Mathematical Modeling of the Mathematical Modeling of the TwoTwo--Part Type Machine Part Type Machine

Young Jae JangYoung Jae JangMassachusetts Institute of Massachusetts Institute of

TechnologyTechnologyMay 12, 2004May 12, 2004

(Courtesy of Young Jang. Used with permission.)

May 12, 2004 Copy Right by Youngjae Jang, 2004 2

ContentsContents

IntroductionIntroductionPrevious workPrevious workIssues on multipleIssues on multiple--partpart--type linetype lineTwo machine lineTwo machine lineDecomposition for type one and type twoDecomposition for type one and type twoHeuristicsHeuristicsAlgorithmAlgorithmComparisons with simulationComparisons with simulationFuture work and summaryFuture work and summary

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IntroductionIntroductionMultipleMultiple--PartPart--Type Processing LineType Processing LineExample of TwoExample of Two--PartPart--Type Processing LineType Processing Line

Supply Machines: Supply Machines: MMs1s1, M, Ms2s2

Demand Machines: Demand Machines: MMd1d1, M, Md2d2

Processing Machine: Processing Machine: MMii

Homogeneous Buffer: Homogeneous Buffer: BBi,ji,jSize: Size: NNijij, , Number of Parts at Number of Parts at BijBij at time t = at time t = nnijij(t(t))

Priority Rule: Type one has a priority over type twoPriority Rule: Type one has a priority over type twoWhy supply and demand machines are needed?Why supply and demand machines are needed?

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Introduction Introduction -- ParametersParametersDiscrete time ModelDiscrete time ModelIdentical processing timeIdentical processing timeOne time unit = processing time for one partOne time unit = processing time for one part

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Introduction Introduction –– ImportanceImportanceMost of processing machines these days are able Most of processing machines these days are able to process several different part typesto process several different part typesLines are usually processing more than one type Lines are usually processing more than one type of productsof productsIn LCD or Semiconductor FAB multiIn LCD or Semiconductor FAB multi--loop loop processing is commonprocessing is commonBetter scheduling for multiple part type lineBetter scheduling for multiple part type lineGM NeedsGM NeedsIt will be a part of analysis of CPP of multiple part It will be a part of analysis of CPP of multiple part type linetype line

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Previous WorkPrevious WorkJoe Joe NemecNemec: MIT PhD Thesis 1998: MIT PhD Thesis 1998

Single Failure ModeSingle Failure ModeToo complicatedToo complicatedLines are not very well convergedLines are not very well converged

Diego Diego SyrowiczSyrowicz: MIT MS Thesis 1999: MIT MS Thesis 1999Multiple failure modeMultiple failure modeDid not complete decomposition equationsDid not complete decomposition equations

TulioTulio TolioTolio, 2003, 2003Two machine two buffer building block, discrete time, multiTwo machine two buffer building block, discrete time, multi--failure modelfailure modelToo complicated, some equations are not clear Too complicated, some equations are not clear

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ApproachApproachObserversObservers

Think that machine is working only single part typeThink that machine is working only single part typeType2 guy thinks that machine is down, when the machine is workiType2 guy thinks that machine is down, when the machine is working on part type oneng on part type one

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Issues on MultipleIssues on Multiple--PartPart--Type Line Type Line –– Idleness FailureIdleness Failure

Idleness failure: Failure while machine is starved or blockedIdleness failure: Failure while machine is starved or blockedThere is no idleness failure in a single part type caseThere is no idleness failure in a single part type caseExampleExample1.1. MM33 downdown2.2. nn2,12,1= N= N2,12,1, n, n2,22,2 <= <= NN2,22,2, n, ns,1s,1 > 0, n> 0, ns,2s,2 > 0> 03.3. MM22 blocked for type 1, starts working type twoblocked for type 1, starts working type two4.4. MM22 goes to down, but goes to down, but nn2,12,1 = N= N2,12,1, n, ns,1s,1 > 0> 0Observer for part type one sees that Observer for part type one sees that MM11 goes to down when it is goes to down when it is blocked for part type one!blocked for part type one!

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Issues on MultipleIssues on Multiple--PartPart--Type Line Type Line –– Failure Mode ChangeFailure Mode Change

Failure mode shift might be detected by an observerFailure mode shift might be detected by an observerThere is no failure mode change in a single part type lineThere is no failure mode change in a single part type lineExampleExample

MM44 downdownnn3,13,1 = = NN3,13,1, , nn3,23,2 < < NN3,23,2MM33 is blocked for type one and start working type twp partis blocked for type one and start working type twp partMM33 is down while working on type twois down while working on type twoMM44 is up and is up and nn3,13,1 < < NN3,13,1MM33 is still down is still down

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Two Machine Line ParametersTwo Machine Line ParametersTwo machine lineTwo machine line

SingleSingle--up, multipleup, multiple--down, failure mode changing Markov down, failure mode changing Markov chainchainαα(t)(t) : machine states at time : machine states at time ttϒϒ : Machine is up, : Machine is up, ∆∆ii: : Machine is down at mode Machine is down at mode i i

[{ ( 1) } |{ ( ) }]

[{ ( 1) } |{ ( ) }]

[{ ( 1) } |{ ( ) } { ( ) }]

[{ ( 1) } |{ ( ) } { ( ) 0}]

u u u u uj j

d d d d dk k

u u u u uj j

d d d d dk k

r Pr t t

r Pr t t

p Pr t t n t N

p Pr t t n t

α α

α α

α α

α α

= + = ϒ = ∆

= + = ϒ = ∆

= + = ∆ = ϒ <

= + = ∆ = ϒ >

∩

∩

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Two Machine Line Two Machine Line –– Markov ModelMarkov ModelTwo machine lineTwo machine line

