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Simulation Operations Prof. Juran

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**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

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**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 Operations -- Prof. Juran

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**Operations -- Prof. Juran**

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**Operations -- Prof. Juran**

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**Operations -- Prof. Juran**

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**Operations -- Prof. Juran**

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Origins of Monte Carlo Operations -- Prof. Juran

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**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? Operations -- Prof. Juran

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**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 Operations -- Prof. Juran

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**Operations -- Prof. Juran**

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**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 Operations -- Prof. Juran

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**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 Operations -- Prof. Juran

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**Some Random Results Sample means from 15 trials:**

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**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? Operations -- Prof. Juran

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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 Operations -- Prof. Juran

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**Operations -- Prof. Juran**

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**Running an @Risk simulation: Define input distribution(s) **

Define output(s) Simulation settings Start simulation Operations -- Prof. Juran

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**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. Operations -- Prof. Juran

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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”. Operations -- Prof. Juran

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**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. Operations -- Prof. Juran

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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

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**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. Operations -- Prof. Juran

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3. Simulation Settings Click on the “Settings” button. Specify the number of iterations. Operations -- Prof. Juran

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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. Operations -- Prof. Juran

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**4. Run the Simulation @Risk displays a graph for each output cell.**

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Analyzing the Results Excel Reports: download and save results in Excel Browse Results: interactive graphs Summary: detailed output for each “special” cell Operations -- Prof. Juran

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Analyzing the Results Operations -- Prof. Juran

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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 Operations -- Prof. Juran

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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. Operations -- Prof. Juran

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**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. Operations -- Prof. Juran

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**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%. Operations -- Prof. Juran

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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. Operations -- Prof. Juran

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**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). Operations -- Prof. Juran

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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

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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). Operations -- Prof. Juran

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**Select cell B9 and click on the Define Distribution button. **

Note that we have used cell references for the mean and standard deviation. Operations -- Prof. Juran

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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. Operations -- Prof. Juran

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**Operations -- Prof. Juran**

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**Now click on the Simulation Settings button, and set the number of iterations.**

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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: Operations -- Prof. Juran

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**The @Risk Output Results report (a new worksheet created automatically):**

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**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 ). Operations -- Prof. Juran

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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). Operations -- Prof. Juran

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**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 Operations -- Prof. Juran

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**Random Number Generator**

Needs a “seed” to get started Each random number becomes the seed for its successor Operations -- Prof. Juran

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**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

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