Transcript Chapter 18
A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 18 Simulation Chapter Contents 18.1 18.2 18.3 18.4 18.5 What is Simulation? Monte Carlo Simulation Random Number Generation Excel Add-Ins Dynamic Simulations 18-2 Chapter 18 Simulation Chapter Learning Objectives (LO’s) LO18-1: LO18-2: LO18-3: LO18-4: LO18-5: LO18-6: LO18-7: List characteristics of situations where simulation is appropriate. Distinguish between stochastic and deterministic variables. Explain how Monte Carlo simulation is used and why it is called static. Explain how to generate random data by using a discrete or continuous CDF. Use Excel to generate random data for several common distributions. Describe functions and features of commercial modeling tools for Excel. Explain the main reasons for using dynamic simulation and queuing models. 18-3 Chapter 18 LO18-1 18.1 What is Simulation? • • • • 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. Simulations helps to quantify relationships among variables that are too complex to analyze mathematically. If the simulation’s predictions differ from what really happens, refine the model in a systematic way until its predictions are in close enough agreement with reality. LO18-1: List characteristics of situations where simulation is appropriate. • In general, consider simulation when - The system is complex - Uncertainty exists in the variables - Real experiments are impossible or costly - The processes are repetitive - Stakeholders can’t agree on policy 18-4 Chapter 18 LO18-2 18.1 What is Simulation? LO18-2: Distinguish between stochastic and deterministic variables. Components of a Simulation Model • • • • • Deterministic variables are nonrandom and fixed. Stochastic variables are random. The distribution must be hypothesized. There are two broad areas of simulation: dynamic and static. 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. 18-5 Chapter 18 LO18-2 18.1 What is Simulation? Components of a Simulation Model Table 18.1 18-6 Chapter 18 LO18-2 18.1 What is Simulation? Components of a Simulation Model Table 18.1 18-7 Chapter 18 LO18-3 18.2 Monte Carlo Simulation LO18-3: Explain how Monte Carlo simulation is used and why it is called static. • • • • • • • The Monte Carlo method is used for static simulation. The computer creates the values of the stochastic random variables. The distribution and its parameters are specified. Samples are repeatedly drawn from each distribution. Each sample yields one possible outcome for each stochastic variable. 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. Which Distribution? • Any distribution can be used for a stochastic input variable. For example: normal, triangular, uniform, exponential etc. 18-8 Chapter 18 LO18-5 18.3 Random Number Generation LO18-5: Use Excel to generate random data for several common distributions. Creating Random Data in Excel 18-9 Chapter 18 18.3 Random Number Generation Other Ways to Get Random Data • (Also With EXCEL): Tools > Data Analysis > Random Number Generation • (With MegaStat): MegaStat > Random Numbers • (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. Bootstrap Method • • • • The 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. 18-10 Chapter 18 18.3 Random Number Generation 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. Calculate the statistic of interest for each sample. The average of these statistics is the bootstrap estimator. The standard deviation from these estimates is the bootstrap standard error. 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 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. 18-11 Chapter 18 LO18-6 18.4 Excel Add-Ins LO18-6: Describe functions and features of commercial modeling tools for Excel. • • 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. Using @Risk Add-In • • 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. 18-12 Chapter 18 LO18-6 18.4 Excel Add-Ins 18-13 Chapter 18 LO18-7 18.5 Dynamic Simulation LO18-7: Explain the main reasons for using dynamic simulation and queuing models. Discrete Event Simulation • • • • 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. A snapshot of the system state at any given moment is observed. The emphasis in discrete event simulation is on measurements such as - Arrival rates - Service rates - Length of queues - Waiting time - Capacity utilization - System throughput 18-14 Chapter 18 18.5 Dynamic Simulation LO18-7 Queuing • • • • • 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 (firstcome, first-served). 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). Assuming that < then the following may be demonstrated 18-15 Chapter 18 LO18-7 18.5 Dynamic Simulation Queuing Models Figure 18.15 18-16