Small two part-type line2 5-04

Small two part-type line2 5-04 - Mathematical Modeling of...

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Unformatted text preview: Mathematical Modeling of the Two-Part Type Machine Young Jae Jang Young Massachusetts Institute of Technology youngjae@mit.edu May 12, 2004 Contents Introduction Previous work Issues on multiple-part-type 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 Multiple-Part-Type Processing Line Example of Two-Part-Type 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 multi-loop 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, multi-failure 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 Multiple-Part-Type 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 Multiple-Part-Type 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 Single-up, multiple-down, 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 Single-up, 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 Two-Machine Line – Efficiency E u= E d May 12, 2004 Copy Right by Youngjae Jang, 2004 14 Heuristics Two pseudo-machines: 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 re-entrance flow CPP with multiple-part 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 two-machine 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, MultiplePart-Type, Multiple-Failure-Mode Production Line, Massachusetts Institute of Part-Type, Technology, SM Thesis 1998 Technology, May 12, 2004 Copy Right by Youngjae Jang, 2004 26 ...
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This note was uploaded on 02/24/2012 for the course ENGINEERIN 2.852 taught by Professor Stanleyb.gershwin during the Spring '04 term at MIT.

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