Simulation Discrete Variables What is it l A

Simulation Discrete Variables

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

Why Simulate? l Safety – flight simulator l Cost – easier to simulate adding a new runway and find out effects than to implement in reality and then find out l 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

How does it work? l Simulation requires you to know l l 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). l Step 2: Create a rule to map the random numbers to values of the variable desired in the right proportion, and apply the rule. l

Example – coin toss l Variable to be simulated is “Outcome of a coin toss”. It takes on values “Heads” and “Tails”, each with 0. 5 probability. l Generate 100 random numbers (100 tosses of coin). l Make a rule like – if random number > 0. 5, then “Heads”, else “Tails”. This will create the right distribution of outcomes.

Example 2: Machine Failures l 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

Simulating Machine Failures, contd. Create the following cumulative probability table. Number of Frequency Probability Failures per (# of months this occurred) 0 1 2 3 Total 36 20 3 1 60 0. 600 0. 333 0. 050 0. 016 1. 00 Cumulative Probability 0. 600 0. 933 0. 983 1. 000

Simulating Machine Failures, contd. l Now map the random numbers between 0 and 1 using the cumulative prob. Column as the cutoffs. 0 failures 0 l Random 1 failure 0. 60 0. 93 2 3 failures 0. 98 numbers between 0 and 0. 6 represent 0 failures, between 0. 6 and 0. 933 represent 1 failure, and so on.

Solution – Random Number Mapping The random numbers are now mapped to number of failures as follows. Random Number # of Failures 0. 345 0 0. 008 0 0. 985 3 0. 878 1