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being able to keep children motivated long enough to complete
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· Do not simply put summaries together one by one or put all of
your Annotated Bibliographies together
· Include an introduction section
· Include a conclusion section
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writing
Sterman
I 1
ISBN : 007238915X
TITLE: BUSINESS DYNAMICS : SYSTEMS THINKING I 1
t
RIAL : 291000
CLASS: BUSINESS
EXHIB :
Business Dynamics
Systems Thinking and
Modeling for a Complex World
John D. Sterman
Massachusetts Institute of Technology
Sloan School of Management
Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San
Francisco St. Louis
Bangkok Bogota Caracas Lisbon London Madrid
Mexico City Milan New Delhi Seoul Singapore Sydney Taipei
Toronto
McGraw-Hill Higher Education
A Division of f i e McGraw-Hill Companies
BUSINESS DYNAMICS
SYSTEMS THINKING AND MOOELING FOR A COMPLEX
WORLD
Copyright 0 2000 by The McGraw-Hill Companies, Inc. AU
rights reserved. Printcd in the United '
States of America. Except as permitted under the United States
Copyright Act of 1976, no part of
this publication may be reproduced or dismbuted in any form or
by m y means, or stored in a
database or retrieval system, without the prior written
permission of the publisher.
This book is printed on acid-free paper. .h
4 5 6 7 8 9 0 KGPiKGP 0 9 8 7 6 5 4 3 2
ISBN 0-07-231135-5
Publisher: Jeffrey J. Shelsfud
Senior sponsoring editor: Scott Isenberg
Marketing manager: Zina Cra3
Senior project manager: Gladys True
Senior production supervisor: Lori Koetters
Freelance design coordinator: A4av L. Christianson
Freelance cover designer: The Wsrrul
Cover image: 0 Sonia Delaumy/L & M Services, Amsterdaflute
Gulleo, LondodArt Resource, NY
Compositor: GAChdianapolis
Typeface: 11/13 Ernes Roman
Printer: Quebecor Printing Book Group/Kingsport
Library of Congress Cataloging-in-Publication Data
Sterman, John.
Business dynamics : systems thinking and modeling for a
complex world I John D. Sterman.
Includes hibliographical references and index.
ISBN 0-07-231135-5 (alk. paper)
1. Indusmal management. 2. System theory. 3. Management
information systems. I.
p. cm.
Title.
HD30.2.S7835 2000
658.4'038'011Ldc21
99-056030
http://www.mhhe.com
For Cindy
ABOUT THE AUTHOR
John D. Sterman is J. Spencer Standish Professor of
Management at the Sloan
School of Management of the Massachusetts Institute of
Technology and Director
of MIT’s System Dynamics Group. His research centers on the
development of
practical methods for systems thinking and dynamic modeling
of complex sys-
tems, with applications to organizational learning and change,
operations manage-
ment, corporate strategy, and nonlinear dynamics in a wide
range of systems, from
supply chains to scientific revolutions. He has pioneered the
development of man-
agement flight simulators of corporate and economic systems.
These flight simu-
lators are used in research to understand and improve
managerial decision making
in complex dynamic systems; more importantly, they are now
widely used by cor-
porations and universities around the world for teaching,
problem solving, and pol-
icy design. Professor Sterman discovered system dynamics
modeling in high
school, studied it as an undergraduate at Dartmouth College,
and received his PhD
from MIT. He has been awarded the Jay W. Forrester Prize,
given for the best pub-
lished work in the field of system dynamics over the prior five
years, and has four
times won awards for teaching excellence from the students of
the Sloan School.
vi
Preface
Accelerating economic, technological, social, and
environmental change challenge
managers and policy makers to learn at increasing rates, while
at the same time the
complexity of the systems in which we live is growing. Many of
the problems we
now face arise as unanticipated side effects of our own past
actions. All too often
the policies we implement to solve important problems fail,
make the problem
worse, or create new problems.
Effective decision making and learning in a world of growing
dynamic com-
plexity requires us to become systems thinkers-to expand the
boundaries of our
mental models and develop tools to understand how the
structure of complex sys-
tems creates their behavior.
This book introduces you to system dynamics modeling for the
analysis of pol-
icy and strategy, with a focus on business and public policy
applications. System
dynamics is a perspective and set of conceptual tools that
enable us to understand
the structure and dynamics of complex systems. System
dynamics is also a rigor-
ous modeling method that enables us to build formal computer
simulations of com-
plex systems and use them to design more effective policies and
organizations.
Together, these tools allow us to create management flight
simulators-micro-
worlds where space and time can be compressed and slowed so
we can experience
the long-term side effects of decisions, speed learning, develop
our understanding
of complex systems, and design structures and strategies for
greater success.
The field of system dynamics is thriving. Over the past decade,
many top com-
panies, consulting firms, and governmental organizations have
used system dy-
namics to address critical issues. More innovative universities
and business
schools are teaching system dynamics and finding enthusiastic
and growing en-
rollments. Hundreds of primary and secondary schools, from
kindergarten to high
school, are integrating systems thinking, system dynamics, and
computer simula-
tion into their curricula. Tools and methods for system
dynamics modeling, the li-
brary of successful applications, and insights into the effective
use of the tools with
executives and organizations are all expanding rapidly.
vii
viii Preface
FEATURES AND CONTENT
University and graduate-level texts, particularly those focused
on business and
public policy applications, have not kept pace with the growth
of the field. This
book is designed to provide thorough coverage of the field of
system dynamics to-
day, by examining
Systems thinking and the system dynamics worldview;
Tools for systems thinking, including methods to elicit and map
the
structure of complex systems and relate those structures to their
dynamics;
Tools for modeling and simulation of complex systems;
Procedures for testing and improving models;
Guidelines for working with client teams and successful
implementation.
You will learn about the dynamics of complex systems,
including the structures
that create growth, goal-seeking behavior, oscillation and
instability, S-shaped
growth, overshoot and collapse, path dependence, and other
nonlinear dynamics.
Examples and applications include
Corporate growth and stagnation,
The diffusion of new technologies,
The dynamics of infectious disease such as HIV/AIDS,
Business cycles,
Speculative bubbles,
The use and reliability of forecasts,
The design of supply chains in business and other organizations,
Service quality management,
Transportation policy and traffic congestion,
Project management and product development,
and many others.
