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

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

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

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

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

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. 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. Without this constraint, the wafer lots can be scheduled to A, B, or C.

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

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

Related Work Kimms (1996) proposed a mixed-integer programming model for optimally scheduling 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.

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

Modeling the Dedicated Machine Constraint-RSEM Steps j Wafer lots i W1W1 R3 R3 R3 R3 W2W2 R 3 R3R3 W3W3 R 2 R2R2 R2R2 R2R2 W4W4 R4R4 R4R4 W5W5 R 1 A Resource Schedule and Execution Matrix (RSEM):

Modeling the Dedicated Machine Constraint-Breakdown (BD) Matrix Steps j Machines k 12.. R1R R2R R3R R4R A Breakdown Matrix:

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

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: (4) (3) (5) (6)

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

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)

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

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

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

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%.

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%.

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%.

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

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

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