Scheduling for Dedicated Machine Constraint Using Integer Programming

1. Scheduling for Dedicated Machine Constraint Using Integer Programming Huy Nguyen Anh Pham, Arthur Shr, and Peter P. Chen Louisiana State University, Baton Rouge, LA, U.S.A. Alan Liu National Chung Cheng University, Chia-Yi, Taiwan

2. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

3. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

4. Dedicated Machine Constraint in Semiconductor Manufacturing • The operations of semiconductor manufacturing repeatedly make an IC product layer by layer. • The dedicated machine constraint is in the photolithography process area

5. Photolithography Process • The most important process for the yield of IC • “Blue Print” for each photo layer • Alignment of layers • Photo machines could be used in different layers. • Different process time for photo each layer

6. Dedicated Photolithography Machine Constraint • With this constraint, the wafer lots dedicated to machine X need to wait for machine X, even if there is no wafer lot waiting for machine Y. • Without this constraint, the wafer lots can be scheduled to A, B, or C.

7. Issues and Current Solutions • Photo machines might become load unbalanced if the wafers are dispatched randomly to the photo machines at the first photo layer. • Some photo machines will be “lost” for a while and the others will be “run” with heavy load • Switch from the highly congested machines to the idle machines. • Rely on experienced engineers to manually handle alignment problems of the wafer lots

8. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

9. Related Work • Kimms (1996) proposed a mixed-integer programming model for optimallyscheduling machines. • (Miwa, 2005) proposed a load balance allocation function between machines and used this function for a dynamic programming method to schedule them. => Their problems without dedicated machine constraint. • (Shr 2008a; 2008b) proposed a heuristic Load Balancing scheduling method. => The approaches did not attempt to optimize the cost for the overall productivity with the dedicated machine constraint.

10. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

11. Modeling the Dedicated Machine Constraint-RSEM • A Resource Schedule and Execution Matrix (RSEM):

12. Modeling the Dedicated Machine Constraint-Breakdown (BD) Matrix • A Breakdown Matrix:

13. Formulation of the Problem • Variables: Xi,j,k is a binary variable. • The penalty cost for switching: (1) (2)

14. Formulation of the Problem – Cont’d • The production cost for wafer i at step j assigned to machine k: • Minimize the production cost of the assignments: • Minimize the queue time for wafer lots waiting at machines • The objective function: (3) (4) (5) (6)

15. Formulation of the Problem – Cont’d • Constraints: • Wafer i at step Wi,j is assigned to exactly one machine: • Assume that each wafer lot Wihas Pi photo steps. The dedicated machine constraint for wafer i at step Wi, j : for either k = 1, 2, …, orQ. (7) (8)

16. Formulation of the Problem – Cont’d • Because of requirements for a canonical form of an IP formulation, by using the Big M method Equation (8) is transformed into: (9)

17. Since the Big M method Formulation of the Problem – Cont’d • The Integer Formulation: (10)

18. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

19. Experiments • 100, 200, and 300 wafer lots with 10 steps are used. • The number of machines is set equal to 3. • BD[j, k] is randomly given in the range [0, 1000]. • Each experiment uses 30 examples

20. Experiments – Cont’d Experiment of 100 wafer lots, 10 steps and 3 machines. • The average execution time: 127 seconds for each example. • The average percentage cost saved: approximately 41.1%.

21. Experiments – Cont’d Experiment of 200 wafer lots, 10 steps and 3 machines. • The average execution time: 1,186.6 seconds for each example. • The average percentage cost saved: approximately 37.9%.

22. Experiments – Cont’d Experiment of 300 wafer lots, 10 steps and 3 machines. • The average execution time: 2,781 seconds for each example. • The average percentage cost saved: approximately 26.5%.

23. Outline • Dedicated Machine Constraint in Semiconductor Manufacturing • Related Work • Integer Programming Approach • Experiments • Conclusion

24. Conclusion • Provide an Integer Programming (IP) approach to model the dedicated machine constraint • Future work • Improving the performance of IP for on-line scheduling problems • Improving the technique to model the constraint

25. Thank You