The goal of systems thinking and system dynamics modeling is
to improve our
understanding of the ways in which an organization’s
performance is related to its
internal structure and operating policies, including those of
customers, competi-
tors, and suppliers and then to use that understanding to design
high leverage poli-
cies for success. To do so this book utilizes
Process Points that provide practical advice for the successful
application
of the tools in real organizations.
Case studies of System Dynamics in Action that present
successful
applications ranging from global warming and the war on drugs
to
reengineering the supply chain of a major computer firm,
marketing
strategy in the automobile industry, and process improvement in
the
petrochemicals industry.
System dynamics is not a spectator sport. Developing systems
thinking and mod-
eling skills requires the active participation of you, the reader,
via
Preface ix
Challenges. The challenges, placed throughout the text, give
you practice
with the tools and techniques presented in the book and will
stimulate your
original thinking about important real world issues. The
challenges range
from simple thought experiments to full-scale modeling
projects.
Simulation software and models. The accompanying CD-ROM
and web
site (http://www.mhhe.com/sterman) include all the models
developed in
the text along with state-of-the-art simulation software to run
them. There
are several excellent software packages designed to support
system
dynamics modeling. These include ithink, Powersim, and
Vensim. The CD
and website include the models for the text in all three software
formats.
The disk also includes fully functional versions of the ithink,
Powersim, and
Vensim software so you can run the models using any of these
packages
without having to purchase any additional software.
Additionally, the Instructor’s Manual and instructor’s section of
the
web site include suggested solutions for the challenges,
additional
assignments, Powerpoint files with the diagrams and figures
from the text
suitable for transparencies, suggested course sequences and
syllabi, and
other materials.
INTENDED AUDIENCE
The book can be used as a text in courses on systems thinking,
simulation model-
ing, complexity, strategic thinking, operations, and industrial
engineering, among
others. It can be used in full or half-semester courses, executive
education, and
self-study. The book also serves as a reference for managers,
engineers, consul-
tants, and others interested in developing their systems thinking
skills or using sys-
tem dynamics in their organizations.
A NOTE ON MATHEMATICS
System dynamics is grounded in control theory and the modern
theory of nonlin-
ear dynamics. There is an elegant and rigorous mathematical
foundation for the
theory and models we develop. System dynamics is also
designed to be a practical
tool that policy makers can use to help them solve the pressing
problems they con-
front in their organizations. Most managers have not studied
nonlinear differential
equations or even calculus, or have forgotten it if they did. To
be useful, system dy-
namics modeling must be accessible to the widest range of
students and practicing
managers without becoming a vague set of qualitative tools and
unreliable gener-
alizations. That tension is compounded by the diversity of
backgrounds within the
community of managers, students, and scholars interested in
system dynamics,
backgrounds ranging from people with no mathematics
education beyond high
school to those with doctorates in physics.
X Preface
IF YOU DON’T HAVE A STRONG MATHEMATICS
BACKGROUND,
FEAR NOT
This book presents system dynamics with a minimum of
mathematical formalism.
The goal is to develop your intuition and conceptual
understanding, without sacri-
ficing the rigor of the scientific method. You do not need
calculus or differential
equations to understand the material. Indeed, the concepts are
presented using only
text, graphs, and basic algebra. Mathematical details and
references to more ad-
vanced material are set aside in separate sections and footnotes.
Higher mathemat-
ics, though useful, is not as important as the critical thinking
skills developed here.
IF YOU HAVE A STRONG MATHEMATICS BACKGROUND,
FEAR NOT
Realistic and useful models are almost always of such
complexity and nonlinearity
that there are no known analytic solutions, and many of the
mathematical tools you
have studied have limited applicability. This book will help you
use your strong
technical background to develop your intuition and conceptual
understanding of
complexity and dynamics. Modeling human behavior differs
from modeling phy s-
ical systems in engineering and the sciences. We cannot put
managers up on the lab
bench and run experiments to determine their transfer function
or frequency re-
sponse. We believe all electrons follow the same laws of
physics, but we cannot
assume all people behave in the same way. Besides a solid
grounding in the mathe-
matics of dynamic systems, modeling human systems requires
us to develop our
knowledge of psychology, decision malung, and organizational
behavior. Finally,
mathematical analysis, while necessary, is far from sufficient
for successful sys-
tems thinlung and modeling. For your work to have impact in
the real world you
must learn how to develop and implement models of human
behavior in organiza-
tions, with all their ambiguity, time pressure, personalities, and
politics. Through-
out the book I have sought to illustrate how the technical tools
and mathematical
concepts you may have studied in the sciences or engineering
can be applied to the
messy world of the policy maker.
~
FEEDBACK
I welcome your comments, criticisms, and suggestions.
Suggestions for additional
examples, cases, theory, models, flight simulators, and so on, to
make the book
more relevant and useful to you are especially invited. I will
update the website
to incorporate user feedback and new materials. Email
comments to <[email protected]
mit .edu > .
ACKNOWLEDGMENTS
This work benefited immensely from the advice, criticism, and
encouragement of
many colleagues, students, and friends. I owe an immeasurable
debt to my first
system dynamics teachers, Dana Meadows, Dennis Meadows,
and Jay Forrester,
for their integrity, high standards, and passionate commitment.
I’m particularly
indebted to the exceptional students of the MIT Sloan School of
Management.
