5.12 system dynamics: systemic feedback modeling for...

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5.12 SYSTEM DYNAMICS: SYSTEMICFEEDBACK MODELING FOR POLICYANALYSIS

Yaman BarlasIndustrial Engineering Department, Bogvaziçi University, Istanbul, Turkey

CONTENTS

1. Introduction2. Dynamic problems and systemic feedback perspective3. Modeling methodology and tools4. Dynamics of basic feedback structures5. Formulation principles and generic model structures6. Mathematical and technical issues7. Model testing, validity, analysis, and design8. Conclusions and future

Knowledge in depthGlossaryBibliographyAcknowledgments

“Que sais-je?” (“What do I know?”) (Montaigne)

To the memory of my father who somehow taught me to bea student for life. . .

SUMMARY

The world is facing a wide range of increasinglycomplex, dynamic problems in the public and privatearenas alike. System dynamics discipline is an attemptto address such dynamic, long-term policy problems.Applications cover a very wide spectrum, includingnational economic problems, supply chains, projectmanagement, educational problems, energy systems,sustainable development, politics, psychology,medical sciences, health care, and many other areas.This article provides a comprehensive overview ofsystem dynamics methodology, including its concep-tual/philosophical framework, as well as the technicalaspects of modeling and analysis. The articlefrequently refers to other articles in the same Themeas appropriate. Today, interest in system dynamicsand related systemic disciplines is growing very fast.System dynamics can address the fundamental struc-tural causes of the long-term dynamic contemporarysocio-economic problems. Its “systems” perspectivechallenges the barriers that separate disciplines. Theinterdisciplinary and systemic approach of systemdynamics could be critical in dealing with the increas-ingly complex problems of our modern world in thisnew century.

1. INTRODUCTION

At the start of the new century, the world is facing awide range of increasingly complex, dynamic problemsin the public and private arenas alike: nations desireeconomic growth, yet growth also results in environ-mental and ecological – sometimes irreversible –destruction. Chronic high inflation, budget deficits,and simultaneous high unemployment together consti-tute a persistent problem for many developing coun-tries. There is a widening gap between the so-called“north” and “south” nations and also a widening gapbetween the poor and the rich in a given nation.Markets – both commodities and financial – generateall sorts of short, medium, and long-term cycles, theuncertainty of which has been a major problem forprivate and public decision-makers for decades. Thou-sands of small companies are initiated each year, manyenjoying a fast – yet unbalanced – growth, only to befollowed by bankruptcy in a few years. Globalization isposing new challenges in the social, economic, andcorporate domains. With the unprecedented speed ofinternational communication, the “new” economymeans that companies will soon find themselves in a

complex network of relationships and only those thatcan understand its dynamics will be able to compete.We are already experiencing the first worldwide reces-sion of the “new” economy – of the new millennium. Inthe socio-economic domain, there is a tension betweenthe interests of the nation-states and those of interna-tional conglomerates, between increasing interna-tional interaction and increasing micro-nationalism.Many nations worldwide face the dilemma between fulldemocracy/human rights and the special measuresoften needed against terrorism.

The examples listed above have some commondefining characteristics: they are all dynamic, long-term policy problems. “Dynamic” means, “changingover time.” Dynamic problems necessitate dynamic,continuous managerial action. Optimum oil welllocation problem may be a very difficult problem, butit is nevertheless a “static” decision problem. Thedecision is made once, and it is not periodically moni-tored and adjusted depending on the results. Thedynamic problems mentioned above, however, mustbe continuously managed and monitored. Thus, inthe specific context of management and policymaking, dynamic problems are the ones that are ofpersistent, chronic, and recurring nature. We takemanagerial actions, observe the results, evaluate themand take new actions, yielding new results, observa-tions, further actions, and so on, which constitutes a“closed loop.” In other words, most dynamic manage-ment problems are “feedback” problems. Feedbackloops exist not only between the control action andthe system, but also in between the various compo-nents within the system. Therefore, it is also said thatsuch dynamic feedback problems are “systemic” innature, that is, they originate as a result of the compli-cated interactions between the system variables.Finally, since dynamic management necessitates astream of dynamic decisions, the research focusshould not be the individual decisions, but the rulesby which these decisions are made, that is, the “poli-cies.” The individual decisions are the outcomes ofthe application of the adopted policies.

System dynamics discipline emerged in the late1950s, as an attempt to address such dynamic, long-term policy issues, both in the public and corporatedomain. Under the leadership of Jay W. Forrester, agroup of researchers at M.I.T. initiated a new fieldthen named Industrial Dynamics. (see “On thehistory, the present, and the future of system dynam-ics,” EOLSS on-line, 2002). The first application areaof the methodology was the strategic management ofindustrial problems. The main output of this researchwas the publication of Industrial Dynamics, the seminalbook that introduced and illustrated the new method-ology in the context of some classical industrial/busi-ness problems. The next major project was UrbanDynamics, presenting a dynamic theory of how theconstruction of housing and businesses determinethe growth and stagnation in an urban area. With thisapplication, “industrial dynamics” method moved tothe larger domain of socio-economic problems andwas eventually renamed “system dynamics.” Thesecond application in the larger socio-economic

domain was World Dynamics (and Limits to Growth).These models show how population growth andeconomic development policies can interact to yieldovershoot and collapse dynamics, when crowding andoverindustrialization exceed the finite capacity of theenvironment. In a short period of time since the late1970s, applications have expanded to a very widespectrum, including national economic modeling,supply chains, project management, educationalproblems, energy systems, sustainable development,politics, psychology, medical sciences, health care,and many other areas. In 1983, the InternationalSystem Dynamics Society was formed. The currentmembership of the Society is over 1,000. The numberof worldwide practitioners is probably much higherthan this number.

The purpose of the Theme articles (see Knowledgein depth) is to present a thorough exposition of themajor aspects of system dynamics method, includingits philosophical and historical roots, its technicalcomponents, selected exemplary applications, andspecific illustrations of its potential in sustainabilitydiscourse. The Theme articles, written by experts inrespective domains, are grouped under some natural“topics.” We start with “System dynamics in action,”EOLSS on-line, 2002, a small collection of exemplaryapplications to give the reader an idea of how themethodology is applied, illustrating the breadth, aswell as the depth. The second topic consists of articlesdiscussing the historical, conceptual and philosophi-cal foundations of the field. The third topic of theTheme covers some fundamental concepts and toolsof the system dynamics methodology. We next focuson selected technical/mathematical issues in model-ing, numerical problems in simulation and other soft-ware considerations. The fifth topic is about policyimprovement and implementation issues, coveringpublic policy as well as business strategy, plus somegeneral concepts and procedures of implementation.Finally, the last topic consists of six different applica-tions of system dynamics approach and methodologyin areas closely related to sustainable development.

The goal of this particular Theme-level article is toprovide a comprehensive overview of system dynamicsmethodology, including its conceptual/philosophicalframework, as well as the technical aspects of modelconstruction, analysis and policy design.

2. DYNAMIC PROBLEMS AND SYSTEMICFEEDBACK PERSPECTIVE

2.1. Dynamic feedback problems

The term “dynamic,” in its general sense, means “inmotion” or “changing over time.” Dynamic problemsare characterized by variables that undergo signifi-cant changes in time. Inventory and productionmanagers must deal with inventories and orders thatfluctuate; city administrations face increasing levels ofsolid waste, air and water pollution; wildlife managersare concerned about declining species diversityworldwide; citizens raise their voices against escalat-

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ing arms race; at a personal level, we are concernedwhen our blood pressure or our heartbeat is unstable,or when our temperature goes up; national leadersare worried about increasing unemployment andinflation levels; the small company is in danger whena few years of fast growth is followed by a sharpdecline in the market share. In each one of thesecases, there are one or more patterns of dynamicbehavior that must be managed (controlled, alteredor even reversed). Figure 1 illustrates some typicaldynamic patterns observed in real life: explosivelygrowing world population, oscillating commodity(pulp) prices, exponentially declining cocaine prices,and growth-then-collapse in revenues (of Atari, Inc).Yet the defining property of a dynamic problem inour sense (in systemic, feedback sense) is not merely thevariables being dynamic. More critically, in a systemicdynamic problem, the dynamics of the variables mustbe closely associated with the operation of the internalstructure of some identifiable system. It is said that thedynamics is essentially caused by the internal feed-back structure of the system. (endogenous perspective).Thus, oscillating inventories is a systemic, feedbackproblem because oscillations are typically generatedby the interaction of ordering and production poli-cies of managers. It would not be a systemic problem,if the oscillations were determined by an externalforce like fluctuating weather conditions: although itwould still be a dynamic problem in its technicalsense. In this latter case, there is not much the inven-

tory managers can do to eliminate or reduce the oscil-lations. If, as allowed by a new deregulation, a multi-national chain colonizes the grocery market, which inturn causes an unavoidable decline in small grocerystore sales, this would not be a systemic feedbackproblem. There is not much “management” the smallgrocery store owner can do in order to reverse thederegulation or influence the policies of the multina-tional chain. The importance of the distinction is thatif/when the dynamics are dictated by forces externalto the system; there is not much space or possibilityfor managerial control and improvement. Systemicfeedback problems on the other hand, necessitatedynamic, continuous managerial action.

Dynamic management problems in real life aretypically feedback problems: we take managerialactions, observe the results, evaluate them and takenew actions, yielding new results, observations,further actions, and so on, which constitutes a “feed-back loop.” Feedback loops exist not only betweenthe managerial action and the system, but also inbetween the various elements within the system. Thatis, most dynamic management problems are also“systemic” in nature. The main purpose of systemdynamics methodology is to understand the causes ofundesirable dynamics and design new policies toameliorate/eliminate them. Managerial understand-ing, action and control are at the heart of themethod. System dynamics thus focuses on dynamicproblems of systemic, feedback nature.

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(a) World population growth (See “The ECOCOSM Paradox,”EOLSS on-line, 2002).

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Figure 1. Illustrations of some typical dynamic patterns observed in real data(a) Exp. growth, (b) Oscillation, (c) decay, and (d) Boom-then-collapse.

2.2. Systems, problems, and models

The term system refers to “reality” or some aspects ofreality. A system may be defined as a “collection ofinterrelated elements, forming a meaningful whole.”So, it is common to talk about a financial system, asocial system, a political system, a production system,a distribution system, an educational system, or abiological system. Each of these systems consists ofmany elements interacting in a meaningful way, sothat the system can presumably serve its “purpose.”But it is not trivial for a system to serve its purposeeffectively: the global socio-economic system is stillfacing millions literally starving to death – certainlynot an intended result; on the other hand oureconomic development generates solid waste, air andwater pollution levels that threaten the sustainabilityof life on earth; commodity and financial systemsalike generate highly unstable fluctuations worldwide;national economies are often unable to control thesimultaneous problems of budget deficit, inflation,and unemployment; small companies typically enjoy afast growth initially, only to be followed by a suddencollapse – an inevitable result of the unbalancedgrowth itself; as individuals, we must often deal withhealth problems such as high blood pressure orcholesterol that our very life style creates, and so on.In short, systems at all levels, scale and scope, whilesolving one set of problems, simultaneously“produce” other complex challenges.

A common scientific tool used in investigatingproblems and solutions is modeling. A model can bedefined as “a representation of selected aspects of areal system with respect to some specific problem(s).”Thus, we do not build “models of systems,” but buildmodels of selected aspects of systems to study specificproblems. The crucial motivation, purpose that trig-gers modeling is a problem. The problem can bepractical (such as increasing levels of unemployment,declining species population, or collapsing stockprices) or theoretical (such as analyzing a cognitivetheory of how knowledge is acquired, or testing thevalidity of Marx’s theory of class struggle, or whetherlong-term ecological/environmental sustainability ispossible in a capitalist system). In any case, without aproblem-purpose, “modeling a system” is meaning-less. One does not build a model of a national econ-omy, or a species population, or stock market. Onebuilds a model of some selected elements and rela-tions in a national economy that are likely to havecaused a recent upward trend in unemployment; or amodel of selected factors believed to play a strongrole in an unstoppable decline in the population of aspecies of concern; or a model that selectively focuseson those factors that are likely to explain a crash in aspecific stock market in a specific time period. In thetheoretical domain, the principle is the same: onebuilds a model to study a specific problem/theory so as to contribute to the debate in the relevantcommunity.

Models can be of many types: Physical modelsconsist of physical objects (such as scaled models ofairplanes, submarines, architectural models, models

of molecules). Symbolic models consist of abstractsymbols (such as verbal descriptions, diagrams,graphs, mathematical equations). System dynamicsmodels are symbolic models consisting of a combina-tion of diagrams, graphs and equations. In anotherdimension, models can be static, representing staticbalances between variables, assumed to be constant ina time period (such as an architectural model or amathematical equation representing the relationbetween price, supply and demand at a point intime). Or they can be dynamic, representing how thevariables change over time (such as an aircraft simu-lator to train pilots, Newton’s laws of motion, or amathematical model of price fluctuations incommodity markets). Another typical classification ofmodels is: descriptive versus prescriptive. Descriptivemodels describe how variables interact and how theproblems are generated “as is,” they do not state howthe system “should” function in order to eliminate theproblems. Prescriptive (often optimization) modelshowever assume certain “objective functions” andseek to derive the “optimum” decisions that maximizethe assumed objective functions. Nonlinear dynamicfeedback problems are typically mathematicallyimpossible to be represented and solved by optimiza-tion models. System dynamics models are thusdescriptive models. The policy recommendations arederived not by the model, but by the modeler, as aresult of a series of simulation experiments. Finally,dynamic models can be continuous or discrete in time.In time-continuous models, change can occur at anyinstant in time (such as air temperature, humidity, orpopulation of a city), whereas in time-discretemodels, change can only occur at pre-defined discretepoints in time (such as salaries changing in multiplesof months, or student grade point average changingeach semester). Real dynamic systems consist of bothtypes of dynamics. So a system dynamics model can becontinuous, discrete or even hybrid. In practice, if thediscrete time steps associated with the discrete vari-ables is small enough compared to the time horizonof interest (such as a model involving salary dynamics,simulated for decades) one can safely do the time-continuity assumption. The rationale behind thisapproximation is that continuous-discrete hybridmodels can be very cumbersome to build, analyze,and communicate. Continuous system dynamicsmodels are mathematically equivalent to differential(or integral) equations, whereas discrete models aredifference equations. In sum, typical system dynamicsmodels are descriptive, continuous or discrete dynam-ics models, focusing on policy problems involvingfeedback structures.

2.3. Structure and behavior

The structure of a system can be defined as “the total-ity of the relationships that exist between system vari-ables.” In a production system, the structure wouldinclude the material and information flows related toproduction, where and how the various stocks arestored and shipped, how the ordering and produc-tion decisions are made, and so on. The structure of


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1. Constant

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Figure 2. Categories of basic dynamic behavior patterns typicallyseen in systemic feedback problems

the system operates over time so as to produce thedynamic behavior patterns of the system variables. It issaid, “the structure creates the behavior.” The struc-ture of the production department operates over timeand generates production rate dynamics, raw materialorders, oscillating inventories, and depending onsystem boundary, dynamics of product quality andsales. Dynamic behaviors of a few variables observedin different real systems were already illustrated inFigure 1. Figure 2 presents a generic template of mostdynamic behavior patterns typically encountered indynamic feedback problems. As seen in the template,the common dynamic behavior categories are:constant, growth, decline, growth-then-decline,decline-then-growth, and oscillatory. In each categorythere are different variants of the representative cate-gory, as seen in the figure. Furthermore, in realitythese basic dynamic patterns can combine in variousways, such as oscillations followed by collapse orgrowth followed by decline-then-growth. Examples ofsuch dynamic behavior patterns will be seen through-out this Theme.

The structure of a real system can naturally beextremely complicated, hence not exactly orcompletely known. But we do know that there is astructure that operates to produce the dynamics ofsystem variables. The structure of the model is a repre-sentation of those aspects of the real structure that webelieve (or hypothesize) to be important, with respectto the specific problems of concern. In the produc-

tion example, the problem may be large oscillationsin inventories or a persistent decline in product qual-ity. These two different problem-orientations wouldyield different model structures. The structure of themodel comprises only those parts of the real structurethat play a primary role in creating the problematicbehavior patterns. The structure of a productionmodel to study the problem of oscillating inventoriesmust include all variables, factors and relationsresponsible for inventory oscillations

The structure of a system dynamics model consistsof the set of relations between model variables, math-ematically represented in the form of equations. Thatis, the structure of a system dynamics model is a set ofdifferential and/or difference equations. The analyti-cal (mathematical) solution of a dynamic model, ifobtainable, would give the exact formula for thedynamic behaviors of variables. So, one way of obtain-ing the dynamic behavior of a model is solving itanalytically. This is often possible in linear cases, butvery rarely possible in non-linear ones. In such cases,the dynamic behavior of the model is obtained bysimulation. Simulation is essentially a step-by-stepoperation of the model structure over compressedtime. Much like the operation of the real structureover real time, the model structure operates oversimulated time, so that the dynamics of model vari-ables gradually unfold. A crucial notion in systemdynamics method is that the term “model” refersstrictly to the structure, to the set of equationsdescribing it. The dynamic behavior of the model,whether represented in equations or plotted graphi-cally, is not a “model” in system dynamics sense. Thedynamic behavior is the output or the result of amodel, very different conceptually from the modelitself. In some disciplines such as forecasting, an equa-tion representing a dynamic behavior would be legit-imately called a model; but the structure-behaviordistinction is crucial in system dynamics method.