Failure mode change and Idleness failureFailure mode change and Idleness failurezz**

ijij : Failure mode change parameters: Failure mode change parametersqq**

I,jI,j: Idleness failure parameters: Idleness failure parameters(* = u or d)(* = u or d)

These parameters are zeroThese parameters are zero

, ' '

, ' '

1

[{ ( 1) } |{ ( ) } { ( ) }]

[{ ( 1) } |{ ( ) } { ( ) 0}]

[{ ( 1) } |{ ( ) }]

[{ ( 1) } |{ ( ) }]

, Total number of fai

u u u u uj j

d d d d dk k

u u u u uj j j j

d d d d dk k k k

Ju u

jj

q Pr t t n t N

q Pr t t n t

z Pr t t

z Pr t t

Q q J

α α

α α

α α

α α

=

= + = ∆ = ϒ =

= + = ∆ = ϒ =

= + = ∆ = ∆

= + = ∆ = ∆

= =∑

∩

∩

1

lure mode of

, Total number of failure mode of

u

Ld u u

ll

M

Q q L M=

= =∑

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One Machine Markov ModelOne Machine Markov Model

SingleSingle--up, Multipleup, Multiple--down, Failure Mode down, Failure Mode Changing Markov Changing Markov ChainChainExample of 3 down Example of 3 down modes Markov Chainmodes Markov Chain

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Two Machine Line Two Machine Line –– Efficiency of the lineEfficiency of the line

Single part type case (without idleness failure), efficiency is[{ ( ) } { ( ) }][{ ( ) } { ( ) 0}]

u u u

d d d

u d

E Pr t n t NE Pr t n t

E E

α

α

= = ϒ <

= = ϒ >

=

∩∩

However, with idlness failure, [{ ( ) } { ( ) }] [{ ( ) } { ( ) 0}]

Efficiency needs to be derived from the definition,[{ ( 1) } { ( ) }]

with the fact

u u d d

u u u

Pr t n t N Pr t n t

E Pr t n t N

α α

α

= ϒ < ≠ = ϒ >

= + = ϒ <

∩ ∩

∩

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TwoTwo--Machine Line Machine Line –– EfficiencyEfficiencyE E u u = E = E dd

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HeuristicsHeuristicsTwo pseudoTwo pseudo--machines: Lp1 and Lp2machines: Lp1 and Lp2Analyze two lines separately using single part type machine lineAnalyze two lines separately using single part type machine linedecomposition decomposition

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Heuristic ResultsHeuristic ResultsType one production rate % error: 2%Type one production rate % error: 2%Type two production rate % error:Type two production rate % error:

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Decomposition Equations:Decomposition Equations:I donI don’’t want you to get boredt want you to get bored……

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AlgorithmAlgorithmJang DDX algorithmJang DDX algorithm6 unknowns, 13 equations => Over constrained equations6 unknowns, 13 equations => Over constrained equationsVery robust Very robust --> always converges > always converges

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Simulation CaseSimulation Case

Random Number GenerationRandom Number Generation-- Triangular distributionTriangular distribution-- Two processing machines two supply machine and two demand line Two processing machines two supply machine and two demand line casecase

e4= efficiency of M’d1e5= efficiency of M’d2

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EE11 ErrorErrorAbs(E1 error) = 0.80712 %Abs(E1 error) = 0.80712 %Std(E1 error) = 0.8627Std(E1 error) = 0.8627

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EE22 ErrorErrorAbs(E2 error) = 1.24 % Abs(E2 error) = 1.24 % Std = 0.8934Std = 0.8934

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Case 1Case 1Type one supply variesType one supply variesReliable type two machines Reliable type two machines

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Case 2Case 2Type two demand variesType two demand variesReliable type two machines Reliable type two machines

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Case 3Case 3Type 2 demand decreasesType 2 demand decreases

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Future WorkFuture Work

Expand Line to long Expand Line to long processing line with five processing line with five part type casespart type casesApplying reApplying re--entrance entrance flowflowCPP with multipleCPP with multiple--part part typetypeContinuous time lineContinuous time line

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ReferenceReferenceStanley B. Gershwin, Stanley B. Gershwin, Manufacturing SystemsManufacturing Systems EngineeringEngineering, PTR , PTR Prentice Hall, 1994Prentice Hall, 1994

Stanley B. Gershwin, An efficient decomposition methods for the Stanley B. Gershwin, An efficient decomposition methods for the approximate evaluation of tandem queues with finite storage spacapproximate evaluation of tandem queues with finite storage space and e and blocking. blocking. Operations Research,Operations Research, March 1987March 1987

T. T. TolioTolio, S. B. Gershwin, A. , S. B. Gershwin, A. MattaMatta, Analysis of two, Analysis of two--machine line with machine line with multiple failure modes, Technical report, multiple failure modes, Technical report, PolitecnicoPolitecnico didi MilanoMilano, 1998, 1998

T. T. TolioTolio, A. , A. MattaMatta, A method for , A method for performaceperformace evaluation of automated evaluation of automated flow lines, Technical report, flow lines, Technical report, PolitecnicoPolitecnico didi MilanoMilano, 1998, 1998

Diego A. Diego A. SyrowiczSyrowicz, Decomposition Analysis of a Deterministic, , Decomposition Analysis of a Deterministic, MultipleMultiple--PartPart--Type, MultipleType, Multiple--FailureFailure--Mode Production Line, Mode Production Line, Massachusetts Institute of Technology, SM Thesis 1998Massachusetts Institute of Technology, SM Thesis 1998