Blading in Photolithography machines - CNR · Blading in Photolithography machines ... Traveling Salesman Problem ... 2Frans Schalekamp and David B Shmoys.\Algorithms for the universal

$Page 1: Blading in Photolithography machines - CNR · Blading in Photolithography machines ... Traveling Salesman Problem ... 2Frans Schalekamp and David B Shmoys.\Algorithms for the universal$
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


2

Introduction


2

Introduction


3

Blading


4

Blading


5

Traveling Salesman Problem

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


6

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.


7

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.


7

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


8

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.


9

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.


9

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).


10

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).


10

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. 331–343.


11

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 Ω(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. 1–3.

3Igor Gorodezky et al. “Improved lower bounds for the universal and apriori tsp”. In: Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques. Springer, 2010, pp. 178–191.


12

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.


13

Properties

Theorem

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

Corollary

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


13

Properties

Theorem

A tour τ 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 α-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 τ .


14

Properties

Theorem

A tour τ 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 α-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 τ .


14

Implementation

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

min∑k

∑i∈Sk

∑j∈Sk,i 6=j

dijxkij

s.t.∑

i∈Sk,i6=j

xkij = 1, ∀j ∈ Sk,∀k ∈ [m] (1)

∑j∈Sk,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. 326–329. issn: 0004-5411.


15

Implementation

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

min∑k

∑i∈Sk

∑j∈Sk,i 6=j

dijxkij

s.t.∑

i∈Sk,i6=j

xkij = 1, ∀j ∈ Sk,∀k ∈ [m] (1)

∑j∈Sk,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. 326–329. issn: 0004-5411.


15

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. 1–41.


16

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

Implementation









16


Implementation









16


Implementation









16


Implementation









16


Implementation









16


Results

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

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

products according to the model proposed by Driessen6.I This reduction is reflected in the machine data after the

optimization.

6Jan Driessen. “An OEE increase of 10 percent on LITHO equipment”.In: 12th European Advanced Process Control and ManufacturingConference. APC—M, Apr. 2012.


17

Results

77 (of 575) products were optimized (= 58.4% of WIP).I For 67 of the 77 products the total blading was reduced.

I On average the blading was reduced by 19.2 % .I This results in a gain of 1% in total time needed for these


optimization.



17

Results

77 (of 575) products were optimized (= 58.4% of WIP).I For 67 of the 77 products the total blading was reduced.I On average the blading was reduced by 19.2 % .

I This results in a gain of 1% in total time needed for theseproducts according to the model proposed by Driessen6.

I This reduction is reflected in the machine data after theoptimization.



17

Results

77 (of 575) products were optimized (= 58.4% of WIP).I For 67 of the 77 products the total blading was reduced.I On average the blading was reduced by 19.2 % .I This results in a gain of 1% in total time needed for these

products according to the model proposed by Driessen6.

I This reduction is reflected in the machine data after theoptimization.



17

Results

77 (of 575) products were optimized (= 58.4% of WIP).I For 67 of the 77 products the total blading was reduced.I On average the blading was reduced by 19.2 % .I This results in a gain of 1% in total time needed for these


optimization.



17

Conclusions

I For the A priori TSP in the scenario model, there is still alarge gap between the lower bound (1.0242) and bestknown approximation algorithm.

I There is an O(log n)-approximation algorithm and a2m− 1-approximation algorithm known.

I The blading problem, an implementation of a priori TSPin the scenario model, can be solved to optimality in alimited amount of time.

I Optimization reduces the processing time of products inthe semiconductor production facility.

Thank you for your attention!


18

Conclusions







18

Conclusions







18

Conclusions







18

Conclusions







18

Documents

Blading in Photolithography machines - CNR · Blading in Photolithography machines ... Traveling Salesman Problem ... 2Frans Schalekamp and David B Shmoys.\Algorithms for the universal