Example 2.1

Consider a simple population dynamics of a singlespecies on a large, isolated island. The structure ofthis system in real life would consist of males andfemales mating and reproducing, growing up, gettingold, and eventually dying. Assuming that there isplenty of food, no competition, no epidemic, and alarge island, if we start with a small number of animals(males and females), the operation of this structureover time would create an exponential growth in thepopulation.

Assume that our purpose is to study this growth inpopulation, its doubling time and its sensitivity todifferent birth and death rates. In the real system,there would be many details like some membersbeing weak and unable to find food or mate, territo-rial fights, different categories of death (old age,fights, disease) geographical distribution of the popu-lation, weather conditions, seasonal changes in habi-tat, and so on. But for the simple purpose statedabove, these details can perhaps be omitted and asimple model would be:


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dx––– = births – deaths, which states that the rate of dtchange of population is (births – deaths), where x(t)denotes the population of the species at time t, andbirths and deaths must be specified.

The simplest formulation for births and deaths is theassumption that they are proportional to the popula-tion: births=bx and deaths=fx, where b is the birth frac-tion per year and f is the death fraction per year.Thus:

dx––– = bx – fxdt

The above model is a differential equation. Alterna-tively, the same equation can be represented in inte-gral form:

x(t) = x(0) + ∫0t(bx – fx)dt,

which states that the population at time t is given byits value at time 0 plus the sum of all births minus alldeaths between time 0 and time t. Population at time0 is assumed to be known as x(0).

In order to be able to solve this model, we mustmake one final assumption about the parameters band f. In the simplest case, these fractions areassumed to be constant. Then, the simple differentialequation model for population growth becomes:

dx––– = (b – f )x = kx,dt

where we let k = (b – f ), also a constant.

To solve, use separation of variables and integrate:

1∫ (–) dx = ∫kdt, which yields:x

lnx + C1 = kt + C2, where C1 and C2 are arbitraryconstants of integration. Combining:

lnx = kt +C3 , where C3 is another constant. Takingexponential of both sides:

x = ekt+C3 = eC3ekt = Cekt, where C is yet another constant.To evaluate it, use x(0):

x(0) = Ce0, or C = x(0). Thus, the complete solution is:

x(t) = x(0)ekt.

We can plot this solution for possible values of kand obtain the dynamic behaviors shown in Figure3(a). Note that the dynamic behavior of the modeldepends on the value of k = b – f. If b > f, then thebehavior is an exponential growth, if b < f, then thebehavior is a negative exponential decline and if b = f,then the population stays constant.

The structure of this system is represented by thedifferential (or integral) equation given above. Itsdynamic behavior is described by the solution equa-tion (x(t) = x(0)ekt). Alternatively, we typically repre-sent the structure by a stock-flow diagram (andassociated simulation equations). The dynamicbehavior is then obtained by simulation and pre-sented graphically. The stock-flow structure of thesimple population model built in STELLA software

and the simulated dynamic behavior are shown inFigure 3 (b) and (c).

2.4. Principles of “systemic feedback” approach

System dynamics discipline deals with dynamic policyproblems of systemic, feedback nature. As discussedabove, such problems arise from the interactionsbetween system variables and from the feedbacksbetween the managerial actions and the system’s reac-tions. The purpose of a system dynamics study is tounderstand the causes of a dynamic problem, andthen search for policies that alleviate/eliminate them.This specific purpose necessitates the adoption of aparticular philosophy of modeling, analysis anddesign. This philosophy can be called “systemic feed-back” philosophy (or approach or thinking orperspective). Systemic feedback thinking has impor-tant historical links with various disciplines andphilosophies, namely General Systems Theory(Ludwig von Bertalanffy), Systems Theory andSciences (i.e. Kenneth Boulding and Herbert Simon),Systems Approach (i.e. West Churchman), Cybernet-ics (Norbert Wiener) and Feedback Control Theory(Gordon Brown). (see “The role of system dynamicswithin the ‘Systems’ movement,” EOLSS on-line,2002) In a sense, systemic feedback philosophy inte-grates systems theory with cybernetics and feedbackcontrol theories. It is thus a unique systems theory,placing critical emphasis on the dynamic and feed-back (closed-loop) nature of policy problems. Fromthis perspective, it is possible to identify some princi-ples that are absolutely essential to systemic feedbackapproach:

Principle 1: Importance of causal relations (as opposedto mere correlations).The purpose of a system dynamicsstudy (“understanding and improving the dynam-ics”) is very different from short-term prediction(forecasting) of future values of variables. The verypurpose of system dynamics study requires that themodel consist of causal relations, not mere statisticalcorrelations. It is possible to generate excellentshort-term forecasts by non-causal correlationalmodels, but impossible to understand and controldynamic problems. For instance, swimsuit sales andice cream sales are highly positively correlated (bothgoing up in spring-summer season). If we have dataon swimsuit sales (x), we can construct a correla-tional model y = f(x) that would provide excellentforecasts for ice cream demand. For this narrowforecasting problem, the above model is very func-tional. On the other hand, assume that a company isfaced with a consistent decline in ice cream salesafter several years of growth and our objective is tounderstand (and reverse) this problematic dynam-ics. In this case, the correlational model y = f(x)would be of no use, since this model gives no clue asto what “causes” the dynamics of ice cream sales.The input variable (swimsuit sales) has no causalinfluence on ice cream sales (y) whatsoever. Icecream sales did not go down because swimsuit saleswent down and they will certainly not improve byimproving swimsuit sales (which is most likely not


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Figure 3. Illustration of model structure and dynamic behaviour (a) the solution of a simple population model, (b) its structure (stock-flowrepresentation) and (c) its simulated dynamic behavior

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affects births. (The more people, the more birthsthere will be). Thus we have:

Births Population

The above picture says that over time, births deter-mine population and population determines births.Such a circular causality is only possible dynamically;it requires passage of time, because at a fixed instantin time it would be impossible for births to determinepopulation and be determined by it. But over time,cause and effect (births and population) continu-ously trade places. More births mean higher popula-tion, which causes even more births, causing evenhigher population, and so on. The operation of thisloop over time would create exponentially growingpopulation. The feedback loop is in this sense the“engine” of change. (As will be seen later, interactionof multiple loops would constitute a much moresophisticated engine, creating much more sophisti-cated dynamics). In general, for a feedback loop toform between two variables, one or more variableswould intervene. For instance, using the very firstcausality example above:

Caloric intake Weight gain

Concern about weight

The above picture states that the more calories I take,the more weight I gain and as I realize my weight gain,I become concerned about it, which leads me to cutdown on my caloric intake. In the original example,caloric intake was cause and weight gain was the effect.In this dynamic version, we observe that weight can bein time an indirect cause of caloric intake, via the newvariable “concern about weight.” Notice also that thedynamic behavior of this loop would be very differentfrom the births-population loop; it would keep thecaloric intake under control. The behavior of this loopwould be a convergent one, rather than a divergentone. A more technical discussion of these two types ofloops will be given later in the following section.

The above 3–variable loop illustrates anotherimportant sub-principle: A cause-effect arrow in amodel must represent a “direct causality” between twovariables, given the other variables in the model. Inthe above example, it would be wrong to draw a linkfrom weight to caloric intake directly (caloric intake← weight), since the effect must really go through theintervening variable “concern about weight.” As amore typical example, consider Quality of a product,Demand, Sales and Revenues. A simple causaldiagram would be:

Quality Demand Sales Revenues

even part of our business)! Similarly, one can showthat rainfall and skin cancer incidences are nega-tively correlated. (In cloudy regions, rainfall goes upand skin cancer goes down due to reduced sunrays).But rainfall does not cause skin cancer incidence togo down. If we confuse this negative correlation withnegative causation, we would start marketing bottledrainfall for people to rub on their skins three timesa day!

A causal relation y=f(x) means that the input vari-able (x) has some causal influence on the output vari-able (y). Statements like “an increase in caloric intakecauses weight gain” / “increased death rate causes thepopulation to decline” / “increased pesticide applica-tion causes an increase in bird deaths” or “an increasein price causes the demand for a given commodity togo down” all reflect simple cause-effect relations. Ineach of these examples, if the cause variable ischanged, one expects “some degree of change” in theeffect variable. The term “expects” is important in thedefinition of “causality” in systemic feedback model-ing. Whether the expected change actually occurs orwhat kind of change is observed (in the model or inreality) depends on many other factors. Each causalrelation in a system dynamics model is a “ceterisparibus” argument: “other things being equal” thecausality is expected to yield the stated effect. Sincethere are many other varying factors in a dynamicfeedback model (and in reality), ceteris paribuscondition would not/should not typically hold. Thus,an increase in price may not actually result in adecline in demand, because the price of the competi-tor product may have increased even more or thequality of the product may have simultaneously goneup. So the notation X → Y reads “other things beingequal, a change in X causes a change in Y.”

From a philosophical perspective, causality is a verydifficult and debatable notion. There is no universallyaccepted, non-problematic definition of causation inphilosophy. Yet the notion of causality needed forsystem dynamics modeling is an operational, practicalone: non-controversial cause-effect relations, wellestablished either by direct real-life experience or byscientific evidence in the literature. In adopting thisdefinition, the main idea is to avoid using mere statis-tical correlations in lieu of causation. Once this isunderstood, in this operational, practical sense, thereis normally no disagreement on what is “causal” andwhat is merely “correlational.” (The correlational andcausal examples given in the above paragraph shouldmake the point clear. More examples will be seenlater in section 3).

Principle 2: Importance of circular causality (feedbackcausation) over time. Identification of one-way causalrelations described above is only the first step indynamic feedback model conceptualization. Thenext crucial phase is the identification of dynamic,circular causalities (feedback loops) over time. One-way causality is in a sense “static” causality at a point(or small interval) in time. The relation births →population states that births are a cause and popula-tion is an effect. But when examined dynamicallyover time, it is also true that population causally


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The diagram says: higher quality causes an increase indemand, meaning higher sales, hence higherRevenues, (assuming fixed price). In daily talk, oneoften summarizes the above diagram by “qualitycauses increased revenues.” An erroneous causaldiagram would be obtained, if one were to implementthis simplification in the above diagram:

Quality

Demand Sales Revenues

(a) Wrong inclusion of “indirect” causality.

Quality Revenues

(b) “Correct” in the context of given variables.

The conceptualization in (a) is clearly wrong, as it is“double counting” the effect of quality on revenues.Quality has no known direct influence on Revenues,other than influencing it via Demand. Also note thatthe diagram would still be wrong, even if one elimi-nated the arrow from Quality to Demand (to avoiddouble counting). Given that the model already hasthe variables Demand and Sales, it would be illogicalfor the Quality to affect the Revenues directly.However, the diagram in part (b) would be a correctsimplification, given the assumption that Demandand Revenues are not needed in the model. In thiscase, the “indirect” causal arrow from Quality toRevenues is an acceptable simplification. Thus theprinciple of “direct causality” is important in systemdynamics, but it is also relative: “direct causality in thecontext of other variables in the model.”

Principle 3: Dynamic behavior pattern orientation (ratherthan event-orientation. It is crucial to reiterate that thepurpose of a system dynamics study is to understandthe causes of a dynamic problem, and search for poli-cies that alleviate/eliminate them. Dynamic problemsare characterized by undesirable performancepatterns, not by one or more isolated “events.” Inordinary daily life, we all react to events: a suddendrop in the stock prices, a jump in the interest rates,resignation of a cabinet, or the September 11 attackon the World Trade Center are dramatic examples.Dynamic feedback approach argues that such eventscan not be analyzed or understood in isolation fromtheir past dynamics. When isolated by their past histo-ries, and by the dynamics of related variables, eventsseem random, unavoidable, and externally caused.But dynamic systems approach argues that mostimportant events occur as a result of some accumula-tions (often hidden) reaching threshold levels overtime. Short-term event orientation makes it impossi-ble to understand the real historical and structuralcauses of events. Hence, analysis and understandingrequires that we move away from the common event-orientation and focus on historical behavior patternsassociated with potential events. With the proper

behavior-pattern orientation, the goal is to constructa hypothesis (a model structure) that explains whyand how the dynamic patterns of concern are gener-ated. (Typical dynamic behavior patterns were illus-trated in Figures 1 and 2).

Principle 4: Internal structure as the main cause ofdynamic behavior (Endogenous perspective). The struc-ture of a system was already defined as “the totalityof the relationships that exist between system vari-ables.” One way of representing the structure of asystem is diagramming the causal links and (espe-cially) loops that exist between the variables. Asstated above, the interaction of the feedback loopsin a system is the main engine of change for thesystem. That is why it is said: “the structure causesthe behavior of the system.” This principle is criticalin a system dynamics study, because the purpose is tounderstand the causes of an undesirable behaviorand try to improve it. The principle can be betterillustrated by an example.

Example 2.2

Consider the dynamics of the population of a moderncity – perhaps a strong early growth followed by stag-nation. A static and exogenous model for populationwould be:

Population = f(Job availability, Salaries, Expenses, Houseprices, House availability, Crime rate, School quality,Air/water quality. . .)

A pictorial representation of this model is shown inFigure 4(a). This model basically says that it is possi-ble to determine population level, once the values ofthe input variables (eight of them) are given.

This model has some very strong assumptions. Itassumes that the input variables are strictly independ-ent variables. That is, they are not influenced at all byPopulation, or by any of the other seven input vari-ables. (These external inputs are called “forcing”functions in engineering and their existence makesthe system “forced” or “non-autonomous”). Themodel in Figure 4 (a) is of course an extremely unre-alistic situation. In reality, Population would stronglyaffect some of these “independent” variables andsome of them would strongly influence others as well.Such a model therefore is not a causal model of popu-lation dynamics, and as such cannot give any infor-mation on why the population behaves the way itdoes. On the other hand, this model could producevery reliable predictions of the population levels, forgiven values of input variables. Thus, such an exoge-nous model can be useful for short-term populationforecasting, but cannot be used at all for policyanalysis and design purposes.

Contrast the above model with the one shown inFigure 4(b). This second one is a dynamic and endoge-nous account of the dynamics of city population. Notethat in this diagram there is no obvious distinction of“input” and “output.” All major factors in the city influ-ence one another and ultimately their own dynamics,around many feedback loops. This example isdesigned deliberately such that most variables in the


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two models are common, so that the two can be easilycompared. For instance, Job availability in the secondmodel is given by the ratio Jobs/Workers, not an inputvariable. Jobs depend on Businesses and Workers are afraction of the population. Businesses are initiated byentrepreneurs, which depend on Population as well asBusinesses. (A fraction of Population yields potentialentrepreneurs, but how active they actually aredepends also on existing businesses). Finally, Job avail-ability influences the in/out migration, which in turncreates an increase or a decrease in Population. Thus,a feedback loop emerges: Population → Entrepreneurs→ Businesses → Jobs → In/out migration → Popula-tion. This loop says that more people, more entre-preneurs, more businesses, more jobs, a morein-migration. This is probably one of the major loops

that create typical early booms in newly industrializedcities. But this growth is not without opposition; thereare other loops. For instance, more Businesses, moreWaste/Emissions, lower Air/water quality and lowernet birth rate. Thus, Population → Entrepreneurs →Businesses → Waste/Emissions → Air/Water Quality→ Net birth rate → Population loop suppresses theindefinite population growth suggested by the previousloop. In a similar fashion, one can trace several othergrowth loops and balancing loops in the diagram. Inthe end, the dynamics of the population is determinedas a result of complex interactions between these loops.The dynamics could be growth, collapse, growth-and-stagnation, growth-and-collapse, and so on. A technicaldiscussion would require building of a complete modelwith equations, beyond the scope of this section. (For


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Figure 4. Comparing (a) an exogenous, static model of city population with (b) an endogenous, dynamic model

Job availability

House prices/rentals

House availability

Crime rate

School quality

Air/water quality

Salaries

Expenses

Population

(a)

Net birth rate

Air/water quality

Population

Area

Crowding

Crime rate

Housingpressure

House availability

Jobs availability

Governmentinvestment

Houses

Housing fraction

Housing business

AreaJobs

Business

Entrepreneurs

Waste andemissions

Workers

Salaries andexpenses

In/out migration

(b)

Pollutionpressure Crime pressure

Jobs growth

Pop. growth

Businessgrowth

a discussion of the structure of a classical urban dynam-ics model and the resulting growth-stagnation dynam-ics, see “Urban dynamics,” EOLSS on-line, 2002). Thepoint of this example is to illustrate the principle of“internal structure as main cause of dynamic behavior.”The example demonstrates that it is the internal struc-ture (the nature and interaction of the loops, theirrelative strengths, and so on) that determines thedynamic behavior. This model is an endogenous theoryabout the dynamics of city population, because thedynamics is not merely imposed by external forces, it isinternally determined. On such a model, it is possibleto study the causes of, say growth-then-collapse of thepopulation of a modern city and explore alternativepolicies. By contrast, the previous static and exogenousmodel is not a theory about population dynamics; itcan be used for forecasting purposes only.