They constantly challenge me to make the discipline of system
dynamics relevant,
Preface xi
useful, and exciting; I hope they’ve learned as much from me as
I’ve learned from
them. In addition, I thank my colleagues at the Sloan School
and in the system
dynamics community around the world, who helped by
providing data and exam-
ples, reviewing the draft, testing early versions in their courses,
and in countless
other ways. This group includes (but is not limited to) the
following folks and
institutions:
Tarek Abdel-Hamid (Naval Postgraduate School); David
Andersen, George
Richardson (SUNY Albany); Ed Anderson (Univ. of Texas);
Carlos Ariza, Sharon
Els, Ken Cooper, Jim Lyneis, Hank Taylor (Pugh-Roberts
Associates); George
Backus (Policy Assessment Corporation); Bent Bakken
(Norwegian Defense Re-
search Establishment); Yaman Barlas (Bogazici University,
Istanbul); Michael
Bean (Powersim Corp.); Eric Beinhocker, Damon Beyer,
Andrew Doman, Usman
Ghani, Maurice Glucksman, Paul Langley, Norman Marshall
(McKinsey and
Company); Laura Black, John Carroll, Vanessa Colella, Ernst
Diehl, Steve Ep-
pinger, Charlie Fine, Mila Getmansky, Paulo Goncalves, Janet
Gould Wilkinson,
Jim Hines, Nan Lux, Brad Morrison, Tim Nugent, Nelson
Repenning, Ed Roberts,
Scott Rockart, George Roth, Ed Schein, Peter Senge (MIT);
Allen and Jane
Boorstein; Steve Cavaleri (Central Connecticut State Univ.);
Geoff Coyle (Royal
Military College of Science, UK, retired); Brian Dangerfield
(Univ. of Salford);
Pi1 Davidsen (Univ. of Bergen); Jim Doyle, Mike Radzicki,
Khalid Saeed
(Worcester Polytechnic Institute); Bob Eberlein, Tom
Fiddaman, Dan Goldner,
David Peterson, Laura Peterson (Ventana Systems); David Foley
and Judy Berk;
Andy Ford (Washington State Univ.); David Ford (Texas A&M
University);
Nathan Forrester (A. T. Kearney); Rich Goldbach (Metro
Machine Corp.); Chris-
tian Haxholdt, Heather Hazard (Copenhagen Business School);
Jack Homer
(Homer Consulting); Jody House (Oregon Graduate Institute);
Bill Isaacs (Dia-
logos); Sam Israelit (Arthur Andersen); Nitin Joglekar (Boston
Univ. School
of Management); Drew Jones (Sustainability Institute);
Christian Kampmann,
Erik Mosekilde (Technical Univ. of Denmark); Daniel Kim,
Virginia Wiley
(Pegasus Communications); Craig Kirkwood (Arizona State
Univ.); Elizabeth
Krahmer Keating (Northwestern Univ.); Don Kleinmuntz (Univ.
of Illinois,
Urbana-Champaign); David Kreutzer (GKA, Inc.); Robert
Landel (Darden School
of Business, Univ. of Virginia); David Lane (London School of
Economics); Erik
Larsen (City University, London); Winston J. Ledet, Winston P.
Ledet (The Man-
ufacturing Game, Inc.); Ralph Levine (Michigan State Univ.);
Angela Lipinski
(Society for Organizational Learning); Martin GroBmann, Frank
Maier, Peter
Milling (Univ. of Mannheim, Germany); Ali Mashayekhi (Sharif
Univ. of Tech-
nology, Teheran); Nathaniel Mass (GenCorp); Paul Monus
(BP/Amoco), John
Morecroft, Ann van Ackere, Kim Warren (London Business
School); Erling
Moxnes (Norwegian School of Economics and Business
Administration); Rogelio
Oliva (Harvard Business School); Mark Paich (Colorado
College); Steve Peterson,
Barry Richmond (High Performance Systems); Greg Petsch
(Compaq Computer);
Nick Pudar (General Motors); Jack Pugh, Julia Pugh, Roberta
Spencer (System
Dynamics Society), JQrgen Randers (World Wildlife Fund
International); Nancy
Roberts (Leslie College); Jenny Rudolph (Boston College);
Jorge Rufat-Latre
(Strategos); Anjali Sastry, Marshall van Alstyne (University of
Michigan); Bob
Stearns; Susan Sterman; Jim Thompson (Global Prospectus,
LLC); John Voyer
xii Preface
(Univ. of Southern Maine); Lyle Wallis (Decisio, Inc.); Jim
Waters (Waters Busi-
ness Systems); Jason Wittenberg (Harvard Univ.); Eric
Wolstenholme (Leeds Busi-
ness School, UK); Pave1 Zamudio Ramirez (Monitor Company);
the Copenhagen
Business School, The International Network of Resource
Information Centers
(aka the Balaton Group), McKinsey and Company, the
Norwegian School of
Management, Pugh-Roberts Associates, the Society for
Organizational Learning,
the Technical University of Denmark, and, of course, the MIT
Sloan School of
Management.
Special thanks to High Performance Systems, Powersim, SA,
and Ventana
Systems-and their great people-for providing their simulation
software and
translations of the models for the CD and website.
The team at IrwidMcGraw-Hill deserves special mention for
their enthusiasm,
patience, and editorial help, particularly Scott Isenberg, Carol
Rose, Jeff Shelstad,
and Gladys True.
Cara Barber and Kelley Donovan provided important secretarial
support.
Kathy Sullivan went beyond the call of duty on library research,
data collec-
Finally, the love and support of my family have been constant
and essential.
tion, editorial changes, and graphics.
Thanks, Cindy, David, and Sarah.
Table of Contents
Preface vii
PART I PERSPECTIVE AND PROCESS 1
1 Learning in and about Complex Systems 3
1.1 Introduction 3
1.1.1
1.1.2 Causes of Policy Resistance 10
1.1.3 Feedback 12
1.1.4 Process Point: The Meaning of Feedback 14
Challenge: Dynamics of Multiple-Loop Systems
Policy Resistance, the Law of Unintended Consequences,
and the Counterintuitive Behavior of Social Systems 5
14
1.2 Learning Is a Feedback Process 14
1.3 Barriers to Learning 19
1.3.1 Dynamic Complexity 21
1.3.2 Limited Information 23
1.3.3 Confounding Variables and Ambiguity 25
1.3.4 Bounded Rationality and the Misperceptions
of Feedback 26
1.3.5 Flawed Cognitive Maps 28
1.3.6 Erroneous Inferences about Dynamics 29
1.3.7 Unscientific Reasoning: Judgmental Errors
andBiases 30
Challenge: Hypothesis Testing 30
1.3.8 Defensive Routines and Interpersonal Impediments
to Learning 32
1.3.9 Implementation Failure 33
1.4.1 Improving the Learning Process: Virtues
of Virtual Worlds 34
1.4.2 Pitfalls of Virtual Worlds 35
1.4.3 Why Simulation Is Essential 37
1.4 Requirements for Successful Learning in Complex Systems
33
1.5 Summary 39
xiii
xiv Contents
2 System Dynamics in Action 41
2.1
2.2
2.3
2.4
2.5
Applications of System Dynamics 41
Automobile Leasing Strategy: Gone Today, Here Tomorrow
2.2.1 Dynamic Hypothesis 44
2.2.2 Elaborating the Model 48
2.2.3 Policy Analysis 5 1
2.2.4 Impact and Follow-up 54
On Time and Under Budget: The Dynamics
