1. Simulation
Operations Prof. Juran

2. Overview Monte Carlo Simulation @Risk Examples
Basic concepts and history
@Risk
Probability Distributions
Uniform, Normal, Gamma
Distribution and Output cells
Simulation Settings
Output Analysis
Examples
Coin Toss, TSB Account
Operations -- Prof. Juran

3. Monte Carlo Simulation
Using theoretical probability distributions to model real-world situations in which randomness is an important factor.
Differences from other spreadsheet models
No optimal solution
Explicit modeling of random variables in special cells
Many trials, all with different results
Objective function studied using statistical inference
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11. Origins of Monte Carlo
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12. Example: Coin Toss Imagine a game where you flip a coin once.
If you get “heads”, you win $3.00
If you get “tails”, you lose $1.00
The coin is not fair; it lands on “heads” 35% of the time
What is the expected value of this game?
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13. Simulation “By Hand” Set up a spreadsheet model
Add an element of randomness
Excel built-in random number generator
Use F9 key to create repetitive iterations of the random system (“realizations”)
Keep track of the results
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15. What Does =RAND() Do? Uniform random number between 0 and 1
Never below 0; never above 1
All values between 0 and 1 are equally likely
P(X<0.35) = 0.35
0.35
0.65
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16. What Does =IF Do? Evaluates a logical expression (true or false)
Gives one result for true and a different result for false
In our “coin” model, RAND and IF work together to generate heads and tails (and profits and losses) from a specific probability distribution
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17. Some Random Results Sample means from 15 trials:
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18. Problems with this Model
Hitting F9 thousands of times is tedious
Keeping track of the results (and summary statistics) is even more tedious
What if we want to simulate something other than a uniform distribution between 0 and 1?
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19. Simulation
Special cells for random variables (Distributions)
Special cells for objective functions (Outputs)
Simulation Settings
Number of trials
Random number seed
Sampling method
Output Analysis
Studying outputs
Extracting data
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21. Running an @Risk simulation: Define input distribution(s)
Define output(s)
Simulation settings
Start simulation
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22. 1. Define Input Distribution
First make sure there is a number in cell A4 (it cannot be blank or contain a formula). Then move the cursor to cell A4 and click on “Define Distributions” button. Choose the uniform distribution from the list of distributions.
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23. After you select “Uniform”, a graph of the uniform distribution will appear. Set the “Min” of the uniform to 0 and the “Max” to 1. Then press “OK”.
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24. Note the special @Risk function now in cell A4
Note the function now in cell A4. You could have entered this function by hand, or by using – Model – Define Distribution menu.
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25. 2. Define Output Cell
Select cell C4. Then click on “Add Output” button. Give the output variable a name, such as “Profit.” The window should now look as shown below. Press “OK” to return to the spreadsheet.
Operations -- Prof. Juran

26. Note the special @Risk function now in cell C4
Note the function now in cell C4. You could have entered this function by hand, or by using – Model – Add Output menu.
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27. 3. Simulation Settings
Click on the “Settings” button. Specify the number of iterations.
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28. 4. Run the Simulation
Click on “Start Simulation” icon. The “Forecast: profit” window will appear, and the number of trials simulated will show in the bottom left corner of the Excel window.
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29. 4. Run the Simulation @Risk displays a graph for each output cell.
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30. Analyzing the Results
Excel Reports: download and save results in Excel
Browse Results: interactive graphs
Summary: detailed output for each “special” cell
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31. Analyzing the Results
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32. Simulation Results
The simulation gives sample mean profit of $ The number $0.400 is only an estimate of the true mean profit from the coin-flipping game.
The standard error of the mean is
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33. Simulation Results
A 95% confidence interval for the true mean profit is approximately:
0.400  1.96( )
We are 95% confident that the true mean lies somewhere between $ and $
To get a better estimate using simulation, we could increase the number of simulation trials, and continue the simulation run.
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34. Example 2: Tax-Saver Benefit
A TSB (Tax Saver Benefit) plan allows you to put money into an account at the beginning of the calendar year that can be used for medical expenses. This amount is not subject to federal tax — hence the phrase TSB.
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35. Your annual salary is $50,000 and your federal income tax rate is 30%.
As you pay medical expenses during the year, you are reimbursed by the administrator of the TSB until the TSB account is exhausted. From that point on, you must pay your medical expenses out of your own pocket. On the other hand, if you put more money into your TSB than the medical expenses you incur, this extra money is lost to you.
Your annual salary is $50,000 and your federal income tax rate is 30%.
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36. Assume that your medical expenses in a year are normally distributed with mean $2000 and standard deviation $500.
Build model in which the output is the amount of money left to you after paying taxes, putting money in a TSB, and paying any extra medical expenses. Experiment with the amount of money put in the TSB, and identify an amount that is approximately optimal.
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37. First, we set up a spreadsheet to organize all of the information
First, we set up a spreadsheet to organize all of the information. In particular, we want to make sure we’ve identified the decision variable (how much to have taken out of our salary and put into the TSB account — here in cell B1), the output (net income — after tax, and after extra medical expenses not covered by the TSB — which we have here in cell B14), and the random variable (in this case the amount of medical expenses — here in cell B9).
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39. Note (this is important): We will never get a simulation model to tell us directly what is the optimal value of the decision variable (how much to have deducted from our pre-tax pay).
We will try different values (here we have arbitrarily started with $3000 in cell B1) and see how the objective changes.
Through educated trial-and-error, we will eventually come to some conclusion about what is the best amount of money to put into the TSB account.
Operations -- Prof. Juran

40. Now we add the element of randomness by making B9 into a distribution cell. First, enter the mean and standard deviation for the medical expenses random variable (we put them in cells B16 and B17, respectively).
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41. Select cell B9 and click on the Define Distribution button.
Note that we have used cell references for the mean and standard deviation.
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43. Now we need to to keep track of our output cell during all of our simulation runs, so we can see its mean and standard deviation over many trials. Select the net income cell B14 and click on the Add Output button.
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45. Now click on the Simulation Settings button, and set the number of iterations.
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47. Unfortunately, we can’t tell whether $3000 is the optimal amount without trying many other possible amounts. This could entail a long and tedious series of simulation runs, but fortunately it is possible to test many values at once.
We set up numerous columns in the worksheet, so that we can perform simulation experiments on many possible TSB amounts simultaneously:
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48. The @Risk Output Results report (a new worksheet created automatically):
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50. Rework part a, but this time assume a gamma distribution for your annual medical expenses.
Use $0 for the location parameter, $125 for the scale parameter (sometimes symbolized with β), and 16 for the shape parameter (sometimes symbolized with ).
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54. Conclusions
The best amount to put into the TSB is apparently about $1,750 per year.
This result is robust over different distributions of medical costs.
This result is based on sample statistics, not known population parameters.
We have confidence in these sample statistics because of the large sample size (1,000).
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55. Random Number Generator
Built into Excel
RAND() function
Tools – Data Analysis – Random Number Generation
Built into all simulation software
Not really random; correctly called pseudo-random
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56. Random Number Generator
Needs a “seed” to get started
Each random number becomes the seed for its successor
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57. Summary Monte Carlo Simulation @Risk Examples
Basic concepts and history
@Risk
Probability Distributions
Uniform, Normal, Gamma
Distribution and Output cells
Simulation Settings
Output Analysis
Examples
Coin Toss, TSB Account
Operations -- Prof. Juran