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) 01230123 36 20 3 1 Total60

7. Simulating Machine Failures, contd. Number of Failures per month Frequency (# of months this occurred) ProbabilityCumulative Probability 01230123 36 20 3 1 0.600 0.333 0.050 0.016 0.600 0.933 0.983 1.000 Total601.00 Create the following cumulative probability table.

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 0.933 represent 1 failure, and so on. 0.600.930.980 0 failures1 failure2 3 failures

9. Solution – Random Number Mapping Random # Number of Failures 0.345 0.008 0.985 0.878 00310031 The random numbers are now mapped to number of failures as follows.