Simulation Operations -- Prof. Juran. 2 Overview Monte Carlo Simulation –Basic concepts and...

Simulation

Operations -- Prof. Juran

Operations -- Prof. Juran 2

OverviewMonte Carlo Simulation

– Basic concepts and history

@Risk– Probability Distributions

• Uniform, Normal, Gamma– Distribution and Output cells– Simulation Settings– Output Analysis

Examples– Coin Toss, TSB Account


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


Origins of Monte Carlo

Stanislaw M. Ulam (1909 - 1984)

Nicholas Metropolis (1915-1999)


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 $1.00The coin is not fair; it lands on “heads” 35% of the timeWhat is the expected value of this game?


Simulation “By Hand”

Set up a spreadsheet modelAdd 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


1234

A B C D E

Random # 0.2002Outcome HeadProfit $3.00

=RAND()=IF(B2<0.35,"Head","Tail")=IF(B2<0.35,3,-1)


What Does =RAND() Do?

Uniform random number between 0 and 1Never below 0; never above 1All values between 0 and 1 are equally likely

P(X<0.35) = 0.35

0.35 0.65


What Does =IF Do?

Evaluates a logical expression (true or false)Gives one result for true and a different result for falseIn our “coin” model, RAND and IF work together to generate heads and tails (and profits and losses) from a specific probability distribution

1234

A B C D E

Random # 0.7438Outcome TailProfit -$1.00

=RAND()=IF(B2<0.35,"Head","Tail")=IF(B2<0.35,3,-1)


Some Random Results

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1011121314151617

A B C D E FRandom # Outcome Profit

0.4619 Tail -$1.000.4118 Tail -$1.000.5815 Tail -$1.000.9792 Tail -$1.000.2852 Head $3.000.9064 Tail -$1.000.9855 Tail -$1.000.9988 Tail -$1.000.2206 Head $3.000.0986 Head $3.000.9696 Tail -$1.000.8026 Tail -$1.000.8189 Tail -$1.000.7137 Tail -$1.000.9258 Tail -$1.00

-$0.20=AVERAGE(C2:C16)

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1011121314151617

A B C D E FRandom # Outcome Profit

0.1979 Head $3.000.9185 Tail -$1.000.4688 Tail -$1.000.6670 Tail -$1.000.0902 Head $3.000.3757 Tail -$1.000.1492 Head $3.000.4518 Tail -$1.000.8503 Tail -$1.000.1392 Head $3.000.1924 Head $3.000.0179 Head $3.000.4799 Tail -$1.000.5064 Tail -$1.000.3051 Head $3.00

$0.87=AVERAGE(C2:C16)

Sample means from 15 trials:


Problems with this Model

Hitting F9 thousands of times is tediousKeeping track of the results (and summary statistics) is even more tediousWhat if we want to simulate something other than a uniform distribution between 0 and 1?


Simulation with @RiskSpecial 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


1234567

A B C D ECOIN.XLS

Random # Outcome Profit0.702 Tail -1

=RAND() =IF(A4<0.35,"Head","Tail")

=IF(A4<0.35,3,-1)


Running an @Risk simulation:1. Define input distribution(s) 2. Define output(s) 3. Simulation settings4. Start simulation


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 the @Risk “Define Distributions” button. Choose the uniform distribution from the list of distributions.


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”.


Note the special @Risk function now in cell A4. You could have entered this function by hand, or by using the @Risk – Model – Define Distribution menu.

1234567

A B C DCOIN.XLS


=RiskUniform(0,1)


2. Define Output Cell

Select cell C4. Then click on the @Risk “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.


Note the special @Risk function now in cell C4. You could have entered this function by hand, or by using the @Risk – Model – Add Output menu.

123456

A B C D E F GCOIN.XLS


=RiskUniform(0,1)

=RiskOutput()+IF(A4<0.35,3,-1)


3. Simulation SettingsClick on the “Settings” button. Specify the number of iterations.


4. Run the SimulationClick on the @Risk “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.


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


Analyzing the Results

Excel Reports: download and save results in Excel

Browse Results: interactive graphs

Summary: detailed output for each “special” cell


Analyzing the Results


Simulation Results

The 10,000-trial @Risk simulation gives sample mean profit of $0.400. 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 0.01908. 𝑠𝑋ത = 𝑠𝑥ξ𝑛

= 1.908ξ10,000

= 0.01908


Simulation ResultsA 95% confidence interval for the true mean profit is approximately:

0.400 1.96(0.01908)

We are 95% confident that the true mean lies somewhere between $0.3626 and $0.4374.

To get a better estimate using simulation, we could increase the number of simulation trials, and continue the simulation run.


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.


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%.


Assume that your medical expenses in a year are normally distributed with mean $2000 and standard deviation $500.

Build an @Risk 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.


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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A B C DTSB Amount (Decision Variable) 3,000.00$

Annual Salary 50,000.00$ Tax Rate 30%After TSB Income 47,000.00$ Taxes Owed 14,100.00$ Net Income Before Medical Expenses 32,900.00$

Total Medical Expenses 2,000.00$ Amount in TSB 3,000.00$ Expenses Not Covered (Must Be Paid Out-Of-Pocket) -$ Money Left Over in TSB (Lost) 1,000.00$

Net Income After Medical Expenses (Objective) 32,900.00$

=B3-B1

=B5*B4

=B5-B6

This will be a random variable.

=B1

=MAX(B9-B10,0)

=MAX(B10-B9,0)

=B7-B11


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.