of Project Management 55
2.3.1 The Claim 56
2.3.2 Initial Model Development 57
2.3.3 Dynamic Hypothesis 58
2.3.4 The Modeling Process 61
2.3.5 Continuing Impact 64
Playing the Maintenance Game 66
2.4.1 Dynamic Hypothesis 67
2.4.2 The Implementation Challenge 74
2.4.3 Results 76
2.4.4 Transferring the Learning: The Lima Experience 77
42
Summary: Principles for Successful Use of System Dynamics 79
3.1 The Purpose of Modeling: Managers as Organization
Designers 84
3.2 The Client and the Modeler 84
3.3 Steps of the Modeling Process 85
3.4 Modeling Is Iterative 87
3.5 Overview of the Modeling Process 89
3 The Modeling Process 83
3.5.1 Problem Articulation: The Importance of Purpose 89
3.5.2 Formulating a Dynamic Hypothesis 94
3.5.3 Formulating a Simulation Model 102
3.5.4 Testing 103
3.5.5 Policy Design and Evaluation 103
4 Structure and Behavior of Dynamic Systems 107
3.6 Summary 104
4.1
4.2
4.3
Fundamental Modes of Dynamic Behavior
4.1.1 Exponential Growth 108
4.1.2 Goal Seeking 11 1
4.1.3 Oscillation 114
4.1.4 Process Point 116
Challenge: Identifying Feedback Structure
from System Behavior 117
Interactions of the Fundamental Modes
4.2.1 S-Shaped Growth 118
4.2.2 S-Shaped Growth with Overshoot 121
Challenge: Identifying the Limits to Growth
4.2.3 Overshoot and Collapse 123
Other Modes of Behavior 127
4.3.1 Stasis, or Equilibrium 127
108
11 8
121
Contents xv
4.3.2 Randomness 127
4.3.3 Chaos 129
4.4 Summary 133
PART I1 TOOLS FOR SYSTEMS THINKING 135
5 Causal Loop Diagrams 137
5.1
5.2
5.3
5.4
5.5
5.6
Causal Diagram Notation 137
Guidelines for Causal Loop Diagrams
5.2.1 Causation versus Correlation 141
5.2.2 Labeling Link Polarity 142
Challenge: Assigning Link Polarities 143
5.2.3 Determining Loop Polarity 143
Challenge: Identifying Link and Loop Polarity
Challenge: Employee Motivation 147
5.2.4 Name YourLoops 148
5.2.5 Indicate Important Delays in Causal Links 150
5.2.6 Variable Names 152
5.2.7 Tips for Causal Loop Diagram Layout 153
5.2.8 Choose the Right Level ofAggregation 154
5.2.9 Don’t Put All the Loops into One Large Diagram
5.2.10 Make the Goals of Negative Loops Explicit 155
5.2.11 Distinguish between Actual
Process Point: Developing Causal Diagrams
from Interview Data 157
Challenge: Process Improvement 158
Conceptualization Case Study: Managing Your Workload
5.4.1 Problem Definition 159
5.4.2 IdentiJLing Key Variables 160
5.4.3 Developing the Reference Mode 160
5.4.4 Developing the Causal Diagrams 163
5.4.5 Limitations of the Causal Diagram 166
Challenge: Policy Analysis with Causal Diagrams
Adam Smith’s Invisible Hand and the
Feedback Structure of Markets 169
Challenge: The Oil Crises of the 1970s 172
Challenge: Speculative Bubbles 173
Challenge: The Thoroughbred Horse Market
5.5.1 Market Failure, Adverse Selection,
Challenge: The Medigap Death Spiral 176
Explaining Policy Resistance: Traffic Congestion
5.6.1 Mental Models of the Traffic Problem 178
5.6.2 Compensating Feedback: The Response
to Decreased Congestion 18 1
5.6.3 The Mass Transit Death Spiral 185
5.6.4 Policy Analysis: The Impact of Technology 188
5.6.5 Compensating Feedback: The Source
of Policy Resistance 189
141
145
154
and Perceived Conditions 156
159
168
174
and the Death Spiral 174
177
xvi Contents
Challenge: Identifying the Feedback Structure
of Policy Resistance 190
5.7 Summary 190
6.1 Stocks, Flows, and Accumulation 191
6 Stocks and Flows 191
6.1.1 Diagramming Notation for Stocks and Flows 192
6.1.2 Mathematical Representation of Stocks and Flows 193
6.1.3 The Contribution of Stocks to Dynamics 195
6.2.1 Units of Measure in Stock and Flow Networks 198
6.2.2 The Snapshot Test 199
Challenge: Identifying Stocks and Flows 201
6.2.3 Conservation of Material in
Stock and Flow Networks 201
6.2.4 State-Determined Systems 202
6.2.5 Auxiliary Variables 202
6.2.6 Stocks Change Only through Their Rates 204
6.2.7 Continuous Time and Instantaneous Flows 206
6.2.8 Continuously Divisible versus Quantized Flows 207
6.2.9 Which Modeling Approach Should You Use? 208
6.2.10 Process Point: Portraying Stocks and Flows
in Practice 209
When Should Causal Loop Diagrams Show
Stock and Flow Structure? 210
6.2 Identifying Stocks and Flows 197
6.3 Mapping Stocks and Flows 210
6.3.1
Challenge: Adding Stock and Flow Structure
to Causal Diagrams 2 11
Challenge: Linking Stock and Flow Structure with Feedback
6.3.2 Aggregation in Stock and Flow Mapping 213
Challenge: Modifying Stock and Flow Maps
Challenge: Disaggregation 214
6.3.3 Guidelines for Aggregation 216
6.3.4 System Dynamics in Action:
6.3.5 Setting the Model Boundary:
6.3.6 System Dynamics in Action: Automobile Recycling 225
212
213
Modeling Large-Scale Construction Projects
“Challenging the Clouds’’ 222
2 18
6.4 Summary 229
7.1 Relationship between Stocks and Flows 232
7 Dynamics of Stocks and Flows 231
7.1.1 Static and Dynamic Equilibrium 232
7.1.2 Calculus without Mathematics 232
7.1.3 Graphical Integration 234
Challenge: Graphical Integration 239
7.1.4 Graphical Diflerentiation 239
Challenge: Graphical Differentiation 24 1
7.2 System Dynamics in Action: Global Warming 241
Contents xvii
7.3 System Dynamics in Action: The War on Drugs 250
7.4 Summary 262
Dynamics of Simple Structures 263
7.3.1 The Cocaine Epidemic after 1990 258
8 Closing the Loop:
8.1
8.2
8.3
8.4
8.5
8.6
First-Order Systems 263
Positive Feedback and Exponential Growth
8.2.1 Analytic
Solution
for the Linear First-Order System 265
8.2.2 Graphical

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 How can a teacher be encouraging and motivating to students durin

  • 1. How can a teacher be encouraging and motivating to students during challenging times ie. (COVID)? The problem is not being able to keep children motivated long enough to complete and submit assignments · Do not simply put summaries together one by one or put all of your Annotated Bibliographies together · Include an introduction section · Include a conclusion section · Consider creating themes · You will need subheadings · You must use APA format · You must have a reference page (use hanging indents...See APA format) · Use the correct in-paper citations · Include different types of literature/resources · Try to avoid using first person "I" (use scholarly, unbiased writing Sterman I 1 ISBN : 007238915X TITLE: BUSINESS DYNAMICS : SYSTEMS THINKING I 1 t RIAL : 291000 CLASS: BUSINESS EXHIB :
