Prediction Model for Long-term Performance Indicators of Semiconductor Wafer Fabrication Facility Based on Gaussian Process Regression Zhihong Min ， Li Li Presenter Zhihong Min
OUTLINE • MOTIVATION & FORMULATION I • MODEL & ANALYSIS II • CONCLUSION & DISCUSSION III 2
Motivation Background Semiconductor Manufacturing Cell phone Time... Flash Disk Pad Computer X-Box Router Factory Customers Information society 3
Motivation Challenges and Opportunities Challenges Opportunities Big Data Multiple re-entrant feature Prediction Method (LR) Complicated working conditions KPI Prediction Platform Cloud Manufacturing Draw plan by experience Customer Factory 4
Motivation Background A bridge for factory and customers Make production Main concern: plan according to When does my the current proposed working condition order could be and order request delivered? Queue On-time Through Work in Cycle Length Delivery put Process Time Rate Equipm Move ent ment Utility Real-time collected by the real fab line Short-term KPI Long-term KPIs Build a prediction Model To choose proper supplier according to To make daily feeding plans from the perspective the prediction results of overall optimization and satisfy customer requirements 5
Motivation Previous Work Multi-linear Regression It is an approach for modeling the relationship between a scalar dependent • variable y and one or more explanatory variables (or independent variables) denoted X. Define Loss function • To obtain the parameters which makes loss function minimum. • 6
OUTLINE • MOTIVATION & FORMULATION I • MODEL & ANALYSIS II • CONCLUSION & DISCUSSION III 7
MODEL & ANALYSIS Process Overview STEP 2 STEP 1 STEP 3 STEP 4 Training Set GPR VS. LR Database Data Set Data Pre-Processing Prediction Method Short-term KPI Experiment Selection Process • Data cleansing • Pearson • Test & • Statistics for • Gaussian Process correlation Comparison of KPIs Regression coefficients GPR & LR (long/short) • Selection Process • Error Analysis Test Set 8
MODEL Data Pre-processing Data cleansing and filtration • one-year daily production database ， 6 sampling times everyday • 31 sheets everyday ， covering almost all information throughout the whole fab • Delete wrong data Statistics for KPI （ long-term & short-term) • Compute long-term KPI according to the production type • Computer short-term KPI everyday 9
Model Short-term KPI Selection Process Pearson correlation coefficients Work In Process To measure the correlation between KPIS. Being fabricated or waiting for further processing in a queue or a buffer storage Queue Length Move Number of lots queued An important KPI to at a particular station measure the overall during a period of time performance of Fab Equipment Utility higher processing capability of semi-conductor • wafer fab Rate of the operation time of higher utility of equipment • some equipment during a period more completed processing tasks. • of time 10
Model & Analysis Short-term KPI Selection Process Ranking results of Pearson correlation coefficients QL-MOVE Coefficient WIP-MOVE Coefficient 1.2 1 1.2 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 0 -0.2 1 2 3 4 5 6 7 8 9 10 11 12 The impact put by a single variable on Move when WIP is higher, is not greater than when WIP is low. • The coefficient for EQP UTI and Move is smaller than that for WIP and Move and that for QL and Move , which clearly indicates that WIP and QL have greater impact on the key short-term performance Move . 11
