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* A list of the early publications in the series is at the end of the book *
Design and Analysis of Simulation Experiments
Jack P.C. Kleijnen Tilburg University, Tilburg, the Netherlands
Jack P.C. Kleijnen Tilburg University Tilburg, The Netherlands
Series Editor: Fred Hillier Stanford University Stanford, CA, USA
Library of Congress Control Number: 2007926589
ISBN-13: 978-0-387-71812-5 e-ISBN-13: 978-0-387-71813-2
Printed on acid-free paper.
© 2008 by Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 1 springer.com
To my wife, Wilma Preface
This book is the successor of several other books that I wrote on (roughly) the same topic. My first book consisted of two volumes, and was published in 1974/1975 (and translated into Russian in 1978). Its successor was pub- lished in 1987. In 1992, Willem van Groenendaal and I wrote a more general book on simulation, which included an update of parts of my 1987 book. So I thought that it was high time to write down all I know about the statistical Design and Analysis of Simulation Experiments, which I abbre- viate to DASE (and pronounce as the girl’s name Daisy). This acronym is inspired by DACE, which stands for Design and Analysis of Computer Experiments; the acronym DACE is popular in deterministic simulation. In this book, I will focus on those DASE aspects that I have a certain expertise in—I think. Though I focus on DASE for discrete-event simulation (which includes queueing and inventory simulations), I also discuss DASE for deterministic simulation (applied in engineering, physics, etc.). I discuss both computationally expensive and cheap simulations. I assume that the readers already have a basic knowledge of simulation; e.g., they know concepts such as terminating simulation and steady-state simulation. They should also have a basic understanding of mathematical statistics, including concepts such as distribution functions, averages, and variances. This book contains more than four hundred references. Yet, I have tried to eliminate older references that are mentioned in more recent references— unless the older reference is the origin of some important idea (so the readers may get a historical perspective). To improve the book’s readability, viii Preface
I try to collect references at the end of paragraphs—as much as seems reasonable. I recommend that the first three chapters be read in their implied order. The next chapters, however, are independent of each other, so they may be read in the order that best suits the interest of the individual reader. I wrote this book in a foreign language (namely English, whereas Dutch is my mother tongue), so style, spelling, etc. may sometimes not be perfect: my apologies. Concerning style, I point out that I place redundant infor- mation between parentheses; the em-dash (or —) signals nonredundant, extra information. To enable readers to browse through the various chap- ters, I repeat the definition of an abbreviation in a given chapter—even if that abbreviation has already been defined in a preceding chapter. The book contains paragraphs starting with the word “Note”, which upon first reading may be skipped. Each website address is displayed on a separate line, because a website address may be so long that it either runs over into the right margin of the page or it must be hyphenated—but then the hyphen may be interpreted as part of the address. A comma or a period at the end of the address is not part of the address! I wrote this book in Scientific Workplace, which also helped me (through its MuPAD computational engine) to solve some of the exercises that I for- mulated in this book. Winfried Minnaert (Tilburg University) introduced me to the basics of that text processor; Jozef Pijnenburg (Tilburg Univer- sity) helped me with some more advanced features. I received valuable comments on preliminary versions of various chap- ters from the following colleagues: Ebru Ang˝un(Galatasaray University, Istanbul), Russell Barton (Pennsylvania State), Victoria Chen (University of Texas at Arlington), Gabriella Dellino (Politecnico di Bari), Dick den Hertog (Tilburg University), Tony Giunta (Sandia), Yao Lin (Georgia In- stitute of Technology), Carlo Meloni (Politecnico di Bari), Barry Nelson (Northwestern), William Notz (Ohio State), Huda Abdullah Rasheed (al- Mustansiriyah University, Baghdad), Wim van Beers (Tilburg University), Willem van Groenendaal (Tilburg University), Jim Wilson (North Carolina State), and Bernard Zeigler (Arizona State). I used a preliminary draft of this book to teach a course called “Simula- tion for Logistics” for the “Postgraduate International Program in Logistics Management Systems” at the Technical University Eindhoven. This helped me to improve parts of the book. Students solved the exercises 1.6, 2.13, 2.15. The names of these students are: Nicolas Avila Bruckner, Olla Gabali, Javier Gomes, Suquan Ju, Xue Li, Mar´ıa Eugenia Martelli, Kurtulus Oner,¨ Anna Ot´ahalov´a, Pimara Pholnukulkit, Shanshan Wang, and Wei Zhang. I especially thank Xue Li and Shanshan Wang. For that course, I also prepared PowerPoinT (PPT) slides that may also be downloaded in Portable Document Format (PDF) format from my web page: Preface ix
http://center.uvt.nl/staff/kleijnen/simwhat.html. My website also offers an update of this book, including corrections, new references, new exercises: visit http://center.uvt.nl/staff/kleijnen/ and click “Publications”. Contents
1 Introduction 1 1.1 What is simulation? ...... 1 1.2WhatisDASE?...... 7 1.3DASEsymbolsandterms...... 10 1.4 Solutions for exercises ...... 12
