Mont

1
Monté Carlo Simulation

• MGS 3100 Chapter 9

2
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).

3
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

4
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).

5
Simulation Methodology

• Estimate probabilities of future events
• Assign random number ranges to percentages
  (probabilities)
• Obtain random numbers
• Use random numbers to simulate events.

6
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 0-5.99 6-6.99 7-7.99 8 or more
Tallies
Frequency 4 10 4 2
20 50 20 10
RN Intervals 01-20 21-70 71-90 91-100
7
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!)

8
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!)

9
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

10
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)

11
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.

12
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.

13
Simulation Applications

• Machine Breakdown problems
• Queuing problems
• Inventory problems
• Many other applications

14
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.

15
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!).

16
Simulation Advantages (contd)

• 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.

17
Simulation Advantages (contd)

• 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!).

18
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.