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).

1617

A BMean 2,000.00$ Standard Deviation 500.00$


Select cell B9 and click on the Define Distribution button.

Note that we have used cell references for the mean and standard deviation.


1234567891011121314151617





Mean 2,000.00$ Standard Deviation 500.00$

=RiskNormal(B16,B17,RiskStatic(2000))


Now we need to tell @Risk 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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1011121314151617





Mean 2,000.00$ Standard Deviation 500.00$

=RiskNormal(B16,B17,RiskStatic(2000))

=RiskOutput()+B7-B11


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


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:1

23456789101112131415161718

A B C D E F G H I J KTSB Amount (Decision Variable) 1,000$ 1,250$ 1,500$ 1,750$ 2,000$ 2,250$ 2,500$ 2,750$ 3,000$ 3,250$

Annual Salary 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ 50,000$ Tax Rate 30% 30% 30% 30% 30% 30% 30% 30% 30% 30%After TSB Income 49,000$ 48,750$ 48,500$ 48,250$ 48,000$ 47,750$ 47,500$ 47,250$ 47,000$ 46,750$ Taxes Owed 14,700$ 14,625$ 14,550$ 14,475$ 14,400$ 14,325$ 14,250$ 14,175$ 14,100$ 14,025$ Net Income Before Medical Expenses 34,300$ 34,125$ 33,950$ 33,775$ 33,600$ 33,425$ 33,250$ 33,075$ 32,900$ 32,725$

Total Medical Expenses 2,000.00$ Amount in TSB 1,000.00$ 1,250.00$ 1,500.00$ 1,750.00$ 2,000.00$ 2,250.00$ 2,500.00$ 2,750.00$ 3,000.00$ 3,250.00$ Expenses Not Covered (Must Be Paid Out-Of-Pocket) 1,000.00$ 750.00$ 500.00$ 250.00$ -$ -$ -$ -$ -$ -$ Money Left Over in TSB (Lost) -$ -$ -$ -$ -$ 250.00$ 500.00$ 750.00$ 1,000.00$ 1,250.00$

Net Income After Medical Expenses (Objective) 33,300.00$ 33,375.00$ 33,450.00$ 33,525.00$ 33,600.00$ 33,425.00$ 33,250.00$ 33,075.00$ 32,900.00$ 32,725.00$

Mean 2,000.00$ Standard Deviation 500.00$ =RiskOutput("1250",A14,2)+C7-C11

=RiskOutput("1500",A14,3)+D7-D11

=RiskOutput("1750",A14,4)+E7-E11


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

@RISK Output ResultsPerformed By: adminDate: Saturday, J anuary 25, 2014 2:54:48 PM

Name Cell Graph Min Mean Max 5% 95% Errors

Range: Net Income After Medical Expenses (Objective)

1000 B14 $ 31,332.38 $ 33,295.75 $ 34,300.00 $ 32,477.41 $ 34,122.42 0

1250 C14 $ 31,407.38 $ 33,360.35 $ 34,125.00 $ 32,552.41 $ 34,125.00 0

1500 D14 $ 31,482.38 $ 33,408.34 $ 33,950.00 $ 32,627.41 $ 33,950.00 0

1750 E14 $ 31,557.38 $ 33,426.10 $ 33,775.00 $ 32,702.41 $ 33,775.00 0

2000 F14 $ 31,632.38 $ 33,400.53 $ 33,600.00 $ 32,777.41 $ 33,600.00 0

2250 G14 $ 31,707.38 $ 33,326.10 $ 33,425.00 $ 32,852.41 $ 33,425.00 0

2500 H14 $ 31,782.38 $ 33,208.34 $ 33,250.00 $ 32,927.41 $ 33,250.00 0

2750 I14 $ 31,857.38 $ 33,060.35 $ 33,075.00 $ 33,002.41 $ 33,075.00 0

3000 J 14 $ 31,932.38 $ 32,895.75 $ 32,900.00 $ 32,900.00 $ 32,900.00 0

3250 K14 $ 32,007.38 $ 32,724.00 $ 32,725.00 $ 32,725.00 $ 32,725.00 0


TSB Simulation Analysis Results

$32,500

$32,600

$32,700

$32,800

$32,900

$33,000

$33,100

$33,200

$33,300

$33,400

$33,500

$1000 $1250 $1500 $1750 $2000 $2250 $2500 $2750 $3000 $3250

Amount Put Into TSB Account

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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 ).


Gamma vs. Normal

$0 $1,000 $2,000 $3,000 $4,000 $5,000

Medical Expense

Pro

ba

bili

ty

Gamma Distribution Normal Distribution


TSB Simulation Analysis Results

$32,500

$32,600

$32,700

$32,800

$32,900

$33,000

$33,100

$33,200

$33,300

$33,400

$33,500

$1000 $1250 $1500 $1750 $2000 $2250 $2500 $2750 $3000 $3250

Amount Put Into TSB Account

Me

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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).


Built into Excel• RAND() function• Tools – Data Analysis – Random Number

GenerationBuilt into all simulation softwareNot really random; correctly called pseudo-random

Random Number Generator


Needs a “seed” to get started

Each random number becomes the seed for its successor

Random Number Generator


SummaryMonte Carlo Simulation

– Basic concepts and history

@Risk– Probability Distributions

• Uniform, Normal, Gamma– Distribution and Output cells– Simulation Settings– Output Analysis

Examples– Coin Toss, TSB Account

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