Monte Carlo Simulation Random Number Generation

Slides:



Advertisements
Similar presentations
McGraw-Hill/Irwin Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. A PowerPoint Presentation Package to Accompany Applied Statistics.
Advertisements

Part 3 Probabilistic Decision Models
Modeling and Simulation By Lecturer: Nada Ahmed. Introduction to simulation and Modeling.
Design of Experiments Lecture I
Introduction into Simulation Basic Simulation Modeling.
Chapter 18 If mathematical analysis is too difficult, we can try each possibility out on paper. That way we can find which alternative appears to work.
IE 429, Parisay, January 2003 Review of Probability and Statistics: Experiment outcome: constant, random variable Random variable: discrete, continuous.
Modeling & Simulation. System Models and Simulation Framework for Modeling and Simulation The framework defines the entities and their Relationships that.
Sampling Distributions (§ )
Engineering Economic Analysis Canadian Edition
1 Simulation Lecture 6 Simulation Chapter 18S. 2 Simulation Simulation Is …  Simulation – very broad term  methods and applications to imitate or mimic.
Simulation.
Chapter 14 Simulation. Monte Carlo Process Statistical Analysis of Simulation Results Verification of the Simulation Model Computer Simulation with Excel.
Introduction to modelling Basic concepts and simple modelling techniques 7/12/20151.
Robert M. Saltzman © DS 851: 4 Main Components 1.Applications The more you see, the better 2.Probability & Statistics Computer does most of the work.
Lab 01 Fundamentals SE 405 Discrete Event Simulation
Monté Carlo Simulation MGS 3100 – Chapter 9. Simulation Defined A computer-based model used to run experiments on a real system.  Typically done on a.
Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,
1 Doing Statistics for Business Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer Chapter 11 Regression.
Introduction to Discrete Event Simulation Customer population Service system Served customers Waiting line Priority rule Service facilities Figure C.1.
Chapter 1 Introduction to Simulation
1 Validation & Verification Chapter VALIDATION & VERIFICATION Very Difficult Very Important Conceptually distinct, but performed simultaneously.
Simulation Examples ~ By Hand ~ Using Excel
1 Performance Evaluation of Computer Networks: Part II Objectives r Simulation Modeling r Classification of Simulation Modeling r Discrete-Event Simulation.
Chapter 10. Simulation An Integrated Approach to Improving Quality and Efficiency Daniel B. McLaughlin Julie M. Hays Healthcare Operations Management.
Introduction to Operations Research
Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Risk Simulation Lecture No. 40 Chapter.
Discrete Distributions The values generated for a random variable must be from a finite distinct set of individual values. For example, based on past observations,
Engineering Economic Analysis Canadian Edition
McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. 1 Simulation.
SUPPLEMENT TO CHAPTER NINETEEN Irwin/McGraw-Hill © The McGraw-Hill Companies, Inc., 1999 SIMULATION 19S-1 Chapter 19 Supplement Simulation.
Outline of Chapter 9: Using Simulation to Solve Decision Problems Real world decisions are often too complex to be analyzed effectively using influence.
Simulation is the process of studying the behavior of a real system by using a model that replicates the behavior of the system under different scenarios.
McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. 1.
Discrete Event (time) Simulation. What is a simulation? “Simulation is the process of designing a model of a real system and conducting experiments with.
Chapter 10 Verification and Validation of Simulation Models
Simulation is the process of studying the behavior of a real system by using a model that replicates the system under different scenarios. A simulation.
1 1 Slide Simulation Professor Ahmadi. 2 2 Slide Simulation Chapter Outline n Computer Simulation n Simulation Modeling n Random Variables and Pseudo-Random.
Monte-Carlo based Expertise A powerful Tool for System Evaluation & Optimization  Introduction  Features  System Performance.
Simulation Chapter 16 of Quantitative Methods for Business, by Anderson, Sweeney and Williams Read sections 16.1, 16.2, 16.3, 16.4, and Appendix 16.1.
System Analysis System – set of interdependent elements that interact in order to accomplish a one or more final outcomes. Constrained and affected by:
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Chapter 18 Management of Waiting Lines.
Simulation Modeling.
Simulasi sistem persediaan
Computer Simulation Henry C. Co Technology and Operations Management,
Simulation Copyright (c) 2008 by The McGraw-Hill Companies. This spreadsheet is intended solely for educational purposes by licensed users of LearningStats.
OPERATING SYSTEMS CS 3502 Fall 2017
McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.
Prepared by Lloyd R. Jaisingh
Two-Sample Hypothesis Testing
Sampling Distributions and Estimation
Modeling and Simulation (An Introduction)
McGraw-Hill/Irwin ©2009 The McGraw-Hill Companies, All Rights Reserved
Simulation Department of Industrial Engineering Anadolu University
Management of Waiting Lines
Simulation - Introduction
Modeling and Simulation CS 313
DSS & Warehousing Systems
Project Management for Software Engineers (Summer 2017)
Chapter 10 Verification and Validation of Simulation Models
Prepared by Lee Revere and John Large
Professor S K Dubey,VSM Amity School of Business
Simulation Modeling.
More Explanation of an example in chapter4
Discrete Event Simulation - 8
DECISION MODELING WITH Prentice Hall Publishers and
Ch13 Empirical Methods.
Sampling Distributions (§ )
SIMULATION IN THE FINANCE INDUSTRY BY HARESH JANI
Simulation Supplement B.
Presentation transcript:

Monte Carlo Simulation Random Number Generation 18 Simulation Chapter What is Simulation? Monte Carlo Simulation Random Number Generation Excel Add-Ins Dynamic Simulation McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

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 to 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.