Principle 5: Systems perspective. For the system dynam-ics methodology to be applicable to a problem, thedynamics of the variables must be closely associatedwith the operation of the internal structure of asystem. But what if the dynamics in the real problemare dictated by external variables? There are twopossibilities: Rarely, it may be indeed true that by itsvery nature, the real system may be too vulnerable toexternal influences. We already mentioned such anexample: If a multinational grocery chain opens asuper-store next to a small grocery store, which inturn causes an unavoidable decline in small storesales, this would not be a systemic feedback problem.There is not much “management” the small grocerystore owner can do in order to influence the policiesof the multinational chain, economies of scale, regu-lations, and so on. Such extreme cases are simply “ill-chosen” system dynamics problems and there is notmuch the methodology can do. The problem itself is“too open” by definition. But more often, the argu-ment that the dynamics of the system are caused byexternal forces is a result of narrow system conceptu-alization, not a property of the real problem. Forinstance, in the population example above, the firstmodel was static and exogenous whereas the secondone was dynamic and endogenous. But assume nowthat, lacking systems perspective, the second modelomitted Businesses and related variables. In this case,Jobs availability, Water/air quality and Housing avail-ability would all become external input variables. Thedynamics of Population would then be determinedessentially by these external input variables in thesecond model as well. In general, the modeler facesthe challenge of adopting proper systemic perspectivesuch that the major forces and interactions areincluded in the internal structure. This is the critical“model boundary” determination issue in systemdynamics method. From systems perspective, themodel boundary must be wide enough so as to havean internal structure rich enough to provide anendogenous account of the dynamics of concern.

The importance of the endogenous/exogenousdistinction is that, if external forces dictate the dynam-ics, there is not much possibility for managerial controland improvement. In real life, perhaps as a self-defensemechanism, we tend to give much more weight to

external forces compared to the internal factors. At apersonal level, when I am behind in my research, I putall the blame on the university bureaucracy, too muchunnecessary committee work, inefficient assistants, andeven bad luck, all of which means that I have no fault.The failing manager blames the unfavorable macroeconomic conditions, vicious/unethical competitors,unprofessional suppliers and yes, bad luck too. Theprime minister blames the opposition parties, thechamber of commerce, worker unions, the president(if from another party), hostile foreign governments,extraordinary weather conditions, and so on. “Theenemy is out there.” The major problem with this atti-tude is that it frees the management of the system(whether private life or a professional organization)from responsibility, which means no critical self-evaluation and no prospect for improvement. One ofthe challenges for a system dynamics study is toconvince people that this type of defensive attitude isunproductive. From a systemic perspective, thedynamic problem is caused neither by the “external”enemy nor by the manager. There is no single personor entity to blame. The cause of the dynamic problemlies in a system structure that cannot cope with unfa-vorable external conditions. (The famous mass-springsystem oscillates, when pushed by an external force. Itdoes not oscillate because it is pushed. It oscillatesbecause it has a spring, a structure ready to oscillatewhen touched). The cause is not the external enemy,but the way our system relates to/deals with the “exter-nal enemy.” The internal structure of the model mustinclude not only the relationships between the internalelements of a system, but also how the systemrelates/reacts to its environment, to the externalforces. That is why “model boundary” determination isa subtle and critical task, quite dependent on theperspective and skills of the modeler. The modelboundary is not automatically dictated by some natural“system boundary;” the modeler determines it.

2.5. Complexity of dynamic systems and necessity ofmodeling and simulation

It is often said that dynamic systems are “complex.”This is especially true for non-linear feedback systemsinvolving human actors – typical subject matter ofsystem dynamics studies. Here are some of the mainreasons why such problems are complex:

Dynamics

Dynamic problems are naturally harder than staticproblems. Variables change over time as they interact.The changes are not straightforward to predict.There are time delays involved between causes andeffects and between actions and reactions. Dynamicsof systems may be hard to predict by intuition evenwith only a few variables.

Feedback

The problem is further complicated when dynamicsare created by operation of feedback loops. It means


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that which way the system will move is not easilypredictable; the evolution path unfolds gradually andcontinuously determines its own path into the future.(Path-dependent dynamics). Feedback dynamics areharder to predict by intuition, because they requiremental simulation of interactions of several loopssimultaneously.

Non-linearity

Most system dynamics problems are non-linear. Thismeans that the cause-effect relations between vari-ables are not proportional. Non-linear effects aresubtle, because a certain effect observed in one rangemay not be valid at all in another range. Non-linearityfurthermore often means that there are “interactioneffects” between variables. That is, doubling theadvertising may increase the demand by 20 percentwhen the price is around US$100, but the same effectmay be only 5 percent when the price is aroundUS$125. Non-linearity is very hard to analyze not onlyintuitively, but also mathematically, especially whenembedded in a dynamic feedback context.

Scale

As the number of variables increases, the complexityof the problem increases nonlinearly. With onlythree or four variables, even a non-linear feedbackproblem can be analyzed in most cases mathemati-cally and perhaps intuitively. But even “small size”policy problems involve tens of variables. At thisscale, a non-linear feedback problem immediatelybecomes impossibly hard to track – mathematicallyand intuitively.

Human dimension

Typical system dynamics problems involve humanactors. So we must model not only the physics of thesystem (including information flows), but also howpeople react to situations, make decisions, set goals,make plans, and so on. This “human dimension” addsyet another layer of complexity. Human elements aremuch harder to model than the mechanical/physicalaspects. There are no established, tested laws of howpeople behave, react, or make decisions. Quite often,the modeler must create his/her own theory of howthe human actors would behave in the specific contextof a given task and environment. Human dimensionmakes the study harder not only in modeling, but alsoin testing and analysis phases.

Cause and effect separated in time and space

The result of the above facts is that in a non-lineardynamic feedback model with several variables, thecause-effect relations become detached in time andspace. When an action is applied at point A in themodel with an expected immediate result at point B,this result may never be obtained and furthermore,some unintended effect may be observed at a distantpoint C, after some significant time delay. We make a

change in our marketing activity to boost sales, butnothing substantial happens at first and then, aftermany months, we may observe some unintendedconsequences in our production department.

Intuitive inadequacy

We human beings are not naturally equipped to dealwith this type of detached cause-effect relation. Ababy touches the stove with his/her index finger, andthe index finger burns, and it burns now. The childlearns immediately that touching the stove burns thehand; the causality is easy to extract, because thecause and effect are close in time and space. Like allanimals, we can immediately learn this type of cause-effect relation. Our very survival depends on this intu-itive skill. Now assume a strange hypothetical worldwhere when a baby touches the stove, nothinghappens to his hand, but weeks later his nose startsbleeding. It would be close to impossible for the babyto be able to link the effect to its cause in this strangeworld, because cause and effect are detached in timeand space. What is worse, the baby would probablylink the effect to some wrong causes, as he surely didmany other things involving his nose just a few days,hours, or minutes before the bleeding. Organiza-tions, socio-economic systems, are unfortunately likethis strange world. A new decision in one sector of theeconomy can have several unexpected effects in othersectors, after many months or even years. With ourtime and space-constrained intuition of causality, weare prone to make wrong causality inferences abouteffectiveness of critical managerial, public or personaldecisions. Finally, our intuitive ability is furtherimpeded by delays, errors, omissions, and bias indata/information that we observe in real life.

All of the above complexities lead to the followingconclusion: Large scale, non-linear, dynamic feed-back models are too complex to be even partiallyanalyzed and understood by our natural intuition. Sowe need help. The help that we obtain is two-fold:first, formal modeling. We build a formal model inorder to make our mental model explicit, rigorouslyanalyzable and testable, making scientific improve-ment possible. Our mental models have some majordrawbacks: they are vague, implicit, often biased,ambiguous, and non-testable. (see “Mental models ofdynamic systems,” EOLSS on-line, 2002). A formalmodel, in contrast is explicit, precise, less biased,unambiguous and testable. So, a well-constructedformal model can eliminate most of the weaknesses ofmental models. But this is only half of the story. Theother major issue is analysis of the formal model. Theexact and most general scientific method is, ofcourse, mathematical analysis. But the required math-ematics is unfortunately almost always impossible forlarge, non-linear dynamic feedback models. Thus, thesecond major assistance we seek is computer simulation.Simulation is an experimental way of analyzing theproblem, which is another standard method of analy-sis in science. But in simulation, instead of experi-menting with the real system, we experiment with amodel of the real problem. In simulation, the model


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structure operates over simulated time, just like theoperation of the real structure over real time, so thatthe dynamics of the variables gradually unfold. Thus,a set of carefully designed experiments can be carriedon the simulation model, yielding some results aboutthe dynamic properties of the system, the causes ofthe problematic dynamics, and how they can beimproved. The validity of the model is of course vitallyimportant in this approach. In system dynamics stud-ies, simulation experimentation is often the only feasi-ble scientific method of analysis. The other twoclassical methods, mathematical analysis and experi-menting in the real system are unfortunately toooften impossible. The impossibility of real systemexperimentation comes from huge risks, costs, timedelays and other practical impossibilities involved inexperimenting with socio-economic systems. Simula-tion combines the cost and risk advantages of mathe-matical modeling/analysis, with the mathematicallyunconstrained power and flexibility of experimentalanalysis.

3. MODELING METHODOLOGY ANDTOOLS

3.1. Steps of the system dynamics method

A typical system dynamics study goes through somestandard steps. Although there will be variationsdepending on the nature of the problem and style ofthe modeler, main steps can be nevertheless summa-rized as follows.

Problem identification and definition (purpose)

A system dynamics project is done to study adynamic problem (applied or theoretical). Selectionand articulation of a meaningful dynamic feedbackproblem is critical for the success of the project. Theproblem must be not only dynamic, but also of feed-back nature. Externally driven dynamic problemsare not meaningful system dynamics topics.Dynamic problems are characterized by behaviorpatterns that may be observed in plotted data or theymay be deduced from available qualitative informa-tion. In a theoretical study, the dynamics would be ahypothetical plot. Some of the sub-steps of problemidentification are:

• Plot all the available dynamic data and examinethe dynamic behaviors.

• Determine the time horizon (into the future andinto the past) and basic time unit of the problem.

• Determine the reference dynamic behavior: Whatare the basic patterns of key variables? What issuggested by data and what is hypothesized ifthere is no data? What is expected in the future?

• Write down a specific, precise statement of whatthe dynamic problem is and how the study isexpected to contribute to the analysis and solu-tion of the problem. Keep in mind that thispurpose statement will guide all the other stepsthat will follow.

Dynamic hypothesis and model conceptualization

The objective of this step is to develop a hypothesis, atheory that explains the causes behind the problematicdynamics. This must of course be an “endogenous”explanation. Later, this hypothesis will be converted toa formal simulation model and the validity of thehypothesis will be tested. Dynamic hypothesis can alsobe called a conceptual model; a model that describesour hypothesis, not yet in a formal, testable form. Thisstep involves the following main activities:

• Examine the real problem and/or the relevanttheoretical information in the literature.

• List all variables playing a potential role in thecreation of the dynamics of concern.

• Identify the major causal effects and feedbackloops between these variables.

• Construct an initial causal loop diagram andexplore alternative hypotheses.

• Add and drop variables as necessary and fine-tunethe causal loop diagram.

• Identify the main stock and flow variables.• Use other conceptualization tools (like sector or

policy diagrams) as suitable.• Finalize a dynamic hypothesis as a concrete basis

for formal model construction.

Formal model construction

In this step, the formal simulation model is built. Thisinvolves the following sub-steps:

• Construct the stock-flow diagram; the structure ofthe model

• Write down mathematical formulations thatdescribe cause-effect relations for all variables

• Estimate the numerical values of parameters andinitial values of stocks

• Test the consistency of the model internally andagainst the dynamic hypothesis (verification).

Model credibility (validity) testing

Is the model an adequate representation of the realproblem with respect to the study purpose? Modelcredibility has two aspects: first, Structural: is the struc-ture of the model a meaningful description of the realrelations that exist in the problem of interest? Andsecond, Behavioral: are the dynamic patterns gener-ated by the model close enough to the real dynamicpatterns of interest? (see “Model testing and validity,”EOLSS on-line, 2002). Examples of structural tests are:having experts evaluate the model structures, dimen-sional consistency with realistic parameter definitions,and robustness of each equation under extreme condi-tions and extreme condition simulations. Behaviortests are designed to compare the major pattern compo-nents in the model behavior with the pattern compo-nents in the real behavior. Such pattern measuresinclude slopes, maxima and minima, periods andamplitudes of oscillations (autocorrelation functionsor spectral densities), inflection points, and so on. Twoprinciples are critical: first, unless structural validity is


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line, 2002). There are some important new develop-ments that seek to enhance implementation success,such as group model building (see “Group model build-ing, EOLSS on-line, 2002) and interactive learning envi-ronments (see “Modeling for learning and interactiveenvironments,” EOLSS on-line, 2002).

3.2. Stock and flow variables

In system dynamics models, it is essential to distin-guish between two types of variables: Stocks and flows.

Stocks

They represent results of accumulations over time.Their values are “levels” of the accumulations. Theyare also called “states” as they collectively representthe state of the system at time t. The standardsymbolic shape for a stock is a rectangle.Stock Examples:

• population• cash balance• inventory of goods• weight• anger level• knowledge level• temperature• glucose in blood

These examples show that stocks can be physical enti-ties as well as information entities; they can be mana-gerial control variables as well as natural/biologicalvariables. Stocks are universal; they represent the crit-ical accumulations on which the very existence of allsorts of life forms depend.

Flows

They directly flow in and out of the stocks, thuschanging their values. They represent the “rate ofchange” of stocks. The symbol for a flow is an arrow(representing the direction of flow) and a valve (TT orX–– representing the fact that the flow quantity is beingregulated).Flow examples:

• births, deaths• income, expenses• production, sales• caloric intake, calories burned• increase and decrease in anger level• learning and forgetting (or obsolescence rate)• heat in, heat out• glucose intake, glucose consumption

Note that the flow examples above are naturallypaired with the stock examples above. For instance,Births are an inflow and Deaths an outflow for thePopulation stock. The symbolic representation:

established to begin with, behavior validity is meaning-less in system dynamics, and second, behavior testingdoes not involve point-by-point comparison of modelbehavior with real behavior; it involves comparing ofpatterns involved in the two.

Analysis of the model

The purpose in this step is to understand the importantdynamic properties of the model. This can be done veryrarely (sometimes partially) by mathematical/analyticalmethods. Although it is impossible to find the solutionequations of system dynamics models mathematically,we can sometimes find the constant equilibrium levelsand determine their stability. (see “Equilibrium andstability analysis,” EOLSS on-line, 2002). More typically,analysis is done by simulation experiments. A series oflogically related simulation runs can provide quite reli-able (although not exact) information about the prop-erties of the model. These simulation runs are alsocalled sensitivity tests, as they try to assess how much theoutput behavior changes as a result of changes inselected parameters, inputs, initial conditions, functionshapes, or other structural changes. (see “Sensitivityanalysis,” EOLSS on-line, 2002).

Design improvement

Once the model is fully tested and its propertiesunderstood, the final step is to test alternative newpolicies to see to what extent they can improve thedynamics of the model. A policy is a decision rule, ageneral way of making decisions. Thus, a productionpolicy would be represented by a set of equations thatdescribe how the productions decisions are beingmade in the company. (An outcome of these equa-tions would be a production decision). In this last step,alternative policies are designed and then tested bysimulation runs. Policy improvement is a complicatedtask: The recommended policy must be realistic,considering the environment in which it will beimplemented; the policy must be robust, in the sensethat it should work under different environmentalconditions and scenarios; interactions (positive ornegative) of policies must be considered; transitiondynamics – the response of the system during thetransition from old policy to the new one – must beexplicitly analyzed by simulation experiments as well.(see “Intellectual roots and philosophy of systemdynamics,” EOLSS on-line, 2002).

Implementation

This step is applicable if the system dynamics study is anapplied one. Needless to say, it is a vitally important step,since the ultimate success of an applied system dynam-ics project means a demonstrable and sustainablesystem improvement. A successful implementation insome sense depends so much on project specifics that itcannot be prescribed general rules or procedures. Onthe other hand, there are some general aspects of imple-mentation success that system dynamics researchershave attacked. (see “Implementation issues,” EOLSS on-


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Population

DeathsBirths

the equations for flows must be specified. For exam-ple, in the simple population model, the simplestspecification for births and deaths would be: Births =b*Population and Deaths = d*Population, where b:constant birth fraction and d: constant death fraction.The stock-flow diagram would then be:

The “effect” arrows in this diagram show that Births= f (Population, Birth Fraction) and Deaths = f (Population,Death Fraction).

Finally, in writing equations for flows, one oftenneeds to define some intermediate variables, calledauxiliary or converter variables. For examples,consider a situation where Birth Fraction is notconstant, but depends on the “crowding level” of thepopulation, defined as Crowding = Population/Capacity. Birth Fraction would then be f (Crowding),where f (.) is typically a monotonically decreasingfunction. A stock-flow diagram including such anauxiliary variable (Crowding) is shown below:

In the above diagram, the flow is “net births” (births– deaths), represented as a “bi-flow” that flows in whenits value is positive and flows out when it is negative.This formulation is obtained by defining the BirthFraction function such that when Crowding < 1 it is posi-tive (births > deaths) and when Crowding > 1 it is negative(births < deaths). (This model will be further studiedbelow).

How to identify stocks and flows?

“Bathtub” is an excellent metaphor for stock variables.Water in the bathtub is the stock; water flowing in andflowing out are its flows. If a variable in the real problemfits this bathtub metaphor, it is a potential stock. Note

Mathematically, the above diagram states:

dPop–––– = births – deaths, in a continuous model.dt

Or, to stress the accumulation nature of a stock:

Pop(t) = Pop(0) + ∫0t(births – deaths)dt, Pop(0) given.

