1. Mathematical Modeling of the Two-Part Type Machine Young Jae Jang Massachusetts Institute of Technology youngjae@mit.edu May 12, 2004

2. Copy Right by Youngjae Jang, 2004 2 Contents Introduction Introduction Previous work Previous work Issues on multiple-part-type line Issues on multiple-part-type line Two machine line Two machine line Decomposition for type one and type two Decomposition for type one and type two Heuristics Heuristics Algorithm Algorithm Comparisons with simulation Comparisons with simulation Future work and summary Future work and summary

3. May 12, 2004 Copy Right by Youngjae Jang, 2004 3 Introduction Multiple-Part-Type Processing Line Multiple-Part-Type Processing Line Example of Two-Part-Type Processing Line Example of Two-Part-Type Processing Line  Supply Machines: M s1, M s2  Demand Machines: M d1, M d2  Processing Machine: M i  Homogeneous Buffer: B i,j Size: N ij, Size: N ij, Number of Parts at Bij at time t = n ij (t) Number of Parts at Bij at time t = n ij (t) Priority Rule: Type one has a priority over type two Priority Rule: Type one has a priority over type two Why supply and demand machines are needed? Why supply and demand machines are needed?

4. May 12, 2004 Copy Right by Youngjae Jang, 2004 4 Introduction - Parameters Discrete time Model Discrete time Model Identical processing time Identical processing time One time unit = processing time for one part One time unit = processing time for one part

5. May 12, 2004 Copy Right by Youngjae Jang, 2004 5 Introduction – Importance Most of processing machines these days are able to process several different part types Most of processing machines these days are able to process several different part types Lines are usually processing more than one type of products Lines are usually processing more than one type of products In LCD or Semiconductor FAB multi-loop processing is common In LCD or Semiconductor FAB multi-loop processing is common Better scheduling for multiple part type line Better scheduling for multiple part type line GM Needs GM Needs It will be a part of analysis of CPP of multiple part type line It will be a part of analysis of CPP of multiple part type line

6. May 12, 2004 Copy Right by Youngjae Jang, 2004 6 Previous Work Joe Nemec: MIT PhD Thesis 1998 Joe Nemec: MIT PhD Thesis 1998  Single Failure Mode  Too complicated  Lines are not very well converged Diego Syrowicz: MIT MS Thesis 1999 Diego Syrowicz: MIT MS Thesis 1999  Multiple failure mode  Did not complete decomposition equations Tulio Tolio, 2003 Tulio Tolio, 2003  Two machine two buffer building block, discrete time, multi- failure model  Too complicated, some equations are not clear

7. May 12, 2004 Copy Right by Youngjae Jang, 2004 7 Approach Observers 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

8. May 12, 2004 Copy Right by Youngjae Jang, 2004 8 Issues on Multiple-Part-Type Line – Idleness Failure Idleness failure: Failure while machine is starved or blocked Idleness failure: Failure while machine is starved or blocked There is no idleness failure in a single part type case There is no idleness failure in a single part type case Example Example  M 3 down  n 2,1 = N 2,1, n 2,2 0, n s,2 > 0  M 2 blocked for type 1, starts working type two  M 2 goes to down, but n 2,1 = N 2,1, n s,1 > 0 Observer for part type one sees that M 1 goes to down when it is blocked for part type one! Observer for part type one sees that M 1 goes to down when it is blocked for part type one!

9. May 12, 2004 Copy Right by Youngjae Jang, 2004 9 Issues on Multiple-Part-Type Line – Failure Mode Change Failure mode shift might be detected by an observer Failure mode shift might be detected by an observer There is no failure mode change in a single part type line There is no failure mode change in a single part type line Example Example  M 4 down  n 3,1 = N 3,1, n 3,2 < N 3,2  M 3 is blocked for type one and start working type twp part  M 3 is down while working on type two  M 4 is up and n 3,1 < N 3,1  M 3 is still down

10. May 12, 2004 Copy Right by Youngjae Jang, 2004 10 Two Machine Line Parameters Two machine line Two machine line  Single-up, multiple-down, failure mode changing Markov chain   (t) : machine states at time t   : Machine is up, Δ i : Machine is down at mode i

11. May 12, 2004 Copy Right by Youngjae Jang, 2004 11 Two Machine Line – Markov Model Two machine line 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

