Unformatted text preview: Mathematical Modeling of the
TwoPart Type Machine
Young Jae Jang
Young
Massachusetts Institute of Technology
[email protected]
May 12, 2004 Contents Introduction Previous work Issues on multipleparttype line Two machine line Decomposition for type one and type two Heuristics Algorithm Comparisons with simulation Future work and summary
May 12, 2004 Copy Right by Youngjae Jang, 2004 2 Introduction MultiplePartType Processing Line
Example of TwoPartType Processing Line Supply Machines: Ms1, Ms2
Supply Demand Machines: Md1, Md2
Demand Processing Machine: Mi
Processing Homogeneous Buffer: Bii,j
Homogeneous
,j Size: Nij, Number of Parts at Bij at time t = nij(t) Priority Rule: Type one has a priority over type two Why supply and demand machines are needed? May 12, 2004 Copy Right by Youngjae Jang, 2004 3 Introduction  Parameters Discrete time Model
Identical processing time
One time unit = processing time for one part May 12, 2004 Copy Right by Youngjae Jang, 2004 4 Introduction – Importance Most of processing machines these days are able to
Most process several different part types
process Lines are usually processing more than one type of
Lines
products
products In LCD or Semiconductor FAB multiloop processing is
In
common
common Better scheduling for multiple part type line GM Needs It will be a part of analysis of CPP of multiple part type
It
line
line
May 12, 2004 Copy Right by Youngjae Jang, 2004 5 Previous Work Joe Nemec: MIT PhD Thesis 1998 Diego Syrowicz: MIT MS Thesis 1999 Single Failure Mode
Too complicated
Lines are not very well converged
Multiple failure mode
Did not complete decomposition equations Tulio Tolio, 2003 Two machine two buffer building block, discrete time, multifailure
Two
model
model
Too complicated, some equations are not clear
Too May 12, 2004 Copy Right by Youngjae Jang, 2004 6 Approach Observers Think that machine is working only single part type Type2 guy thinks that machine is down, when the machine is working on part type one May 12, 2004 Copy Right by Youngjae Jang, 2004 7 Issues on MultiplePartType Line
Issues
– Idleness Failure
Idleness Idleness failure: Failure while machine is starved or blocked
There is no idleness failure in a single part type case
Example
1.
2.
3.
4. M3 down
n2,1= N2,1, n2,2 <= N2,2, ns,1 > 0, ns,2 > 0
M2 blocked for type 1, starts working type two
blocked
M2 goes to down, but n2,1 = N2,1, ns,1 > 0 Observer for part type one sees that M1 goes to down when it is blocked for
Observer
part type one!
part May 12, 2004 Copy Right by Youngjae Jang, 2004 8 Issues on MultiplePartType Line
Issues
– Failure Mode Change
Failure Failure mode shift might be detected by an observer
There is no failure mode change in a single part type line
Example M4 down
n3,1 = N3,1, n3,2 < N3,2
M3 is blocked for type one and start working type twp part
M3 is down while working on type two
M4 is up and n3,1 < N3,1
M3 is still down May 12, 2004 Copy Right by Youngjae Jang, 2004 9 Two Machine Line Parameters Two machine line Singleup, multipledown, failure mode changing Markov chain
α(t) : machine states at time t
ϒ : Machine is up, Δi: Machine is down at mode i rju = Pr[{α u (t + 1) = ϒu }{α u (t ) = ∆ uj }]
rkd = Pr[{α d (t + 1) = ϒ d }{α d (t ) = ∆ d }]
k
p u = Pr[{α u (t + 1) = ∆ uj }{α u (t ) = ϒ u } I{n(t ) < N }]
j
d
pk = Pr[{α d (t + 1) = ∆ d }{α d (t ) = ϒ d } I{n(t ) > 0}]
k May 12, 2004 Copy Right by Youngjae Jang, 2004 10 Two Machine Line – Markov Model Two machine line Failure mode change and Idleness failure
z*ij : Failure mode change parameters
q*I,j: Idleness failure parameters
(* = u or d)
(* These parameters are zero q u = Pr[{α u (t + 1) = ∆ uj } {α u (t ) = ϒu } I{n(t ) = N }]
j
d
