A comprehensive analytical approach for policy analysis of system dynamics models

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Formal tools to link system dynamics model's structure to the system modes of behavior have recently become available. In this paper, we aim to expand the use of these tools to perform the model's policy analysis in a more structured and
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  Decision Support A comprehensive analytical approach for policy analysis of system dynamics models Mohamed Saleh a,1 , Rogelio Oliva b, * , Christian Erik Kampmann c,2 , Pål I. Davidsen d,3 a Decision Support Department, Faculty of Computers and Information, Cairo University, 5 Ahmed-Zwail Street, Orman–Giza, P.O. Box 12613, Egypt  b Mays Business School, Texas A&M University, 301C Wehner – TAMU 4217, College Station, TX 77843, USA c Department of Innovation and Organizational Economics, Copenhagen Business School, Kilevej 14A-B, Room 3.82, DK-2000 Frederiksberg, Denmark d System Dynamics Group, Department of Geography, University of Bergen, P.O. Box 7800, 5020 Bergen, Norway a r t i c l e i n f o  Article history: Received 24 April 2008Accepted 15 September 2009Available online 20 September 2009 Keywords: System dynamicsLinear model analysisEigenvalue analysisLeverage points a b s t r a c t Formal tools to link system dynamics model’s structure to the system modes of behavior have recentlybecome available. In this paper, we aim to expand the use of these tools to perform the model’s policyanalysis in a more structured and formal way than the exhaustive exploratory approaches used to date.Weconsiderhowapolicyintervention(aparameterchange)affectsaparticularbehaviormodebyaffect-ing the gains of particular feedback loops as well as how it affects the presence of that mode in the var-iableofinterest.Thepaperdemonstratestheutilityofconsideringbothoftheseaspectssincetheanalysisprovidesanassessmentoftheoverall impactofapolicyonavariableandexplains why theimpact occursin terms of structural changes in the model. Particularly in the context of larger models, this methodenablesamuchmoreefficientsearchforleveragepolicies,byrankingtheinfluenceofeachmodelparam-eter without the need for multiple simulation experiments. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction The purpose of a system dynamics (SD) intervention is to identify how structure and decision policies generate system behavior iden-tified as problematic, so that structural and policy oriented solutions can be identified and implemented (Forrester, 1961; Sterman, 2000).The approach relies on formal simulation models to capture the detailed complexity of the problemsituation and to make reliable behav-ioralinferences.Thefieldhasdevotedagreatdealofattentiontomodelvalidationandthekindsofexplicittestsamodelneedstopass(e.g.,Barlas, 1989; Barlas and Carpenter, 1990; Forrester and Senge, 1980; Oliva, 2003; Sterman, 2000). However, since SD modeling is problemdriven, the discipline also takes a functional perspective: validation is considered an iterative process of gradually building confidence inthe model as a ‘‘useful” representation of the problem at hand and the theoretical assumptions taken (van Horn, 1971). (See alsoGass, 1983; Miser, 1993; Mitroff, 1972; Roy, 1993; Smith, 1993, for evidence of this shift of validation in the OR/OM community).Onceconfidenceinthemodelhasbeenattained,thegenerationof policysolutionsisbasedonexperimentationdrivenbythemodelers’expertise (Forrester, 1961), or exhaustivewhat-if scenarioanalysis(Morecroft, 1988). Theseapproaches relyontrial-and-error simulation, changing parameter values or switching individual links and feedback loops on and off, to discover important systemelements and derivepolicy recommendations. The intuition guiding this effort relies on simple feedback systems with one or a few state variables, where thebehavior is fully understood. A third approach relies on automated optimization software and an explicit objective function to explore themodel’s parameter space (Kleijnen, 1995). (SeeLane and Oliva, 1998, for a description of the SD method and its assumptions). However, each of these