Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Simulation Operations -- Prof. Juran Overview Monte 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 Operations -- Prof. Juran 2 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 3 Operations -- Prof. Juran 4 Operations -- Prof. Juran 5 Operations -- Prof. Juran 6 Operations -- Prof. Juran 7 Operations -- Prof. Juran 8 Operations -- Prof. Juran 9 Operations -- Prof. Juran 10 Origins of Monte Carlo Stanislaw M. Ulam (1909 - 1984) Nicholas Metropolis (1915-1999) Operations -- Prof. Juran 11 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 12 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 13 A B 1 2 Random # 0.2002 3 Outcome Head 4 Profit $3.00 Operations -- Prof. Juran C D E =RAND() =IF(B2<0.35,"Head","Tail") =IF(B2<0.35,3,-1) 14 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 0.35 0.65 P(X<0.35) = 0.35 Operations -- Prof. Juran 15 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 A B 1 2 Random # 0.7438 3 Outcome Tail 4 Profit -$1.00 Operations -- Prof. Juran C D E =RAND() =IF(B2<0.35,"Head","Tail") =IF(B2<0.35,3,-1) 16 Some Random Results Sample means from 15 trials: A B C 1 Random # Outcome Profit 2 0.4619 Tail -$1.00 3 0.4118 Tail -$1.00 4 0.5815 Tail -$1.00 5 0.9792 Tail -$1.00 6 0.2852 Head $3.00 7 0.9064 Tail -$1.00 8 0.9855 Tail -$1.00 9 0.9988 Tail -$1.00 10 0.2206 Head $3.00 11 0.0986 Head $3.00 12 0.9696 Tail -$1.00 13 0.8026 Tail -$1.00 14 0.8189 Tail -$1.00 15 0.7137 Tail -$1.00 16 0.9258 Tail -$1.00 17 -$0.20 D E =AVERAGE(C2:C16) Operations -- Prof. Juran F A B C 1 Random # Outcome Profit 2 0.1979 Head $3.00 3 0.9185 Tail -$1.00 4 0.4688 Tail -$1.00 5 0.6670 Tail -$1.00 6 0.0902 Head $3.00 7 0.3757 Tail -$1.00 8 0.1492 Head $3.00 9 0.4518 Tail -$1.00 10 0.8503 Tail -$1.00 11 0.1392 Head $3.00 12 0.1924 Head $3.00 13 0.0179 Head $3.00 14 0.4799 Tail -$1.00 15 0.5064 Tail -$1.00 16 0.3051 Head $3.00 17 $0.87 D E F =AVERAGE(C2:C16) 17 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 18 Simulation with @Risk 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 19 A B 1 COIN.XLS 2 3 Random # Outcome 4 0.702 Tail 5 6 =RAND() 7 Operations -- Prof. Juran C D Profit -1 E =IF(A4<0.35,3,-1) =IF(A4<0.35,"Head","Tail") 20 Running an @Risk simulation: 1. Define input distribution(s) 2. Define output(s) 3. Simulation settings 4. Start simulation Operations -- Prof. Juran 21 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. Operations -- Prof. Juran 22 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 23 A B C 1 COIN.XLS 2 3 Random # Outcome Profit 4 0.500 Tail -1 5 =RiskUniform(0,1) 6 7 D 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. Operations -- Prof. Juran 24 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. Operations -- Prof. Juran 25 A B C 1 COIN.XLS 2 3 Random # Outcome Profit 4 0.500 Tail -1 5 =RiskUniform(0,1) 6 D E F G =RiskOutput()+IF(A4<0.35,3,-1) 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. Operations -- Prof. Juran 26 3. Simulation Settings Click on the “Settings” button. Specify the number of iterations. Operations -- Prof. Juran 27 4. Run the Simulation Click 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. Operations -- Prof. Juran 28 4. Run the Simulation @Risk displays a graph for each output cell. Operations -- Prof. Juran 29 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 30 Analyzing the Results Operations -- Prof. Juran 31 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. 𝑠𝑋 = = 𝑠𝑥 𝑛 1.908 10,000 = 0.01908 Operations -- Prof. Juran 32 Simulation Results A 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. Operations -- Prof. Juran 33 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 34 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 35 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. Operations -- Prof. Juran 36 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 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A TSB Amount (Decision Variable) B $ 3,000.00 C D =B3-B1 Annual Salary Tax Rate After TSB Income Taxes Owed Net Income Before Medical Expenses $ 50,000.00 30% $ 47,000.00 $ 14,100.00 $ 32,900.00 Total Medical Expenses Amount in TSB Expenses Not Covered (Must Be Paid Out-Of-Pocket) Money Left Over in TSB (Lost) $ 2,000.00 $ 3,000.00 $ $ 1,000.00 Net Income After Medical Expenses (Objective) $ 