12. May 12, 2004 Copy Right by Youngjae Jang, 2004 12 One Machine Markov Model Single-up, Multiple- down, Failure Mode Changing Markov Chain Single-up, Multiple- down, Failure Mode Changing Markov Chain Example of 3 down modes Markov Chain Example of 3 down modes Markov Chain

13. May 12, 2004 Copy Right by Youngjae Jang, 2004 13 Two Machine Line – Efficiency of the line

14. May 12, 2004 Copy Right by Youngjae Jang, 2004 14 Two-Machine Line – Efficiency E u = E d E u = E d

15. May 12, 2004 Copy Right by Youngjae Jang, 2004 15 Heuristics Two pseudo-machines: Lp1 and Lp2 Two pseudo-machines: Lp1 and Lp2 Analyze two lines separately using single part type machine line decomposition Analyze two lines separately using single part type machine line decomposition

16. May 12, 2004 Copy Right by Youngjae Jang, 2004 16 Heuristic Results Type one production rate % error: 2% Type one production rate % error: 2% Type two production rate % error: Type two production rate % error:

17. May 12, 2004 Copy Right by Youngjae Jang, 2004 17 Decomposition Equations: I don’t want you to get bored… I don’t want you to get bored…

18. May 12, 2004 Copy Right by Youngjae Jang, 2004 18 Algorithm Jang DDX algorithm Jang DDX algorithm 6 unknowns, 13 equations => Over constrained equations 6 unknowns, 13 equations => Over constrained equations Very robust -> always converges Very robust -> always converges

19. May 12, 2004 Copy Right by Youngjae Jang, 2004 19 Simulation Case e4= efficiency of M’ d1 e5= efficiency of M’ d2 Random Number Generation - Triangular distribution - Two processing machines two supply machine and two demand line case

20. May 12, 2004 Copy Right by Youngjae Jang, 2004 20 E 1 Error Abs(E1 error) = 0.80712 % Abs(E1 error) = 0.80712 % Std(E1 error) = 0.8627 Std(E1 error) = 0.8627

21. May 12, 2004 Copy Right by Youngjae Jang, 2004 21 E 2 Error Abs(E2 error) = 1.24 % Abs(E2 error) = 1.24 % Std = 0.8934 Std = 0.8934

22. May 12, 2004 Copy Right by Youngjae Jang, 2004 22 Case 1 Type one supply varies Type one supply varies Reliable type two machines Reliable type two machines

23. May 12, 2004 Copy Right by Youngjae Jang, 2004 23 Case 2 Type two demand varies Type two demand varies Reliable type two machines Reliable type two machines

24. May 12, 2004 Copy Right by Youngjae Jang, 2004 24 Case 3 Type 2 demand decreases Type 2 demand decreases

25. May 12, 2004 Copy Right by Youngjae Jang, 2004 25 Future Work Expand Line to long processing line with five part type cases Expand Line to long processing line with five part type cases Applying re-entrance flow Applying re-entrance flow CPP with multiple-part type CPP with multiple-part type Continuous time line Continuous time line

26. May 12, 2004 Copy Right by Youngjae Jang, 2004 26 Reference Stanley B. Gershwin, Manufacturing Systems Engineering, PTR Prentice Hall, 1994 Stanley B. Gershwin, Manufacturing Systems Engineering, PTR Prentice Hall, 1994 Stanley B. Gershwin, An efficient decomposition methods for the approximate evaluation of tandem queues with finite storage space and blocking. Operations Research, March 1987 Stanley B. Gershwin, An efficient decomposition methods for the approximate evaluation of tandem queues with finite storage space and blocking. Operations Research, March 1987 T. Tolio, S. B. Gershwin, A. Matta, Analysis of two-machine line with multiple failure modes, Technical report, Politecnico di Milano, 1998 T. Tolio, S. B. Gershwin, A. Matta, Analysis of two-machine line with multiple failure modes, Technical report, Politecnico di Milano, 1998 T. Tolio, A. Matta, A method for performace evaluation of automated flow lines, Technical report, Politecnico di Milano, 1998 T. Tolio, A. Matta, A method for performace evaluation of automated flow lines, Technical report, Politecnico di Milano, 1998 Diego A. Syrowicz, Decomposition Analysis of a Deterministic, Multiple- Part-Type, Multiple-Failure-Mode Production Line, Massachusetts Institute of Technology, SM Thesis 1998 Diego A. Syrowicz, Decomposition Analysis of a Deterministic, Multiple- Part-Type, Multiple-Failure-Mode Production Line, Massachusetts Institute of Technology, SM Thesis 1998