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McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4 th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

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18-2 Simulation Chapter Contents 18.1 What is Simulation? 18.2 Monte Carlo Simulation 18.3 Random Number Generation 18.4 Excel Add-Ins 18.5 Dynamic Simulations Chapter 18

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18-3 Chapter Learning Objectives (LOs) LO18-1: List characteristics of situations where simulation is appropriate. LO18-2: Distinguish between stochastic and deterministic variables. LO18-3: Explain how Monte Carlo simulation is used and why it is called static. LO18-4: Explain how to generate random data by using a discrete or continuous CDF. LO18-5: Use Excel to generate random data for several common distributions. LO18-6: Describe functions and features of commercial modeling tools for Excel. LO18-7: Explain the main reasons for using dynamic simulation and queuing models. Chapter 18Simulation

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18-4 18.1 What is Simulation? A simulation is a computer model that attempts to imitate the behavior of a real system or activity.A simulation is a computer model that attempts to imitate the behavior of a real system or activity. Models are simplifications that try to include the essentials while omitting unimportant details.Models are simplifications that try to include the essentials while omitting unimportant details. Simulations helps to quantify relationships among variables that are too complex to analyze mathematically.Simulations helps to quantify relationships among variables that are too complex to analyze mathematically. If the simulations predictions differ from what really happens, refine the model in a systematic way until its predictions are in close enough agreement with reality.If the simulations predictions differ from what really happens, refine the model in a systematic way until its predictions are in close enough agreement with reality. Chapter 18 In general, consider simulation whenIn general, consider simulation when - The system is complex - Uncertainty exists in the variables - Real experiments are impossible or costly - The processes are repetitive - Stakeholders cant agree on policy LO18-1 LO18-1: List characteristics of situations where simulation is LO18-1: List characteristics of situations where simulation is appropriate. appropriate.

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18-5 Deterministic variables are nonrandom and fixed.Deterministic variables are nonrandom and fixed. Stochastic variables are random. The distribution must be hypothesized.Stochastic variables are random. The distribution must be hypothesized. There are two broad areas of simulation: dynamic and static.There are two broad areas of simulation: dynamic and static. In dynamic simulation models, events occur sequentially over time. Specialized software is required.In dynamic simulation models, events occur sequentially over time. Specialized software is required. In static simulation models time is not explicit and the analysis can be done in Excel spreadsheets.In static simulation models time is not explicit and the analysis can be done in Excel spreadsheets. Components of a Simulation Model Components of a Simulation Model Chapter 18 18.1 What is Simulation? LO18-2 LO18-2: Distinguish between stochastic and deterministic variables.

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18-6 Components of a Simulation Model Components of a Simulation Model Table 18.1 Chapter 18 18.1 What is Simulation? LO18-2

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18-7 Components of a Simulation Model Components of a Simulation Model Table 18.1 Chapter 18 18.1 What is Simulation? LO18-2

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18-8 The Monte Carlo method is used for static simulation.The Monte Carlo method is used for static simulation. The computer creates the values of the stochastic random variables.The computer creates the values of the stochastic random variables. The distribution and its parameters are specified.The distribution and its parameters are specified. Samples are repeatedly drawn from each distribution.Samples are repeatedly drawn from each distribution. Each sample yields one possible outcome for each stochastic variable.Each sample yields one possible outcome for each stochastic variable. For each output variable, look at percentiles as well as the mean.For each output variable, look at percentiles as well as the mean. For each input variable, look at a histogram to verify that we are sampling from the desired distribution.For each input variable, look at a histogram to verify that we are sampling from the desired distribution. Chapter 18 18.2 Monte Carlo Simulation Which Distribution? Which Distribution? Any distribution can be used for a stochastic input variable. For example: normal, triangular, uniform, exponential etc.Any distribution can be used for a stochastic input variable. For example: normal, triangular, uniform, exponential etc. LO18-3 LO18-3: Explain how Monte Carlo simulation is used and why it is called static. called static.