qk = Pr[{α d (t + 1) = ∆ d } {α d (t ) = ϒ d } I{n(t ) = 0}]
k z u, j ' = Pr[{α u (t + 1) = ∆ uj ' }{α u (t ) = ∆ uj }]
j
d
d
d
zk ,k ' = Pr[{α d (t + 1) = ∆ k '}{α d (t ) = ∆ k }]
J Q = ∑ qu ,
j
u J = Total number of failure mode of M u j =1
L Q = ∑ qlu ,
d L = Total number of failure mode of M u l =1 May 12, 2004 Copy Right by Youngjae Jang, 2004 11 One Machine Markov Model May 12, 2004 Singleup, Multipledown, Failure Mode
down,
Changing Markov
Chain
Chain
Example of 3 down
Example
modes Markov Chain
modes Copy Right by Youngjae Jang, 2004 12 Two Machine Line – Efficiency of the line
Single part type case (without idleness failure), efficiency is
E u = Pr[{α u (t ) = ϒu } I{n(t ) < N }]
E d = Pr[{α d (t ) = ϒ d } I{n(t ) > 0}]
Eu = E d However, with idlness failure,
Pr[{α u (t ) = ϒu } I{n(t ) < N }] ≠ Pr[{α d (t ) = ϒ d } I{n(t ) > 0}]
Efficiency needs to be derived from the definition,
E u = Pr[{α u (t + 1) = ϒu } I{n(t ) < N }]
with the fact May 12, 2004 Copy Right by Youngjae Jang, 2004 13 TwoMachine Line – Efficiency E u= E d May 12, 2004 Copy Right by Youngjae Jang, 2004 14 Heuristics Two pseudomachines: Lp1 and Lp2
Analyze two lines separately using single part type machine line
Analyze
decomposition May 12, 2004 Copy Right by Youngjae Jang, 2004 15 Heuristic Results Type one production rate % error: 2%
Type two production rate % error: May 12, 2004 Copy Right by Youngjae Jang, 2004 16 Decomposition Equations: I don’t want you to get bored… May 12, 2004 Copy Right by Youngjae Jang, 2004 17 Algorithm Jang DDX algorithm
6 unknowns, 13 equations => Over constrained equations
Very robust > always converges
Very May 12, 2004 Copy Right by Youngjae Jang, 2004 18 Simulation Case
Random Number Generation
 Triangular distribution
 Two processing machines two supply machine and two demand line case e4= efficiency of M’d1
e5= efficiency of M’d2 May 12, 2004 Copy Right by Youngjae Jang, 2004 19 E1 Error Abs(E1 error) = 0.80712 %
Std(E1 error) = 0.8627 May 12, 2004 Copy Right by Youngjae Jang, 2004 20 E2 Error Abs(E2 error) = 1.24 %
Abs(E2
Std = 0.8934 May 12, 2004 Copy Right by Youngjae Jang, 2004 21 Case 1 Type one supply varies
Reliable type two machines
Reliable May 12, 2004 Copy Right by Youngjae Jang, 2004 22 Case 2 Type two demand varies
Reliable type two machines
Reliable May 12, 2004 Copy Right by Youngjae Jang, 2004 23 Case 3 Type 2 demand decreases May 12, 2004 Copy Right by Youngjae Jang, 2004 24 Future Work Expand Line to long
Expand
processing line with five
part type cases
part
Applying reentrance flow
CPP with multiplepart
CPP
type
type
Continuous time line May 12, 2004 Copy Right by Youngjae Jang, 2004 25 Reference Stanley B. Gershwin, Manufacturing Systems Engineering, PTR Prentice Hall,
Stanley
Manufacturing
Engineering PTR
1994
1994 Stanley B. Gershwin, An efficient decomposition methods for the approximate
Stanley
evaluation of tandem queues with finite storage space and blocking.
Operations Research, March 1987
Operations T. Tolio, S. B. Gershwin, A. Matta, Analysis of twomachine line with
T.
multiple failure modes, Technical report, Politecnico di Milano, 1998
multiple T. Tolio, A. Matta, A method for performace evaluation of automated flow
T.
lines, Technical report, Politecnico di Milano, 1998
lines, Diego A. Syrowicz, Decomposition Analysis of a Deterministic, MultiplePartType, MultipleFailureMode Production Line, Massachusetts Institute of
PartType,
Technology, SM Thesis 1998
Technology, May 12, 2004 Copy Right by Youngjae Jang, 2004 26 ...
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 Spring '04
 StanleyB.Gershwin
 Systems Engineering, Supply And Demand, Markov chain, Andrey Markov, Youngjae Jang

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