In numerical simulation, the same equation isapproximately represented as:

Pop(t) = Pop(t – dt) + . . . for t = dt, 2dt, 3dt. . .

Finally, if the model described a discrete dynamicsystem, then:

Pop(t) = Pop(t – 1) + (births – deaths), for t=1,2,3. . .

In any case, we see that the standard stock equation isa basic conservation equation over time. To representthe conservation more explicitly, the flow Births wouldflow out of some stock (Babies in conception) andDeaths would flow in some stock (dead people). In theabove diagram, “clouds” represent the implicit stocks“Babies in conception” and “Dead people.” The cloudsymbol means that these two stocks are outside the modelboundary, so we do not need to track them. (“Cloud”usage is standard in “information” stock-flows such as“knowledge increase,” because an increase in myknowledge does not decrease somebody else’s knowl-edge: information flows typically are not conserved).More generally, in a stock-flow diagram there would bemultiple stocks and flows, such as:

The above example says that there can be somesales directly from raw material inventory and somescraps from finished inventory. The computationalstock equations in this case would be:

RawMatrlInv(t) = RawMatrlInv(t – dt) + dt * (Receiving– Processing – RawMatlSale)

FinishedInv(t) = FinishedInv(t – dt) + dt * (Processing– Shipping – Sraps)

RetailInv(t) = RetailInv(t – dt) + dt * (Shipping)

In general, if a stock has j flows, the continuous stockequation is:

Stock(t) = Stock(0) + ∫0t(∑ flows)dt

j

If a model has n stocks, it is said to be of order n. Thisis mathematically equivalent to a set of n 1st orderdifferential equations, or a single nth order differen-tial equation (that is, highest derivative involved is oforder n).

The above examples do not represent completemodels. To be solvable (analytically or by simulation),

RawMatrl Inv

ProcessingReceiving Shipping

Retail InvFinished Inv

SrapsRawMatlSale


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Births Population

Birth fraction

Deaths

Death fraction

~

BirthsPopulation

Crowding

Birth fraction

Capacity

-- -

- --

that the stock examples listed above all fit the metaphor.A quick and effective test to identify the stocks is to“freeze” time and motion, and see which variables stillpersist. Since stocks are accumulated quantities, theyare well defined, even if there is no time and motion.Population, inventory, cash balance, knowledge level. .. the stocks still persist; but their flows vanish: there canbe no births, no deaths, no production, no sales, noexpenditures, and no learning without passage of time.If a given stock were measured in certain units (saypeople, liras, items. . .) its flows would be in units/timeperiod (people/year, liras/month, items/hour . . .).That is why flows become undefined without passage oftime, but stocks persist.

The above test is important and presents a necessarycondition for a variable to be a stock, but the conditionis not sufficient. Not all variables that pass the testwould be modeled as a stock; many would be insteadmodeled as auxiliary variables. Identifying a variable asa stock means deciding to model its flow variables. Thiswould not only make the modeling more complicated,but addition of stock-flow structures would also compli-cate testing, analysis, and design phases, because stock-flow structures add to the dynamic and scalecomplexity of the model. Thus, the second importantprinciple is that stock variables identified in a problemare believed to be especially important accumulations thatdecisions in the real system try to control or dependon. In the simple example above, although the vari-able Crowding passes the “freeze” test, we did notmodel it as an accumulation, because it was notbelieved to be important enough to justify the addedcomplexity of formulating its flows in and out. Simi-larly, many such variables are approximated by auxil-iary variables. Stock variables are those accumulationsthat seem most important with respect to the dynamicproblem definition. Finally, potential stocks that varytoo slowly compared to the time horizon of the modelare modeled as constants. Selections of stocks aretherefore also relative to the time horizon and timeunits of the problem. In some cases, even the stock-flowdistinction can be relative. Cars traveling from Ankarato Istanbul can be a stock variable in one problem defi-nition (short-term, micro dynamics), but it would be aflow variable (cars traveling per day) in a longer term,macro problem.

Importance of stocks

Stocks play a central role in dynamic feedbackmanagement problems for several reasons. Theircontrol is often the primary responsibility ofmanagers, since the survival of the system is often crit-ically dependent on them: water in reservoirs, grainstocks, cash in banks, food stock in our kitchen,glucose in blood, and pollutant level in air. We notonly control these variables, but also use their valuesas basis for action in managing other variables. Yet,controlling stocks is subtle and dynamically complexby their very nature:

• Stocks can be changed only via their flows: If wewant to change the value of an inventory, we

cannot directly and immediately change it. It canonly be changed by changing its inflow and/or itsoutflow (production rate and/or sales). We candirectly manipulate our caloric intake by chang-ing our eating (flow), but cannot directly manip-ulate our weight. If changing our eating habit ishard, changing our weight will be indirect,delayed and much harder.

• Stocks have inertia: Since they have historicallyaccumulated values, they cannot be easilychanged. We can suddenly increase the inflowinto the bathtub by 100 percent, but if it wasalready half-full, it will be a long time before thestock in the batch tub increases by just 10percent. If I have a negative opinion of a politicalparty, it would take quite an effort (and time) tochange it.

• Multiple flows make the control even harder:Stocks and their flows may move in oppositedirections. We may have increasing revenues, butthe cash stock will still go down, if the expenses(outflow) are greater than revenues. We may havedecreasing births and increasing deaths, but thepopulation will still increase if births are never-theless greater than deaths. In general, control-ling the dynamics of a stock requires taking intoaccount all of its flows simultaneously, which isnot a trivial task, theoretically and practically.

• Stocks are the source of endogenous dynamics:Stocks make it possible for the inflows and outflowsto differ. The differences between the flows accu-mulate in the stock. There are several water reser-voirs around the city of Istanbul. Water in thesereservoirs is a stock, streams feeding into the reser-voirs are the major inflow and the water consump-tion of the city is the major outflow. Withoutreservoirs, the consumption of the city would haveto equal the stream flows at all times, which meansthe consumption would fluctuate the same asstream flows do. Such wild fluctuations in wateravailability would make life in the city impossible.The reservoirs allow smooth water consumption(outflow), by buffering against seasonal fluctua-tions in the stream flows. The stocks influencetheir own flows in various ways. Since stocks are ofsuch critical importance we continuously monitortheir values and take actions to keep their values insome desired band. Stocks also cause time delays insystems. In the raw material supply chain diagramshown above, the fact that there are two stocksbetween the receiving of raw material and the ship-ments of goods to the retailers introduces a timedelay between “Receiving” and “Shipping” flows.Similarly, information (or perception) delays alsoinvolve stocks. In sum, stocks play a central role inthe creation of the endogenous dynamics in asystem.

Basic stock-flow dynamics

Stocks integrate the flows, modifying and complicatingthe dynamics involved. Figures 5, 6 and 7 illustratesome basic flow-stock dynamics. Since stock integrates


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direction”). When unemployment in a country goesup, immigration goes down, ceteris paribus.Conversely, when unemployment goes down, immi-gration goes up. The same reasoning is true for thePrice→ Demand and Frustration → Studying exam-ples. But the second example, being a flow-stock rela-tion, is again somewhat different: when deaths go up,population either goes down or it increases less than itwould otherwise have been (this would be true if, in spiteof increased deaths, births were still higher). Thus,the term “in the opposite direction” in this casemeans, “an increase (decrease) in x causes y todecrease (increase) below (above) what it wouldotherwise have been.”

Positive and negative feedback loops

As seen before, a feedback loop is a succession ofcause-effect relations that start and end with the samevariable. It constitutes a circular causality, only mean-ingful dynamically, over time. The sign (or polarity) ofa loop is the algebraic product of all signs around theloop. If the resulting sign is +, the loop is “positive” or“compounding” or “reinforcing.”

+

Births Population

+The above picture says that more births mean

higher population, which causes even more births,causing even higher population, and so on. The oper-ation of this loop over time would create exponen-tially growing population. This feedback loopreinforces or compounds an initial change. Not allpositive loops create growth; some create collapse aswill be seen later.

If the resulting sign of the loop is negative, then theloop is called “negative” or “balancing” or “goal-seeking.” This type of feedback loop seeks a balanceor a goal.

Caloric intake Weight gain

Concern about weight

The above picture states that the more calories Itake, the more weight I gain and as I realize my weightgain, I become concerned about it, which leads me tocut down on my caloric intake. This loop tries to keepthe caloric intake (hence the weight) under control.The behavior of this loop would be a convergent one,rather than a divergent one. In this example, theperson tries to control her weight around some“desired” (“goal”) weight. In some other negativeloops, there is no explicit or deliberate goal. Forinstance, consider a tank of water emptying itself outof a hole punched in the bottom. The outflow would

it flows, observe in Figure 5 that when the inflow andthe outflow are constant, stock is linearly increasingor decreasing (or constant when inflow = outflow).

In Figure 6, we see the dynamics of the stock whenthe inflow and/or the outflow are changing linearly.Note that the stock is increasing or decreasing non-linearly (quadratic), due to integration.

Finally in Figure 7, the inflow is oscillating and theoutflow is constant. The purpose of this example is toillustrate how integration creates time delay. Notethat the stock imitates the inflow, except that it lagsbehind the inflow with phase lag of π/2, since integralof cosine is sine and sin (ß) = cos (ß-π/2). Alsoobserve that amplitude of the oscillations in stock ismore than the amplitude of the inflow (amplifica-tion).

3.3. Positive and negative causal effects andfeedback loops

Positive and negative effects

We already mentioned that causal relation x → ymeans the input variable (x) has some causal influenceon the output variable (y). A positive influence means:“a change in x, ceteris paribus, causes y to change inthe same direction.” Examples are:

Population ––––>+ Housing DemandBirths ––––>+ PopulationPesticide ––––>+ Bird deathsMotivation ––––>+ Productivity

The “+” symbol denotes positive causality. (Someauthors use the symbol “s” denoting “same direc-tion”). When population goes up, housing demandgoes up, ceteris paribus. (The ceteris paribus condi-tion will be true in all causal relations, but will beomitted from now on to avoid repetition).Conversely, when population goes down, housingdemand goes down. The same reasoning is true forthe Pesticide → Bird deaths and Motivation →Productivity examples. But the second example issomewhat different: when births go up, populationeither goes up or it decreases less than it would otherwisehave been (this would be true if, in spite of increasedbirths, deaths were still higher). This situation is possi-ble when the cause-effect relation is a flow-stock rela-tion. Thus, the term “in the same direction” is subtlewhen flow-stock relation is involved. It means, “anincrease (decrease) in x causes y to increase(decrease) above (below) what it would otherwisehave been.”

A negative influence means: “a change in x, ceterisparibus, causes y to change in the opposite direction.”Examples are:

Unemployment ––––>- ImmigrationDeaths ––––>- PopulationPrice ––––>- DemandFrustration ––––>- Studying

The “-” symbol denotes negative causality. (Someauthors prefer the symbol “o” denoting “opposite


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+

Figure 5. Dynamics of a stock: (a) with constant inflow only, (b) with constant outflow only and (c) with constant inflow (k1) and constantoutflow (k2), when k1>k2, k1=k2 and k1<k2 respectively

k

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be approximately proportional to the water level, sothe diagram would be:

–Water level Outflow

+In the above example, there is no conscious goal

seeking. The tank would empty itself out and it would“decay” gradually to zero level. (One could say thatthe “effective” or implicit “goal” of this system is toreach zero water levels).

Positive and negative feedback loops are basicbuilding blocks of dynamic structures. In reality,many such loops interact together. The feedbackloops in interaction are displayed together in causalloop diagrams. (See for instance Figure 4 (b)).

4. DYNAMICS OF BASIC FEEDBACKSTRUCTURES

4.1. Single linear positive feedback loop

As mentioned above, positive feedback loops createexponential growth or crash. Figure 8 (a) illustratesthe simple linear positive feedback structure and itsbehavior. This model is of course solvable (alreadysolved in the beginning of this article). The solutionequation is:

Pop (t) = Pop (0)*Expon (BirthFraction*t), BirthFractiona constant.

A very important property of the exponential curveis that it doubles its value in constant time intervals.The formula for the doubling time can be derived tobe: Td = ln2/(BirthFraction) ≈ 0.70/BirthFraction.Thus, once the growth constant (BirthFraction) is


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in out

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(c)

Figure 6. Dynamics of a stock (a) when the inflow is increasinglinearly (outflow constant), (b) when the outflow is increasinglinearly (inflow constant) and (c) when both flows change linearlyand cross

Time

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Inflow Outflow

Stock

Figure 7. Inflow is oscillating and the outflow is constant. Observethat the oscillations in the stock are lagging behind the inflowoscillations, with a phase lag of π/2: integration creates delays

given, the process doubles itself every Td, independ-ent of Pop (0). The doubling time in Figure 8 (a) isapproximately fourteen time units. Figure 8 (b) is aless common version of linear positive feedback loop.It illustrates how positive feedback can lead to crashrather than growth. Consider the value of a stock inthe stock market and assume that there is a criticalvalue C, below which panic selling starts. Further-more, in panic selling, the more the value drops themore people sell, the more it drops, and so on. Weuse the following flow equation to describe thisprocess:

Loss = fract*(C-Stock), where fract is a constant per unittime and C is the critical value.

Observe that the resulting dynamic behavior is acrash or collapse, which is a decline at increasing rates.

(Constant “doubling time” still applies, except thatwhat doubles is the discrepancy (C-Stock), rather thanthe stock itself).

4.2. Single linear negative feedback loop

As mentioned earlier, negative feedback loops creategoal seeking or balancing dynamics. Figure 9 (a) illus-trates the atomic linear negative feedback structureand its decaying behavior. This linear model is easilysolvable and the solution equation is:

Pop (t) = Pop (0)*Expon (– DeathFraction*t), DeathFractiona constant.

Analogous to exponential growth, an importantproperty of exponential decay is that it halves its valuein constant time intervals. The formula for the half-lifecan be derived to be: Th = ln2/(DeathFraction) ≈ 0.70/DeathFraction. Thus, once the decay constant (Death-Fraction) is given, the process halves itself every Th,independent of Pop (0). The half-life in Figure 9 (a)is approximately 7 time units. Figure 9 (b) illustratesanother typical application of linear negative feed-back loop. It is about the automatic heating/coolingof a room with a thermostat. There is a desiredtemperature and if the room temperature is below it,the thermostat activates the heater and if the roomtemperature is above the desired one then the ther-mostat activates the cooler. We use the following flowequation to describe this process:

Temperature Change = Discrepancy/Adjustment Time, where Discrepancy = (Desired Temperature – Room

Temperature)

and Adjustment Time is a constant (in unit time) thatrepresents how fast the thermostat reacts.

Observe that the resulting dynamic behavior is a goalseeking one. Whether the room temperature startsabove or below the desired one, it gradually reaches it.


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500

00.00 9.00 18.00 27.00 36.00

Time

1

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(b)

Figure 8(a) A simple linear positive feedback structure and resulting exponential growth; (b) A linear positive feedback example that wouldyield exponential crash

11

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00.00 6.25 12.50 18.75 25.00

Time

Births Population

Birth fraction

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Loss rate

Fraction

C

(a)

Also note that the pattern of behavior is an exponentialdecay, characteristic of a negative feedback loop. (The“decay” is less obvious when the room temperature startsbelow the desired one, but in both cases what decaysexponentially in time is the discrepancy = desired –room temperature). Finally, constant “half life” stillapplies, except that what halves is the discrepancy(desired – room temperature), rather than the temper-ature itself. (The adjustment time in these runs is 4 timeunits, so that the half-life is 0.70/0.25 ≈ 2.8 time units).

4.3. Simple coupling of linear loops

Linear positive loop with a constant outflow

When a model is subject to an external (independ-ent) force (such as external temperature, rain, inter-est rates, competitor price), the model is said to be“non-autonomous” (or non-homogeneous). In thesetwo sub-sections, we examine what happens to dynam-ics of linear feedback structures when they are non-homogeneous. When a positive loop is subject to aconstant outflow, there are three possible dynamics.(Figure 10(a)). First, if the outflow K is equal to initialinflow = fract*S (0), in this trivial case the stockremains forever at the initial equilibrium S (0).Second, if the constant outflow K < fract*S (0), then itis also straightforward to guess that the dynamics willbe an exponential growth, as the inflow will become

greater and greater compared to the outflow. Finally,if the outflow K > fract*S (0), then we obtain some-what less obvious dynamics: The positive feedbackloop cannot create growth; instead, it yields a crash-ing behavior: the decline in a stock subject to aconstant outflow is linear; but when positive feedbackloop is involved, the decline becomes an exponentialcrash, not an immediately intuitive result.

Linear negative loop with a constant inflow

A similar case is a negative loop subject to a constantinflow. (Figure 10(b)). Again, if the inflow K is initiallyequal to the outflow = fract*S (0), then the variables stayat equilibrium. Second, if the inflow K is greater thanthe initial outflow fract*S (0), then the Stock increasesby definition. But as it grows, its outflow becomes largerand larger, eventually equating the inflow K, reachingan equilibrium Se (fract*Se = K, yielding Se = K/fract).Finally, if the inflow K is less than the initial outflowfract*S (0), then the stock decays gradually to a non-zero equilibrium, again at Se = K/fract.

