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Blading in Photolithography machinesAn application of the a priori TSP problem

Teun Janssen

Joined work withJan 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

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Introduction

Blading in Photolithography machines January, 2016

2

Introduction

Blading in Photolithography machines January, 2016

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Blading

Blading in Photolithography machines January, 2016

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Blading

Blading in Photolithography machines January, 2016

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Traveling Salesman Problem

Goal: Find an ordering of the cities such that the salesmanvisits all cities exactly once and distance travelled is minimized.

Blading in Photolithography machines January, 2016

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Blading

An ASCII-file defines the positions of the blades.

This ASCII file is used every time a certain product goestrough the machine, but not every blade position is used.

Blading in Photolithography machines January, 2016

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Blading

An ASCII-file defines the positions of the blades.

This ASCII file is used every time a certain product goestrough the machine, but not every blade position is used.

Blading in Photolithography machines January, 2016

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Blading

Layer IDLithography steps

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Position A xPosition B xPosition C xPosition D xPosition E x x x xPosition F x x x x x x x x xPosition G xPosition H x x x x x x x x xPosition I x x x x x x x x xPosition J x x x x x x x x xPosition K x x x x x x x x xPosition L xPosition M x x x x x x x x xPosition N x x x x x x x x xPosition O x xPosition P x xPosition Q x xPosition R x xTotal steps 2 2 1 1 5 5 7 7 7 7 7 7 7 8 8

Blading in Photolithography machines January, 2016

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Adjusted Traveling Salesman Problem

Find an ordering of the cities such that the salesmen visit allcities exactly once and the sum of all distances traveled isminimized.

Blading in Photolithography machines January, 2016

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Adjusted Traveling Salesman Problem

Find an ordering of the cities such that the salesmen visit allcities exactly once and the sum of all distances traveled isminimized.

Blading in Photolithography machines January, 2016

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A priori Traveling Salesman Problem

Given:

I A complete weighted graph G = (V,E) (metric).

I A set of scenarios S = {S1, . . . , Sm} 2V .I 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 itminimizes the expected length of tours on the scenarios(second-stage tour).

Blading in Photolithography machines January, 2016

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A priori Traveling Salesman Problem

Given:

I A complete weighted graph G = (V,E) (metric).

I A set of scenarios S = {S1, . . . , Sm} 2V .I 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 itminimizes the expected length of tours on the scenarios(second-stage tour).

Blading in Photolithography machines January, 2016

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Known Results

The problem has been considered in 2 cases.

The independent decision model:

I Shmoys and Talwar1 show that a sample-and-augmentapproach gives an 4-approximation.

1David Shmoys and Kunal Talwar. A constant approximationalgorithm for the a priori traveling salesman problem. In: IntegerProgramming and Combinatorial Optimization. Springer, 2008,pp. 331343.

Blading in Photolithography machines January, 2016

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Known Results

The problem has been considered in 2 cases.

The black-box model:

I Schalekamp and Shmoys2 show that for the black-boxmodel there is a randomized O(log n)-approximationwithout sampling.

I There is an (log n) lower bound for deterministicalgorithms3.

2Frans Schalekamp and David B Shmoys. Algorithms for the universaland a priori TSP. . In: Operations Research Letters 36.1 (2008), pp. 13.

3Igor Gorodezky et al. Improved lower bounds for the universal and apriori tsp. In: Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques. Springer, 2010, pp. 178191.

Blading in Photolithography machines January, 2016

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Properties

Theorem

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

Corollary

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

Blading in Photolithography machines January, 2016

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Properties

Theorem

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

Corollary

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

Blading in Photolithography machines January, 2016

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Properties

Theorem

A tour can be constructed, that is a 2m 1-approximationfor a priori TSP in the scenario model, where m 2 is thenumber of scenarios.

Construction:

I For each scenario, find an -approximate tour and sort thescenarios on their resulting tour lengths Tj . Rename thescenarios such that T1 T2 . . . Tm.

I Traverse the tours 1, 2, . . . ,m, while skipping alreadyvisited vertices, resulting in tour .

Blading in Photolithography machines January, 2016

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Properties

Theorem

A tour can be constructed, that is a 2m 1-approximationfor a priori TSP in the scenario model, where m 2 is thenumber of scenarios.

Construction:

I For each scenario, find an -approximate tour and sort thescenarios on their resulting tour lengths Tj . Rename thescenarios such that T1 T2 . . . Tm.

I Traverse the tours 1, 2, . . . ,m, while skipping alreadyvisited vertices, resulting in tour .

Blading in Photolithography machines January, 2016

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Implementation

Goal: Test what the possible gain could be if we used an ILPformulation4 to optimize the blading.

mink

iSk

jSk,i 6=j

dijxkij

s.t.

iSk,i6=j

xkij = 1, j Sk,k [m] (1)

jSk,i6=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 ProgrammingFormulation of Traveling Salesman Problems. In: J. ACM 7.4 (Oct.1960), pp. 326329. issn: 0004-5411.

Blading in Photolithography machines January, 2016

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Implementation

Goal: Test what the possible gain could be if we used an ILPformulation4 to optimize the blading.

mink

iSk

jSk,i 6=j

dijxkij

s.t.

iSk,i6=j

xkij = 1, j Sk,k [m] (1)

jSk,i6=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 ProgrammingFormulation of Traveling Salesman Problems. In: J. ACM 7.4 (Oct.1960), pp. 326329. issn: 0004-5411.

Blading in Photolithography machines January, 2016

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Implementation

Goal: Test what the possible gain could be if we used an ILPformulation 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 thesubproblems.

4. The solutions of these subproblems where combined in oneoptimal ordering and published.

5. The optimal ordering was then used to chance the originaljob.

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. 141.

Blading in Photolithography machines January, 2016

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http://mpc.zib.de/index.php/MPC/article/view/4

Implementation

Goal: Test what the possible gain could be if we used an ILPformulation 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 thesubproblems.

4. The solutions of these subproblems where combined in oneoptimal ordering and published.

5. The optimal ordering was then used to chance the originaljob.

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. 141.

Blading in Photolithography machines January, 2016

16

http://mpc.zib.de/index.php/MPC/article/view/4

Implementation

Goal: Test what the possible gain could be if we used an ILPformulation 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 thesubproblems.

4. The solutions of these subproblems where combined in oneoptimal ordering and published.

5. The optimal ordering was then used to chance the originaljob.

5Tobias Achterberg. SCIP: Solving constraint integer programs. In:Mathematical Programming Compu

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