1. Simulation
Discrete Variables

2. What is it? A mathematical model Probabilistic
Uses the entire range of possible values of a variable in the model

3. 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

4. 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.

5. 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 > 0.5, then “Heads”, else “Tails”. This will create the right distribution of outcomes.

6. Example 2: Machine Failures
Simulate machine failures based on this historical data
Number of Failures per month
Frequency
(# of months this occurred) 1 2 3 36 20
Total 60

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

8. 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 represent 1 failure, and so on.
0.60
0.93
0.98
0 failures
1 failure 2
3 failures

9. 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 3 1