1. Lunchtime Webinar Series:
Monte Carlo Simulation
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2. Agenda
• What is Monte Carlo Simulation?
• An example of Monte Carlo Simulation
• A Case Study
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3. Definition
Monte Carlo Simulations are computational routines to
simulate physical or mathematical performance using repeated
random sampling.
What does it mean???
Let us look at simple example…
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4. Example
We are going to build a tower of 5 square blocks.
We need to produce blocks of the following sizes: 5x5 cm, 4x4
cm, 3x3 cm, 2x2 cm and 1x1 cm. E
D
We need to make the tower like this:
C
We produce blocks and we verify that on
average dimensions are as required. B
What is the size of the tower?
A
Answer: 15 cm, but……
On average!
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5. Example
We observe variation in dimensions of blocks we produce.
We study the nature of variation and we find out that the
actual dimensions follow normal distribution with standard
deviation of 0.1 cm for each block.
How likely is it that the size of the tower will be between 14.8
cm and 15.2 cm?
This problem can be solved using analytical skills but the
simpler way to answer is to apply Monte Carlo.
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6. Example
We are going to randomly generate a number for the size of
block A from normal distribution with average of 5 and
standard deviation of 0.1.
We do same for blocks B to E.
We add the randomly generated numbers and calculate size of
tower.
Then we randomly select the next set of sizes for blocks A-E
and calculate size of tower.
And we do it over and over again thousands of times.
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7. Example
In Minitab we can do it like this:
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8. Example
The result will look like this:
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9. Example
Now we can calculate the probability of having a tower size
between 14.8 cm and 15.2 cm. We can do it like this:
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10. Example
The result will look like this:
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11. Example - Summary
Although on average size our tower is 15 cm there is a 37.76%
probability that it will be outside of the specification (14.8 cm –
15.2 cm)
If you did this exercise on your own computer, the results could
be slightly different (due to random sampling), although if we
keep increasing sample size results on different computers will
become closer and closer (due to Law of Large Numbers)
An understanding of the nature of variation of each process
step is the key to applying Monte Carlo.
Now let’s practice…..
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12. Case Study
The process consists of 5 consecutive process steps.
We collected data on the time it takes to complete each
process step in minutes (Cycle Time). You can find the results in
the Excel data file that was distributed to you before Webinar.
Question 1: What is the average processing time?
Question 2: How likely is it that we will complete the process in
under 180 minutes? (this is the requirement from customer)
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13. Case Study – Question 1
We need to calculate the average in each data set and add it
together.
One of the way to do it (in Minitab) is the following:
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14. Question 1 - Solution
The answer for Question 1 is the following:
Average time of the process is:
30.144 (Average of Process Step 1) +
31.100 (Average of Process Step 2) +
30.476 (Average of Process Step 3) +
30.268 (Average of Process Step 4) +
30.102 (Average of Process Step 5) =
152.09 minutes
This was actually an easy question, we can answer it even with
just Excel.
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15. Case Study – Question 2
The second question is much more difficult to answer.
In order to answer this question we need to understand the
nature of variation in each process step.
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16. Case Study – Question 2
We see that data from Process Step 1, Process Step 3 and
Process Step 4 are normally distributed, so for each of the
process steps we can generate random data from normal
distribution with respective averages and standard deviations.
Unfortunately Process Step 2 and Process Step 5 are NOT
normally distributed, so we can NOT generate random data
from normal distribution.
What we have to do, is to understand from which distribution,
both data sets are coming from.
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17. Case Study – Question 2
We can do it in the following way:
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18. Case Study – Question 2
After we randomly generate data into all 5 columns and make a
simple formula in sixth column, the result may look like this:
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19. Question 2 - Solution
If we perform capability analysis for the normal data on our
sixth column with an Upper Specification Limit of 180 minutes,
the result will be the following:
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20. Monte Carlo Simulation - Summary
 Monte Carlo is used to simulate performance using repeated
random sampling.
 We need to understand function y = f(x1, x2…).
 We need to understand the nature of variation in each x and
what are the underlying distributions the data is coming
from.
 We usually sample thousands of data points for each x using
appropriate distributions in order to produce thousands of
data points for y.
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21. Learn More
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More information: http://www.bmgi.com/training
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22. Questions & Answers
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