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18-9 Creating Random Data in Excel Creating Random Data in Excel Chapter 18 LO18-5: Use Excel to generate random data for several common distributions. distributions. LO18-5 18.3 Random Number Generation

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18-10 Other Ways to Get Random Data Other Ways to Get Random Data (Also With EXCEL): Tools > Data Analysis > Random Number Generation (Also With EXCEL): Tools > Data Analysis > Random Number Generation (With MegaStat): (With MegaStat): MegaStat > Random Numbers (With MINITAB): (With MINITAB): Calc > Random Data For general Monte Carlo simulation, it is best to use a specialized package such as @Risk or Crystal Ball that offers many built-in functions to create random data and keep track of your simulation results. Chapter 18 18.3 Random Number Generation Bootstrap Method bootstrap methodThe bootstrap method resample to estimate unknown parameters. This method can be applied to just about any parameter. It requires specialized software. Bootstrap principle: The sample reflects everything we know about the population.

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18-11 Bootstrap Method Bootstrap Method From a sample of n observations, use Monte Carlo random integers to take repeated samples of n items with replacement from the sample.From a sample of n observations, use Monte Carlo random integers to take repeated samples of n items with replacement from the sample. Calculate the statistic of interest for each sample.Calculate the statistic of interest for each sample. The average of these statistics is the bootstrap estimator.The average of these statistics is the bootstrap estimator. The standard deviation from these estimates is the bootstrap standard error.The standard deviation from these estimates is the bootstrap standard error. The distribution of these repeated estimates is the bootstrap distribution.The distribution of these repeated estimates is the bootstrap distribution. The percentiles of the resulting distribution of sample estimator provide the bootstrap confidence interval.The percentiles of the resulting distribution of sample estimator provide the bootstrap confidence interval. The accuracy of the bootstrap estimator increases with the number of resample. The bootstrap method is an excellent choice when data are badly skewed. There are bootstrap estimators for most common statistics. Chapter 18 18.3 Random Number Generation

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18-12 Random data can be generated by using Excel, however, Excel does not keep track of your results.Random data can be generated by using Excel, however, Excel does not keep track of your results. Excel add-ins offer more features such as calculating probabilities and permitting Monte Carlo simulation.Excel add-ins offer more features such as calculating probabilities and permitting Monte Carlo simulation. Chapter 18 18.4 Excel Add-Ins LO18-6: Describe functions and features of commercial modeling tools for Excel. tools for Excel. Using @Risk Add-In Intuitive and easy to use, @Risk input functions can be pasted directly into cells in and Excel spreadsheet.Intuitive and easy to use, @Risk input functions can be pasted directly into cells in and Excel spreadsheet. The input cell becomes active and will change each time you update the spreadsheet by pressing F9.The input cell becomes active and will change each time you update the spreadsheet by pressing F9. LO18-6

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18-13 Chapter 18 18.4 Excel Add-Ins LO18-6

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18-14 In a dynamic simulation, stochastic variables may be discrete (measured only at regular time intervals) or continuous (changing smoothly over time).In a dynamic simulation, stochastic variables may be discrete (measured only at regular time intervals) or continuous (changing smoothly over time). Discrete event simulation assesses the system state by a clock at distinct points in time.Discrete event simulation assesses the system state by a clock at distinct points in time. A snapshot of the system state at any given moment is observed.A snapshot of the system state at any given moment is observed. Discrete Event Simulation Discrete Event Simulation Chapter 18 18.5 Dynamic Simulation The emphasis in discrete event simulation is on measurements such asThe emphasis in discrete event simulation is on measurements such as - Arrival rates - Service rates - Length of queues - Waiting time - Capacity utilization - System throughput LO18-7 LO18-7: Explain the main reasons for using dynamic simulation and queuing models. queuing models.

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18-15 Queuing theory is the study of waiting lines (the length of customer queues, mean waiting times, facility utilization, etc.).Queuing theory is the study of waiting lines (the length of customer queues, mean waiting times, facility utilization, etc.). In a single-server facility, customers form a single, well-disciplined queue (first- come, first-served).In a single-server facility, customers form a single, well-disciplined queue (first- come, first-served). The arrivals are from an infinite source and are Poisson distributed with mean (customer arrivals per unit of time).The arrivals are from an infinite source and are Poisson distributed with mean (customer arrivals per unit of time). The service times are exponentially distributed with mean 1/ (customers served per unit of time).The service times are exponentially distributed with mean 1/ (customers served per unit of time). Queuing Queuing Chapter 18 18.5 Dynamic Simulation Assuming that < then the following may be demonstratedAssuming that < then the following may be demonstratedLO18-7

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18-16 Queuing Models Queuing Models Figure 18.15 Chapter 18 18.5 Dynamic Simulation LO18-7

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