approaches suffers from inherent limitations, both in the model development and the policy analysis phase.Automaticoptimizationmethodsdonotreadilyofferanintuitiveinterpretationoftheresults,hencethereisariskthatthemodelistreatedasablackbox.Thisisalimitationinlightoftheemphasisinthefieldonusingmodelsaslearningandcommunicationtools.Conversely,theintuitive approaches have limitations in large scale models withperhaps hundreds of state variables. In practice, model building and anal-ysis is often done using a ‘nested’ partial model testing approach where one goes from the level of small pieces of structure to entire sub-systems of the model, with frequent re-use of known formulations and partial models (e.g.,Homer, 1983; Oliva, 2003). Although this 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2009.09.016 * Corresponding author. Tel.: +1 979 862 3744; fax: +1 979 845 5653. E-mail addresses: saleh@salehsite.info(M. Saleh),roliva@tamu.edu(R. Oliva),cek.ino@cbs.dk(C.E. Kampmann),pal.davidsen@geog.uib.no(P.I. Davidsen). 1 Tel.: +20 2 33350 178; fax: +20 2 33350 109. 2 Tel.: +45 4083 8444; fax: +45 3815 2540. 3 Tel.: +47 55 58 41 34; fax +47 55 58 30 99. European Journal of Operational Research 203 (2010) 673–683 Contents lists available atScienceDirect European Journal of Operational Research journal homepage:www.elsevier.com/locate/ejor  approach does carry a long way, it can be very difficult to discover feedback mechanisms that transcend model substructures in ways notanticipated by the modeler in the srcinal dynamic hypothesis. In addition to being time consuming, the approach therefore carries thedanger that observed behavior is falsely attributed to certain feedback mechanisms when in fact another set of feedbacks is driving theoutcome, and the derivedpolicy recommendations might be counterproductive. Clearly, a morerigorous theory for the link between feed-back structure and behavior in general large-scale systems would be of great value.Recently, formal tools to articulate precise theories in SD models have become available (seeKampmann and Oliva, 2008, 2009, for areviewof this literature). This work has focused on linkingmodel structureto the systemmodes of behavior, expressed as the eigenvaluesof the linearized model. 4 In particular, in what has been dubbed loop eigenvalue elasticity analysis (LEEA) , the work has shown that there is aclose relationship between behavior modes and the strength of feedback loops, so that one may decompose the effects of structural changes interms of individual feedback loop contributions (seeKampmann, 1996; Kampmann and Oliva, 2006). In this paper, we aimto expand the use of these formal tools to performstructured policy analysis. So far, little effort has been devotedtoexploringtheuseofthesetoolsforpolicydesign.Inpart,thismaybeduetothecomputationalintensityrequiredtoperformtheanalysisandthelackofintegrationofthemethodintomainstreammodelingtools.Beyondthesetechnicalchallenges,however,ithasbeendifficultto interpret the results because eigenvalues define the characteristics of the system’s behavior modes (e.g., exponential growth, exponen-tial decay, expanding oscillations, damped oscillations), but these behavior modes are not equally manifested in the time path of a partic-ular model variable, making it difficult to link the eigenvalue analysis directly to the observed simulated behavior (Kampmann and Oliva,2006). We argue that for policy analysis, it is also necessary to consider this latter aspect.Linearsystemstheorydemonstrateshowthebehaviorofagivenvariablecanbeexpressedasaweightedsumofallthesystembehaviormodes. The weights, which we have dubbed dynamic decomposition weights (DDW) , determine the manifestation of a particular behaviormodeinthevariableofinterestandarerelatedtothesystemeigenvectorsandthecurrentstateofthemodel.Notethattheterm‘‘dynamic”refers tothefact that thevalueof aweightchangesasthestateof themodel changes. Thisisamanifestationof thefact that, eveninlinearmodels, in the transient phase, the contributions of the various modes of behavior to the total behavior evolve with time.A number of SD scholars have indeed begun to consider the relative weights of