Monte Carlo Simulation with
Minitab® Statistical Software
The Monte Carlo method is often used in
Design for Six Sigma (DFSS) to analyze
the sensitivity of a prototype system,
and to predict yields and/or Cp and Cpk
values.
A product design should ideally have a
small degree of sensitivity to process
variation so its performance remains well
within specification limits.
If you use Minitab® Statistical Software,
then
you
already
have
all
the
functionalities needed to run these types
of studies.
THE MONTE CARLO APPROACH
The Monte Carlo method is a probabilistic
technique based on generating a large
number of random samples.
Simulations are particularly useful in the
design phases of product development
because they unravel the uncertainty or
variability of a complex system.
An example from the semiconductor
industry
This example shows how Monte Carlo
methods may be implemented in a
manufacturing process environment, to
study the sensitivity of a process to
variations.
In the semiconductor industry, Chemical
Mechanical Polishing (CMP) processes
are used to produce wafers that are as
flat as possible to maintain high yield
rates.
A design of experiments is carried out to
increase Removal Rates, leading to
shorter cycle times. Six factors of the
wafer
manufacturing
process
are
investigated: Down Force, Back Force,
Oscillations, Type of PAD, Carrier
Velocity, and Table Velocity.
The results from the Minitab DOE
analysis show that, of the six factors,
Down Force (DF), Carrier (CV) and Table
(TV) Velocities, along with the Down
Force*
Carrier
Velocity
interaction
(DF*CV), have a significant impact on
the Removal Rate (RR).
The optimization tool in Minitab is used
to identify the optimal settings to
maximize the removal rate (shorter cycle
times): Carrier Velocity needs to be kept
low, whereas the levels of Down Force
and Table Velocity have to be increased.
Optimal solutions are not enough,
though. The robustness of the process
window to manufacturing variations also
needs to be taken into account.
A sensitivity analysis based on a Monte
Carlo simulation should be considered to
estimate yields and Cpk values under
standard operating conditions.
The Monte Carlo method may be divided
into several steps:
STEP 1
First, using the results of the DOE,
identify the process inputs that have a
statistically significant effect:
Removal Rate = 253 + 3.49 DF –
4.98 CV + 1.58 TV + 0.033 DF*CV
In order to conduct the Monte Carlo
simulation, we need to identify the
associated parametric distributions of the
inputs.
The objective is to generate a sample for
each one of these variables from a
distribution already identified.
We expect the Down Force to follow a
Normal distribution with a mean of 250
and a standard deviation of 10.
We expect the Carrier Velocity to follow
a Normal distribution with a mean of
14.5 and a standard deviation of 3.
Finally, suppose that the Table Velocity
follows a Weibull distribution with a
shape of 4 and a scale of 11.
STEP 2
Random samples are then generated for
the inputs, according to their underlying
distributions. Random sampling may be
performed using Minitab’s random data
functionality.
Go to Calc > Random Data, and select
the desired distribution.
Then specify the number of random
variables you want to generate (100 000
in this example) and the supposed
parameters for the distribution to store
the results in columns in a Minitab
worksheet:
STEP 3
We then need to calculate numeric
values for the output from the simulated
inputs now stored in columns of the
Minitab worksheet.
Use Minitab’s calculator Calc > Calculator
to introduce the transfer function, which
is based on the final model from the DOE
analysis:
Removal Rate = 253 + 3.49 DF –
4.98 CV + 1.58 TV + .033 DF*CV*VC
Diagram 1. Monte Carlo Simulation
It is now possible to analyze the
distribution of the simulated output
variable.
A variety of statistical and graphical tools
are available within Minitab to help you
study the distribution of the output
according to your objectives.
For example, you can perform a process
capability analysis to assess the
expected behavior of the output (Stat >
Quality Tools > Capability Analysis).
This capability analysis provides a better
understanding of how much variability in
the output can be expected in normal
operating conditions.
Diagram 2. Analysis of the simulated
output: a Capability Analysis
Whenever the output remains too
sensitive, and no satisfactory solution
can be found to limit the effects of input
variations, process engineers need to
focus their attention on controlling the
process inputs that are directly linked to
the output.
Transfer Function :
Y = 253 + 3.49 DF – 4.98 CV
+ 1.58 TV + 0.033 DF*CV
Output ?
Variation Propagation ?
Random Sampling
DF Normal
Mean : 250
Standard
deviation : 10
Random
Sampling
TV
Weibull
Shape : 4 Scale : 11
CV Normal
Mean : 14.5
Standard deviation : 3
MONTE CARLO SIMULATION
CONCLUSION
Minitab’s ability to quickly generate a
very large number of simulated values,
along with its integrated statistical and
graphic capabilities, make it a very
powerful tool for Monte Carlo simulation
methods.
For
another
Technical
article
on
performing Monte Carlo simulations in
Minitab, visit:
http://www.minitab.com/Published-
Articles/Doing-Monte-Carlo-Simulation-
in-Minitab-Statistical-Software/
Bruno Scibilia
Minitab Sarl Technical Training Specialist
Visit
www.minitab.com.
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