Computer Simulation

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# Computer Simulation - PowerPoint PPT Presentation

Computer Simulation. The Essence of Computer Simulation. A stochastic system is a system that evolves over time according to one or more probability distributions.

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Presentation Transcript

### Computer Simulation

The Essence of Computer Simulation
• A stochastic system is a system that evolves over time according to one or more probability distributions.
• Computer simulation imitates the operation of such a system by using the corresponding probability distributions to randomly generate the various events that occur in the system.
• Rather than literally operating a physical system, the computer just records the occurrences of the simulated events and the resulting performance of the system.
• Computer simulation is typically used when the stochastic system ivolved is too complex to be analyzed satisfactorily by analytical models.

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Example 1: A Coin-Flipping Game
• Rules of the game:
• Each play of the game involves repeatedly flipping an unbiased coin until the difference between the number of heads and tails tossed is three.
• To play the game, you are required to pay \$1 for each flip of the coin. You are not allowed to quit during the play of a game.
• You receive \$8 at the end of each play of the game.
• Examples:

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Computer Simulation of Coin-Flipping Game
• A computer cannot flip coins. Instead it generates a sequence of random numbers.
• A number is a random number between 0 and 1 if it has been generated in such a way that every possible number within the interval has an equal chance of occurring.
• An easy way to generate random numbers is to use the RAND() function in Excel.
• To simulate the flip of a coin, let half the possible random numbers correspond to heads and the other half to tails.
• 0.0000 to 0.4999 correspond to heads.
• 0.5000 to 0.9999 correspond to tails.

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Freddie the Newsboy
• Freddie runs a newsstand in a prominent downtown location of a major city.
• Freddie sells a variety of newspapers and magazines. The most expensive of the newspapers is the Financial Journal.
• Cost data for the Financial Journal:
• Freddie pays \$1.50 per copy delivered.
• Freddie charges \$2.50 per copy.
• Freddie’s refund is \$0.50 per unsold copy.
• Sales data for the Financial Journal:
• Freddie sells anywhere between 40 and 70 copies a day.
• The frequency of the numbers between 40 and 70 are roughly equal.

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Application of Crystal Ball
• Four steps must be taken to use Crystal Ball on a spreadsheet model:
• Define the random input cells.
• Define the output cells to forecast.
• Set the run preferences.
• Run the simulation.

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Step 1: Define the Random Input Cells
• A random input cell is an input cell that has a random value.
• An assumed probability distribution must be entered into the cell rather than a single number.
• Crystal Ball refers to each such random input cell as an assumption cell.
• Procedure to define an assumption cell:
• Select the cell by clicking on it.
• If the cell does not already contain a value, enter any number into the cell.
• Click on the Define Assumption button.
• Select a probability distribution from the Distribution Gallery.
• Click OK to bring up the dialogue box for the selected distribution.
• Use the dialogue box to enter parameters for the distribution (preferably referring to cells on the spreadsheet that contain these parameters).
• Click on OK.

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Step 2: Define the Output Cells to Forecast
• Crystal Ball refers to the output of a computer simulation as a forecast, since it is forecasting the underlying probability distribution when it is in operation.
• Each output cell that is being used to forecast a measure of performance is referred to as a forecast cell.
• Procedure for defining a forecast cell:
• Select the cell.
• Click on the Define Forecast button on the Crystal Ball tab or toolbar, which brings up the Define Forecast dialogue box.
• This dialogue box can be used to define a name and (optionally) units for the forecast cell.
• Click on OK.

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Step 3: Set the Run Preferences
• Setting run preferences refers to such things as choosing the number of trials to run and deciding on other options regarding how to perform the simulation.
• This step begins by clicking on the Run Preferences button on the Crystal Ball tab or toolbar.
• The Run Preferences dialogue box has five tabs to set various types of options.
• The Trials tab allows you to specify the maximum number of trials to run for the computer simulation.

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Step #4: Run the Simulation
• To begin running the simulation, click on the Start Simulation button.
• Once started, a forecast window displays the results of the computer simulation as it runs.
• The following can be obtained by choosing the corresponding option under the View menu in the forecast window display:
• Frequency chart
• Statistics table
• Percentiles table
• Cumulative chart
• Reverse cumulative chart

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Choosing the Right Distribution
• A continuous distribution is used if any values are possible, including both integer and fractional numbers, over the entire range of possible values.
• A discrete distribution is used if only certain specific values (e.g., only some integer values) are possible.
• However, if the only possible values are integer numbers over a relatively broad range, a continuous distribution may be used as an approximation by rounding any fractional value to the nearest integer.

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A Popular Central-Tendency Distribution: Normal
• Some value most likely (the mean)
• Values close to mean more likely
• Symmetric (as likely above as below mean)
• Extreme values possible, but rare

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The Uniform Distribution
• Fixed minimum and maximum value
• All values equally likely

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The Discrete Uniform Distribution
• Fixed minimum and maximum value
• All integer values equally likely

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A Distribution for Random Events: Exponential
• Widely used to describe time between random events (e.g., time between arrivals)
• Events are independent
• Rate = average number of events per unit time (e.g., arrivals per hour)

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Yes-No Distribution
• Describes whether an event occurs or not
• Two possible outcomes: 1 (Yes) or 0 (No)

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Distribution for Number of Times an Event Occurs: Binomial
• Describes number of times an event occurs in a fixed number of trials (e.g., number of heads in 10 flips of a coin)
• For each trial, only two outcomes are possible
• Trials independent
• Probability remains the same for each trial

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