behavior modes in specific system variables (see, e.g.,Gonçalves, 2009; Güneralp, 2006; Saleh, 2002; Saleh et al., 2005). The innovation of the present paper is twofold. First, we demonstratehowthedynamicdecompositionweightsanalysis(DDWA)andtheLEEAinawayarecomplementstoeachother.WhileLEEAisconcernedwith changing the eigenvalues, DDWA is concerned with enhancing or suppressing the behavior modes for particular system variables.Second, weexplicitlylinktheenhancement (or suppression)of behaviormodesinsystemvariablestopolicydesign. Weexplorethepolicydesign space by assessing the elasticity of the presence behavior modes to changes in system parameters.The paper is structured as follows. In Section2,we outline the analytical framework, with most emphasis on the DDW analysis (since LEEA is well documented in previous work) and the link to policy analysis and model testing. In Section3we explore how the methodapplies to a simple inventory-workforce model (Sterman, 2000), which aims to provide an endogenous explanation of inventory and pro-ductionoscillationsinamanufacturingsetting,beginningwithadiscussionofthesignificanceofoscillationasageneralproblemphenom-enon and the ways one might formalize policy criteria for improvement. The analysis not only shows the impact of parameter changes onbehaviormodesandweights,butitalsoshowshowtheLEEA/DDWAmethodscanaidanunderstandingof  why thepolicychangeshavetheeffect they do. We conclude the paper with reflections on the general utility of the method. The mathematical results underlying the ap-proach are either documented in previous work or they are standard results from linear systems theory. The reader is referred to the on-lineAppendixlisted at the end of the paper for details of the derivations. 2. Analytical framework  A schematic representation of the analytical framework is provided inFig. 1. The aim of the analysis is to provide an understanding of the link between three elements: the model structure, the policy parameters in the model, and the resulting time behavior of key systemvariables of interest (the three oval items inFig. 1). As the figure further indicates, the method involves three analytical components: lin-earization of the model, LEE analysis, and DDW analysis. These three components complement each other and provide the basis for a cir-cular process of model testing, policy analysis, and policy interpretation for implementation. Linearization LEEABDWAPolicy interpretation Policy analysis  OriginalmodelLinearizedmodelLink gainsPolicyparametersLoop gainsDDW(mode weights)Eigenvalues(modes)Eigenvectors,Reference pointSystemBehavior  Model testing  Fig. 1. Schematic representation of analytical process. 4 In the following, we shall be somewhat informal in using the terms behavior mode and eigenvalue interchangeably. Strictly speaking, though, the eigenvalue k corresponds tothe behavior mode e k t  . 674 M. Saleh et al./European Journal of Operational Research 203 (2010) 673–683   2.1. Linearization Mathematically, asystemdynamicsmodelisasetofnonlinearordinarydifferentialequations.Onemayapproximatethemodelarounda particular point in time t  0 by a set of time-invariant linear differential equations _  x  ð t  Þ ¼ Gx  ð t  Þ þ Bu ð t  Þ þ b x  ð t  0 Þ ¼ x  0 ; ð 1 Þ where  x  ; u arecolumnvectorsofthe n statevariables(levels),and  p exogenousvariables,respectively, _  x  isthevectoroffirsttimederivatives(rates), t  isthesimulatedtime, G  and B areconstantmatrices,and b aconstantvectoroftheappropriatedimension(see,e.g.,DialloandRahn,1990). The LEE and DDWanalyses are both based upon this linearized system. Since the main aimof our analysis is initiallyconcerned withthe endogenous response of the system, we focus here on the endogenous dynamics by assuming that the exogenous variables are zero orconstant ð u ð t  Þ ¼ 0 Þ . (SeeKampmann and Oliva, 2006, for a discussion of when such an approximation is appropriate and useful). In the absence of changes in exogenous inputs, the resulting behavior of any given state variable x ð t  Þ can be written as a weighted sumof a set of behavior modes,  x ð t  Þ ¼ w  0 þ w  1 e k 1 t  þ Á Á Á þ w  n e k n t  ; ð 2 Þ wherethe k s are theeigenvalues of thesystemJacobianmatrix G  , expressedby thecharacteristic polynomial P  ð k Þ ¼ det ð k I À G  Þ ¼ 0, andtheweights w areafunctionoftheeigenvectorsof  G  andthevector b in(1)(Chen,1970).Forrealeigenvalues,thebehaviormode e k t  amountstoan exponential growth ð k > 0 Þ or adjustment ð k < 0 Þ . Complex eigenvalues appear in conjugate pairs d Æ i x , leading to terms of the form e d t  sin ð x t  þ h Þ , correspondingtoexpandingor dampedoscillations (if  d > 0or d < 0, respectively). The weights w determinehowmucheachof these modes is expressed in a particular system variable.  