Coupling of linear positive and negative loops

When linear positive and negative loops are coupled,the resulting behavior will be either exponential growth,or decay, depending on which loop is “stronger” (domi-nates). An example of such a structure was already


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Figure 9. A simple linear negative feedback structure and resulting exponential decay; (b) A linear negative feedback example that seeksan explicit non-zero goal

1

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shown in Figure 3 (b) and the resulting possible behav-iors in 3 (c). If the positive loop dominates (birth frac-tion larger than death fraction), the system yieldsexponential growth, if the reverse is true, then expo-nential decay and if the two fractions are equal, popula-tion remains of course unchanged. (Because of“superposition” property of linear models, combininglinear structures cannot create novel behaviors).

4.4. A basic non-linear coupling of loops: density-dependent growth

The paragraph above shows that the linear coupling offeedback loops is not very subtle. A more interesting,non-linear coupling of positive and negative loops isquite common in different disciplines such as ecology,marketing, and epidemiology. Figure 11 illustrates thisstructure in the context of population dynamics.Assume that births are greater than deaths so that thepopulation is growing. In linear coupling, this expo-nential growth would continue indefinitely. This is ofcourse not realistic, as all growth must eventually facesome limits (area, food, water, resource, air. . .). Tomodel such a limiting process, we define “crowding” aspopulation/maximum capacity. The maximum capacity(often called “carrying capacity” in ecology) is an aggre-gate summary measure representing various limits(area, food, etc), so that a given region cannot sustain

more animals (or people) than capacity. Thus, “crowd-ing” has a negative effect on birth fraction as seen inFigure 11, introducing a new feedback loop: Higherpopulation means increased crowding, which lowers thebirth fraction and which in turn creates a negative influ-ence back on population: Population→ Crowding →Birth fraction → Population loop is a balancing one; itsuppresses indefinite growth of population. As crowdinggets larger and larger, birth fraction keeps falling andeventually approaches death fraction, stopping thepopulation growth. The equations of the model are:

Deaths = Death fraction* Population, where Death fractionis constant (per time)

Crowding = Population/CapacityBirth fraction = f (Crowding)Births = Birth fraction*Population

The model is non-linear because Crowding = Population/Capacity, Birth fraction = f (Crowding), which means Births= f (Population)*Population, a non-linear flow formula-tion.

In the simple version of density-dependent model,Capacity is assumed constant. The standard conven-tion in these models is to specify the birth fraction = f (Crowding) function such that when Crowding = 1.0,the function yields birth fraction = death fraction. Sincethe death fraction is assumed constant, this specifica-tion is not difficult. A general birth fraction function


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1:S10 000

5 000

00.00 12.50 25.00 37.50 50.00

Time

1

2:S 3:S

(a)

Figure 10. (a) Positive feedback structure subject to a contant outflow and possible behaviors; (b) Negative feedback structure subject toa constant inflow and possible behaviors

Fraction

In K

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is shown in Figure 11. This function says that whenCrowding is rather low, increasing it would not havemuch effect on birth fraction. So, as Crowding becomeshigh enough to disrupt the basic needs of the popu-lation, birth fraction becomes more and more affectedby it. Note finally that the function is specified so thatit yields birth fraction = death fraction (0.06), whenCrowding = 1.0. (The simpler formulation of Birthfraction is linear: Birth fraction = a – b*Crowding, alinearly decreasing function of Crowding. But notethat this does not make the model linear!)

The dynamic behaviors of this model are shown inFigure 11. In these runs, Capacity is specified as 200, sothat population eventually seeks this level, regardless ofwhere it starts. If the Population starts above 200, Crowd-ing is greater than 1.0, yielding a birth fraction smallerthan death fraction, thus population declining graduallytoward 200. If population is below 200, then birth fraction> death fraction, so there is growth. Note that there aretwo different growth possibilities. The most informativeis the case when Population (0) is low enough, yieldingan early exponential growth. It means that the netgrowth rate (births-deaths) is getting bigger and bigger.But as the population reaches a certain level (of Crowd-ing), note that the net growth rate (births – deaths)starts becoming smaller and smaller (although it is stillpositive). Thus, as Crowding gets larger, the differencebetween births and deaths starts diminishing. This isthe effect of the non-linear negative feedback loopdescribed above. Hence, due to this non-linearity, thepopulation growth in a density-dependent model canhave two different phases: an exponential growthphase, followed by a goal-seeking growth phase. Thefirst phase is caused by the dominance of the simple posi-tive loop (Population → Births → Population) and thesecond phase is caused by the shift of loop dominance tothe negative density dependence loop. Owing to thisnon-linear structure, the density-dependent model cangenerate the famous S-shaped (or logistic) growthdynamics. The significance of this dynamics is that it ispossible for a process to start with exponential growth,but then later shift to goal-seeking behavior, as certainlimits are approached. This type of dynamics can beseen in the spread of new innovations (see “R&D, tech-nological innovations and diffusion,” EOLSS on-line,2002), new ideas; certain epidemic dynamics, and vari-ous ecological processes. (Note finally in Figure 11, thatif the initial population is already near enough itscapacity, then the exponential growth phase does notexist at all. The growth is pure goal seeking from thestart, because birth fraction is already dropping fast andapproaching death fraction).

4.5. Beyond S-shaped growth: overshoot-then-decline

In the basic density-dependent structure above, thelimiting capacity was assumed to be constant (such asfixed land). In many situations, the limiting factor isitself some resource stock (food) that the growth vari-able (population) depletes. Figure 12 illustrates such asituation where the population is consuming a foodresource and the death rate of the population itselfdepends on the food availability (Food per Capita). In

this example, as an illustration, the density-dependenteffect is on death fraction (instead of birth fraction asin the previous example). Thus, Population → Food perCapita → Death fraction → Population loop acts as adensity-dependent limit, just as in the previous exam-ple. What is new in this case is the fact that the “capac-ity” (Food) is a variable, being depleted by Population.(The specific functions Effect_of_Food_on_Df andEffect_of_Food_on_Consumption will be discussed in thefollowing section. At this point, just note that DeathFraction is a decreasing function of Food per capita andConsumption is an increasing function of Food per Capitaas well as Population). As the population grows expo-nentially, the consumption rate increases so much thatat some point it causes a collapse in the food stock(Figure 13). Shortly after the food stock collapses, thepopulation collapses too, due to increased death ratescaused by starvation (as measured by Food per Capita).

The complete set of equations and functions arelisted in Figure 13. Some of these equations will bediscussed later in the following section. A couple ofobservations about the dynamics of this model: First,growth-then-collapse behavior would not have beenpossible in a single-stock model. This is mathematicallyprovable, but also intuitively deducible: if there were asingle stock, once the stock reached its peak and a slope(rate of change) of zero, then by definition, it wouldstay there, at its equilibrium. There needs to be someother source of energy (like another stock food not yetat its equilibrium) to push this stock away (down) fromits own equilibrium. It is thus true in general that first-order, continuous time; autonomous systems canexhibit only monotonous dynamics. Second, this popula-tion-food model may generate other behaviors(decline only or oscillations) with other parameter andfunctions. For instance, it may yield declining popula-tion from start, if the initial Population is high enoughrelative to initial Food. Or, it may yield oscillations fordifferent parameter values and/or functions. (see“Ecological interactions: predator and prey dynamicson the Kaibab plateau,” EOLSS on-line, 2002).

Overshoot-then-collapse is observed in various sectorsand problems in real life. (For a business application see“Market growth, collapse and failures to learn frominteractive simulation games,” EOLSS on-line, 2002).Another example that can yield overshoot-then-declinebehavior is the dynamics of epidemics. Assume a conta-gious disease that can spread by contact between twogroups of people: Susceptible and Infected. The causalloop and stock-flow diagrams are shown in Figure 14. Asimple formulation for infection rate is obtained bybasing it on “all possible contacts” (Susceptible*Infected):

Infection Rate = InfectFract*Contacts, where Infect-Fract represents what fraction of contacts results ininfection

Contacts = ContactFract*Susceptible*Infected, whereContactFract represents what fraction of all possiblecontacts actually occurs per time unit (month).

The other flows are simple: There is an outflow fromthe Infected population (Removal), representing bothdeaths and recoveries. (It is assumed that recoveredpeople gain permanent immunity, hence they do not


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Figure 11. Density-dependent growth: (a) Causal loop diagram, (b) stock-flow structure and (c) possible dynamic behaviors

Births Population Deaths

Death fraction

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Input Output0.000 0.0800.167 0.0790.333 0.0780.500 0.0750.667 0.0710.833 0.0661.000 0.0601.167 0.0531.333 0.0441.500 0.0351.667 0.0241.833 0.0132.000 0.000

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00.00 62.50 125.00 187.50 250.00

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~

flow back into the susceptible pool). Finally, it isassumed that the susceptible group has a constantinflow (such as a net immigration). When the modelis run with ContactFract = 0.1 per month and InfectFract= 0.01, Infected group exhibits the overshoot-then-collapse behavior seen in Figure 14 (c). Thus, in thiscase there is an epidemic outbreak, which eventuallysettles down to an equilibrium level (of a relatively fewnumber of people). The equilibrium is obtainedwhen all flows equal (In = Infection Rate = Removals). Ina second experiment, when the model is run with alower infectivity (InfectFract = 0.002), then we observe“epidemic oscillations.” With even lower infectivityvalues and/or higher initial values of the infectedstock, there would be no epidemics at all – Infectedwould decline monotonically over time.

5. FORMULATION PRINCIPLES ANDGENERIC MODEL STRUCTURES

5.1. General principles of formulation

System dynamics models, being causal-descriptive ones,consist of causal cause-effect equations. As such, theseequations must obey some fundamental principles:

Equations must have real-life meaning

Equations must correspond to real processes. Theymust have defendable meaning. It is unacceptable touse an equation just because it fits data or providesexcellent predictions. (This point was already discussedabove, in the section on causality versus correlation).


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Figure 12. A population-food interaction model: causal loop and stock-flow diagrams

(–)Normal death

fraction

Food per capitaFood Regeneration

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Food

Normal food per capita

Effect of food on consumption

Normal consumption per capita

ConsumptionRegeneration

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Population

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Equations

Food (t) = Food (t - dt) + (Regeneration – Consumption) * dtINIT Food = 2 000Regeneration = Regeneration_fraction * FoodConsumption = Effect_of_food_on_consumption * Normal_consumption_per_capita * PopulationPopulation (t) = Population (t – dt) + (Births – Deaths) * dtINIT Population = 50Births = Birth_fraction * PopulationDeaths = Death_fraction * PopulationBirth_fraction = 0.2Death_fraction = Effect_of_food_on_Df * Normal_DfFood_per_capita = Food/PopulationNormal_consumption_per_capita = 15Normal_Df = 0.05Normal_Food_per_capita = 45Regeneration_fraction = 0.5

0.000

1.00

0.000 1.000

Input Output0.000 0.0000.091 0.3650.182 0.6200.273 0.7750.364 0.8750.455 0.9300.545 0.9800.636 0.9900.727 0.9950.818 0.9980.909 1.0001.000 1.000

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Effect_of_food_on_DfEffect_of_food_on_consumption

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Input Output0.000 9.9900.091 7.4500.182 4.9500.273 3.3500.364 2.4000.455 1.8000.545 1.3000.636 1.1500.727 1.0700.818 1.0200.909 1.0101.000 1.000

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Figure 13. The population-food interaction model: Equations and the resulting overshoot-then-collapse behavior

1

For an equation to be meaningful, all of its variablesand parameters must of course be meaningful. Vari-ables and parameters must correspond to real-lifeconcepts. This includes quantitative variables as well asqualitative ones and it includes applied variables as wellas theoretical ones.

Equations must be dimensionally consistent

If the units of the right hand side variables in a birthrate formulation yield something like 1/people, thisis of course unacceptable (units in reality arepeople/time period). It is equally unacceptable to usea variable like “population effective” formulated as100/population1/2 and then multiply it by some “coef-ficient K” with units people2/month, so as to yieldbirth rate in people/month! The dimensional consis-tency of each equation must be verified, without includ-ing any “dummy” coefficients in order to make theunits consistent. The real-life units of each variableand coefficient must be plugged in and the units ofthe right hand side must “automatically” yield theunits of the output variable, without any “dummy”scaling coefficient.

Equations must yield valid results even in extremeconditions

When the population is assumed zero (a hypotheticalextreme), death rate must also be zero, for we knowthis is necessarily the case in real life. If the death rateformulation yields a non-zero value under the zeropopulation tests, arguing, “zero population is an unre-

alistic extreme” is not a valid defense. When an equa-tion fails under extreme conditions (very small, verylarge, zero, and so on), this often indicates that there issomething illogical in the formulation that is probablyyielding wrong results in the normal operating range ofthe input variables, but in more “hidden” ways.

Realism should not be sacrificed for mathematical simplicity

The model must provide a realistic description of thereal processes with respect to the dynamic problem.Typically, real processes involve non-linear relations,interacting multiple feedback loops and time delays. Eachof these three factors greatly complicates the mathe-matical analysis of the model. A tendency in such casesis to approximate non-linearities with linear ones, toignore the existence of some of the loops and to omittime delays (that increase the order and dynamiccomplexity of the model). None of these simplifyingassumptions is justifiable just to obtain mathematicaltractability. Mathematical (analytical) exactness is notof primary concern in system dynamics; much moreessential is the realism of the model. Since solutions insystem dynamics method are obtained by simulation,non-linearities need not be regarded as a threat totractability and therefore should not be evaded.

Equations must not unrealistically assume optimality orequilibrium

Equations must describe the way the real world works,not the way it “should” work assuming that all actorsbehave optimally, or assuming that variables are at


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Figure 14. Epidemic dynamics: (a) causal loop diagram, (b) stock-flow structure and (c) possible dynamic behaviors

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equilibrium. It is not realistic to assume that mostdecision-makers have access to all data and algorithmsneeded to solve the complex optimization problems.Nor is it realistic to derive equations based on theassumption that variables are at equilibrium. Thepurpose of system dynamics modeling is to investigatethe causes of dynamic behavior and equations derivedfrom equilibrium (i.e. non-dynamic) assumptions donot normally apply in the dynamic mode of operation.

5.2. Linear formulation

A linear equation assumes that the output is propor-tional to the input. The general form is Y=a+bX,where the intercept a and slope b are both constant.Typical examples we have already seen include:

Births = birthfract*PopRecovery = Recovfract*InfectedCrowding = Pop/CapacityTempChange = (DesiredTemp – Temperature)/AdjustTime

The above formulations are all linear, because birth-fract, Recovfract, Capacity, and AdjustTime are allassumed constant. If any of these parameters werefunctions (direct or indirect) of the stock variable(Pop, Infected, Temperature), then the formulationwould be non-linear, because proportionality (orconstant slope) assumption between input andoutput would not hold. Thus, the simple definition ofa non-linear formulation is “a formulation that is notlinear.” This means any mathematical expressionother than a+bX (including any xa, ln(x), ex and anycombination of such functions). Furthermore, in aformulation involving more than one variable, anyproduct or division of variables would be non-linearas well, since this would violate proportionality. Insum, linearity is a pretty restrictive assumption. Insome situations such as the ones mentioned above itmay be reasonable to assume linearity, but it shouldbe kept in mind that these situations are rare insystemic feedback models, as will be seen below.

5.3. Non-linear equation formulation

Prevalence of non-linearity

In a dynamic feedback model, non-linearity is often“natural,” almost a rule. Consider the simplestdensity-dependent population model seen earlier. Wehave:

Crowding = Pop/Capacity andBirthFract = a – b*Crowding

Note that both of these equations are linear, since weassume that Capacity, a and b are all constant. Butwhen we next write Births = Birthfract*Pop, we obtain anon-linear model. Non-linearity comes from the factthat Births = f (Pop)*Pop. What is interesting here isthe fact that we simply write down a succession oflinear equations, but obtain a non-linear model whenthe feedback loop is “closed.” The recognition ofPopulation → Crowding → BirthFract → Births → Popu-lation loop automatically makes the model non-linear.

That is why; most feedback models are non-linear bytheir very nature. To make a dynamic model linear,one must not only assume proportional relationsbetween all pairs of variables, but must also furtheromit most of the feedback loops operating in thesystem.

Non-linear formulation examples and techniques

A basic and standard non-linear formulation we havealready seen is the “product” formulation used in theepidemic dynamics model:

Infection Rate = InfectFract*ContactsContacts = ContactFract*Susceptible*Infected, Contact-

Fract and Infectfract are constants

The Infection flow in the above model is non-linear, asit is a function of the product of the Infected andSusceptible stocks. This formulation has wide applica-bility, for instance in formulating how a new productspreads between people by word of mouth. (see “R&D,technological innovations and diffusion,” EOLSS on-line, 2002). More generally, this type of formulation isused when the existence of a flow requires the simulta-neous existence of two different stocks. For instance, ina predator-prey relation, the rate preying would be f(Predators*Preys). This says that the preying rate must bezero if there are no preys or if there are no predators.(For an actual formulation see “Ecological interactions:predator and prey dynamics on the Kaibab plateau,”EOLSS on-line, 2002).

Multiplicative and additive “effect” formulations

A generalization of the above product formulation isthe “multiplicative effect” formulation. The generalform of the formulation is:

Y = (Effect of X1 on Y)*(Effect of X2 onY)*. . .*(Effect of Xn on Y)*Ynormal

The above equation states that Y is a multiplicativefunction of X1, X2,. . .Xn and the effects of these vari-ables are formulated by “Effect of Xi on Y” functionstimes the “normal” value of Y, Ynormal. Thus, foreach effect, we have:

Effect of Xi on Y = f(Xi/Xi,normal), so that the inputvariables are normalized.