2 Low-order polynomial regression metamodels and their designs: basics 15 2.1 Introduction ...... 16 2.2Linearregressionanalysis:basics...... 19 2.3 Linear regression analysis: first-order polynomials ...... 27 2.3.1 First-order polynomial with a single factor ...... 27 2.3.2 First-order polynomial with several factors ..... 28 2.4 Designs for first-order polynomials: resolution-III ...... 36 2.4.1 2k−p designsofresolution-III...... 36 2.4.2 Plackett-Burman designs of resolution-III ...... 39 2.5 Regression analysis: factor interactions ...... 40 2.6 Designs allowing two-factor interactions: resolution-IV . . . 42 2.7 Designs for two-factor interactions: resolution-V ...... 46 2.8 Regression analysis: second-order polynomials ...... 49 2.9 Designs for second-degree polynomials: Central Composite Designs (CCDs) ...... 50 2.10 Optimal designs and other designs ...... 51 2.11 Validation of metamodels ...... 54 xii Contents
2.11.1 Coefficients of determination and correlation coefficients...... 54 2.11.2 Cross-validation ...... 57 2.12 More simulation applications ...... 63 2.13 Conclusions ...... 66 2.14 Appendix: coding of nominal factors ...... 66 2.15 Solutions for exercises ...... 69
3 Classic assumptions revisited 73 3.1 Introduction ...... 73 3.2 Multivariate simulation output ...... 74 3.2.1 Designs for multivariate simulation output ...... 77 3.3 Nonnormal simulation output ...... 78 3.3.1 Realistic normality assumption? ...... 78 3.3.2 Testing the normality assumption ...... 79 3.3.3 Transformations of simulation I/O data, jackknifing, andbootstrapping...... 80 3.4 Heterogeneous simulation output variances ...... 87 3.4.1 Realistic constant variance assumption? ...... 87 3.4.2 Testing for constant variances ...... 88 3.4.3 Variance stabilizing transformations ...... 89 3.4.4 LS estimators in case of heterogeneous variances . . 89 3.4.5 Designs in case of heterogeneous variances ...... 92 3.5 Common random numbers (CRN) ...... 93 3.5.1 Realistic CRN assumption? ...... 94 3.5.2 Alternative analysis methods ...... 94 3.5.3 Designs in case of CRN ...... 96 3.6 Nonvalid low-order polynomial metamodel ...... 97 3.6.1 Testing the validity of the metamodel ...... 97 3.6.2 Transformations of independent and dependent regressionvariables...... 98 3.6.3 Adding high-order terms to a low-order polynomial metamodel ...... 98 3.6.4 Nonlinear metamodels ...... 99 3.7Conclusions...... 99 3.8 Solutions for exercises ...... 100
4 Simulation optimization 101 4.1 Introduction ...... 101 4.2RSM:classicvariant...... 105 4.3 Generalized RSM: multiple outputs and constraints ..... 110 4.4 Testing an estimated optimum: KKT conditions ...... 116 4.5Riskanalysis...... 123 4.5.1 Latin Hypercube Sampling (LHS) ...... 126 4.6 Robust optimization: Taguchian approach ...... 130 Contents xiii
4.6.1 Case study: Ericsson’s supply chain ...... 135 4.7Conclusions...... 137 4.8 Solutions for exercises ...... 138
5 Kriging metamodels 139 5.1 Introduction ...... 139 5.2 Kriging basics ...... 140 5.3 Kriging: new results ...... 147 5.4 Designs for Kriging ...... 149 5.4.1 Predictor variance in random simulation ...... 151 5.4.2 Predictor variance in deterministic simulation .... 152 5.4.3 Related designs ...... 154 5.5Conclusions...... 155 5.6 Solutions for exercises ...... 156
6 Screening designs 157 6.1 Introduction ...... 157 6.2 Sequential Bifurcation ...... 160 6.2.1 Outline of simplest SB ...... 160 6.2.2 Mathematical details of simplest SB ...... 165 6.2.3 Case study: Ericsson’s supply chain ...... 167 6.2.4 SB with two-factor interactions ...... 169 6.3Conclusions...... 171 6.4 Solutions for exercises ...... 172
7 Epilogue 173
References 175
Index 211 1 Introduction
This chapter is organized as follows. In Section 1.1, I define various types of simulation. In Section 1.2, I define the DASE approach. In Section 1.3, I define DASE symbols and terms. I finish with solutions for the exercises of this chapter.
1.1 What is simulation?
In this section, I give 1. a definition of “simulation models” 2. a simple example of a deterministic simulation model, namely the Net Present Value (NPV) calculation of a loan 3. two simple examples of random or stochastic simulation, namely a single-server queueing model and an inventory management model.
Definition 1.1 A simulation model is a dynamic model that is meant to be solved by means of experimentation.
This definition deserves some comments. A simulation model may be a physical model—e.g., a miniature air- plane in a windtunnel (other examples are automobiles and ships). I, how- ever, ignore such physical models; i.e., I focus on mathematical models. These models are usually converted into computer programs—also called computer codes. 2 1. Introduction
The term dynamic means that time plays an explicit and special role; i.e., the variables in the mathematical model have a time index—e.g., the variable x becomes x(t) where t denotes a point in time. Static models, however, may also be solved through experimentation. For example, the well-known Newton-Ralphson method may be used to find the roots of an equation, and Interior Point (IP) methods may be used to find the optimum solution of a Linear Programming (LP) model (also see Chapter 4). Closely related to simulation are Monte Carlo methods, which I define as methods that use PseudoRandom Numbers (PRNs). PRNs are generated by means of a computer program, so they are not really random, and yet they are assumed to be independently and uniformly distributed on the interval [0, 1]—so Monte Carlo methods involve chance, which explains the name. Monte Carlo methods are used to solve multiple integrals, which arise in physics, mathematical statistics, etc. All the methods discussed in the preceding comments are numerical methods. In this book, I focus on dynamic, random simulation, but I also discuss deterministic simulation explicitly (see the next example). More- over, many DASE methods also apply to static Monte Carlo methods. Also see my previous publications, starting with my 1974 book [181], pp. 3–22 and ending with my 2005 contribution [193]. Example 1.1 Assume that the following data are given: θ the discount factor used by the decision maker; n the length of the planning period mea- sured in years, and xt the cash flow in year t with (t =0,...,n).Then the NPV—also called the Present Value (PV)—(say) y may be computed through the following equation (engineers often use an alternative formula, assuming continuous time—so becomes ,etc.):