2.2. Loop eigenvalue elasticity analysis The Loop Eigenvalue Elasticity Analysis, LEEA, is concerned with what happens to an eigenvalue k when one changes individual ele-ments g  of the matrix G  in (1), often measured as the eigenvalue elasticity , e ¼ ð @  k =@   g  Þð  g  = k Þ . A theorem known as Mason’s Rule showshow the coefficients of  P  ð k Þ can be interpreted as gains of feedback loops (measured as the product of the gains of the links constitutingthe loops), i.e., there is a one-to-one correspondence between loop gains and eigenvalues. In particular, changes in relationships in themodel that are not part of a feedback loop will have no effect upon the system eigenvalues.Kampmann (1996)pointed out a problemin this interpretation: that a given system may potentially contain very large number of feedback loops. Using graph theory, he showedhow one can focus on a much smaller subset of  independent  feedback loops that still capture the full feedback complexity of the systemand support a computation of the loop eigenvalue elasticity . The analysis, therefore, supports an interpretation of the relative importanceof particular feedback loops in generating a particular mode of behavior, where loops with large elasticities are considered importantfor the behavior mode in question.Oliva (2004) and Oliva and Mojtahedzadeh (2004)showed how choosing the shortest independent loopsets allows for relatively more intuitive interpretation of the loops.  2.3. Dynamic decomposition weight analysis The Dynamic Decomposition Weight Analysis, DDWA, is concerned with what happens to the weights w in (2) when changes aremade to the system elements. In contrast to LEEA, all the links in the model are potentially relevant in DDWA. Furthermore, the weightsare specific to each output variable of interest as well as the current state of the system (the reference point from which the lineariza-tion is made).Thepseudo-codefortheDDWalgorithmislistedinFig.2.Thealgorithmconsistsoftwoloops.Theouterlooptraversesthetimehorizonof the study with a time step s – that is, the computations are performed at regular intervals of length s . First, the values of the state vari-ablesareobtainedfromthesimulationdataandthenthe Weights functioniscalledtogeneratethebasevaluesfortheeigenvaluevector k à and the weights matrix W  à . These values are derived from the values of the elements of  G  , which in turn are computed from the currentvalues of the state variables and parameters, and b . Once the base values are computed, in the inner loop each parameter in p (one at atime) is slightly perturbed to numerically compute the eigenvalue and weight elasticities, by comparing the new values to the base case.Once the computations have been performed, the analysis is based on the manipulation and interpretation of the output variables E k and E w . Depending upon the degree of nonlinearity in the model, the analysis should be performed at several points along the simulated tra- jectory of the model. (Kampmann and Oliva (2006)discuss the merits of using linear analysis in nonlinear models).  2.3.1. Behavior decomposition As a first step, it is possible to assess the projection of each of the reference modes in each state variable by expressing the time trajec-toryofthestatevariablesintheformofEq.