Transcript Simulation – Discrete

Simulation
Discrete Variables
What is it?
A mathematical model
 Probabilistic
 Uses the entire range of possible values
of a variable in the model

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

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.
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.

Example 2: Machine Failures

Simulate machine
failures based on
this historical data
Number of
Frequency
Failures per (# of
month
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
month
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.

Now map the random numbers between
0 and 1 using the cumulative prob.
Column as the cutoffs.
0 failures
0

1 failure
0.60
0.93
2
3 failures
0.98
Random 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