  • 2. Business Dynamics Systems Thinking and Modeling for a Complex World John D. Sterman Massachusetts Institute of Technology Sloan School of Management Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogota Caracas Lisbon London Madrid Mexico City Milan New Delhi Seoul Singapore Sydney Taipei Toronto McGraw-Hill Higher Education A Division of f i e McGraw-Hill Companies BUSINESS DYNAMICS SYSTEMS THINKING AND MOOELING FOR A COMPLEX WORLD Copyright 0 2000 by The McGraw-Hill Companies, Inc. AU rights reserved. Printcd in the United ' States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or dismbuted in any form or by m y means, or stored in a database or retrieval system, without the prior written
  • 3. permission of the publisher. This book is printed on acid-free paper. .h 4 5 6 7 8 9 0 KGPiKGP 0 9 8 7 6 5 4 3 2 ISBN 0-07-231135-5 Publisher: Jeffrey J. Shelsfud Senior sponsoring editor: Scott Isenberg Marketing manager: Zina Cra3 Senior project manager: Gladys True Senior production supervisor: Lori Koetters Freelance design coordinator: A4av L. Christianson Freelance cover designer: The Wsrrul Cover image: 0 Sonia Delaumy/L & M Services, Amsterdaflute Gulleo, LondodArt Resource, NY Compositor: GAChdianapolis Typeface: 11/13 Ernes Roman Printer: Quebecor Printing Book Group/Kingsport Library of Congress Cataloging-in-Publication Data Sterman, John. Business dynamics : systems thinking and modeling for a complex world I John D. Sterman. Includes hibliographical references and index. ISBN 0-07-231135-5 (alk. paper) 1. Indusmal management. 2. System theory. 3. Management information systems. I. p. cm.
  • 4. Title. HD30.2.S7835 2000 658.4'038'011Ldc21 99-056030 http://www.mhhe.com For Cindy ABOUT THE AUTHOR John D. Sterman is J. Spencer Standish Professor of Management at the Sloan School of Management of the Massachusetts Institute of Technology and Director of MIT’s System Dynamics Group. His research centers on the development of practical methods for systems thinking and dynamic modeling of complex sys- tems, with applications to organizational learning and change, operations manage- ment, corporate strategy, and nonlinear dynamics in a wide range of systems, from supply chains to scientific revolutions. He has pioneered the development of man- agement flight simulators of corporate and economic systems. These flight simu- lators are used in research to understand and improve managerial decision making in complex dynamic systems; more importantly, they are now widely used by cor-
  • 5. porations and universities around the world for teaching, problem solving, and pol- icy design. Professor Sterman discovered system dynamics modeling in high school, studied it as an undergraduate at Dartmouth College, and received his PhD from MIT. He has been awarded the Jay W. Forrester Prize, given for the best pub- lished work in the field of system dynamics over the prior five years, and has four times won awards for teaching excellence from the students of the Sloan School. vi Preface Accelerating economic, technological, social, and environmental change challenge managers and policy makers to learn at increasing rates, while at the same time the complexity of the systems in which we live is growing. Many of the problems we now face arise as unanticipated side effects of our own past actions. All too often the policies we implement to solve important problems fail, make the problem worse, or create new problems. Effective decision making and learning in a world of growing dynamic com- plexity requires us to become systems thinkers-to expand the boundaries of our mental models and develop tools to understand how the
  • 6. structure of complex sys- tems creates their behavior. This book introduces you to system dynamics modeling for the analysis of pol- icy and strategy, with a focus on business and public policy applications. System dynamics is a perspective and set of conceptual tools that enable us to understand the structure and dynamics of complex systems. System dynamics is also a rigor- ous modeling method that enables us to build formal computer simulations of com- plex systems and use them to design more effective policies and organizations. Together, these tools allow us to create management flight simulators-micro- worlds where space and time can be compressed and slowed so we can experience the long-term side effects of decisions, speed learning, develop our understanding of complex systems, and design structures and strategies for greater success. The field of system dynamics is thriving. Over the past decade, many top com- panies, consulting firms, and governmental organizations have used system dy- namics to address critical issues. More innovative universities and business schools are teaching system dynamics and finding enthusiastic and growing en- rollments. Hundreds of primary and secondary schools, from kindergarten to high school, are integrating systems thinking, system dynamics, and computer simula-
  • 7. tion into their curricula. Tools and methods for system dynamics modeling, the li- brary of successful applications, and insights into the effective use of the tools with executives and organizations are all expanding rapidly. vii viii Preface FEATURES AND CONTENT University and graduate-level texts, particularly those focused on business and public policy applications, have not kept pace with the growth of the field. This book is designed to provide thorough coverage of the field of system dynamics to- day, by examining Systems thinking and the system dynamics worldview; Tools for systems thinking, including methods to elicit and map the structure of complex systems and relate those structures to their dynamics; Tools for modeling and simulation of complex systems; Procedures for testing and improving models; Guidelines for working with client teams and successful implementation. You will learn about the dynamics of complex systems, including the structures that create growth, goal-seeking behavior, oscillation and instability, S-shaped growth, overshoot and collapse, path dependence, and other
