Infineon GOR AG Optimization under Uncertainty Physikalisches Institut

Infineon GOR AG "Optimization under Uncertainty" Physikalisches Institut, Bad Honnef Bonn IFX_0705_AIM/PD Page 1 October 2005 Robust Nominal Plans for Dispatching in Semico Manufacturing n n Causes for stochasticity in semiconductor manufacturing processes Calculation of robust nominal plans Dispatching strategy using nominal plan Outlook Copyright © Infineon Technologies 2005. All rights reserved.

Wafer fabrication logistic model n Physical World: A Front End Manufacturing Facility n Mathematical Model: - Open queueing network with external arrivals - Multiple products and associated routes - Job class changes along routes - Service times dependent on job class and machine; deterministic and random component - Single service and batch service stations - Processing activities require a single machine resource - Routing is in general of 'Pull' type n Bonn IFX_0705_AIM/PD Page 2 Purpose of mathematical model: Find optimal parameters for routing policy Copyright © Infineon Technologies 2005. All rights reserved.

Model and goals n Discrete event simulation model - application of a continuous review policy to realize optimal routing - performance analysis for system characteristics which cannot be captured exactly by mathematical analysis, such as the influence of overtaking in the network on cycle time distributions n Bonn IFX_0705_AIM/PD Page 3 Minimize cycle times and treat customers fairly with respect to holding costs Copyright © Infineon Technologies 2005. All rights reserved.

Breaking down the goals n Perform solid capacity check - for medium and long term (typically stationary case) - for short term (transient case or stationary case) Bonn IFX_0705_AIM/PD Page 4 n Evaluate whether target cycle times per product (under given holding costs) can be achieved n Balance cycle time through robust and efficient Routing policies n Robust routing means – Remove avoidable idleness in a non-scheduled environment under common loading conditions (uptime utilization < 100%) – React smoothly to machine breakdowns varying from long interruptions to short minor stops Copyright © Infineon Technologies 2005. All rights reserved.

Randomness brought upon by basic production data n Service times depend on age of media engaged (e. g. etching rates, duration of photolithgraphic step) n Randomly distributed machine breakdowns n Lot release depends on network state and short term management decisions n Certain process steps are prown to rework n Complex logistics imposed by - Batch service processes - Setup requirements - Machine internal buffer limitations - Time bound sequences of process steps Bonn IFX_0705_AIM/PD Page 5 Copyright © Infineon Technologies 2005. All rights reserved.

Stochasticity imposed by engineering and test Reservoir Preparation of test wafers Cleaning NCC Random variables: 1. Success probability of test 2. Recycling successful Deposition Nitrid 790/400 nm Maximum of N wafers In progress Test with prepared wafer Recycling Bonn IFX_0705_AIM/PD Page 6 Copyright © Infineon Technologies 2005. All rights reserved.

Stochastic influences of SCM Bonn IFX_0705_AIM/PD Page 7 n There is no common understanding about a mathematical functional description of the importance of due date delivery versus throughput maximization n Yield, goodness of chips and customer demand are not exactly predictable and cause short-term changes in mix and volume of lot release Copyright © Infineon Technologies 2005. All rights reserved.

Dispatching for loops n Some resource pools are more involved in the BEOL, e. g. Metallization, Polishing, Protection FEOL Bonn IFX_0705_AIM/PD Page 8 BEOL n WIP-State at FEOL at time n less than target value FEOL production for feedback loop increases n WIP-State at FEOL at time n larger than target value FEOL production for BEOL increases Copyright © Infineon Technologies 2005. All rights reserved.

System dynamics Priority = Relative earliness or lateness / remaining cycle time N, FEOL + + RCT, FEOL - Prio of lots for FEOL + cum. lateness, FEOL + Prio of lots for BEOL Balancing Feedback Loop (+)(-)(+) 1= (+) (-) 1 = (-) 1 Bonn IFX_0705_AIM/PD Page 9 Copyright © Infineon Technologies 2005. All rights reserved.

A principle insight n Balancing feedback loop has chaotic effect on individual priorities and state development( similar to quadratic iterator in logistic equation; Peitgen/Jürgens/Saupe 1992, Scholz-Reiter, Freitag, Middelberg, Industrie Management 20, 2004) n At microscopic level of consideration a minor characteristics such as computational accuracy (number of relevant digits) can have a significant impact on a production scenario Stochasticity is inherent to cyclic production even if all demand service processes were deterministic Bonn IFX_0705_AIM/PD Page 10 Copyright © Infineon Technologies 2005. All rights reserved.

