Simulation

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Simulation

• Discrete Variables

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What is it?

• A mathematical model
• Probabilistic
• Uses the entire range of possible values of a
  variable in the model

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Why Simulate?

• Safety flight simulator
• Cost easier to simulate adding a new runway and
  find out effects than to implement in reality and
  then find out
• Time Boeing uses simulated manufacturing before
  the real thing, with tremendous savings in time
  and money can discover parts that do not fit
  and fix them before actual production

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How does it work?

• Simulation requires you to know
• What variable is to be simulated
• The distribution of the variable values it can
  take on and the probabilities of those values
  occurring.
• Step 1 Generate a variable containing uniformly
  distributed random variables between 0 and 1 (the
  rand() function in Excel).
• Step 2 Create a rule to map the random numbers
  to values of the variable desired in the right
  proportion, and apply the rule.

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Example coin toss

• Variable to be simulated is Outcome of a coin
  toss. It takes on values Heads and Tails,
  each with 0.5 probability.
• Generate 100 random numbers (100 tosses of coin).
  
• Make a rule like if random number gt 0.5, then
  Heads, else Tails. This will create the right
  distribution of outcomes.

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Example 2 Machine Failures

• Simulate machine failures based on this
  historical data

Number of Failures per month Frequency ( of months this occurred)
0 1 2 3 36 20 3 1
Total 60
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Simulating Machine Failures, contd.
Create the following cumulative probability table.
Number of Failures per month Frequency ( of months this occurred) Probability Cumulative Probability
0 1 2 3 36 20 3 1 0.600 0.333 0.050 0.016 0.600 0.933 0.983 1.000
Total 60 1.00
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Simulating Machine Failures, contd.

• Now map the random numbers between 0 and 1 using
  the cumulative prob. Column as the cutoffs.
• Random numbers between 0 and 0.6 represent 0
  failures, between 0.6 and 0.933 represent 1
  failure, and so on.

0 failures
1 failure
2
3 failures
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Solution Random Number Mapping
The random numbers are now mapped to number of
failures as follows.
Random Number of Failures
0.345 0.008 0.985 0.878 0 0 3 1