32,900.00 Operations -- Prof. Juran =B5*B4 =B5-B6 This will be a random variable. =B1 =MAX(B9-B10,0) =MAX(B10-B9,0) =B7-B11 38 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 39 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). A 16 Mean 17 Standard Deviation Operations -- Prof. Juran B $ 2,000.00 $ 500.00 40 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 41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A TSB Amount (Decision Variable) $ Annual Salary Tax Rate After TSB Income Taxes Owed Net Income Before Medical Expenses $ 50,000.00 30% $ 47,000.00 $ 14,100.00 $ 32,900.00 Total Medical Expenses Amount in TSB Expenses Not Covered (Must Be Paid Out-Of-Pocket) Money Left Over in TSB (Lost) $ $ $ $ Net Income After Medical Expenses (Objective) $ 32,900.00 Mean Standard Deviation $ $ Operations -- Prof. Juran B 3,000.00 2,000.00 3,000.00 1,000.00 C D =RiskNormal(B16,B17,RiskStatic(2000)) 2,000.00 500.00 42 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. Operations -- Prof. Juran 43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 A TSB Amount (Decision Variable) $ Annual Salary Tax Rate After TSB Income Taxes Owed Net Income Before Medical Expenses $ 50,000.00 30% $ 47,000.00 $ 14,100.00 $ 32,900.00 Total Medical Expenses Amount in TSB Expenses Not Covered (Must Be Paid Out-Of-Pocket) Money Left Over in TSB (Lost) $ $ $ $ Net Income After Medical Expenses (Objective) $ 32,900.00 Mean Standard Deviation $ $ Operations -- Prof. Juran B 3,000.00 2,000.00 3,000.00 1,000.00 C D =RiskNormal(B16,B17,RiskStatic(2000)) =RiskOutput()+B7-B11 2,000.00 500.00 44 Now click on the Simulation Settings button, and set the number of iterations. Operations -- Prof. Juran 45 Operations -- Prof. Juran 46 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 A TSB Amount (Decision Variable) $ B 1,000 $ D 1,500 $ 50,000 30% 47,750 14,325 33,425 $ $ I 2,750 $ J 3,000 $ 3,250.00 $ $ 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 =RiskOutput("1750",A14,4)+E7-E11 =RiskOutput("1500",A14,3)+D7-D11 47 $ K 3,250 $ 3,000.00 $ $ 1,000.00 $ $ $ 50,000 30% 47,000 14,100 32,900 $ $ 2,750.00 $ $ 750.00 $ $ $ 50,000 30% 47,250 14,175 33,075 $ $ 2,500.00 $ $ 500.00 $ $ $ 50,000 30% 47,500 14,250 33,250 $ $ 2,250.00 $ $ 250.00 $ $ $ $ H 2,500 $ 2,000.00 $ $ - $ $ $ 50,000 30% 48,000 14,400 33,600 G 2,250 $ $ 1,750.00 $ 250.00 $ - Operations -- Prof. Juran $ F 2,000 $ 1,500.00 $ 500.00 $ - $ $ $ 50,000 30% 48,250 14,475 33,775 $ $ 1,250.00 $ 750.00 $ - $ 2,000.00 =RiskOutput("1250",A14,2)+C7-C11 $ 500.00 $ E 1,750 $ 2,000.00 $ 1,000.00 $ 1,000.00 $ - $ $ $ 50,000 30% 48,500 14,550 33,950 $ Total Medical Expenses Amount in TSB Expenses Not Covered (Must Be Paid Out-Of-Pocket) Money Left Over in TSB (Lost) $ $ $ 50,000 30% 48,750 14,625 34,125 $ $ Mean Standard Deviation $ C 1,250 Annual Salary Tax Rate After TSB Income Taxes Owed Net Income Before Medical Expenses $ $ $ 50,000 30% 49,000 14,700 34,300 $ $ $ $ 50,000 30% 46,750 14,025 32,725 The @Risk Output Results report (a new worksheet created automatically): @RISK Output Results Performed By: admin Date: Saturday, January 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 J14 $ 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 Operations -- Prof. Juran 48 TSB Simulation Analysis Results $33,500 $33,400 Mean Net Income $33,300 $33,200 $33,100 $33,000 $32,900 $32,800 $32,700 $32,600 $32,500 $1000 $1250 $1500 $1750 $2000 $2250 $2500 $2750 $3000 $3250 Amount Put Into TSB Account Operations -- Prof. Juran 49 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 50 Operations -- Prof. Juran 51 Gamma vs. Normal Normal Distribution Probability Gamma Distribution $0 $1,000 $2,000 $3,000 $4,000 $5,000 Medical Expense Operations -- Prof. Juran 52 TSB Simulation Analysis Results $33,500 $33,400 Mean Net Income $33,300 $33,200 $33,100 $33,000 $32,900 $32,800 $32,700 $32,600 $32,500 $1000 $1250 $1500 $1750 $2000 $2250 $2500 $2750 $3000 $3250 Amount Put Into TSB Account Operations -- Prof. Juran 53 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 54 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 55 Random Number Generator Needs a “seed” to get started Each random number becomes the seed for its successor Operations -- Prof. Juran 56 Summary Monte 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 Operations -- Prof. Juran 57