Decomposition method n Build fab graph with - Machines = Vertices - Edge between any pair of machines with a common process qualification Bonn IFX_0705_AIM/PD Page 11 n Closed Machine Sets CMS = Connected Components of fab graph n Build service time matrices (job class i, machine j) and arrival rate vectors for each CMS n Aggregate job classes to Routing Job Class Sets where job classes belonging to one particular set have pairwise linear dependant row vectors in service time matrix Copyright © Infineon Technologies 2005. All rights reserved.

Closed machine sets (CMS) Example RTP Oxid and BPSG: Dedication J 1 J 2 J 3 J 4 J 5 J 6 J 7 J 8 J 9 J 10 J 11 J 12 J 13 J 14 M 1 0 8 0 0 7 0 0 0 8 0 6 0 0 0 M 2 0 8 7 7 7 0 0 0 8 9 6 9 0 0 M 3 0 8 7 7 7 13 0 0 8 9 6 9 0 0 M 4 20 0 0 13 0 9 0 0 0 12 M 5 20 0 0 0 9 0 0 0 12 M 6 0 0 0 0 9 0 0 0 12 M 7 0 0 0 13 9 0 0 0 9 0 M 8 0 0 0 9 0 Graph Representation 2 3 1 6 4 8 7, 13 5 7 6 Edge between vertice i and vertice j: load can be balanced between machine i and j, also transitive Bonn IFX_0705_AIM/PD Page 12 Copyright © Infineon Technologies 2005. All rights reserved.

CMS picture in the whole fab CMS 1 CMS 6 CMS 4 Route 1 Route 2 CMS 7 Route 3 CMS 5 CMS 3 Bonn IFX_0705_AIM/PD Page 13 Copyright © Infineon Technologies 2005. All rights reserved.

Limits for and assumptions on interarrival processes n Queueing system M/G/∞ features M M property (N. M. Mirasol, OR, 1962) – Resource pool super server with a large amount of discretionary traffic approaches M M property n . /M/1 . . . . /M/1 has Poisson departure at the n-th station, for large n (T. Mountford, B. Prabharkar, Ann. Appl. Prob. 1995) n Approximation of point processes by renewal processes - Interarrival processes: Markov and Markov Modulated Poisson Processes (MMPP) for environments with Batch service or extensive setup - Two-moment matching for service times Bonn IFX_0705_AIM/PD Page 14 Copyright © Infineon Technologies 2005. All rights reserved.

Interarrival Processes Typical departure stochastic process Departure Process at furnace Departure process At implant Bonn IFX_0705_AIM/PD Page 15 Copyright © Infineon Technologies 2005. All rights reserved.

Service Time Processes n Small Disturbances or Handling: Deterministic + Triangular n Setup systems: Deterministic + Log. Normal: LN(2. 214, 0. 837) Bonn IFX_0705_AIM/PD Page 16 Copyright © Infineon Technologies 2005. All rights reserved.

Dedicated and discretionary traffic Dedicated Traffic Discretionary Traffic Preferred mode for stochastic systems Mixture n Principal Job-to-Machine Qualifications are given n How to guide discretionary traffic such that a CMS acts like a heavy traffic resource pool? Bonn IFX_0705_AIM/PD Page 17 Copyright © Infineon Technologies 2005. All rights reserved.

Resource pooling n Original CMS is split into CMS 1 and CMS 2 when job class Js is taken away from it n CMS fulfills resource pooling condition if discretionary traffic contributed by Js has higher absolute value in work load than difference of dedicated work loads for CMS 1 and CMS 2 CMS 1 Minimum Cut Js CMS 2 Bonn IFX_0705_AIM/PD Page 18 Copyright © Infineon Technologies 2005. All rights reserved.

Nominal plan for a CMS: Routing matrix P For a given CMS let a Vector of arrivals per job class, B Service time matrix, F Indicator function matrix for job class machine qualification, P Branching probability matrix Basic equation system for utilization vector I Index set for job classes I = {1, . . . , m} J Index set for machines J = {1, . . . , n} Bonn IFX_0705_AIM/PD Page 19 Copyright © Infineon Technologies 2005. All rights reserved.

Quadratic optimization problem n Minimize u Average Utilization + v Variance of Utilization inside each resource pool; Operational costs (chemicals, test wafers, energy, labour); Holding costs, penalties for delays; Distribute the load of each job class broadly upon the machines qualified to process it n Constraints: - Every job has to be processed: Sum p(i, . ) = 1, for all i - Utilization of machine j < 1, for all j Bonn IFX_0705_AIM/PD Page 20 n Choose v >> u n Determination of nominal plan is based on rates of stochastic processes only Copyright © Infineon Technologies 2005. All rights reserved.