The meaning of “normal” values is that when all inputvariables are at their normal values, we expect theoutput value Y to be at its normal value Ynormal.(These are also called “reference” values). In a multi-plicative effect formulation this requires that all f (.)must yield 1, when Xi are at their normal valueXi,normal. In short, f (1) = 1 for all multiplicativeeffect functions. Such a normalization is important inbuilding robust models, because if absolute values areused as inputs to these functions, then it would bealmost impossible to experiment with different modelparameters, as the input values would quickly gooutside the ranges of the functions.

Examples of such effect formulations involvingsingle input variables were already seen in the


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population-food interaction model above (see Figure13). We have:

DeathFract = (Effect of Food on Df)*NormalDfNormalDf=0.05Effect of Food on Df = f (FoodperCapita/NormalFood-

perCapita)NormalFoodperCapita=45 (kg/animal)

The above formulation says that when the input vari-able FoodperCapita is at its “normal” value of 45, thedeath rate is at its “normal” value of 0.05. Thus, f(1)=1 by definition. (see the function plotted infigure 13). The specific shape and values of the func-tion depends of course on the specifics of the prob-lem. The function plotted in Figure 13 states thatwhen there is more food than “normal” the deathrate does not get any lower than normal. (The x-axisis therefore cut off at 1). Furthermore, the functionassumes that when FoodperCapita is moderately belowits normal value (between 0.75 and 1), death frac-tion is not affected much by food availability. (Thefunction starts yielding values increasingly greaterthan one, only after the food per capita ratios dropsbelow 0.75).

This formulation is a single-input multiplicativeeffect formulation. The formulation is non-linear intwo ways. First, since Deaths = DeathFract*Pop andDeathFract = f (Food), then in effect we have: Deaths = f(Food)*Pop, a non-linearity due to the product of twostate variables. A second source of non-linearity is thefact that DeathFract is itself a non-linear function ofFood (the curve in Figure 13).

In the same model, a second similar multiplicativeformulation exists:

Consumption = (Effect of Food on Consumption)*NormalConsumptionperCapita*Population

NormalConsumptionperCapita = 15 kg/animal/month

Effect of Food on Consumption = f (FoodperCapita/NormalFoodperCapita)

This formulation is very similar to the previous one.The function must again satisfy f (1)=1. (Figure 13). Itsays that when there is “normal” or above normalfood, the consumption per capita is at its “normal”(15 kg/animal/month). If the food per capita ratiodrops significantly (below about 0.60), then peranimal food consumption starts decreasing at increas-ing rates. One minor technical difference betweenthis example and the previous one is that since thereis only a single input, the effect formulation is embed-ded in the flow equation, as a short cut.

More typical applications of effect formulationwould involve several input variables. For instance,consider sales personnel booking orders. Each salesperson has a “normal” sales productivity, but this maybe affected by at least two obvious factors: Price of theproduct and the advertising done by the company.Thus, we have:

Sales = Sales Productivity*Sales ForceSales Productivity =(Effect of Price on SP)*(Effect of

Advertising on SP)*Normal SP

Effect of Price on SP = f (CompanyPrice/Competitor-Price)

Effect of Advertising on SP = f(CompanyAdvertis-ing/CompetitorAdvertising)

Normal SP = 50 units/person/monthSalesForce, Company Price, CompanyAdvertising, typi-

cally endogenous variablesCompetitorPrice, CompetitorAdvertising, could be vari-

ables or constants

The functions f (CompanyPrice/CompetitorPrice) and f(CompanyAdvertising/CompetitorAdvertising) are shownin Figure 15. Observe that they obey the principlesoutlined above. They both satisfy f (1)=1. Thus, theSales Productivity (SP) formulation above says thatwhen the price is at its normal value and the advertis-ing is at its normal value, sales productivity is at itsnormal value of fifty. Note that in this problemcontext, “normal” values are defined as competitor priceand competitor advertising. In other words, we assumethat there is a major competitor and if our price is atabout their level, then the price has “no effect” onsales productivity. Similarly, if our advertising(Liras/month) is about at competitor’s level, then ithas neutral effect on sales productivity. If our price isabove the competitor price, then it has the effect ofreducing our sales (function value less than 1). And ifour advertising is below the competitor’s, it has simi-lar negative effect. (See both functions in Figure 15a).Conversely, if our price is below the competitor’s,then the price effect function is greater than 1, mean-ing positive effect on sales productivity. And if ouradvertising is above the competitor’s, then the adver-tising effect function is greater than 1, meaning posi-tive effect on sales productivity. (See Figure 15a). Inthe context of the larger model involved, salespeople, company price, and advertising are mostlikely variables, all taking feedbacks from sales,revenues, salaries, cost-profit ratio, and so on.Furthermore, competitor price and/or advertisingmay be variables too in a model with larger boundary.Thus, it is important to note that all the “normal”values used in effect formulations may be variables aswell as constants. Finally, there can be more effectssuch as “effect of delivery delay” and so on, yieldingSales Productivity=(Effect of Price on SP)*(Effect ofAdvertising on SP)*(Effect of DelivDel)*Normal SP,where the Effect of DelivDel function would again besimilarly constructed.

Multiplicative effect formulations have two impor-tant properties. First, in a multiplicative formulation,extreme values of any of the inputs (if yielding zero)will completely dominate the outcome. Thus, theprevious example assumes that if our price were threetimes higher than the competitor price then nobodywould buy our product (Sales Productivity would bezero), even if we did a lot of advertising. The same maybe true if our delivery delay were five times as high asthe competitor’s, and so on. In such cases, if only oneof the effects is at its extreme worst value, then theoutput would be at its worst value, regardless of howfavorable the other inputs are. This assumption is real-istic in some cases, but not so in others, as will be seen


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later. A second important characteristic of multiplica-tive formulation is, since effects are multiplied, thecombined effect will increase or decrease geometri-cally, quickly yielding very small or large values, ifseveral inputs are involved. If there are eight multi-plicative effects and each has a moderately unfavorablevalue of 0.7, then the combined effect would be 0.78

(0.05), an extreme reduction in the output. It is there-fore risky to use more than four to five multiplicativeeffects, because controlling and validating such unsta-ble multivariable functions is extremely difficult.

It is therefore desirable in some cases to use additiveeffect formulations. Consider the urban populationdynamics problem discussed briefly earlier. Thecausal loop diagram is shown in Figure 4. Accordingto this conceptual model, in/out migration fractiondepends on job availability, house availability andcrime rate. The hypothesis is that high job and houseavailability values and low crime rate would encour-age in-migration, whereas low job and house avail-ability values and high crime rates would induceout-migration. We will give an additive effect formula-tion to represent this migration process. Since hous-ing and jobs have very similar effects, we will illustratethe formulation by focusing on jobs and crime rate.The following equations present such a formulation:

In_outMigration = MigrFract*PopulationMigrFract = NormalFract + (Effect of Jobs) + (Effect of

Crimes)

NormalFract = constant {can be 0 or some positive ornegative fraction}

Effect of Jobs = f (JobsperLabor/NormalJobsperLabor) Effect of Crimes = f(CrimeRate/NormalCrimeRate)NormalJobsperLabor, NormalCrimeRate, typically con-

stantsJobsperLabor, CrimeRate, other endogenous variables.

The In-outMigration formulation is a “bi-flow” whichmeans that if MigrFract is positive we have in-migrationand if MigrFract is negative we have out-migration. Inthe above additive formulation, NormalFract would beset 0, if we assume that when the jobs and crime rateare at their “normal” values, there is normally no in-migration to or out-migration from the city. (Alterna-tively, if NormalFract was say – 0.05 per year, then weassume that even if jobs and crime rate are at theirnormal values, there are some other not-includedfactors such as schools, which would create out-migra-tion from the city).

The functions f (JobsperLabor/NormalJobsperLabor)and f (CrimeRate/NormalCrimeRate) are shown inFigure 15(b). Note that their properties are differentthan the multiplicative ones. First, note that in anadditive formulation “effects” have units, 1/time inthe above example, while they are dimensionless inmultiplicative formulation. Second, at their “normal”values they yield 0 rather than one: f (1) = 0. Thisproperty is necessary for the addition to returnNormalFract, when both JobsperLabor and CrimeRate are


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Figure 15. (a) Multiplicative and (b) Additive effect function examples

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Input Output0.000 2.5000.250 1.9250.500 1.5500.750 1.2351.000 1.0001.250 0.7751.500 0.6251.750 0.4752.000 0.3252.250 0.2152.500 0.1252.750 0.0503.000 0.000

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at their normal values. Third, the additive effects can(often must) have negative values, since they add toor subtract from a “normal” value. Thus, the MigrFractformulation above says that when JobsperLabor is abovesome given “normal” JobsperLabor, then it has theeffect of promoting in-migration (function returninga positive fraction). And if CrimeRate is below Normal-CrimeRate, it has similar positive effect on in-migration. (See both functions in Figure 15b).Conversely, if JobsperLabor is below NormalJobsperLabor,then the jobs effect function is less than 0, meaning apush for out-migration. And if CrimeRate is aboveNormalCrimeRate, it has similar negative effect. (SeeFigure 15b). Note that there can be no domination of asingle factor in such an additive formulation. Even ifCrimeRate had an extremely negative effect, if Jobsper-Labor was very high, the effects could cancel eachother, or people could even migrate-in, according tothe additive formulation. In a multiplicative formula-tion, it is possible to shut off completely the in--migration if JobsperLabor is extremely low, regardlessof how low the crime rate is. This would be impossiblein an additive formulation. On the other hand,observe that the above “bi-flow” (in or out) formula-tion of migration would be impossible with multi-plicative effects. If the multiplicative effects areallowed to become negative, then one would getmeaningless results if both effects are negative: prod-uct of two negative effects would yield a positiveeffect, hence creating an in-migration! (Also, whenone effect is negative and the other one is positive, ina multiplicative formula, the larger is the positiveeffect, the more negative would be the combinedeffect – again meaningless). In practice, whethermultiplicative or additive effect formulation is suit-able depends on the specifics of the problem. (Alsonote that in an additive formulation “effects” haveunits, 1/time in the above example, while they aredimensionless in multiplicative formulation).

5.4. Time delay formulations

Time delays often play an important role in thedynamics of systems. Significant time lags may inter-vene between causes and their effects. There are twogeneral categories of time delays in system dynamics:material delays and information delays.

Simple material delay

Such delays exist on conserved (material) stock-flowchains. For instance, orders do not immediatelycreate arrivals of goods; this delay is often called “leadtime” of orders. When new workers are hired, there isa training period before they join the actual produc-tion force. Sometimes delays are in the very definitionof variables: normally, four years must pass beforefreshers can graduate from university. Two materialdelay examples are shown in Figure 16(a). In the firstexample, it takes an average of three years (SeedlingDelay) before seeds planted can create seedlings. Inthe second example, there is a production delaybefore a production decision can create finished

goods. Observe that in both cases delays exist on aconserved (material) stock-flow chain. The genericstock-flow diagram is also shown in Figure 16(c).Since delays involve stock-flow structures, they intro-duce phase lags between the inputs and outputs.(Figure 7 provides a clear visual illustration of how astock in between two flows introduces a phase lag).There are different ways of formulating such delays,depending on the specifics of the problem. Forinstance, in one extreme, when there is a change inthe inflow, it is assumed that there is no change at allin the output for t < Del and then the entire change isobserved at t = Del. In other words, OUT (t + Del) =IN(t). This is called a discrete delay and assumes thatthe delay Del is very rigid and applies to each entity inthe process exactly. The assumption is proper in rela-tively micro-level modeling. In more macro-levelmodels, the delay is an average, with a rather largevariance. The process involves an aggregation ofmany different types of entities so that some may beprocessed much faster than Del and some muchslower than Del. In these cases, under the assumptionof a continuous and homogeneous mixing of the differ-ent entities, a suitable formulation of the outflow is:

OUT = Stock/Del

Thus, the above formulation says that the inflow INaccumulates in the stock and then we expect 1/Del ofthe stock to flow out per unit time. If it takes an averageof three years for seeds to yield seedlings, we expectSeeds/3 seedlings per year. In the production example,the average Finishing rate is (Goods_In_Process)/ProdDelay. In the above formulation, when the inflow INis suddenly stepped up or down by a constant quantity,the output OUT gradually changes toward the newvalue of IN. (We assume that initially, IN=OUT, so thatthe structure is initially at equilibrium). Note that OUT ischanging continuously, before Del as well as after Del.(Figure 16(b)). The dynamic behavior of Stock (andOUT) is a goal seeking, negative exponential curve:

Stock (t) = Del*IN + (Stock(0) – Del*IN)*EXPON(– t/Del),

OUT = Stock/Del

The term Del *IN is the new equilibrium value of thestock, so that when Stocke= Del *IN, the outflow OUTe=Del*IN/ Del = IN, which is consistent at the equilibrium.

Observe in Figure 16 (b) that Del is just an averagedelay. Some entities are processed much faster andsome are processed much slower. Mathematically, Delis the average value of the above exponential func-tion. When t = Del is plugged in, it can be seen that 63percent of the discrepancy between the initial valueand the new equilibrium is covered and 37 percentremains. (In a discrete delay, the entire discrepancywould be covered suddenly at t = Del and no action atall before or after Del L). The continuous, exponen-tial delay formulation is suitable in most systemdynamics problems, since these macro-level modelstypically involve aggregation of many different enti-ties involving different time constants. But one majordrawback of this simple exponential delay is the factthat when IN is changed, the largest response in OUT


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is obtained in the next instant in time and the shapeof the OUT curve has a monotonically decreasingderivative (in magnitude). In some cases, it is morerealistic to have no or very small response early in theprocess and increasingly larger responses later on (anS-shaped response). This type of delay behavior –somewhere in between the simple exponential delayand the simple discrete delay – can be obtained byhigher order exponential delay formulation, as will beseen below.

Simple information delay

In some situations, a variable influences another onewith a delay, but not in a material stock-flow chain.This may mean “delayed effect” of a variable onanother one, such as crowding affecting birth fractionafter a time delay. Or it may represent human aware-ness, gradual learning of a changing situation, such as“perceived jobs per labor” as a delayed informationabout real jobs per labor (in which case our migrationdecision would be affected not directly by the latter,but by the “perceived” one). Two information delayexamples are shown in (a) of Plate 6.63–1. The firstexample illustrates the “delayed effect” implementa-tion. In the density-dependent population growthmodel, Crowding had a negative effect on Birth Fraction(Figure 11). Now suppose that a change in Crowdinglevel does not immediately affect Birth Fraction, but itdoes so after some time delay. Thus, Crowding directlyaffects “Implied Birth Fraction” and then after sometime delay, this latter becomes “Actual Birth Frac-tion.” The corresponding stock-flow diagram is shownin (b) of Plate 6.63–1. ImpliedBirthFract is an input tothe information delay structure and the delayedoutput is the ActualBirthFract. Just as in the materialdelay case, this can be represented either as a discretedelay, exhibiting no effect at all for a period of Del andthen the entire effect at Del or as a continuous, grad-ual delay. In the information delay context, discreteeffects are rare; continuous delay structure illustratedin (a) of Plate 6.63–1 is much more common. Theequations corresponding to the first example in (a) ofPlate 6.63–1 are:

ImpliedBirthFract = f (Crowding), exactly as givenalready in Figure 11

Correction = (ImpliedBirthFract – ActualBirthFract)/Correct_Del

ActualBirthFract (t+dt) = ActualBirthFract (t) +dt*(Correction)

In the above formulation, the INPUT to the delaystructure is ImpliedBirthFract and the delayed OUTPUTis ActualBirthFract. The dynamic behavior of thiscontinuous information delay structure is exactly thesame as that of the continuous material delay shownin Figure 16 (b). Initially, we assume that the system isat equlibrium i.e. ActualBirthFract = ImpliedBirthFract,so that the correction flow Correction = 0. As a test,when the INPUT (ImpliedBirthFract) is stepped up ordown, the OUTPUT (ActualBirthFract) approaches theINPUT in an exponential goal-seeking manner. (SeeFigure 16 (b)). At t= Del, 63 percent of the discrep-

ancy between the initial value and the new equilib-rium (the new INPUT) is again covered.