Model Short-term KPI Selection Process Selection Rule 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 • Too much equipment would be used in each month • Varing considerably 18 Equipment 12
Model Training Set WIP UTI CT QL Lot-in-time Product ① Move TH Product ② WIP Product ③ ODR QL Lot-in-time Move 13
Model Prediction Method Gaussian Regression Process • Gaussian distribution over functions, a collection of random variables • Function-space view: • Every point in some input space is associated with normally distributed random variable • Every finite collection of those random variables has a multivariate normal distribution Mean Function: 𝑛 𝑦 = 𝐹 𝑔 𝑦 𝝂 𝚻 For notational simplicity: 𝒬( 𝑛 𝑦 , 𝑙(𝑦, 𝑦 ′ ) 𝑛 𝑦 = 0 Covariance function: − 𝑛(𝑦 ′ ))] 𝑙 𝑦, 𝑦 ′ = 𝐹[(𝑔 𝑦 − 𝑛 𝑦 )(𝑔 𝑦 ′ Squared exponential (SE): = 𝐟𝐲𝐪(− 𝟐 𝟑 𝒚 − 𝑦 ′ ) 𝑙 𝑦, 𝑦 ′ 14
Model Prediction Method Gaussian Regression Process * A consistency requirement: if the GP e.g. specifies ( 𝒛 𝟐 , 𝒛 𝟑 ) ~ N ( 𝝂 , 𝚻 ), then it must also specify property: 𝒛 𝟐 ~ N ( 𝝂 𝟐 , 𝚻 𝟐𝟐 ) , where 𝚻 𝟐𝟐 is the relevant submatrix of 𝚻 . • In predicting problems, when given training data we want to make predictions f* at test points X* . • Joint distribution of f and f* ： • Calculating f* can be simplified to calculating the posterior based on the known observations Predicted value • • 15
OUTLINE • MOTIVATION & FORMULATION I • MODEL & ANALYSIS II • CONCLUSION & DISCUSSION III 16
Experiment Error Rate Comparison Data Analysis • Error Rate Comparison Form for CT • Error Rate Comparison Form for ODR Product GPR LR Product GPR LR B0 9.55% 28.03% B0 3.14% 10.96% B9 10.01% 26.38% B9 2.04% 5.01% 6L1 14.04% 17.57% 6L1 4.30% 20.51% 6M1 14.79% 25.71% 6M1 2.54% 12.96% B1 9.59% 13.50% B1 1.08% 2.79% 6B1 4.25% 24.56% 6B1 12.33% 30.30% 6M0 4.35% 10.69% 6M0 12.98% 25.26% 6M9 2.07% 7.02% 6M9 4.73% 24.87% RESULTS: Error rate improves for every kind of product, compared to linear regression • Accuracy of ODR is much higher than that of CT • Huge fluctuations of cycle time. • 17 Other factors not included •
Experiment Error Rate Comparison Chart Analysis 35.00% 30.00% 30.00% 25.00% 25.00% 20.00% 20.00% 15.00% 15.00% 10.00% 10.00% 5.00% 5.00% 0.00% 0.00% B0 B9 6L1 6M1 B1 6B1 6M0 6M9 B0 B9 6L1 6M1 B1 6B1 6M0 6M9 GPR LR GPR LR CT error rate comparison histogram ODR error rate comparison histogram RESULT Improvement of error rates on CT dataset are less than that on ODR dataset, compared to • multi-linear regression 18
Experiment Radar comparison map Cycle Time CT_6B1 CT_6M0 CT_6M1 CT_6M9 60 60 80 60 30 40 40 40 20 20 20 20 10 0 0 0 0 CT_6L1 CT_B0 CT_B1 CT_B9 10 60 40 40 8 30 30 40 6 20 20 4 20 10 10 2 0 0 0 0 *Blue denotes predicted value. Orange denotes actual value. 19
Experiment Radar comparison map On-time Delivery Rate ODR_6B1 ODR_6M0 ODR_6M1 ODR_6M9 1 1 1.5 1.02 1 0.8 0.95 1 0.98 0.6 0.9 0.96 0.4 0.85 0.5 0.94 0.2 0.8 0.92 0 0.75 0 0.9 ODR_6L1 ODR_B0 ODR_B1 ODR_B9 1 1.05 1 1 0.8 1 0.99 0.95 0.95 0.6 0.98 0.4 0.9 0.9 0.97 0.2 0.85 0 0.8 0.85 0.96 *Blue denotes predicted value. Orange denotes actual value. 20
CONCLUSION & DISCUSSION Conclusion Prediction Model for Long-term Performance Indicators of Semiconductor Wafer Fabrication Facility Short-term KPI Selection Process Gaussian Process Regression Discussion Incorporate More quantifiable factors working condition /equipment utility /resource allocation /staff allocation Incorporate other relations between a few factors Little’s Law... 21
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