  • 8. nonlinear dynamics. Examples and applications include Corporate growth and stagnation, The diffusion of new technologies, The dynamics of infectious disease such as HIV/AIDS, Business cycles, Speculative bubbles, The use and reliability of forecasts, The design of supply chains in business and other organizations, Service quality management, Transportation policy and traffic congestion, Project management and product development, and many others. The goal of systems thinking and system dynamics modeling is to improve our understanding of the ways in which an organization’s performance is related to its internal structure and operating policies, including those of customers, competi- tors, and suppliers and then to use that understanding to design high leverage poli- cies for success. To do so this book utilizes Process Points that provide practical advice for the successful application of the tools in real organizations. Case studies of System Dynamics in Action that present successful applications ranging from global warming and the war on drugs to reengineering the supply chain of a major computer firm, marketing strategy in the automobile industry, and process improvement in
  • 9. the petrochemicals industry. System dynamics is not a spectator sport. Developing systems thinking and mod- eling skills requires the active participation of you, the reader, via Preface ix Challenges. The challenges, placed throughout the text, give you practice with the tools and techniques presented in the book and will stimulate your original thinking about important real world issues. The challenges range from simple thought experiments to full-scale modeling projects. Simulation software and models. The accompanying CD-ROM and web site (http://www.mhhe.com/sterman) include all the models developed in the text along with state-of-the-art simulation software to run them. There are several excellent software packages designed to support system dynamics modeling. These include ithink, Powersim, and Vensim. The CD and website include the models for the text in all three software formats. The disk also includes fully functional versions of the ithink, Powersim, and Vensim software so you can run the models using any of these packages
  • 10. without having to purchase any additional software. Additionally, the Instructor’s Manual and instructor’s section of the web site include suggested solutions for the challenges, additional assignments, Powerpoint files with the diagrams and figures from the text suitable for transparencies, suggested course sequences and syllabi, and other materials. INTENDED AUDIENCE The book can be used as a text in courses on systems thinking, simulation model- ing, complexity, strategic thinking, operations, and industrial engineering, among others. It can be used in full or half-semester courses, executive education, and self-study. The book also serves as a reference for managers, engineers, consul- tants, and others interested in developing their systems thinking skills or using sys- tem dynamics in their organizations. A NOTE ON MATHEMATICS System dynamics is grounded in control theory and the modern theory of nonlin- ear dynamics. There is an elegant and rigorous mathematical foundation for the theory and models we develop. System dynamics is also designed to be a practical tool that policy makers can use to help them solve the pressing problems they con- front in their organizations. Most managers have not studied nonlinear differential equations or even calculus, or have forgotten it if they did. To
  • 11. be useful, system dy- namics modeling must be accessible to the widest range of students and practicing managers without becoming a vague set of qualitative tools and unreliable gener- alizations. That tension is compounded by the diversity of backgrounds within the community of managers, students, and scholars interested in system dynamics, backgrounds ranging from people with no mathematics education beyond high school to those with doctorates in physics. X Preface IF YOU DON’T HAVE A STRONG MATHEMATICS BACKGROUND, FEAR NOT This book presents system dynamics with a minimum of mathematical formalism. The goal is to develop your intuition and conceptual understanding, without sacri- ficing the rigor of the scientific method. You do not need calculus or differential equations to understand the material. Indeed, the concepts are presented using only text, graphs, and basic algebra. Mathematical details and references to more ad- vanced material are set aside in separate sections and footnotes. Higher mathemat- ics, though useful, is not as important as the critical thinking skills developed here.
  • 12. IF YOU HAVE A STRONG MATHEMATICS BACKGROUND, FEAR NOT Realistic and useful models are almost always of such complexity and nonlinearity that there are no known analytic solutions, and many of the mathematical tools you have studied have limited applicability. This book will help you use your strong technical background to develop your intuition and conceptual understanding of complexity and dynamics. Modeling human behavior differs from modeling phy s- ical systems in engineering and the sciences. We cannot put managers up on the lab bench and run experiments to determine their transfer function or frequency re- sponse. We believe all electrons follow the same laws of physics, but we cannot assume all people behave in the same way. Besides a solid grounding in the mathe- matics of dynamic systems, modeling human systems requires us to develop our knowledge of psychology, decision malung, and organizational behavior. Finally, mathematical analysis, while necessary, is far from sufficient for successful sys- tems thinlung and modeling. For your work to have impact in the real world you must learn how to develop and implement models of human behavior in organiza- tions, with all their ambiguity, time pressure, personalities, and politics. Through- out the book I have sought to illustrate how the technical tools and mathematical concepts you may have studied in the sciences or engineering can be applied to the