The second example in (a) of Plate 6.63–1 is anillustration of gradual human awareness, learning orperception. Assume that a certain product has a vary-ing quality (measured by frequency of breakdown)that customers would like to know in deciding topurchase the product or not. Customers in themarket place cannot instantaneously learn the actuallevel of the product quality; they can only learn itgradually, after a perception (learning) delay. Thus,the INPUT of the information delay is Product Qualityand the OUTPUT is Perceived Quality, which would lagbehind the actual Product Quality. The exact behaviorof Perceived Quality depends on the value of the percep-tion delay (Percept_Del). The smaller the perception delaythe faster will be the Correction; hence faster will be thetracking of the INPUT. But the risk of using too small“smoothing” constants is that they may yield unstablebehavior if the INPUT changes too fast. Note that theinformation delay structure given in Plate 6.63–1 is anapplication of the goal-seeking structure shown inFigures 9 and 20. In the information delay structure,the “goal” is the INPUT of the delay structure and theOUTPUT stock seeks this goal. (see the generic struc-ture shown in (b) of Plate 6.63–1). Finally observethat the information delay structure used above is acontinuous version of the exponential smoothingmethod often used in data smoothing for forecasting.The OUTPUT equation for the generic structuregiven in (b) of Plate 6.63–1 is:

OUT(t+dt) = OUT(t) + dt*Correction, whereCorrection = (IN – OUT)/SmoothDel

Thus:

OUT(t+dt) = OUT(t) + dt* (IN(t) – OUT(t))/SmoothDel

The standard exponential smoothing equation is:

OUT(t+1) = OUT(t) + alfa*(IN(t) – OUT(t))

Comparing the last two equations, we see that theyare the same by calling alfa = dt/SmoothDel. If ourmodel is discrete (dt=1), then alfa is just1/SmoothDel. The information delay structure is thusa continuous, exponential averaging process. Thelarger SmoothDel (the smaller alfa) the more smooth-ing is done on the INPUT. The noisy INPUT in (c) ofPlate 6.63–1 illustrates the smoothing action. Notethat OUTPUT is a smoother and time-lagged versionof the INPUT and that the larger SmoothDel, thesmoother the OUTPUT is.

Higher order delays

Delay structures are not direct causal formulations;they are behavioral approximations to the outputs ofmany cause-effect interactions that would make themodel unnecessarily too complicated, if explicitlymodeled. So, instead of modeling all the workforce,schedule and raw material interactions in a produc-tion setting, we simply approximate the resultingdelayed behavior by a material delay formulation. Assuch, it is critical that the delay structure capture the


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Figure 16. First order material delay (a) Examples, (b) Output behavior, (c) Generic structure

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input-output dynamics of the real structure realisti-cally. As mentioned earlier, a major drawback of thesimple exponential delay is the fact that when IN ischanged, the largest response in OUT is obtained inthe next instant in time and has a monotonicallydecreasing derivative. In some cases, it is more real-istic to have no or very small response early in theprocess and increasingly larger responses later on(an S-shaped response). This type of delay behaviorcan be obtained by higher order exponential delayformulation. A higher order delay is simply a serialcascading of several first order exponential delays.In Figure 17 (a), a third-order material delay exam-ple is illustrated, by simply extending the forest seed-planting example already introduced in Figure 16(a). The generic structure of the third-order contin-uous material delay is given in Figure 17 (b). Theequations are identical to those already describedfor the first order material delay. (Note that for thetotal average delay of the entire structure to be Del,the delay time used in each sub-structure must beDel/3). The dynamic behavior of the third-orderdelay structure, when disturbed from equilibrium isshown in Figure 17 (c). Observe that the output hasnow an s-shaped pattern – first increasing slope andthen decreasing slope. The initial period of lowresponse may be important in capturing thedynamic response characteristics of some real lifeprocesses. Similarly, a third order information delayis obtained by cascading three first order informa-tion delay structures, each with delay times of Del /3(Figure 17(d)). The third order delay is a commonillustration of high-order delays. Most software hasbuilt in macro functions for third order delays. Ifmore steep S-shaped response is desired, one caneasily increase the order of the delay by cascadingadditional first or even third order delay structures.In the limit, the discrete delay formulation OUT (t+Del) = IN(t) can be shown to be the nth order contin-uous delay structure, as n → ∞.

5.5. Delays in action: oscillating structures

An important consequence of having delays in struc-tures is that they are potential sources of oscillatorybehavior. For instance, consider the density depend-ent growth structure and its S-shaped behavior givenin Figure 11. In the previous section, the followinginformation delay structure was added to thedensity-loop:

ImpliedBirthFract = f(Crowding), exactly as givenalready in Figure 11

Correction = (ImpliedBirthFract – ActualBirthFract)/Correct_Del

ActualBirthFract (t+dt) = ActualBirthFract (t) +dt*(Correction)

With the addition of the above delay, the populationnow has the potential to oscillate up and downaround the equilibrium (Capacity), before it eventu-ally settles down. Because the effect of Crowding onbirth fraction is “delayed,” population can overshootthe Capacity, then undershoot, and so on.

The stock management structure

A very typical managerial application of delays is inthe context of the standard goal-seeking (stock-adjustment) structure illustrated earlier in Figure 9.The generic version of this stock-adjustment structureis also provided in Figure 20. The equations are:

Discrepancy = Goal – StateControlAction = Discrepancy/AdjustTimeState (t+dt) = State (t) + dt*ControlAction

The behavior of the above model is pure exponentialgoal seeking. But if delays exist anywhere around thegoal-seeking loop, then the system has the potentialto oscillate. The delay can be in measuring the State(an information delay between the State and theDiscrepancy), between the Discrepancy and ControlAction(a delayed action, information delay) or between theControlAction and the State (a material delay before theaction reaches the State). One or more of such delayscan cause the system to oscillate. For instance, anextension of this simple goal-seeking structure is thestock management structure, used in various manage-ment settings (such as inventory management,human resources management, cash management,driving, weight control. . .) We illustrate the generalstock adjustment problem in the context of inventoryorder management in Figure 18. Compared to thebasic goal-seeking structure, there are two majorextensions: first, there is an outflow from the stock(Shipments) and this outflow must be taken intoconsideration in writing the ordering equation.Second, there is a material delay (Supply Line) beforethe ordered goods reach our stock, which must alsobe taken into account. (The Supply Line delay isformulated by the standard material delay equation:Acquisition_rate = Supply_Line/Acquisition _delay).

First, if the outflow Shipments is not considered, it iseasy to prove that the original equationOrderRate = (DesiredStock – Stock)*StockAdjFract will notbring the stock to its desired equilibrium. At the equi-librium, we must have OrderRate = Shipments, whichmeans:

(DesiredStock – Stock)*StockAdjFract = Shipments, so that:Stocke = DesiredStock – Shipments/StockAdjFract, a value

smaller than DesiredStock.

To correct this “steady state error” we simply addShipments to the order equation:

OrderRate = (DesiredStock – Stock)*StockAdjFract +Shipments,

which will yield Stocke= DesiredStock.

But the above formulation is not realistic, as itassumes that we know the actual values of shipmentsat all times. A more realistic formulation must use“estimated” or “expected” shipments instead. Thislatter can be formulated by using the standard expo-nential smoothing (averaging) structure seen earlier:

Expected_shipments(t) = Expected_shipments(t – dt) +(Correction) * dt

Correction = (shipments – Expected_shipments)*alphaalpha = 0.5


1164


1165

Seeds planting

Seeds Seedlings growth Seedlings Youth growth Young treesMaturing rate

Maturing delayYouth delaySeedling delay(a)

Out

Del:3

In

S1 Flow1 S2 Flow2 S3

(b)

(c)

(d)

Figure 17. High (third) order delays: (A) Material delay example, (b) Generic third order material delay structure, (c) Output behavior, (d)Generic third order information delay structure

In

Correct1 Stock1

Correct2 Stock2

Correct3 Out

Del:3

1:In 2:Out 3:In 2 4:Out 2

1 2 3 4 1

21

3

1

3

200

100

00.00 7.50 15.00 22.50 30.00

Months

2

2

3 44

4

And the “desired” stock level can also be tied to theexpected shipments, since higher shipments wouldnormally call for higher desired inventory levels:

Desired_stock = Stock_Level_Factor*Expected_shipmentsStock_Level_Factor = 3

The resulting ordering equation is:

OrderRate = (DesiredStock – Stock)*StockAdjFract +Expected_Shipments

When the model is run with this ordering formula-tion, the stock indeed reaches its desired level (ofthirty), as seen in Figure 19(a). Observe though theoscillations in the inventory and supply line stocks.These oscillations are caused by the existence of thematerial delay (Supply Line) between orders and theinventory. The oscillations are further amplified bythe fact that the above ordering formulationcompletely ignores the existence of the supply linedelay. A better formulation would take into accountthe existence of Supply line (SL) as follows:

order_decision = stock_adjustment + supply_line_adjustment + Expected_shipments

desired_supply_line = acquisition_delay*Expected_shipments

supply_line_adjustment = (desired_supply_line-supply_line)*SL_adj_frac

SL_adj_frac = 0.4

The only new term in the above formulation is“supply line adjustment” which is identical to thestandard “stock adjustment” term defined earlier,except that “desired supply line” must be properlydefined. At the equilibrium, the supply line mustyield an outflow (acquisition) equal to Shipments.Thus,

Supply_line/Acquisition_delay = Shipments,

yielding

Supply_linee = acquisition_delay*Shipments,

and since we do not exactly know the shipments, then:

Desired_supply_line = Supply_linee = acquisition_delay*Expected_shipments.

When the same model is run with the above order-ing policy (all parameters remaining unchanged), theoscillatory behavior disappears as shown in Figure19(b). Inclusion of “goods on order” (supply line) inthe ordering equation prevents unnecessary over-ordering, hence yielding an improved, and morestable inventory system.

There are naturally other oscillatory structures.For instance, the epidemic model described inFigure 14 was demonstrated to exhibit oscillatorybehavior for certain values of infection fraction.Also, when two stocks are coupled with a negativefeedback loop (such as a predator-prey relation),there is potential for oscillatory behavior. (see“Ecological interactions: predator and prey dynam-ics on the Kaibab plateau,” EOLSS on-line, 2002)The generic structure of the 2–stock oscillator isshown in Figure 20.

5.6. Generic structures and dynamics

Some important elementary structures are frequentlyused as basic building blocks in diverse applications.These are sometimes called “generic” structures. Forconvenience, the most basic generic stock-flow struc-tures, their causal loop diagrams and their possibledynamic behaviors are summarized in Figure 20. Eachof these structures was either described in detail or atleast mentioned earlier in this article. Figure 20 isprovided as a quick reference.

6. MATHEMATICAL AND TECHNICALISSUES

6.1. Solutions, equilibrium, and stability

There are no general mathematical solution meth-ods for non-linear, high-order feedback models. Insome situations, approximate or partial solutionscan be obtained. But this often requires that theorder of the model be reduced, some non-linearrelations linearized, and some feedback loopsbroken, in which case there is always the risk oflosing some of the most critical dynamic characteris-tics of the system. Analytical procedures are there-fore very rarely applicable in system dynamicsstudies. A more useful and relevant application ofsuch procedures is in examining the elementarygeneric structures mentioned above. Exact knowl-edge of the dynamic properties of these structures isvery useful in building larger models and carryingout quasi-mathematical analysis.

A relatively more tractable mathematical problemis to determine the equilibrium solutions of the modeland their stability. The equilibrium solutions are thosethat make all derivatives zero. In other words, for eachstock i, the sum of its flows must be zero. The equilib-rium solutions can be obtained by solving the follow-ing set of equations:

∑ flows = 0, for each stock i.j

If there are n stocks, this would give n simultaneousnon-linear equations to be solved for Si. Althoughthese equations are algebraic (not dynamic), solvingnon-linear algebraic equations is still a very difficultproblem in general. In most cases, “some” of theequilibrium points can be found, but no knowledgeof how many global equilibria there are. A veryimportant property of an equilibrium point is itsstability. There are different technical definitions ofstability. For the purpose of this article it is enoughto state that an equilibrium point is stable, if solu-tions that start near it tend toward it or stay in theclose neighborhood of it. The stability of each equi-librium point can be analyzed by linearizing themodel in the close neighborhood of each equilib-rium point, and checking the eigenvalues. (see“Equilibrium and stability analysis,” EOLSS on-line,2002). Finally, some non-linear models can exhibitvery complicated dynamic behaviors such as limitcycles, bifurcations, period-doubling, strange attractors,


1166


1167

Sl adj. frac.

Acquisition delay

Shipments

Stock

Acquisition rateSupply line

Supply lineadjustment

Order decision

Desired supply line

Stock level factor

Desired stock

Expectedshipments

Stock adjustmentfraction

Stock adjustment

++

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Stock adj. frac.

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Sl adj. frac.

Acquisition delay

Stock level factor

Supply line

Acquisition rate

Stock acquisition system

Ordering policy

Desired supply line

Figure 18 The general stock adjustment problem applied to inventory management. (a) causal loop diagram (b) stock-flow structure

(a)

(b)

–

–

Order rate

and chaos. Analysis of such models is not possible bystandard mathematical or even numerical methods.There are special numerical and (limited) analyticaltechniques to analyze complex non-linear dynamics.(See “Non-linear dynamics and chaos,” EOLSS on-line, 2002).

6.2. Discrete versus continuous time

System dynamics models can be continuous-time ordiscrete time. In a continuous model, the stock equa-tions are of the form:

Stock(t) = Stock (0)+ ∫0t(∑ flows)dt

j

In simulation, the above equation is approximated by:

Stock(t+dt) = Stock(t) + dt *∑ flowsj

It is important to note that in the above continuousmodel “dt” is an arbitrarily small computational step.It has no real life meaning. On the other hand, in adiscrete model, the stock equations are:

Stock(t+1) = Stock(t) + ∑ flowsj

So that dt is exactly one time step that corresponds to

the real discrete time period in the real system. (Suchas a month, a decade, a century, a second or fivedays). Dynamics of discrete systems can be very differ-ent than continuous systems. For instance, 1–stockdiscrete systems can oscillate, but 1–stock continuoussystems cannot. First order discrete system can evengenerate chaotic dynamics. (See “Non-linear dynam-ics and chaos,” EOLSS on-line, 2002). If the model iscontinuous, care must be exercised in choosingproper dt value, because a large dt value may producespurious dynamic behaviors as a result of numericalsolution errors.

6.3. Numerical integration and software

Proper selection of dt is not the only issue in obtain-ing correct simulated behaviors. There are numerousnumerical integration methods, ranging from simpleand fast ones to very sophisticated yet slow ones. Thesimplest is Euler’s method, which is also used as thedefault form of stock equation:

(Stock (t+dt) = Stock (t) + dt*)∑ flows).j

There are higher order methods such as high orderTaylor methods, Runge-Kutta methods and variousadaptive, self-correcting methods. For some methods(like Euler’s) small enough dt selection is crucial,while other methods vary their own dt adaptively asneeded, during simulation. (See “Simulation softwareand numerical considerations,” EOLSS on-line,2002). Finally, numerical analysis of non-linear struc-tures producing complex non-linear dynamics (suchas bifurcations, period-doubling, chaos) necessitatesspecial numerical procedures. (see “Non-lineardynamics and chaos,” EOLSS on-line, 2002)

Several user-friendly modeling and simulation soft-ware exist for system dynamics. The most popularones are STELLA, Vensim, Powersim and DYNAMO.These types of software are all equipped with differ-ent numerical simulation procedures, as well as someother modeling and analysis support tools. (see“Simulation software and numerical considerations,”EOLSS on-line, 2002).

6.4. Noise (randomness) in models

System dynamics models can be deterministic as wellas stochastic. Randomness can be included in param-eters (like delay constant or productivity coefficient)or external input variables (such as demand or stockprices). In any case, the common practice is toinclude the randomness in the model late in theprocess, after the model has been completed andtested. Dynamics of a systemic feedback model areprimarily created by the structure of the model.Premature inclusion of noise in model parameterswill unnecessarily complicate the construction, test-ing and analysis of the model. After the model is fullytested, the purpose of including noise in selectedparameters and variables is to see if there are some“dormant” modes of behaviors (such as oscillations atdifferent frequencies) that would reveal themselves


1168

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Figure 19. The behavior of the stock adjustment structure: (a)Goods in supply line are ignored in the order decision (b) supplyline information is taken into account. All other parameters are thesame

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once excited with resonant noise frequencies. Large-scale feedback models are typically extremely insensi-tive to independent (“white”) noise sequences. Suchnon-correlated noise sequences are immediatelyabsorbed by the feedback loops of the model so thatthe dynamic behaviors remain essentially unaltered.In order to be able to excite the model with noisyinput, auto correlated noise (“pink” noise) must beused. Auto correlated noise can be obtained by pass-ing white noise through an exponential smoothingfilter. In general, the larger the smoothing time, themore auto correlated the noise will become (see Plate6.63–1 for an illustration). How much autocorrela-tion is needed in the noise input depends on thestructure and natural oscillations of a given model.

7. MODEL TESTING, VALIDITY,ANALYSIS, AND DESIGN

7.1. Model testing and validity

As mentioned earlier in the steps of system dynamicsmethodology, the model must be tested thoroughly,before policy design and implementation phases. Themodel must first be tested for internal consistency: Isthe simulation model an accurate representation ofthe conceptual model (the dynamic hypothesis)?Does the model do what the modeler intends it to do?This question is known as the model “verification”problem in simulation. The purpose is to assure thatthere are no inconsistencies between the model andthe dynamic hypothesis. Verification tests also includechecking the dimensional consistency of the modeland to make sure that the simulation step dt is smallenough to yield acceptable numerical accuracy.

Model validity (credibility) testing is the next step.The purpose here is to establish that the model is anacceptable description of the real system with respectto the dynamic problem of interest. Model credibilityis established by two types of tests:

• Structure tests: is the structure of the model ameaningful description of the real relations thatexist in the problem of interest?

• Behavior tests: are the dynamic patterns gener-ated by the model close enough to the realdynamic patterns of interest? (see “Model testingand validity,” EOLSS on-line, 2002).

