Monté Carlo Simulation

MGS 3100 – Chapter 9

Simulation DefinedA 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.

Seconds 0-5.996-6.997-7.998 or more

Tallies Frequency41042

You then count the tallies and make a frequency distribution.

%20502010

Then convert the frequencies into percentages.

Finally, use the percentages to develop the random number intervals.

RN Intervals01-2021-7071-9091-100

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 DistributionsA 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.