  • 13. messy world of the policy maker. ~ FEEDBACK I welcome your comments, criticisms, and suggestions. Suggestions for additional examples, cases, theory, models, flight simulators, and so on, to make the book more relevant and useful to you are especially invited. I will update the website to incorporate user feedback and new materials. Email comments to <[email protected] mit .edu > . ACKNOWLEDGMENTS This work benefited immensely from the advice, criticism, and encouragement of many colleagues, students, and friends. I owe an immeasurable debt to my first system dynamics teachers, Dana Meadows, Dennis Meadows, and Jay Forrester, for their integrity, high standards, and passionate commitment. I’m particularly indebted to the exceptional students of the MIT Sloan School of Management. They constantly challenge me to make the discipline of system dynamics relevant, Preface xi useful, and exciting; I hope they’ve learned as much from me as I’ve learned from them. In addition, I thank my colleagues at the Sloan School
  • 14. and in the system dynamics community around the world, who helped by providing data and exam- ples, reviewing the draft, testing early versions in their courses, and in countless other ways. This group includes (but is not limited to) the following folks and institutions: Tarek Abdel-Hamid (Naval Postgraduate School); David Andersen, George Richardson (SUNY Albany); Ed Anderson (Univ. of Texas); Carlos Ariza, Sharon Els, Ken Cooper, Jim Lyneis, Hank Taylor (Pugh-Roberts Associates); George Backus (Policy Assessment Corporation); Bent Bakken (Norwegian Defense Re- search Establishment); Yaman Barlas (Bogazici University, Istanbul); Michael Bean (Powersim Corp.); Eric Beinhocker, Damon Beyer, Andrew Doman, Usman Ghani, Maurice Glucksman, Paul Langley, Norman Marshall (McKinsey and Company); Laura Black, John Carroll, Vanessa Colella, Ernst Diehl, Steve Ep- pinger, Charlie Fine, Mila Getmansky, Paulo Goncalves, Janet Gould Wilkinson, Jim Hines, Nan Lux, Brad Morrison, Tim Nugent, Nelson Repenning, Ed Roberts, Scott Rockart, George Roth, Ed Schein, Peter Senge (MIT); Allen and Jane Boorstein; Steve Cavaleri (Central Connecticut State Univ.); Geoff Coyle (Royal Military College of Science, UK, retired); Brian Dangerfield (Univ. of Salford); Pi1 Davidsen (Univ. of Bergen); Jim Doyle, Mike Radzicki,
  • 15. Khalid Saeed (Worcester Polytechnic Institute); Bob Eberlein, Tom Fiddaman, Dan Goldner, David Peterson, Laura Peterson (Ventana Systems); David Foley and Judy Berk; Andy Ford (Washington State Univ.); David Ford (Texas A&M University); Nathan Forrester (A. T. Kearney); Rich Goldbach (Metro Machine Corp.); Chris- tian Haxholdt, Heather Hazard (Copenhagen Business School); Jack Homer (Homer Consulting); Jody House (Oregon Graduate Institute); Bill Isaacs (Dia- logos); Sam Israelit (Arthur Andersen); Nitin Joglekar (Boston Univ. School of Management); Drew Jones (Sustainability Institute); Christian Kampmann, Erik Mosekilde (Technical Univ. of Denmark); Daniel Kim, Virginia Wiley (Pegasus Communications); Craig Kirkwood (Arizona State Univ.); Elizabeth Krahmer Keating (Northwestern Univ.); Don Kleinmuntz (Univ. of Illinois, Urbana-Champaign); David Kreutzer (GKA, Inc.); Robert Landel (Darden School of Business, Univ. of Virginia); David Lane (London School of Economics); Erik Larsen (City University, London); Winston J. Ledet, Winston P. Ledet (The Man- ufacturing Game, Inc.); Ralph Levine (Michigan State Univ.); Angela Lipinski (Society for Organizational Learning); Martin GroBmann, Frank Maier, Peter Milling (Univ. of Mannheim, Germany); Ali Mashayekhi (Sharif Univ. of Tech- nology, Teheran); Nathaniel Mass (GenCorp); Paul Monus
  • 16. (BP/Amoco), John Morecroft, Ann van Ackere, Kim Warren (London Business School); Erling Moxnes (Norwegian School of Economics and Business Administration); Rogelio Oliva (Harvard Business School); Mark Paich (Colorado College); Steve Peterson, Barry Richmond (High Performance Systems); Greg Petsch (Compaq Computer); Nick Pudar (General Motors); Jack Pugh, Julia Pugh, Roberta Spencer (System Dynamics Society), JQrgen Randers (World Wildlife Fund International); Nancy Roberts (Leslie College); Jenny Rudolph (Boston College); Jorge Rufat-Latre (Strategos); Anjali Sastry, Marshall van Alstyne (University of Michigan); Bob Stearns; Susan Sterman; Jim Thompson (Global Prospectus, LLC); John Voyer xii Preface (Univ. of Southern Maine); Lyle Wallis (Decisio, Inc.); Jim Waters (Waters Busi- ness Systems); Jason Wittenberg (Harvard Univ.); Eric Wolstenholme (Leeds Busi- ness School, UK); Pave1 Zamudio Ramirez (Monitor Company); the Copenhagen Business School, The International Network of Resource Information Centers (aka the Balaton Group), McKinsey and Company, the Norwegian School of Management, Pugh-Roberts Associates, the Society for Organizational Learning,
  • 17. the Technical University of Denmark, and, of course, the MIT Sloan School of Management. Special thanks to High Performance Systems, Powersim, SA, and Ventana Systems-and their great people-for providing their simulation software and translations of the models for the CD and website. The team at IrwidMcGraw-Hill deserves special mention for their enthusiasm, patience, and editorial help, particularly Scott Isenberg, Carol Rose, Jeff Shelstad, and Gladys True. Cara Barber and Kelley Donovan provided important secretarial support. Kathy Sullivan went beyond the call of duty on library research, data collec- Finally, the love and support of my family have been constant and essential. tion, editorial changes, and graphics. Thanks, Cindy, David, and Sarah. Table of Contents Preface vii PART I PERSPECTIVE AND PROCESS 1 1 Learning in and about Complex Systems 3
  • 18. 1.1 Introduction 3 1.1.1 1.1.2 Causes of Policy Resistance 10 1.1.3 Feedback 12 1.1.4 Process Point: The Meaning of Feedback 14 Challenge: Dynamics of Multiple-Loop Systems Policy Resistance, the Law of Unintended Consequences, and the Counterintuitive Behavior of Social Systems 5 14 1.2 Learning Is a Feedback Process 14 1.3 Barriers to Learning 19 1.3.1 Dynamic Complexity 21 1.3.2 Limited Information 23 1.3.3 Confounding Variables and Ambiguity 25 1.3.4 Bounded Rationality and the Misperceptions of Feedback 26 1.3.5 Flawed Cognitive Maps 28 1.3.6 Erroneous Inferences about Dynamics 29 1.3.7 Unscientific Reasoning: Judgmental Errors andBiases 30 Challenge: Hypothesis Testing 30 1.3.8 Defensive Routines and Interpersonal Impediments to Learning 32 1.3.9 Implementation Failure 33 1.4.1 Improving the Learning Process: Virtues of Virtual Worlds 34 1.4.2 Pitfalls of Virtual Worlds 35
