Blading in Photolithography machines An application of the a priori - - PowerPoint PPT Presentation

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

Layer ID Lithography steps 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 = {S1, . . . , Sm} ⊆ 2V . ◮ A probability pk per scenario Sk of being the active set,

with

k pk = 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 = {S1, . . . , Sm} ⊆ 2V . ◮ A probability pk per scenario Sk of being the active set,

with

k pk = 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 Talwar1 show that a sample-and-augment

approach gives an 4-approximation.

1David 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 Shmoys2 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

algorithms3.

2Frans Schalekamp and David B Shmoys. “Algorithms for the universal

and a priori TSP”. . In: Operations Research Letters 36.1 (2008), pp. 1–3.

3Igor 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 |Sk| ≤ 4 for all k.

Blading in Photolithography machines January, 2016 13

Properties

Theorem

A priori TSP is NP-complete when |Sk| ≤ 4 for all k.

Corollary

Under the Unique Games Conjecture, there is no 1.0242 approximation for a priori TSP when |Sk| ≤ 4 for all k.

Blading in Photolithography machines January, 2016 13

Properties

Theorem

A tour τ can be constructed, that is a 2m − 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 2m − 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 Tj. Rename the scenarios such that T1 ≤ T2 ≤ . . . ≤ Tm.

◮ 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 formulation4 to optimize the blading.

• 4C. 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 formulation4 to optimize the blading.

min

• k
• i∈Sk
• j∈Sk,i=j

dijxkij s.t.

• i∈Sk,i=j

xkij = 1, ∀j ∈ Sk, ∀k ∈ [m] (1)

• j∈Sk,i=j

xkij = 1, ∀i ∈ Sk, ∀k ∈ [m] (2) ui − uj + nxkij ≤ n − 1,∀i ∈ Jk, ∀j ∈ Sk \ {i}, ∀k ∈ [m] (3) xkij ∈ {0, 1}, ∀i ∈ Sk, ∀j ∈ Sk \ {i}, ∀k ∈ [m] 1 ≤ ui ≤ n − 1, ∀i ∈ Sk (4)

• 4C. 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.

5Tobias 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.

5Tobias 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.

5Tobias 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 SCIP5 was used to optimize the

subproblems.

5Tobias 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 SCIP5 was used to optimize the

subproblems.

• 4. The solutions of these subproblems where combined in one
• ptimal ordering and published.

5Tobias 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 SCIP5 was used to optimize the

subproblems.

• 4. The solutions of these subproblems where combined in one
• ptimal ordering and published.
• 5. The optimal ordering was then used to chance the original

job.

5Tobias 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

Results

77 (of 575) products were optimized (= 58.4% of WIP).

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.

In: 12th European Advanced Process Control and Manufacturing

• Conference. APC—M, Apr. 2012.

Blading in Photolithography machines January, 2016 17

Results

77 (of 575) products were optimized (= 58.4% of WIP).

◮ For 67 of the 77 products the total blading was reduced.

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.

In: 12th European Advanced Process Control and Manufacturing

• Conference. APC—M, Apr. 2012.

Blading in Photolithography machines January, 2016 17

Results

77 (of 575) products were optimized (= 58.4% of WIP).

◮ For 67 of the 77 products the total blading was reduced. ◮ On average the blading was reduced by 19.2 % .

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.

In: 12th European Advanced Process Control and Manufacturing

• Conference. APC—M, Apr. 2012.

Blading in Photolithography machines January, 2016 17

Results

77 (of 575) products were optimized (= 58.4% of WIP).

◮ For 67 of the 77 products the total blading was reduced. ◮ On average the blading was reduced by 19.2 % . ◮ This results in a gain of 1% in total time needed for these

products according to the model proposed by Driessen6.

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.

In: 12th European Advanced Process Control and Manufacturing

• Conference. APC—M, Apr. 2012.

Blading in Photolithography machines January, 2016 17

Results

77 (of 575) products were optimized (= 58.4% of WIP).

◮ For 67 of the 77 products the total blading was reduced. ◮ On average the blading was reduced by 19.2 % . ◮ This results in a gain of 1% in total time needed for these

products according to the model proposed by Driessen6.

◮ This reduction is reflected in the machine data after the

• ptimization.

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.

In: 12th European Advanced Process Control and Manufacturing

• Conference. APC—M, Apr. 2012.

Blading in Photolithography machines January, 2016 17

Conclusions

◮ For the A priori TSP in the scenario model, there is still a

large gap between the lower bound (1.0242) and best known approximation algorithm.

Blading in Photolithography machines January, 2016 18

Conclusions

◮ For the A priori TSP in the scenario model, there is still a

large gap between the lower bound (1.0242) and best known approximation algorithm.

◮ There is an O(log n)-approximation algorithm and a

2m − 1-approximation algorithm known.

Blading in Photolithography machines January, 2016 18

Conclusions

◮ For the A priori TSP in the scenario model, there is still a

large gap between the lower bound (1.0242) and best known approximation algorithm.

◮ There is an O(log n)-approximation algorithm and a

2m − 1-approximation algorithm known.

◮ The blading problem, an implementation of a priori TSP

in the scenario model, can be solved to optimality in a limited amount of time.

Blading in Photolithography machines January, 2016 18

Conclusions

◮ For the A priori TSP in the scenario model, there is still a

large gap between the lower bound (1.0242) and best known approximation algorithm.

◮ There is an O(log n)-approximation algorithm and a

2m − 1-approximation algorithm known.

◮ The blading problem, an implementation of a priori TSP

in the scenario model, can be solved to optimality in a limited amount of time.

◮ Optimization reduces the processing time of products in

the semiconductor production facility.

Blading in Photolithography machines January, 2016 18

Conclusions

◮ For the A priori TSP in the scenario model, there is still a

large gap between the lower bound (1.0242) and best known approximation algorithm.

◮ There is an O(log n)-approximation algorithm and a

2m − 1-approximation algorithm known.

◮ The blading problem, an implementation of a priori TSP

in the scenario model, can be solved to optimality in a limited amount of time.

◮ Optimization reduces the processing time of products in

the semiconductor production facility. Thank you for your attention!

Blading in Photolithography machines January, 2016 18