mathematical modeling of the two-part type machine young jae jang massachusetts institute of...

Mathematical Modeling of Mathematical Modeling of the the

Two-Part Type Machine Two-Part Type Machine

Young Jae JangYoung Jae JangMassachusetts Institute of TechnologyMassachusetts Institute of Technology

[email protected]@mit.eduMay 12, 2004May 12, 2004

May 12, 2004Copy Right by Youngjae Jang, 2004

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ContentsContents

IntroductionIntroduction Previous workPrevious work Issues on multiple-part-type lineIssues on multiple-part-type line Two machine lineTwo machine line Decomposition for type one and type twoDecomposition for type one and type two HeuristicsHeuristics AlgorithmAlgorithm Comparisons with simulationComparisons with simulation Future work and summaryFuture work and summary


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IntroductionIntroduction Multiple-Part-Type Processing LineMultiple-Part-Type Processing Line Example of Two-Part-Type Processing LineExample of Two-Part-Type 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,j i,j Size: NSize: Nijij, , Number of Parts at Bij at time t = nNumber of Parts at Bij at time t = n ijij(t)(t)

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


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Introduction - ParametersIntroduction - Parameters Discrete time ModelDiscrete time Model Identical processing timeIdentical processing time One time unit = processing time for one partOne time unit = processing time for one part


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Introduction – ImportanceIntroduction – Importance Most of processing machines these days are Most of processing machines these days are

able to process several different part typesable to process several different part types Lines are usually processing more than one Lines are usually processing more than one

type of productstype of products In LCD or Semiconductor FAB multi-loop In LCD or Semiconductor FAB multi-loop

processing is commonprocessing is common Better scheduling for multiple part type lineBetter scheduling for multiple part type line GM NeedsGM Needs It will be a part of analysis of CPP of multiple It will be a part of analysis of CPP of multiple

part type linepart type line


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Previous WorkPrevious Work Joe Nemec: MIT PhD Thesis 1998Joe Nemec: MIT PhD Thesis 1998

Single Failure ModeSingle Failure Mode Too complicatedToo complicated Lines are not very well convergedLines are not very well converged

Diego Syrowicz: MIT MS Thesis 1999Diego Syrowicz: MIT MS Thesis 1999 Multiple failure modeMultiple failure mode Did not complete decomposition equationsDid not complete decomposition equations

Tulio Tolio, 2003Tulio Tolio, 2003 Two machine two buffer building block, discrete time, multi-failure Two machine two buffer building block, discrete time, multi-failure

modelmodel Too complicated, some equations are not clear Too complicated, some equations are not clear


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ApproachApproach ObserversObservers

Think that machine is working only single part typeThink that machine is working only single part type Type2 guy thinks that machine is down, when the machine is working on part type oneType2 guy thinks that machine is down, when the machine is working on part type one


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Issues on Multiple-Part-Type Line Issues on Multiple-Part-Type Line – Idleness Failure– Idleness Failure Idleness failure: Failure while machine is starved or blockedIdleness failure: Failure while machine is starved or blocked There is no idleness failure in a single part type caseThere is no idleness failure in a single part type case ExampleExample

1.1. MM33 down down2.2. nn2,12,1= N= N2,12,1, n, n2,22,2 <= N <= N2,22,2, 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 > 0

Observer for part type one sees that Observer for part type one sees that MM11 goes to down when it is block goes to down when it is blocked for part type one!ed for part type one!


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Issues on Multiple-Part-Type Line Issues on Multiple-Part-Type Line – Failure Mode Change– Failure Mode Change Failure mode shift might be detected by an observerFailure mode shift might be detected by an observer There is no failure mode change in a single part type lineThere is no failure mode change in a single part type line ExampleExample

MM44 down down nn3,13,1 = N = N3,13,1, , nn3,23,2 < N < N3,23,2 MM33 is blocked for type one and start working type twp part is blocked for type one and start working type twp part MM33 is down while working on type two is down while working on type two MM44 is up and is up and nn3,13,1 < N < N3,13,1 MM33 is still down is still down


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Two Machine Line ParametersTwo Machine Line Parameters Two machine lineTwo machine line

Single-up, multiple-down, failure mode changing Markov chaiSingle-up, multiple-down, failure mode changing Markov chainn

(t)(t) : machine states at time : machine states at time tt : Machine is up, : Machine is up, ΔΔii: : 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


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Two Machine Line – Markov Two Machine Line – Markov ModelModel Two machine lineTwo machine line

Failure mode change and Idleness failureFailure mode change and Idleness failure zz**

ijij : Failure mode change parameters : Failure mode change parameters qq**

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

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

Single-up, Multiple-Single-up, Multiple-down, Failure Mode down, Failure Mode Changing Markov Changing Markov ChainChain

Example of 3 down Example of 3 down modes Markov modes Markov ChainChain


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Two Machine Line – Efficiency of the Two Machine Line – Efficiency of the lineline

Single part type case (without idleness failure), efficiency is

[{ ( ) } { ( ) }]

[{ ( ) } { ( ) 0}]

u u u

d d d

u d

E Pr t n t N

E Pr t n t

E E

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


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Two-Machine Line – EfficiencyTwo-Machine Line – Efficiency E E u u = E = E dd


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HeuristicsHeuristics Two pseudo-machines: Lp1 and Lp2Two pseudo-machines: Lp1 and Lp2 Analyze two lines separately using single part type machine Analyze two lines separately using single part type machine

line decomposition line decomposition


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Heuristic ResultsHeuristic Results Type 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 don’t want you to get bored…I don’t want you to get bored…


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AlgorithmAlgorithm Jang DDX algorithmJang DDX algorithm 6 unknowns, 13 equations => Over constrained equations6 unknowns, 13 equations => Over constrained equations Very robust -> always converges Very robust -> always converges


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Simulation CaseSimulation Case

e4= efficiency of M’d1

e5= efficiency of M’d2

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


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


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


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


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


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Case 3Case 3 Type 2 demand Type 2 demand

decreasesdecreases


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Future WorkFuture Work

Expand Line to long Expand Line to long processing line with processing line with five part type casesfive part type cases

Applying re-entrance Applying re-entrance flowflow

CPP with multiple-CPP with multiple-part typepart type

Continuous time lineContinuous time line


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ReferenceReference Stanley B. Gershwin, Stanley B. Gershwin, Manufacturing SystemsManufacturing Systems EngineeringEngineering, PTR Prentice , PTR Prentice

Hall, 1994Hall, 1994

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

T. Tolio, S. B. Gershwin, A. Matta, Analysis of two-machine line with mulT. Tolio, S. B. Gershwin, A. Matta, Analysis of two-machine line with multiple failure modes, Technical report, Politecnico di Milano, 1998tiple failure modes, Technical report, Politecnico di Milano, 1998

T. Tolio, A. Matta, A method for performace evaluation of automated floT. Tolio, A. Matta, A method for performace evaluation of automated flow lines, Technical report, Politecnico di Milano, 1998w lines, Technical report, Politecnico di Milano, 1998

Diego A. Syrowicz, Decomposition Analysis of a Deterministic, Multiple-PDiego A. Syrowicz, Decomposition Analysis of a Deterministic, Multiple-Part-Type, Multiple-Failure-Mode Production Line, Massachusetts Institutart-Type, Multiple-Failure-Mode Production Line, Massachusetts Institute of Technology, SM Thesis 1998e of Technology, SM Thesis 1998

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