Copyright © Infineon Technologies 2005. All rights reserved. Bonn IFX_0705_AIM/PD Page 1 October...

Copyright © Infineon Technologies 2005. All rights reserved.

Bonn IFX_0705_AIM/PDPage 1

October 2005

Infi

neo

nGOR AG "Optimization under Uncertainty"Physikalisches Institut, Bad Honnef

Robust Nominal Plans for Dispatching in SemiconductorManufacturing

Causes for stochasticity in semiconductor manufacturing processes Calculation of robust nominal plans Dispatching strategy using nominal plan Outlook



Wafer fabrication logistic model

Physical World: A Front End Manufacturing Facility

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 machineresource

- Routing is in general of 'Pull' type

Purpose of mathematical model: Find optimal parameters for routing policy



Model and goals

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

Minimize cycle times and treat

customers fairly with respect to

holding costs



Breaking down the goals

Perform solid capacity check

- for medium and long term (typically stationary case)

- for short term (transient case or stationary case)

Evaluate whether target cycle times per product

(under given holding costs) can be achieved

Balance cycle time through robust and efficient Routing

policies

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



Randomness brought upon by basic production data

Service times depend on age of media engaged(e.g. etching rates, duration of photolithgraphic step)

Randomly distributed machine breakdowns

Lot release depends on network state and short term management decisions

Certain process steps are prown to rework

Complex logistics imposed by

- Batch service processes

- Setup requirements

- Machine internal buffer limitations

- Time bound sequences of process steps



Stochasticity imposed by engineering and test

Preparation of test wafers

Cleaning NCC

DepositionNitrid 790/400 nm

Test with prepared wafer

Recycling

Reservoir

Maximum of N wafersIn progress

Random variables:1. Success probability

of test2. Recycling successful



Stochastic influences of SCM

There is no common understanding about a mathematical functional description of the importance of due date delivery versus throughput maximization

Yield, goodness of chips and customer demand are not exactly predictable and cause short-term changes in mix and volume of lot release



Some resource pools are more involved in the BEOL,e.g. Metallization, Polishing, Protection

WIP-State at FEOL at time n less than target value FEOL production for feedback loop increases

WIP-State at FEOL at time n larger than target value FEOL production for BEOL increases

Dispatching for loops

FEOL BEOL



System dynamics

Priority = Relative earliness or lateness / remaining cycle time

N, FEOL

RCT, FEOL cum. lateness, FEOL

Prio of lots for FEOL Prio of lots for BEOL

+ +

+ -

- +

Balancing Feedback Loop (+)(-)(+) 1= (+) (+) (-) 1 = (-) 1



Stochasticity is inherent to cyclic productioneven if all demand and service processes were deterministic

Stochasticity is inherent to cyclic productioneven if all demand and service processes were deterministic

A principle insight

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)

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



Decomposition method

Build fab graph with

- Machines = Vertices

- Edge between any pair of machines with a common process qualification

Closed Machine Sets CMS = Connected Components of fab graph

Build service time matrices (job class i, machine j) and arrival rate vectors for each CMS

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



Edge between vertice i and vertice j: load can be balanced between machine i and j, also transitive

Edge between vertice i and vertice j: load can be balanced between machine i and j, also transitive

Example RTP Oxid and BPSG: Dedication

Example RTP Oxid and BPSG: Dedication

Closed machine sets (CMS)

Graph RepresentationGraph Representation

M1 M2 M3 M4 M5 M6 M7 M8J1 0 0 0 20 20 0 0 0J2 8 8 8 0 0 0 0 0J3 0 7 7 0 0 0 0 0J4 0 7 7 0 0 0 0 0J5 7 7 7 0 0 0 0 0J6 0 0 13 13 0 0 13 0J7 0 0 0 0 0 0 9 9J8 0 0 0 9 9 9 0 0J9 8 8 8 0 0 0 0 0J10 0 9 9 0 0 0 0 0J11 6 6 6 0 0 0 0 0J12 0 9 9 0 0 0 0 0J13 0 0 0 0 0 0 9 9J14 0 0 0 12 12 12 0 0

6

7, 13

1

2

3

4

5

6

7

8



CMS picture in the whole fab

Route 1

Route 2

Route 3

CMS 1

CMS 2

CMS 3

CMS 4

CMS 5

CMS 6

CMS 7



Limits for and assumptions on interarrival processes

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

./M/1 ./M/1 . . . ./M/1 has Poisson departure at then-th station, for large n (T. Mountford, B. Prabharkar, Ann. Appl. Prob. 1995)

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



Interarrival Processes

Typical departure stochastic process

Departure Process at furnace

0 2 4 6 8 10 12 14 16

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7

Departure processAt implant



Service Time Processes

Small Disturbances or Handling: Deterministic + Triangular

Setup systems: Deterministic + LogNormal

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100RZmin

100

200

300

400

500

Häufigkeit

LogNormal: LN(2.214,0.837)



Dedicated and discretionary traffic

M1 M2J1 1 1J2 1 1

M1 M2J1 1 0J2 0 1

Dedicated Traffic Discretionary Traffic

Principal Job-to-Machine Qualifications are given

How to guide discretionary traffic such that a CMS acts like a heavy traffic resource pool?

