GOR AG "Optimization under Uncertainty"
Physikalisches Institut, Bad Honnef
October 2005
Robust Nominal Plans for Dispatching in Semiconductor
Manufacturing
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Bonn IFX_0705_AIM/PD
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Causes for stochasticity in semiconductor manufacturing processes
Calculation of robust nominal plans
Dispatching strategy using nominal plan
Outlook
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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 machine
resource
- Routing is in general of 'Pull' type
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Purpose of mathematical model:
Find optimal parameters for routing policy
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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

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Minimize cycle times and treat
customers fairly with respect to
holding costs
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Breaking down the goals

Perform solid capacity check
- for medium and long term (typically stationary case)
- for short term (transient case or stationary case)
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Evaluate whether target cycle times per product
(under given holding costs) can be achieved
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Balance cycle time through robust and efficient Routing
policies
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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
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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
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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
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Stochastic influences of SCM
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
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
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Dispatching for loops

Some resource pools are more involved in the BEOL,
e.g. Metallization, Polishing, Protection
FEOL
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BEOL

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
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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
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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
Stochasticity is inherent to cyclic production
even if all demand and service processes were deterministic
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Decomposition method

Build fab graph with
- Machines = Vertices
- Edge between any pair of machines with a common process
qualification
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Closed Machine Sets CMS = Connected Components of fab
graph
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Build service time matrices (job class i, machine j) and arrival
rate vectors for each CMS
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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
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Closed machine sets (CMS)
Example RTP Oxid and BPSG:
Dedication
M1
J1 0
J2 8
J3 0
J4 0
J5 7
J6 0
J7 0
J8 0
J9 8
J10 0
J11 6
J12 0
J13 0
J14 0
M2
0
8
7
7
7
0
0
0
8
9
6
9
0
0
M3
0
8
7
7
7
13
0
0
8
9
6
9
0
0
M4
20
0
0
0
0
13
0
9
0
0
0
0
0
12
M5
20
0
0
0
0
0
0
9
0
0
0
0
0
12
M6
0
0
0
0
0
0
0
9
0
0
0
0
0
12
M7
0
0
0
0
0
13
9
0
0
0
0
0
9
0
M8
0
0
0
0
0
0
9
0
0
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
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CMS picture in the whole fab
CMS 1
CMS 6
CMS 4
Route 1
Route 2
CMS 2
CMS 7
Route 3
CMS 5
CMS 3
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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 the
n-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
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Interarrival Processes
Typical departure
stochastic process
Departure Process
at furnace
Departure process
At implant
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Service Time Processes
 Small Disturbances or Handling: Deterministic + Triangular
 Setup systems: Deterministic + LogNormal
LogNormal: LN(2.214,0.837)
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Dedicated and discretionary traffic
Dedicated Traffic
Discretionary Traffic
Preferred mode for stochastic
systems
Mixture
 Principal Job-to-Machine Qualifications are given
 How to guide discretionary traffic such that a CMS acts like a
heavy traffic resource pool?
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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
Minimum Cut
Js
CMS 2
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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}
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Quadratic optimization problem
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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
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Constraints:
- Every job has to be processed: Sum p(i,.) = 1, for all i
- Utilization of machine j < 1, for all j
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Choose v >> u
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Determination of nominal plan is based on rates of
stochastic processes only
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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
0.2
0.25
0.5
0
0.75
10
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
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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  1for all machines
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Linear problem
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
Minimize the maximum utilization in resource pool
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Variables to be determined: Branching probability matrix
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Linear problem
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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
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Resulting branching probability matrix:

M3 is now single equipment for job class J2!
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Linear solutions and robustness
Any point on red line
minimizes maximum
utilization
Most robust solution
Robust solution
LP solutions
LP solutions
LP optimization provides
a solution with
least robustness
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Quadratic programming solution
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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
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Resulting branching probability matrix:
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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
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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
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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
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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
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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
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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:
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
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Polling principles
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
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)
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Special issues on CMS with batch servers
Example Configuration
509-601
BVT 103
thr11
thr21
BVT104
BVT105
thr22
thr32
509-619
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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)
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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
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Server efficiencies and threshold tables
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
Choose thresholds with respect to target cycle times for job
classes

Calculate jps for each job machine combination using the
queuing Model M/D [r,K ] /1-S with one job class

Calculate jps for each job machine combination using a new
result for queuing Model M/D [r,K ] /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
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Example: One machine, four job classes, K = 8
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
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)
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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
0.2
0.1
Class1
Class2
Class3
Class1
Class2
Class3
Class4
Class4
0.1
r/K
0
0
Threshold = 3
Threshold = 4
0.7
0.8
0.6
0.6
0.5
Class1
0.4
Class1
Class2
Class3
Class4
Class2
Class3
Class4
0.4
0.3
0.2
0.2
0.1
0
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0
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Server efficiency: Gain over lower bound r/K
Estimated and simulated
Threshold = 5
Threshold = 6
0.8
0.8
Class1
0.6
Class1
Class2
Class3
Class4
0.4
0.2
0.2
0
0
Threshold = 7
0.8
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Class3
Class4
Threshold = 8
Class1
Class2
Class3
0.6
0.4
Class1
Class2
Class2
Class3
Class4
Class4
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
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Optimal Threshold Values (2,1,1,1)
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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
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- Parameterization of non-anticipative set up strategies
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Never stop thinking.
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