Blading in Photolithography machines An application of the a priori TSP problem Teun Janssen Joined work with Jan Driessen (NXP), Martijn van Ee (VU Amsterdam), Leo van Iersel (TU Delft) & Rene Sitters (VU Amsterdam) Delft University of Technology January, 2016 Blading in Photolithography machines January, 2016 1
Introduction Blading in Photolithography machines January, 2016 2
Introduction Blading in Photolithography machines January, 2016 2
Introduction Blading in Photolithography machines January, 2016 3
Blading Blading in Photolithography machines January, 2016 4
Blading Blading in Photolithography machines January, 2016 5
Traveling Salesman Problem Goal: Find an ordering of the cities such that the salesman visits all cities exactly once and distance travelled is minimized. Blading in Photolithography machines January, 2016 6
Blading An ASCII-file defines the positions of the blades. Blading in Photolithography machines January, 2016 7
Blading An ASCII-file defines the positions of the blades. This ASCII file is used every time a certain product goes trough the machine, but not every blade position is used. Blading in Photolithography machines January, 2016 7
Blading Lithography steps Layer ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Position A x Position B x Position C x Position D x Position E x x x x Position F x x x x x x x x x Position G x Position H x x x x x x x x x Position I x x x x x x x x x Position J x x x x x x x x x Position K x x x x x x x x x Position L x Position M x x x x x x x x x Position N x x x x x x x x x Position O x x Position P x x Position Q x x Position R x x Total steps 2 2 1 1 5 5 7 7 7 7 7 7 7 8 8 Blading in Photolithography machines January, 2016 8
Adjusted Traveling Salesman Problem Blading in Photolithography machines January, 2016 9
Adjusted Traveling Salesman Problem Find an ordering of the cities such that the salesmen visit all cities exactly once and the sum of all distances traveled is minimized. Blading in Photolithography machines January, 2016 9
A priori Traveling Salesman Problem Given: ◮ A complete weighted graph G = ( V, E ) (metric). ◮ A set of scenarios S = { S 1 , . . . , S m } ⊆ 2 V . ◮ A probability p k per scenario S k of being the active set, with � k p k = 1 . Blading in Photolithography machines January, 2016 10
A priori Traveling Salesman Problem Given: ◮ A complete weighted graph G = ( V, E ) (metric). ◮ A set of scenarios S = { S 1 , . . . , S m } ⊆ 2 V . ◮ A probability p k per scenario S k of being the active set, with � k p k = 1 . Goal: Find a tour on all vertices (first-stage tour), such that it minimizes the expected length of tours on the scenarios (second-stage tour). Blading in Photolithography machines January, 2016 10
Known Results The problem has been considered in 2 cases. The independent decision model: ◮ Shmoys and Talwar 1 show that a sample-and-augment approach gives an 4-approximation. 1 David Shmoys and Kunal Talwar. “A constant approximation algorithm for the a priori traveling salesman problem”. In: Integer Programming and Combinatorial Optimization . Springer, 2008, pp. 331–343. Blading in Photolithography machines January, 2016 11
Known Results The problem has been considered in 2 cases. The black-box model: ◮ Schalekamp and Shmoys 2 show that for the black-box model there is a randomized O (log n ) -approximation without sampling. ◮ There is an Ω(log n ) lower bound for deterministic algorithms 3 . 2 Frans Schalekamp and David B Shmoys. “Algorithms for the universal and a priori TSP”. . In: Operations Research Letters 36.1 (2008), pp. 1–3. 3 Igor Gorodezky et al. “Improved lower bounds for the universal and a priori tsp”. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques . Springer, 2010, pp. 178–191. Blading in Photolithography machines January, 2016 12
Properties Theorem A priori TSP is NP-complete when | S k | ≤ 4 for all k . Blading in Photolithography machines January, 2016 13
Properties Theorem A priori TSP is NP-complete when | S k | ≤ 4 for all k . Corollary Under the Unique Games Conjecture, there is no 1.0242 approximation for a priori TSP when | S k | ≤ 4 for all k . Blading in Photolithography machines January, 2016 13
Properties Theorem A tour τ can be constructed, that is a 2 m − 1 -approximation for a priori TSP in the scenario model, where m ≥ 2 is the number of scenarios. Blading in Photolithography machines January, 2016 14
Properties Theorem A tour τ can be constructed, that is a 2 m − 1 -approximation for a priori TSP in the scenario model, where m ≥ 2 is the number of scenarios. Construction: ◮ For each scenario, find an α -approximate tour and sort the scenarios on their resulting tour lengths T j . Rename the scenarios such that T 1 ≤ T 2 ≤ . . . ≤ T m . ◮ Traverse the tours 1 , 2 , . . . , m , while skipping already visited vertices, resulting in tour τ . Blading in Photolithography machines January, 2016 14
Implementation Goal: Test what the possible gain could be if we used an ILP formulation 4 to optimize the blading. 4 C. E. Miller, A. W. Tucker, and R. A. Zemlin. “Integer Programming Formulation of Traveling Salesman Problems”. In: J. ACM 7.4 (Oct. 1960), pp. 326–329. issn : 0004-5411. Blading in Photolithography machines January, 2016 15
Implementation Goal: Test what the possible gain could be if we used an ILP formulation 4 to optimize the blading. � � � min d ij x kij k i ∈ S k j ∈ S k ,i � = j � s.t. x kij = 1 , ∀ j ∈ S k , ∀ k ∈ [ m ] (1) i ∈ S k ,i � = j � x kij = 1 , ∀ i ∈ S k , ∀ k ∈ [ m ] (2) j ∈ S k ,i � = j u i − u j + nx kij ≤ n − 1 , ∀ i ∈ J k , ∀ j ∈ S k \ { i } , ∀ k ∈ [ m ] (3) x kij ∈ { 0 , 1 } , ∀ i ∈ S k , ∀ j ∈ S k \ { i } , ∀ k ∈ [ m ] 1 ≤ u i ≤ n − 1 , ∀ i ∈ S k (4) 4 C. E. Miller, A. W. Tucker, and R. A. Zemlin. “Integer Programming Formulation of Traveling Salesman Problems”. In: J. ACM 7.4 (Oct. 1960), pp. 326–329. issn : 0004-5411. Blading in Photolithography machines January, 2016 15
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 1. Machine data was converted to a table. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 1. Machine data was converted to a table. 2. Using Matlab the input was split in smaller subproblems. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 1. Machine data was converted to a table. 2. Using Matlab the input was split in smaller subproblems. 3. The ILP solver SCIP 5 was used to optimize the subproblems. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 1. Machine data was converted to a table. 2. Using Matlab the input was split in smaller subproblems. 3. The ILP solver SCIP 5 was used to optimize the subproblems. 4. The solutions of these subproblems where combined in one optimal ordering and published. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
Implementation Goal: Test what the possible gain could be if we used an ILP formulation to optimize the blading. 1. Machine data was converted to a table. 2. Using Matlab the input was split in smaller subproblems. 3. The ILP solver SCIP 5 was used to optimize the subproblems. 4. The solutions of these subproblems where combined in one optimal ordering and published. 5. The optimal ordering was then used to chance the original job. 5 Tobias Achterberg. “SCIP: Solving constraint integer programs”. In: Mathematical Programming Computation 1.1 (2009). http://mpc.zib.de/index.php/MPC/article/view/4 , pp. 1–41. Blading in Photolithography machines January, 2016 16
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