Resource pool optimization B= a= Contours Function to be minimized 1 0. 8 Surface 0. 6 2 Minimum 1 1 0. 75 0 0 0. 5 0. 25 0. 4 0. 25 0. 75 10 0 0 0. 2 0. 4 0. 6 0. 8 1 Optimum: put 47% of job class 1 and 59% of job class 2 on machine 2 Bonn IFX_0705_AIM/PD Page 21 Copyright © Infineon Technologies 2005. All rights reserved.

Linear versus quadratic optimization n A Heavy Traffic Example - Service time matrix - Arrival vector left side limit - Load balancing possible with either LP and QP, such that U 1 for all machines Bonn IFX_0705_AIM/PD Page 22 Copyright © Infineon Technologies 2005. All rights reserved.

Linear problem Bonn IFX_0705_AIM/PD Page 23 n Minimize the maximum utilization in resource pool n Variables to be determined: Branching probability matrix Copyright © Infineon Technologies 2005. All rights reserved.

Linear problem Bonn IFX_0705_AIM/PD Page 24 n Multiple solutions: LP Solver gives a solution which is an extreme point of the constraint set n Number of nonzero branching probabilities is at most m + n -1 (Harrison, López, QS 33 (1999); from above example: 3 + 3 - 1 = 5 n Resulting branching probability matrix: n M 3 is now single equipment for job class J 2! Copyright © Infineon Technologies 2005. All rights reserved.

Linear solutions and robustness Any point on red line minimizes maximum utilization Most robust solution Robust solution LP solutions LP optimization provides a solution with least robustness Bonn IFX_0705_AIM/PD Page 25 Copyright © Infineon Technologies 2005. All rights reserved.

Quadratic programming solution n Bonn IFX_0705_AIM/PD Page 26 Distribute Job Classes amongst machines such - General load level is minimal - Variance of machine utilization vector is minimal - Distribution of individual job class on different machines is as homogeneous as possible n Resulting branching probability matrix: n QP solution does not contain any single equipments - Dispatching is robust against random interference such as machine breakdowns, machine specific processing problems - Significantly improved normalized waiting time Copyright © Infineon Technologies 2005. All rights reserved.

Resource pooling effect Resource Pooling effect is approximately described by the factor Σi λi number of basic activities for class i / Σ λi : = k normalized waiting time: 1/k U/(1 -U) Using QP factor k is approximately 3. 25 times higher than with LP solution when applied in typical semiconductor front end resource pools Bonn IFX_0705_AIM/PD Page 27 Copyright © Infineon Technologies 2005. All rights reserved.

RPO: Summary n Calculation of optimal branching probabilities which allow the highest possible degree of resource pooling for each CMS n Avoids aggressive job to machine allocations where a flexible machine takes too many jobs from a flexible job class n Each job class is guaranteed an appropriate portion of the whole capacity of resource pool Benefit: maximum utilization in a CMS is up to 30% less as compared to the case where popular distribution mechanisms are used such as 'least flexible job / least flexible machine' or a Speed Ranking Rule Bonn IFX_0705_AIM/PD Page 28 Copyright © Infineon Technologies 2005. All rights reserved.

A continuous review policy How to approach ideal system behaviour? Implementation of Multiple job multiple machine polling with a cycle stealing mechanism for starvation avoidance Bonn IFX_0705_AIM/PD Page 29 Copyright © Infineon Technologies 2005. All rights reserved.

Polling principles n Polling is used to put optimum branching probabilities into practice n Perspective of job classes: Each job class has an associated polling cycle, which is the ordered set of machines to be polled by the job class; the frequency of appearance of a particular machine in a particular polling cycle is in accordance with the (job class, machine) - branching probability n Perspective of machines: analog n Example: Machine 1: Machine 2: (7, 1, 3, 1, 1, 7, 1, 3, 1) (3, 7) machine 1 polls at the very first time job class 7, then 1, 3, again 1 and so forth; a pointer memorizes the last poll; the last element of a polling cycle is connected to its first element Bonn IFX_0705_AIM/PD Page 30 Copyright © Infineon Technologies 2005. All rights reserved.

Polling principles Bonn IFX_0705_AIM/PD Page 31 n The Polling Table for job classes is used when a dispatch event is triggered by an arriving job, because one or more machines are idling (prevalent under low utilization) n The Polling Table for machines is used when a dispatch event is triggered by a machine finishing service (prevalent under high utilization) n Literature Boxma, Levy, Westrate, QS 9 (1991), Perf. Eval. 18 (1993) Copyright © Infineon Technologies 2005. All rights reserved.

