labels of thestates.
◮ Therefore, the steady-state probability 3
distributions of the two systems are the same, except for the labels of the states.
◮ The relationship between the labels of the 3
states is:
(n1 A , n2
A) ⇐⇒ (n1 T ,N2 − n2
T ) 3
◮ Therefore, in steady state,
µ3
µ3
µ3
µ
µ
µ
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
µ1 µ1
prob(n1 A , n2
A) = prob(n1 T ,N2 − n2
T )
c2.852 Manufacturing Systems Analysis 24/41 Copyright �2010 Stanley B. Gershwin.
� �
Equivalence Assembly System Production Rate
Production rate = rate of flow of material into M1
1 3
= µ1 p(n1, n2) n1=0 n2=0
µ2
1N = 2
µ1
µ3
N = 3 2
µ3
µ3
µ3
µ3
µ3
µ3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
00 10
01 11
1202
03 13 23
22
21
20
µ1 µ1
c2.852 Manufacturing Systems Analysis 25/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Assembly System Production RateTransfer Line Production Rate
Production rate = rate of flow of material into M1
= µ1
1 �
n1=0
3 �
n2=0
p(n1, n2)
1N = 2
2N = 3
µ3
µ1
µ2
µ3
µ3
µ3
µ3
µ3
µ3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
03 13
02 12
1101
00 10 20
21
22
23
µ1 µ1
c2.852 Manufacturing Systems Analysis 26/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Assembly Systtem Production RatEqual Produc ion Rates
Therefore
PA PT =
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� � � �
� �
Equivalence Assembly System n̄1
2 3 2 3
n̄1 = n1p(n1, n2) = n1 p(n1, n2)
µ
µ
µ
µ1 µ1 n1=0 n2=0 n1=0 n2=0
µ2
1N = 2
µ1
µ3
N = 3 2
3
3
3
µ3
µ3
µ3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
00 10
01 11
1202
03 13 23
22
21
20
c2.852 Manufacturing Systems Analysis 28/41 Copyright �2010 Stanley B. Gershwin.
� �
Equivalence Assembly System n̄1Transfer Line n̄1
2 3 2 3� � � �
n̄1 = n1p(n1, n2) = n1 p(n1, n2) µ1 µ1 n1=0 n2=0 n1=0 n2=0
N = 2 N = 31 2
µ µ µ1 2 3
3
3
3
µ3
µ3
µ3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
03 13
02 12
1101
00 10 20
21
22
23
c2.852 Manufacturing Systems Analysis 29/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Assembly System Production RateEqual n̄1
Therefore
n̄1 A = n̄1
T
c2.852 Manufacturing Systems Analysis 30/41 Copyright �2010 Stanley B. Gershwin.
� � � �
� �
Equivalence Assembly System n̄2
2 3 3 2
n̄2 = n2p(n1, n2) = n2 p(n1, n2) µ µ1 1 n1=0 n2=0 n2=0 n1=0
µ2
1N = 2
µ1
µ3
N = 3 2
µ µ3
µ µ3
µ µ3
3
3
3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
00 10
01 11
1202
03 13 23
22
21
20
c2.852 Manufacturing Systems Analysis 31/41 Copyright �2010 Stanley B. Gershwin.
� � � �
� �
Equivalence Assembly System n̄2Transfer Line n̄2
2 3 3 2
n̄2 = n2p(n1, n2) = n2 p(n1, n2) µ µ1 1 n1=0 n2=0 n2=0 n1=0
µ µ3 N = 2 N = 3
1 2
µ µ3
µ µ µ1 2 3
µ µ3
3
3
3
µ 2
µ 2
µ 2
µ 2
µ 2
µ 2
µ1
µ1 µ1
µ1
µ1
µ1
µ3
µ3
µ3
03 13
02 12
1101
00 10 20
21
22
23
c2.852 Manufacturing Systems Analysis 32/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Assembly System Production RateComplementary n̄1
Therefore
n̄2 A = N2 − n̄2
T
c2.852 Manufacturing Systems Analysis 33/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Theorem
◮ Notation: Let j be a buffer. Then the machine upstream of the buffer is u(j) and the machine downstream of the buffer is d(j).
◮ Theorem:◮ Assume
′ ◮ Z and Z are two exponential A/D networks with the same number of
machines and buffers. Corresponding machines and buffers have the ′ same parameters; that is, µi = µi , i = 1, ..., kM and
′ Nb = Nb , b = 1, ..., kB . ′
◮ There is a subset of buffers Ω such that for j 6∈ Ω, u (j) = u(j) and ′ ′ ′ d (j) = d(j); and for j ∈ Ω, u (j) = d(j) and d (j) = u(j). That is,
there is a set of buffers such that the direction of flow is reversed in the two networks.
′◮ Then, the transition equations for network Z are the same as those of
Z , except that the buffer levels in Ω are replaced by the amounts of space in those buffers.
c2.852 Manufacturing Systems Analysis 34/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Theorem
′ ◮ That is, the transition (or balance) equations of Z can be written by
transforming those of Z .
◮ In the Z equations, replace nj by Nj − nj for all j ∈ Ω.
c2.852 Manufacturing Systems Analysis 35/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Theorem
Corollary: ◮ Assume:
◮ The initial states s(0) and s ′(0) are related as follows: nj ′(0) = nj (0)
′for j 6∈ Ω, and nj (0) = Nj − nj (0) for j ∈ Ω.
◮ Then
P ′ (n ′ (0)) = P(n(0))
n̄b′ (n ′ (0)) = n̄b(n(0)), for j 6∈ Ω
n̄b′ (n ′ (0)) = Nb − n̄b(n(0)), for j ∈ Ω
c2.852 Manufacturing Systems Analysis 36/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Theorem
Corollary: That is,
◮ the production rates of the two systems are the same,
◮ the average levels of all the buffers in the systems whose direction of flow has not been changed are the same,
◮ the average levels of all the buffers in the systems whose direction of flow has been changed are complementary; the average number of parts in one is equal to the average amount of space in the other.
c2.852 Manufacturing Systems Analysis 37/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Equivalence class of three-machine systems
N N1 2
µ1
2N
µ2
µ3
µ1
1N
2 N
2B
1B
µ2
µ3
2
1
1
2
3
N
N
µ
µ
µ
1N
2B
1B
µ µ µ3 2 1
c2.852 Manufacturing Systems Analysis 38/41 Copyright �2010 Stanley B. Gershwin.
Equivalence Equivalence classes of four-machine systems
Representative members
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Equivalence Example of equivalent loops
2
2
1
1
4
4
3
3
4
3
3
2
2
1
4
1
Ω= (3, 4)
(a) A Fork/ Join Network (b) A Closed Network
c2.852 Manufacturing Systems Analysis 40/41 Copyright �2010 Stanley B. Gershwin.
Equivalence To come
◮ Loops and invariants
◮ Two-machine loops
◮ Instability of A/D systems with infinite buffers
c2.852 Manufacturing Systems Analysis 41/41 Copyright �2010 Stanley B. Gershwin.
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