Monté Carlo
Simulation
MGS 3100 – Chapter 9
Simulation Defined
A computer-based model used to run
experiments on a real system.
 Typically
done on a computer.
 Determines reactions to different operating
rules or change in structure.
 Can be used in conjunction with traditional
statistical and management science
techniques (such as waiting line problems,
when the basic assumptions do not hold, or
where problems involve multiple phases).
Differences Between
Optimization and Simulation
Optimization models
 Yield decision variables as outputs
 Promise the best (optimal) solution to the
model
Simulation models
 Require the decision variables as inputs
 Give only a satisfactory answer
Types of Simulation Models

Continuous
 Based on mathematical equations.
 Used for simulating continuous values
for all points in
time.
 Example: The amount of time a person spends in a
queue.

Discrete
 Used for simulating specific values or specific points.
 Example: Number of people in a waiting line (queue).
Simulation Methodology:
Estimate probabilities of future events
 Assign random number ranges to
percentages (probabilities)
 Obtain random numbers
 Use random numbers to “simulate”
events.

Data Collection and Random
Number Interval Example
Suppose you timed 20 athletes running the 100-yard dash
and tallied the information into the four time intervals below.
You then count the tallies and make a frequency distribution.
Then convert the frequencies into percentages.
Finally, use the percentages to develop the random number intervals.
Seconds Tallies
0-5.99
6-6.99
7-7.99
8 or more
Frequency
4
10
4
2
%
20
50
20
10
RN Intervals
01-20
21-70
71-90
91-100
Sources of Event Probabilities and
Random Numbers
Event Probabilities


From historical data (assuming the future will be like
the past)
From expert opinion (if the future will be unlike the
past, or no data is available)
Random Numbers



From probability distributions that ‘fit’ the historical
data or can be assumed (use Excel functions)
From manual random number tables
From your instructor (for homework and tests, so we
all get the same answer!)
Probability Distributions
A probability distribution defines the behavior of
a variable by defining its limits, central tendency
and nature
Mean
 Standard Deviation
 Upper and Lower Limits
 Continuous or Discrete

Examples are:
Normal Distribution (continuous)
 Binomial (discrete)
 Poisson (discrete)
 Uniform (continuous or discrete)
 Custom (create your own!)

Normal Distribution

Conditions:
Uncertain variable is symmetric about the mean
 Uncertain variable is more likely to be in vicinity of the
mean than far away


Use when:
Distribution of x is normal (for any sample size)
 Distribution of x is not normal, but the distribution of
sample means (x-bar) will be normally distributed with
samples of size 30 or more (Central Limit Theorem)


Excel function: NORMSDIST() – returns a random
number from the cumulative standard normal
distribution with a mean of zero and a standard deviation
of one [e.g., NORMSDIST(1) = .84]
Uniform Distribution


All values between minimum and maximum
occur with equal likelihood
Conditions
Minimum Value is Fixed
 Maximum Value is Fixed
 All values occur with equal likelihood


Excel function: RAND() – returns a uniformly
distributed random number in the range (0,1)
Note on Random Numbers in Excel
Spreadsheets


Once entered in a spreadsheet, a random
number function remains “live.” A new random
number is created whenever the spreadsheet is
re-calculated. To re-calculate the spreadsheet,
use the F9 key. Note, almost any change in the
spreadsheet causes the spreadsheet to be
recalculated!
If you do not want the random number to
change, you can freeze it by selecting: tools,
options, calculations, and checking “manual.”
Evaluating Results

Conclusions depend on the degree to
which the model reflects the real system

The only true test of a simulation is how
well the real system performs after the
results of the study have been
implemented.
Simulation Applications

Machine Breakdown problems

Queuing problems

Inventory problems

Many other applications
Many Computer Games
Are Simulations!
SimCity, SimFarm, SimIsle, SimCoaster, and
others in this family of games have elaborate
Monte Carlo models underlying the game
exterior. Microsoft has recently released Train
Simulator, for which there are numerous
additional scenarios available on the Internet.
Strategy games such as Civilization and
Railroad Tycoon are also based on simulation
modeling. Most of these games contain
editors, in which the user can create new
scenarios, new terrain, and even control the
likelihoods of events.
Advantages of Simulation





Simulation often leads to a better understanding
of the real system.
Years of experience in the real system can be
compressed into seconds or minutes.
Simulation does not disrupt ongoing activities of
the real system.
Simulation is far more general than
mathematical models.
Simulation can be used as a game for training
experience (safety!).
Simulation Advantages (cont’d)




Simulation can be used when data is hard to
come by.
Simulation can provide a more realistic
replication of a system than mathematical
analysis.
Simulation can be used to analyze transient
conditions, whereas mathematical techniques
usually cannot.
Simulation considers variation and can
calculate confidence intervals of model
results.
Simulation Advantages (cont’d)




Simulation can model a system with multiple
phases
Simulation can model a system when it is
already in a steady-state (i.e., can initialize
the system with the beginning queue,
beginning inventory, etc.!).
Simulation can also test a “range” of inputs to
perform what-if/sensitivity analysis.
Many standard simulation software packages
are available commercially (and Excel works
fine too!).
Disadvantages of Simulation






There is no guarantee that the model will, in fact,
provide good answers.
There is no way to prove reliability.
Simulation may be less accurate than
mathematical analysis because it is randomly
based.
Building a simulation model can take a great deal
of time (but if the payoff is great, then it is worth it!).
A significant amount of computer time may be
needed to run complex models (old concern - no
longer an issue!).
The technique of simulation still lacks a
standardized approach.