Preferred mode for stochasticsystems

M1 M2J1 1 1J2 0 1

Mixture



Resource pooling

Original CMS is split into CMS 1 and CMS 2 when job class

Js is taken away from it

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

CMS 2

Minimum Cut Js



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}

uj i1m aipijfijbij



Quadratic optimization problem

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

Constraints:

- Every job has to be processed: Sum p(i,.) = 1, for all i

- Utilization of machine j < 1, for all j

Choose v >> u

Determination of nominal plan is based on rates of

stochastic processes only



Optimum:put 47% of job class 1 and 59% of job class 2 on machine 2

Optimum:put 47% of job class 1 and 59% of job class 2 on machine 2

Resource pool optimization

1.16 0.6 00 1 1

Function to be minimized

Surface

Contours

a = B =1.71.2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Minimum1.71.2

00.25

0.5

1 0

0.25

0.5

0.75

1

0

1

2

0.75

1



Linear versus quadratic optimization

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

M1 M2 M3J1 1 2 0J2 0 2 1J3 1 0 1

J1 56

J2 56

J3 56



Linear problem

Minimize the maximum utilization in resource pool

Variables to be determined: Branching probability matrix

M1 M2 M3J1 1 x x 0J2 0 1 y yJ3 z 0 1 z



Linear problem

Multiple solutions: LP Solver gives a solution which is an extreme point of the constraint set

Number of nonzero branching probabilities is at most m + n -1 (Harrison, López, QS 33 (1999);from above example: 3 + 3 - 1 = 5

Resulting branching probability matrix:

M3 is now single equipment for job class J2!

M1 M2 M3J1 2

535

0

J2 0 0 1

J3 45

0 15



Linear solutions and robustness

Robust solution

LP solutions

0

0.2

0.4

0.6

x

0

0.2

0.4

0.6y

0.2

0.4

0.6

0.8

z

0

0.2

0.4

0.6

x

0

0.2

0.4

0.6y

Most robust solution

LP solutions

LP optimization provides a solution with least robustness

Any point on red line minimizes maximum utilization



Quadratic programming solution

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

Resulting branching probability matrix:

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

M1 M2 M3J1 7

10310

0

J2 0 310

710

J3 12

0 12



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 applied in typical semiconductor front end resource pools

Using QP factor k is approximately 3.25 times higher than with LP solution when applied applied in typical semiconductor front end resource pools



RPO: Summary

Calculation of optimal branching probabilities which allow the highest possible degree of resource pooling for each CMS

Avoids aggressive job to machine allocations where a flexible machine takes too many jobs from a flexible job class

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



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



Polling principles

Polling is used to put optimum branching probabilities into practice

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

Perspective of machines: analog

Example:Machine 1: (7,1,3,1,1,7,1,3,1)

Machine 2: (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



Polling principles

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)

The Polling Table for machines is used when a dispatch event is

triggered by a machine finishing service (prevalent under high

utilization)

Literature

Boxma, Levy, Westrate, QS 9 (1991), Perf. Eval. 18 (1993)



Example ConfigurationExample Configuration

Special issues on CMS with batch servers

BVT 103

BVT104

BVT105

thr11

thr21

thr22

thr32

509-601

509-619



Special issues on batch servers

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

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)



Queueing modelling of batch servers with many job classes

M/D/1 with batch service and threshold server starting strategy

Infinitely many job classes Equal mix of single arrivals and batch

arrivals First arrival (respectively departure) of each

service cycle is single Different Batch-ID for each job class

Service discipline Round Robin:Closed formula for jps



Server efficiencies and threshold tables

Choose thresholds with respect to target cycle times for job classes

Calculate jps for each job machine combination using the queuing Model M/D /1-S with one job class

Calculate jps for each job machine combination using a new result for queuing Model M/D /1 with infinitely many job classes

Interpolate results of the two analyzes using entropy of job for a given machine

Trade-Off between holding costs and operative costs

Use an iterative scheme to determine cost optimum

[r,K ]

[r,K ]



Example: One machine, four job classes, K = 8

Probabilities of appearances of Job Classes:(0.5, 0.25, 0.125, 0.125)

Overall utilization under full batch policy: 0.25

Entropy of Job Class 1.75

Maximum entropy with four Job Classes given: 2(each job class appears with equal probability)

Optimum thresholds: (2,1,1,1)



r/K

Server efficiency: Gain over lower bound r/KEstimated and simulated

Class1 Class2 Class3 Class4

0

0.1

0.2

0.3

0.4


0

0.1

0.2

0.3

0.4

0.5


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


0

0.2

0.4

0.6

0.8

Threshold = 2Threshold = 1




Server efficiency: Gain over lower bound r/KEstimated and simulated


0

0.2

0.4

0.6

0.8


0

0.2

0.4

0.6

0.8


0

0.2

0.4

0.6

0.8


0

0.2

0.4

0.6

0.8





Optimal Threshold Values (2,1,1,1)


0

0.1

0.2

0.3

0.4

0.5

Thresholds 2,1,1,1



Outlook

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.

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

Review policy is effective, efficient and robust

Methods from Quadratic Optimization and Queueing Theory have been

combined in the treatment of the important batch service environments

Future enhancements:

- Processing activities which require more than one resource

- Parameterization of non-anticipative set up strategies



Never stop thinking.

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