Examples of structural tests are: having experts evalu-ate the model structures, and robustness of eachequation under extreme conditions and extremecondition simulations. Behavior tests are designed tocompare the major pattern components in the modelbehavior with the pattern components in the realbehavior. Such pattern measures include slopes,maxima and minima, periods and amplitudes of oscil-lations (autocorrelation functions or spectral densi-ties), inflection points, and so on. Behavior testingdoes not involve point-by-point comparison of modelbehavior with real behavior; it involves comparing ofpatterns involved in the two. It is important to notethat unless structural validity is established first,

behavior validity is meaningless in system dynamicsmodeling. (see “Model testing and validity,” EOLSSon-line, 2002).

7.2. Model analysis and design

Model analysis can be defined as “understanding whyand how the model behaves the way it does.” Analysiscan be done mathematically when possible, asmentioned above (analytical solution, equilibriumpoints, stability analysis). Since this is often impossi-ble, analysis is more typically done by sensitivity analy-sis: how and to what extent the behavior of the modelchanges, as a result of changes in its parametersand/or structures. Sensitivity analysis can be of differ-ent types (with respect to structures and with respectto parameter values; numerical sensitivity and behav-ior pattern sensitivity) and there are various methodsto do sensitivity analysis. (see “Sensitivity analysis,”EOLSS on-line, 2002).

Policy analysis is about the sensitivity of modelbehavior to the policy parameters and/or policy struc-tures. Policies represent rules for conscious humancontrol. They may be characterized by:

• set of parameter values (numeric)• function values (numeric)• function shapes (structural)• forms of policy equations (structural)

Policy analysis and design may involve altering one ormore of the above model characteristics and examin-ing the resulting behavior. Like sensitivity analysis,policy analysis can also be numerical or pattern-oriented. Pattern-oriented policy analysis is naturallymuch more important, since the purpose of systemdynamics studies is to improve undesirable dynamicbehavior patterns. Policy design is determining whatchanges in the model structure and parameters(“improved policy”) would lead to an improvedmodel behavior.

No matter how scientific and exact the policy designphase is, naturally the ultimate success depends onimplementation. There are some general guidelinesfor implementation success that system dynamicsresearchers have developed. (see “Implementationissues,” EOLSS on-line, 2002). There are also newimportant developments that seek to enhance imple-mentation success, such as group model building (see“Group model building,” EOLSS on-line, 2002) andinteractive learning environments (see “Modeling forlearning and interactive environments,” EOLSS on-line, 2002).

8. CONCLUSIONS AND FUTURE

System dynamics discipline emerged in the late 1950s,as an attempt to address dynamic, long-term policyissues, both in the public and corporate domain.Since then, applications have expanded to a very widespectrum, including national economic problems,supply chains, project management, educationalproblems, energy systems, sustainable development,


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politics, psychology, medical sciences, health care,and many other areas. Today, interest in systemdynamics and related systemic disciplines is growingvery fast. At present there is a bottleneck in formaleducation for system dynamics expertise. As Jay W.Forrester puts it: “new fields, like system dynamics,that cut across boundaries of existing fields, but donot lie within any one of those fields, lack a home anda supporting constituency in universities.” A majorchallenge for the next decade or so is to design andset up educational institutions that can meet thegrowing need for system dynamics and systems think-ing education. These institutions should cover theentire spectrum; from K–12 to post graduate educa-tion. There is already evidence that great success canbe obtained in K–12 education by combining systemdynamics with “learner-directed” learning. (see “Onthe history, the present and the future of systemdynamics,” EOLSS on-line, 2002).

The second major challenge for system dynamics isto overcome the “implementation” hurdles. Tradi-tional system dynamics studies have been in the “exter-nal consultant” mode, in which a modeling expert(consultant) studies a problem and comes up withrecommendations. Unfortunately, the implementationsuccess record in using this approach has been ratherslim. There are numerous important causes of theimplementation failure beyond the scope of thissection. (see “Intellectual roots and philosophy ofsystem dynamics,” and “Implementation issues,”EOLSS on-line, 2002). But they can perhaps be summa-rized as: “the collective mindset, the jargon of the real-system participants and that of the external consultantdo not match.” The real system therefore cannotinternalize the design recommendations made by theexternal expert. To overcome this hurdle, there havebeen some significant developments in the areas of“group model building” and “modeling for learning.”In “group model building,” system participants takepart in all the phases of the system dynamics project.The idea is to make full use of the “mental models” ofthe system participants in building the system dynam-ics model, as well as to give them time and opportunityto modify their mental models so that internalizationof model recommendations is possible. (see “Groupmodel building,” EOLSS on-line, 2002). In the “model-ing for learning” approach, models are typically small,they are built in a group process (including non-tech-nical participants) and their primary use is to enhance(organizational) learning. Another use of systemdynamics modeling that often supports “modeling forlearning” is interactive simulation gaming (“manage-ment flight simulators”). In this mode, players makesome decisions interactively during the course of thesimulation. Such interactive games can be used insetting up “organizational learning laboratories.” (see“Modeling for learning and interactive learning envi-ronments,” EOLSS on-line, 2002).

System dynamics has the potential to make revolu-tionary contributions to education and learning ingeneral. In management education, the traditional“case study approach” is already being complem-ented by system dynamics models (“model-based case

studies”). In model-based case studies, students canexperiment on simulation models of the case, whichgives them a chance to rigorously test their own theoriesof how the problems in the case could be avoided. Asimilar approach can be used in various disciplines rang-ing from psychology to political science, from physics tomedicine, from environmental science to economics.

System dynamics has a unique set of tools to offeras we enter the twenty-first century. It can address thefundamental structural causes of the long-termdynamic contemporary socio-economic problems. Its“systems” perspective challenges the barriers thatseparate scientific disciplines. The problems in thereal world have no “discipline barriers,” they havesocial, economic, psychological, political, and techni-cal aspects, all interacting in a complex manner. Manyeducation experts believe that the current heavilycompartmentalized education will change into amore integrated education in the 21st Century. Theinterdisciplinary and systemic approach of systemdynamics could be critical in dealing with the increas-ingly complex problems of our modern world: fromhunger to ecological problems; from globalization tounemployment, from wars to education.

KNOWLEDGE IN DEPTH

In depth knowledge of this subject is available inseveral chapters in the on-line Encyclopedia of LifeSupport Systems, organized as follows:

Systems Dynamics: Systemic Feedback Modeling forPolicy Analysis

1. Systems Dynamics in Action: Selected ExamplesUrban dynamics, Louis Edward Alfeld, USASupply-chain dynamics, the “Beer Distribution Game”

and misperceptions in dynamic decision-making,John Sterman, Sloan School, MIT, Cambridge, Mass., USA

Market growth, collapse and failures to learn frominteractive simulation games, John Sterman, SloanSchool, MIT, Cambridge, Mass., USA

A dynamic model of cocaine prevalence, Jack B.Homer, Homer Consulting, USA

Ecological interactions: predator and prey dynamicson the Kaibab plateau, Andrew Ford, Program in Envi-ronmental Science and Regional Planning, WashingtonState University, USA

R&D, technological innovations and diffusion, PeterM. Milling and Frank H. Maier, Industrieseseminar derUniversitat Mannheim, Mannheim University, Germany.

2. Conceptual and Philosophical Foundations, YamanBarlas, Department of Industrial Engineering, BogvaziçiUniversity, Turkey

On the history, the present and the future of systemdynamics, Jay W. Forrester, MIT System DynamicsGroup, Sloan School of Management, MassachusettsInstitute of Technology, USA

Different philosophical positions and system dynam-ics, Yaman Barlas, Department of Industrial Engineer-ing, Bogvaziçi University, Turkey


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Intellectual roots and philosophy of system dynamics,Willard R. Fey, Ecocosm Dynamics Ltd., USA

The role of system dynamics within the systems move-ment, Markus S. Schwaninger, Institute of Management(IfB), University of St. Gallen, Switzerland

System thinking, modeling and organizational learn-ing, P. Senge, Society for Organizational Learning, USA.

3. Methodology for Systematic Feedback ModelingStep by step illustration of the methodology, James H.

Hines, M.I.T., Cambridge, Mass., USA.Mental models of dynamic systems, James K. Doy, David

N. Ford, Michael J. Radzicki, and W. Scott TreesKnowledge elicitation, Jac A.M. Vennix, Policy Sciences,

Nymegen University, HollandQualitative and quantitative modeling in system

dynamics, R. Geoff Coyle, Cranfield University, UKModel testing and validation, Yaman Barlas, Depart-

ment of Industrial Engineering, Bogvaziçi University,Turkey

Sensitivity analysis, Andrew Ford, Program in Environ-mental Science and Regional Planning, WashingtonState University, USA

4. Technical Issues in Modeling and SimulationEquilibrium and stability analysis, Javier Aracil, Escuela

Superior de Ingenieros, Universidad de Sevilla, Spain;Francisco Gordillo, Escuela Superior de Ingenieros,Universidad de Sevilla, Spain

Complex non-linear dynamics and chaos, Erik Mosek-ilde

Simulation software and numerical issues, AlexanderL. Pugh, Sloan School of Management, MassachusettsInstitute of Technology, USA

Optimization of systemic feedback models, BrianCharles Dangerfield

5. Policy Improvement and Implementation IssuesPublic policy, Ali N. Mashayekhi, Department of Indus-

trial Engineering, Sharif University of Technology andInstitute for Research in Planning and Development, Iran

Strategic management, systems thinking and model-ing, Erich Zahn, Department of Business Administra-tion, Universitat Stuttgart, Germany

Implementation issues, James H. Hines; James M. LyneisGroup model building, Jac A.M. Vennix, Policy Sciences,

Nymegen University, HollandModeling for learning and interactive environments,

Pal I. Davidsen

6. System Dynamics and Sustainable DevelopmentA pervasive duality in economic systems: implications

for development planning, Khalid Saeed, SocialScience and Policy Studies, Worcester Polytechnic Insti-tute, USA

The ECOCOSM paradox, Willard R. Fey, EcocosmDynamics Ltd., USA; Ann C. W. Lam, Ecocosm Dynam-ics Ltd., USA

Lessons from electric utility modeling, Andrew Ford,Program in Environmental Science and Regional Plan-ning, Washington State University, USA

Regional development and irrigation projects, A.Saysel, University of Bergen, Norway and Yaman Barlas,Department of Industrial Engineering, Bogvaziçi Univer-sity, Turkey

Public infrastructure/public finance, Ali N. Mashayekhi,Department of Industrial Engineering, Sharif University ofTechnology and Institute for Research in Planning andDevelopment, Iran

System dynamics for discerning developmental prob-lems, Khalid Saeed, Social Science and Policy Studies,Worcester Polytechnic Institute, USA

GLOSSARY

Dynamic behavior: dynamic performance patterns of thevariables created by the operation of the structure of thesystem over time.

Dynamic problems: problems characterized by variablesthat undergo significant changes in time. Being chronicin nature, they necessitate continuous managerial moni-toring, control, and action.

Endogenous: the dynamics are essentially caused by theinternal feedback structure of the system.

Feedback: a succession of cause and effect relations thatstarts and ends with the same variable. Also called “loop”or “circular” causality.

Flows: They directly flow in and out of the stocks, thuschanging their values. They represent the “rate ofchange” of stocks.

Generic structures: fundamental compact model structuresfrequently used as basic building blocks in diverse appli-cations.

Model: a representation of selected aspects of a real systemwith respect to some specific problem(s).

Negative loop: a feedback loop that balances, counteractsan initial change in any of its variables. It produces a goal-seeking behavior.

Non-linearity: any relationship where the output is notpurely proportional to the input; any relationship that isnot linear.

Positive loop: a feedback loop that reinforces, compoundsan initial change in any of its variables. It produces anexponentially growing or collapsing behavior.

Simulation: a step-by-step operation of the model structureover compressed time; imitation of the operation of thereal system.

Stocks: They represent the important accumulations overtime. They are also called “states” as they collectivelyrepresent the state of the system at time t.

Structure: the totality of the relationships that exist betweensystem variables.

Systemic: resulting from the complicated interactionsbetween many variables in the system.

Time delay: a time duration that intervenes before a causecan reach its effect. Delay can exist in physical flows or ininformational cause-effect relations.

BIBLIOGRAPHY

BARLAS, Y. 1996. Formal Aspects of Model Validity and Vali-dation in System Dynamics. System Dynamics Review, Vol.12,No.3, pp. 183–210. [A comprehensive overview of modelvalidity and validity testing in system dynamics, includingphilosophical foundations as well as specific tests.]

COYLE, R G 1996, System Dynamics Modelling: A PracticalApproach, London, Chapman and Hall, now availablefrom CRC Press. [A treatment of both qualitative andquantitative system dynamics modeling, as well as theoptimization of models.]


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FORD, A. 1999. Modeling the Environment, Island Press. [Anintroductory textbook on system dynamics, as well as envi-ronmental and energy applications.]

FORRESTER, J. W. 1961. Industrial Dynamics. Waltham, Mass.,Pegasus Communications. 464 pp. [The book establishesthe foundation of system dynamics as a new discipline. Itcontains conceptual foundations, several business appli-cations, and very useful technical appendices.]

–––––.1969. Urban Dynamics, Waltham, Mass., PegasusCommunications. 285 pp. [Presents a classical model ofurban growth, stagnation, and possible revival. Simulationsshow why popular policies for improving cities can fail.]

–––––.1975. Collected Papers of Jay W. Forrester. Waltham,Mass., Pegasus Communications. 284 pp.[ Seventeen clas-sical papers on management, corporate organization,urban policy, and the global environment.]

GUTIERREZ, L. T.; FEY, W. R. 1980. Ecosystem Succession.Cambridge, Mass., The MIT Press. [This is one of themost extensive system dynamics analysis of a large-scale,non-human living system (grassland biome). It illustratesthe important variables and feedback loops that createmost biomes’ dynamic patterns and the philosophy ofstudying non-human systems.]

MEADOWS, D. H.; MEADOWS, D. L.; RANDERS, J.; BEHRENS

III. W. W. 1972. The Limits to Growth. New York, UniverseBooks. 205 pp. [A popular book on world dynamics thathas been translated into many languages, and has soldmillions of copies.]

MORECROFT, J. D. W.; STERMAN, J. D. (eds.). 1994. Modelingfor learning organizations. Portland, Ore., ProductivityPress. [This book contains a collection of papersdiscussing the rationale behind modeling for learning,issues like knowledge elicitation, and examples of groupmodel building in action.]

RICHARDSON, G. P. 1999. Feedback Thought in Social Scienceand Systems Theory. Waltham, MA., Pegasus Communica-tions. [A comprehensive text that traces the history offeedback thinking in the social sciences and systemstheory.]

ROBERTS, N, ET.AL. 1983. Introduction to Computer Simulation:A System Dynamics Approach, Portland, Ore., ProductivityPress. [A very useful introductory book on systems think-ing and system dynamics, especially for high school andnon-technical university students.]

SAEED, K. 1998. Towards Sustainable Development, 2nd Edition:Essays on System Analysis of National Policy. Aldershot, UK,Ashgate Publishing Company. [Readings on system dynam-ics applications to sustainable developmental agendas.]

SENGE, P. M. 1990. The fifth discipline. The art and practice ofthe learning organization. New York, Doubleday/Currency.[A very popular text that introduces system dynamics andsystems thinking to the business world, placing them inthe context of organizational learning, together withsister disciplines.]

STERMAN, J. D. 2000. Business Dynamics: Systems Thinking andModeling for a Complex World. New York, Irwin. McGraw-Hill. 982 pp. [An excellent, very comprehensive text bookfor the study of system dynamics.]

System Dynamics Review. 1985. Chichester, UK, John Wiley.[Main scientific journal for system dynamics, publishingmethodological articles as well as a wide range of appli-cations.]

VENNIX, J. A. M. 1996. Group model-building: facilitating teamlearning using system dynamics. Chichester, UK, John Wiley.[This book focuses on Group Model Building, includingtheoretical/conceptual chapters as well as real applica-tion projects.]

ACKNOWLEDGMENTS

I am grateful to my student-assistants Özge Karanfiland Özge Özdemir for their meticulous help inpreparing the figures.

BIOGRAPHICAL SKETCH

Yaman Barlas received his B.S., M.S. and Ph.D.degrees in Industrial and Systems Engineering, fromMiddle East Technical University, Ohio Universityand Georgia Institute of Technology respectively. In1985, upon receiving his doctoral degree, he joinedMiami University of Ohio as an assistant professor ofSystems Analysis where he was granted tenure in1990. He worked as a guest researcher at MIT in thesummer of 1990. In 1992, he gave seminars in Istan-bul, as a United Nations TOKTEN consultant. Hejoined Bogv aziçi University in 1993, where he iscurrently working as a professor of Industrial Engi-neering and directing the SESDYN research labora-tory (http://www.ie.boun.edu.tr/labs/sesdyn/). Hespent his sabbatical leave at University of Bergen inNorway in 2000.

His interest areas are validation of simulationmodels, system dynamics methodology, modeling/analysis of socio-economic problems and simulation asa learning/training platform. He has publishednumerous articles in journals, proceedings and books,and offered academic and professional seminarsnationally and internationally in these areas. Heteaches simulation, system dynamics, dynamics ofsocio-economic systems and advanced dynamic systemsmodeling. He is a founding member of the SystemDynamics Society and member of several other inter-national and national professional organizations.Professor Barlas was the Chair of the 15th InternationalSystem Dynamics Conference, held in Istanbul in 1997.He is the editor of System Dynamics Review for short arti-cles, an invited Honorary Editor of the Encyclopedia ofLife Support Systems and a former President of theSystem Dynamics Society.


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