  • 19. 1.4.3 Why Simulation Is Essential 37 1.4 Requirements for Successful Learning in Complex Systems 33 1.5 Summary 39 xiii xiv Contents 2 System Dynamics in Action 41 2.1 2.2 2.3 2.4 2.5 Applications of System Dynamics 41 Automobile Leasing Strategy: Gone Today, Here Tomorrow 2.2.1 Dynamic Hypothesis 44 2.2.2 Elaborating the Model 48 2.2.3 Policy Analysis 5 1 2.2.4 Impact and Follow-up 54 On Time and Under Budget: The Dynamics of Project Management 55 2.3.1 The Claim 56 2.3.2 Initial Model Development 57 2.3.3 Dynamic Hypothesis 58 2.3.4 The Modeling Process 61 2.3.5 Continuing Impact 64
  • 20. Playing the Maintenance Game 66 2.4.1 Dynamic Hypothesis 67 2.4.2 The Implementation Challenge 74 2.4.3 Results 76 2.4.4 Transferring the Learning: The Lima Experience 77 42 Summary: Principles for Successful Use of System Dynamics 79 3.1 The Purpose of Modeling: Managers as Organization Designers 84 3.2 The Client and the Modeler 84 3.3 Steps of the Modeling Process 85 3.4 Modeling Is Iterative 87 3.5 Overview of the Modeling Process 89 3 The Modeling Process 83 3.5.1 Problem Articulation: The Importance of Purpose 89 3.5.2 Formulating a Dynamic Hypothesis 94 3.5.3 Formulating a Simulation Model 102 3.5.4 Testing 103 3.5.5 Policy Design and Evaluation 103 4 Structure and Behavior of Dynamic Systems 107 3.6 Summary 104 4.1 4.2 4.3 Fundamental Modes of Dynamic Behavior 4.1.1 Exponential Growth 108
  • 21. 4.1.2 Goal Seeking 11 1 4.1.3 Oscillation 114 4.1.4 Process Point 116 Challenge: Identifying Feedback Structure from System Behavior 117 Interactions of the Fundamental Modes 4.2.1 S-Shaped Growth 118 4.2.2 S-Shaped Growth with Overshoot 121 Challenge: Identifying the Limits to Growth 4.2.3 Overshoot and Collapse 123 Other Modes of Behavior 127 4.3.1 Stasis, or Equilibrium 127 108 11 8 121 Contents xv 4.3.2 Randomness 127 4.3.3 Chaos 129 4.4 Summary 133 PART I1 TOOLS FOR SYSTEMS THINKING 135 5 Causal Loop Diagrams 137 5.1 5.2 5.3
  • 22. 5.4 5.5 5.6 Causal Diagram Notation 137 Guidelines for Causal Loop Diagrams 5.2.1 Causation versus Correlation 141 5.2.2 Labeling Link Polarity 142 Challenge: Assigning Link Polarities 143 5.2.3 Determining Loop Polarity 143 Challenge: Identifying Link and Loop Polarity Challenge: Employee Motivation 147 5.2.4 Name YourLoops 148 5.2.5 Indicate Important Delays in Causal Links 150 5.2.6 Variable Names 152 5.2.7 Tips for Causal Loop Diagram Layout 153 5.2.8 Choose the Right Level ofAggregation 154 5.2.9 Don’t Put All the Loops into One Large Diagram 5.2.10 Make the Goals of Negative Loops Explicit 155 5.2.11 Distinguish between Actual Process Point: Developing Causal Diagrams from Interview Data 157 Challenge: Process Improvement 158 Conceptualization Case Study: Managing Your Workload 5.4.1 Problem Definition 159 5.4.2 IdentiJLing Key Variables 160 5.4.3 Developing the Reference Mode 160 5.4.4 Developing the Causal Diagrams 163 5.4.5 Limitations of the Causal Diagram 166 Challenge: Policy Analysis with Causal Diagrams Adam Smith’s Invisible Hand and the Feedback Structure of Markets 169 Challenge: The Oil Crises of the 1970s 172
  • 23. Challenge: Speculative Bubbles 173 Challenge: The Thoroughbred Horse Market 5.5.1 Market Failure, Adverse Selection, Challenge: The Medigap Death Spiral 176 Explaining Policy Resistance: Traffic Congestion 5.6.1 Mental Models of the Traffic Problem 178 5.6.2 Compensating Feedback: The Response to Decreased Congestion 18 1 5.6.3 The Mass Transit Death Spiral 185 5.6.4 Policy Analysis: The Impact of Technology 188 5.6.5 Compensating Feedback: The Source of Policy Resistance 189 141 145 154 and Perceived Conditions 156 159 168 174 and the Death Spiral 174 177
  • 24. xvi Contents Challenge: Identifying the Feedback Structure of Policy Resistance 190 5.7 Summary 190 6.1 Stocks, Flows, and Accumulation 191 6 Stocks and Flows 191 6.1.1 Diagramming Notation for Stocks and Flows 192 6.1.2 Mathematical Representation of Stocks and Flows 193 6.1.3 The Contribution of Stocks to Dynamics 195 6.2.1 Units of Measure in Stock and Flow Networks 198 6.2.2 The Snapshot Test 199 Challenge: Identifying Stocks and Flows 201 6.2.3 Conservation of Material in Stock and Flow Networks 201 6.2.4 State-Determined Systems 202 6.2.5 Auxiliary Variables 202 6.2.6 Stocks Change Only through Their Rates 204 6.2.7 Continuous Time and Instantaneous Flows 206 6.2.8 Continuously Divisible versus Quantized Flows 207 6.2.9 Which Modeling Approach Should You Use? 208 6.2.10 Process Point: Portraying Stocks and Flows in Practice 209 When Should Causal Loop Diagrams Show Stock and Flow Structure? 210 6.2 Identifying Stocks and Flows 197 6.3 Mapping Stocks and Flows 210
  • 25. 6.3.1 Challenge: Adding Stock and Flow Structure to Causal Diagrams 2 11 Challenge: Linking Stock and Flow Structure with Feedback 6.3.2 Aggregation in Stock and Flow Mapping 213 Challenge: Modifying Stock and Flow Maps Challenge: Disaggregation 214 6.3.3 Guidelines for Aggregation 216 6.3.4 System Dynamics in Action: 6.3.5 Setting the Model Boundary: 6.3.6 System Dynamics in Action: Automobile Recycling 225 212 213 Modeling Large-Scale Construction Projects “Challenging the Clouds’’ 222 2 18 6.4 Summary 229 7.1 Relationship between Stocks and Flows 232 7 Dynamics of Stocks and Flows 231 7.1.1 Static and Dynamic Equilibrium 232 7.1.2 Calculus without Mathematics 232 7.1.3 Graphical Integration 234 Challenge: Graphical Integration 239 7.1.4 Graphical Diflerentiation 239 Challenge: Graphical Differentiation 24 1
  • 26. 7.2 System Dynamics in Action: Global Warming 241 Contents xvii 7.3 System Dynamics in Action: The War on Drugs 250 7.4 Summary 262 Dynamics of Simple Structures 263 7.3.1 The Cocaine Epidemic after 1990 258 8 Closing the Loop: 8.1 8.2 8.3 8.4 8.5 8.6 First-Order Systems 263 Positive Feedback and Exponential Growth 8.2.1 Analytic Solution
  • 27. for the Linear First-Order System 265 8.2.2 Graphical