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Stochastic Modeling & Simulation Spring 2015 Course Introduction

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Welcome What is this course about? One part spreadsheet modeling We’ll be using Excel every day. Bring your computer or we can just use the lab Mac vs PC – we’ll need PC but I use a PC emulator (vmFusion) on my Mac. Two parts stochastic simulation of static models Random sampling, or “Monte Carlo,” methods. Two parts stochastic simulation of dynamic models Discrete event simulation. We’ll stick with Excel but other software packages (such as Arena and several others) are more common in practice. More focused on simulations of stochastic models, than stochastic models per se. 2

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About me… I think some of you (most?) have had courses with me before… But I still want to know more about you / fill me in on how things have been. Stop by this week and chat with me for 5 min or so. 3

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Course Details Full disclosure: this course is very much in flux. 1. I wont lie to you, it’ll be rough. But it’ll also be a learning experience for all of us. Simulation methods are very cool and VERY useful. 2. The textbook sucks. Sorry. I’m not a fan. (long story) I’ll be providing handouts and notes as the semester progresses. Syllabus details. 1. The course will be very application based. The diversity of problems these methods can used to solved is impressive. We’ll just focus on some of the business applications, but they’re much more general. For instance, Monte Carlo is used often in Statistics and Numerical Integration. 4

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Course Details: Grades Grades: what you stress over but shouldn’t 1. Homework ~30% 2. Exams (x2) ~ 40% These will almost surely be take home exams (individually based) where you a fixed period of time to work on it. 3. Final project ~ 30% I’m open to either group or individual projects Probably group, but there are pros and cons to both I’ll provide a handout later, but topics can include anything CMUQ or EC based (or broader if you can convince me). Find a system that you want to model to try to improve. You’ll build a simulation model of it and present your results. 5

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Why Excel? Why are we using Excel rather than a more powerful statistical or computation engine? The wisdom of Willie Sutton Who is Willie Sutton? “…because that’s where the money is.” There are lots of great statistical packages (I’m a fan of R) But there are many benefits of being comfortable with Excel IMO those benefits outweigh the pain of using Excel for something it wasn’t really designed to do. Excel is actually very flexible and powerful with use of VBA If you’re unfamiliar with Excel, I would highly suggest reviewing videos via Lynda.com: http://www.cmu.edu/lynda/

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Stochastic Simulation methods AKA “Monte Carlo” methods Yes, it is named after the Monte Carlo casino(s) in Monaco… It’s not just about gambling; it’s about a computational method to estimate a distribution of a quantity of interest. Some people reserve the phrase “Monte Carlo” for a specific method and some use it more generally. Basically Stochastic Simulation methods use random sampling (or random number generation) to estimate a much harder problem. Generally they follow the following sequence: 1. Define the domain of the problem 2. Randomly sample from a specified probability distribution 3. Plug in that sample and perform some computation 4. Repeat many many times and aggregate the results. 7

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Monte Carlo example: π Here’s a simple way to use Monte Carlo methods to estimate π 1. First call that the area of a circle is πr 2 2. And the area of the square is (2r) 2 Then the ratio of 1 to 2 is just π / 4 So how do we estimate that ratio? 8

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Monte Carlo example: π We sample many points from inside the square and see how many are inside the circle. This gives us an estimate of π. You’ll be asked to do this on your first homework. 9

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Motivating Problem Suppose you need to order 2016 calendars to sell next year Each one you order will cost you $7 Demand is uncertain but suppose that you have historical data and you average around 200 when you price at $10 But it varies and actually looks somewhat symmetric and Normal-shaped with a standard deviation of 15 Let’s suppose you price at $10 How many do you order? What’s your expected profit? 10

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Motivating Problem Common method: Assume the expected value of demand is the actual demand, i.e., 200. Given you have a positive margin ($3), you order 200. Given this order amount, the expected profit is $600 WRONG. Actual expected profit, ordering 200 units, is closer to $540ish. How? Simulation… If there are numbers in your spreadsheet that are not known with certainty, putting in estimated average values can lead to terrible results! 11

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Modeling Why model? Many times we’re trying to make better forecasts (profits next quarter, budget or project risk, etc.) or improve a system (inventory management, airport security queuing, etc.) We do that my simplifying the problem and modeling it. ‘A model is a lie that helps us understand the truth’ -- unknown. ‘All models are wrong but some are useful.’ – George Box ‘Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise.’ – John Tukey 12

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Seven Modeling Steps: Step One Define the Problem What is the objective? Minimize, maximize, increase, decrease… Is there more than one problem? Who identified the problem? Is that classification person-specific? Is the objective coherent? Often not clear Note: Common homework dialogue: Student: “I have a question about the homework.” TA: “What’s your question?” Student: “I don’t get it.” What decisions/actions/choices are possible? Why is that the set of possible actions? How is it limited? 13

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Seven Modeling Steps: Step Two Observe the System and Collect Data What do you know about the problem/system? What is the baseline? Has anyone even measured performance over time? Has the problem/system changed over time? Do you actually know how good or bad it is now? What data are important? How reliable is the data? 14

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Seven Modeling Steps: Step Three Formulate a Model What type of model should you create? How will the model be used? Who will use the model? Does the it need to be integrated with other models? What kind of output do you want the model to produce? 15

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Seven Modeling Steps: Step Four Verify the Model and Use it for Prediction Does the model’s output make sense? How would you know if the model’s output is correct? Is there historical data to use for testing? Can the model “back-cast”? Cross validation? 16

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Seven Modeling Steps: Step Five Selection of an Alternative Exact or Approximate solution Some problems are sufficiently complicated to solve that only “good enough” solutions can be found Heuristic methods are often used to find “good” solutions quickly to very complex problems Dealing with multiple solutions Is have multiple solutions a good or a bad thing? Does the solution achieve the objective? How are you measuring success? If not, what would have to change about the problem in order to achieve it? 17

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Seven Modeling Steps: Step Six Presentation of the Results “The big problem with management science models is that managers practically never use them” -- John Little, 1970 Resistance to quantitative models: ``I go with my gut” I don’t understand it I don’t trust it It doesn’t sufficiently address the problem (it’s too abstract) I don’t know how it works (the “black box”) “Your objective is not my objective – you solved the wrong problem” Anticipating questions: what if… 18

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Seven Modeling Steps: Step Seven Implementation If a model is not implemented, it is useless. Is it easy to use? Does it require significant readjustment by the client? Does your solution suggest an implementation plan? Modeler: “After months of study, we have determined that you should do X” Client: “OK. How do I get from where we are now to X?” Modeler: “Uh…that wasn’t part of the study.” Client: “Go away.” 19

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A Final Word on Modeling 20

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