What is Simulation? A Versatile Tool Simulation is - a rehearsal - planning - a behavioral tool - not just a quantitative tool for operations research specialists - a general device to help people think clearly

What is Simulation? Applications Whether simple or complex, simulation studies have improved - Passenger flows at Vancover International Airport - Hospital surgery scheduling at Henry Ford Hospital - Traffic flows in metropolitan Oakland County - Waiting lines at Disney World - Just-in-time scheduling in G.M. auto assembly plants

What is Simulation? When Do We Simulate? 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

What is Simulation? When Do We Simulate? Conversely, we are less inclined to simulate when - The system is simple - Variables are stable or non-stochastic - Real experiments are cheap and no disruptive - The event will only happen once - Stakeholders agree on policy

What is Simulation? Advantages of Simulation In a deterministic model, variables can’t vary. Simulation lets key variables change in random but specified ways. Simulation helps us understand the range of possible outcomes and their probabilities. Simulation allows sensitivity analysis.

What is Simulation? Advantages of Simulation Simulation is useful because it - Is less disruptive than real experiments - Forces us to state our assumptions clearly - Helps us visualize the implications of our assumptions - Reveals system interdependencies - Quantifies risk by showing probabilities of events - Helps us see a range of possible outcomes - Promotes constructive dialogue among stakeholders

What is Simulation? Advantages of Simulation A simulation project has the following phases: Phase I (design) – identify the problem, set objectives, design the model, collect data. Phase II (execution) – empirical modeling, specify the variables, validate the model, execute the simulation, prepare reports. Phase III (communication) – explain the findings to decision-makers.

What is Simulation? Risk Assessment Risk assessment means thinking about a range of outcomes and their probabilities. Variation is inevitable. Knowing the 95% range of possible values for the decision variable as well as the “most likely” value m, is the point of risk assessment. Risk assessment is useful when the model is complex.

What is Simulation? Components of a Simulation Model Deterministic variables are nonrandom and fixed. Stochastic variables are random. The distribution must be hypothesized.

What is Simulation? Components of a Simulation Model 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.

What is Simulation? Components of a Simulation Model Table 18.1

What is Simulation? Components of a Simulation Model Table 18.1

Monte Carlo Simulation 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.

Monte Carlo Simulation 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.

Monte Carlo Simulation Which Distribution? Any distribution can be used for a stochastic input variable. Four probability distributions are used more with static simulation because they correspond to real life and can be easily simulated in Excel.

Monte Carlo Simulation Which Distribution?

Monte Carlo Simulation Which Distribution?

Monte Carlo Simulation Which Distribution?

Monte Carlo Simulation Which Distribution?

Monte Carlo Simulation Simulation Setup for Revenue Calculation PB PC Table 18.3

Random Number Generation Basic Concept: Inverse CDF Random x from Continuous CDF Random x from Discrete CDF

Random Number Generation Creating Random Data in Excel Table 18.5

Random Number Generation Other Ways to Get Random Data Tools > Data Analysis > Random Number Generation Figure 18.4

Random Number Generation Other Ways to Get Random Data Using MegaStat Figure 18.5

Random Number Generation Other Ways to Get Random Data Using Learning Stats Figure 18.6

Random Number Generation Other Ways to Get Random Data Using MINITAB Figure 18.7

Random Number Generation 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.

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.

Random Number Generation Bootstrap Method 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.

Random Number Generation Bootstrap Method 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.

Excel Add-Ins 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 @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.

Excel Add-Ins @Risk Functions Table 18.6

Excel Add-Ins @Risk Illustration: Bob’s Stochastic Balance Sheet Figure 18.8

Excel Add-Ins @Risk Illustration: Distributions Used in Bob’s Stochastic Balance Sheet Table 18.7

Excel Add-Ins @Risk Illustration: Buttons to Set Up the Simulation Figure 18.9

Excel Add-Ins @Risk Illustration: Typical Simulation Settings Figure 18.10

Excel Add-Ins @Risk Illustration: Typical Simulation Settings Figure 18.10

Excel Add-Ins @Risk Illustration: Selecting a Graph Figure 18.11

Excel Add-Ins @Risk Illustration: Histogram of 500 Iterations of Net Worth Figure 18.12

Excel Add-Ins @Risk Illustration: Distribution of Net Worth for 500 Iterations Figure 18.13

Excel Add-Ins @Risk Illustration: Tornado Graph for Sensitivity Analysis Figure 18.14

Dynamic Simulation 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.

Dynamic Simulation Discrete Event Simulation The emphasis in discrete event simulation is on measurements such as - Arrival rates - Service rates - Length of queues - Waiting time - Capacity utilization - System throughput

Dynamic Simulation 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 (first-come, first-served). The arrivals are from an infinite source and are Poisson distributed with mean a (customer arrivals per unit of time). The service times are exponentially distributed with mean s (customers served per unit of time).

Dynamic Simulation Queuing Assuming that a < s then the following may be demonstrated

Dynamic Simulation Queuing Models Figure 18.15

Applied Statistics in Business & Economics End of Chapter 18 18-50