Special issues on CMS with batch servers Example Configuration 509 -601 BVT 103 BVT 104 BVT 105 thr 11 thr 22 thr 32 509 -619 Bonn IFX_0705_AIM/PD Page 32 Copyright © Infineon Technologies 2005. All rights reserved.

Special issues on batch servers n Since dispatching is non-anticipative a threshold policy is used for deciding when to build a batch for a given job class on a given machine of some CMS n Calculation of utilization according to u = a b/K under full batch policy u = a b/(jps K) under threshold policy with K the maximum batch size, jps the average number of jobs per machine start (server efficiency) Bonn IFX_0705_AIM/PD Page 33 Copyright © Infineon Technologies 2005. All rights reserved.

Queueing modelling of batch servers with many job classes n M/D/1 with batch service and threshold server starting strategy n Infinitely many job classes n Equal mix of single arrivals and batch arrivals n First arrival (respectively departure) of each service cycle is single n Different Batch-ID for each job class Service discipline Round Robin: Closed formula for jps Bonn IFX_0705_AIM/PD Page 34 Copyright © Infineon Technologies 2005. All rights reserved.

Server efficiencies and threshold tables Bonn IFX_0705_AIM/PD Page 35 n Choose thresholds with respect to target cycle times for job classes n Calculate jps for each job machine combination using the queuing Model M/D [r, K ] /1 -S with one job class n Calculate jps for each job machine combination using a new result for queuing Model M/D [r, K ] /1 with infinitely many job classes n Interpolate results of the two analyzes using entropy of job for a given machine n Trade-Off between holding costs and operative costs n Use an iterative scheme to determine cost optimum Copyright © Infineon Technologies 2005. All rights reserved.

Example: One machine, four job classes, K = 8 Bonn IFX_0705_AIM/PD Page 36 n Probabilities of appearances of Job Classes: (0. 5, 0. 25, 0. 125) n Overall utilization under full batch policy: 0. 25 n Entropy of Job Class 1. 75 n Maximum entropy with four Job Classes given: 2 (each job class appears with equal probability) n Optimum thresholds: (2, 1, 1, 1) Copyright © Infineon Technologies 2005. All rights reserved.

Server efficiency: Gain over lower bound r/K Estimated and simulated Threshold = 1 Threshold = 2 0. 4 0. 5 0. 3 0. 4 0. 3 0. 2 0. 1 r/K Class 1 Class 2 Class 3 Class 4 0. 1 0 0 Threshold = 3 Threshold = 4 0. 7 0. 8 0. 6 0. 5 Class 1 0. 4 0. 3 0. 2 Class 1 Class 2 Class 3 Class 4 0. 2 0. 1 0 Bonn IFX_0705_AIM/PD Page 37 0 Copyright © Infineon Technologies 2005. All rights reserved.

Server efficiency: Gain over lower bound r/K Estimated and simulated Threshold = 5 Threshold = 6 0. 8 Class 1 0. 6 Class 1 Class 2 Class 3 Class 4 0. 2 0 0 Threshold = 7 0. 8 Bonn IFX_0705_AIM/PD Page 38 Class 3 Class 4 Threshold = 8 Class 1 Class 2 Class 3 0. 6 0. 4 Class 1 Class 2 Class 4 Class 2 Class 3 Class 4 0. 8 0. 6 0. 4 0. 2 0 0 Copyright © Infineon Technologies 2005. All rights reserved.

Optimal Threshold Values (2, 1, 1, 1) Bonn IFX_0705_AIM/PD Page 39 Copyright © Infineon Technologies 2005. All rights reserved.

Outlook n Robustness of optimal nominal plan is achieved by selecting a solution for routing problem which is robust against higher-order data, like variance of interarrival times, service times, interdependances of stochastic processes etc. n QP Programming on decomposition model allows routing optimization for a real fab with hundreds of machines, fifty and more different product routes, and up to 400 steps per route in reasonable computational time n Review policy is effective, efficient and robust n Methods from Quadratic Optimization and Queueing Theory have been combined in the treatment of the important batch service environments n Future enhancements: - Processing activities which require more than one resource Bonn IFX_0705_AIM/PD Page 40 - Parameterization of non-anticipative set up strategies Copyright © Infineon Technologies 2005. All rights reserved.

Never stop thinking. Bonn IFX_0705_AIM/PD Page 41 Copyright © Infineon Technologies 2005. All rights reserved.