(2).Thisrepresentationimmediatelyrevealsthedurationandintensityoftheprojectionofeach eigenvalueintheoverallbehaviorofthestatevariable.Thisrepresentationhelpsfocustheanalysistothebehaviorpatternsthatneedtobeaddressed – either to increase or decrease their projection depending on whether the behavior pattern is desirable or not.  2.3.2. Policy (parameter) analysis Themethodthenproceedstopolicyanalysisbyconsideringhowspecificinterventions(parameterchanges)inthemodelaffectthesys-tem’s response. An exploration of the policy design space can be achieved by assessing the influence of model parameters on the dynamicdecomposition weights. By focusing on the weights of the behavior modes for the variable of interest we can identify leverage points toincrease or decrease the influence of a behavior mode on the variable. While changes to model parameters might influence several statevariables simultaneously, parameters reflect policies and various ‘‘physical” realities in the system and as such represent intuitive inter-vention points. By assessing the parameter influence, E  w , in the DDWs, it is possible to quickly identify scaling parameters (parameters M. Saleh et al./European Journal of Operational Research 203 (2010) 673–683 675  that affect the scale but not the behavior mode of state variables) and the parameters with high leverage on the desired (or undesired)reference modes.Changesinparameters,however, notonlyimpactthedynamicdecompositionweights,butalsochangetheeigenvaluesthemselves(ex-pressed by the measure E  k ). This dual impact of parameter changes introduces a challenge in developing policy recommendations sincechangesnotonlyaffectthewayabehaviormodeisprojectedinthetrajectoryofastatevariable,butalsochangesthebehaviormodeitself.After completing the analysis for all state variables and time instants of interest, it is possible to generate a set of recommendations forthe policymaker in terms of changes in parameter values and explain the effects of these parameter changes by analyzing their roles inchangingthe feedback structure withthe help fromthe LEEA results. By referring back to the srcinal model, the analyst is affordeda dee-perunderstandingofwhytheparticularpolicyinterventionsworkthewaytheydo,whichcanbethebasisofreal-worldinterpretationandexplanation and model validation and testing.BoththeLEEAandDDWAmethodshavebeenimplementedinMathematica Ò routinesandareavailableonline(Oliva, 2009),alongwiththe example models in Vensim Ò and text parsing routines that generate the appropriate Mathematica Ò files from a Vensim Ò model file. 3. Analysis of a simple inventory-workforce model In this section we apply the analytical framework to a simple system dynamics model of oscillations in a manufacturing system. Pro-duction and inventory oscillations and minimization of inventory carrying costs as well as adjustment costs in changing output are fre-quently studied in the OR/OM literature (e.g.,Chandra and Grabis, 2005; Hoberg et al., 2007; Iglehart, 1963; Sterman, 1989). Thebenefit of an SD model to address this issue is the articulation of an endogenous explanation for these oscillations, i.e., an explanationin terms of variables that are under management control. The formal analysis of the model that we propose here provides a rigorousanddirect identificationof theleversthat aremoresignificant for management purposes. Beforeproceedingtotheanalysis, webrieflydis-cuss criteria for successful policy changes.  3.1. Oscillation and policy criteria Forrester (1982)discusses different measures of stabilizing policies and their possible tradeoffs. This issue, however, is difficult to treatin general, since the policy criteria are linked to the purpose of the model and the problemdefinition, which may involve transient behav-iors like overshoot and collapse – e.g., in the World model (Forrester, 1971) – or the settlement in the system to undesirable end states –e.g.,intheUrbanDynamicsmodel(Forrester,1969).Inthispaper,wefocusonpoliciesthatreducetheoscillatorytendenciesofthesystem,since the model presented is designed to address this issue, and since, as was demonstrated byKampmann and Oliva (2006), it appears tobe one of areas where the eigenvalue analysis shows the most promise.As mentioned above, eigenvalues associated with oscillations appear as complex conjugate pairs d Æ i x . In the context of unwantedinstabilities (oscillations), effective policies are often defined as those that either increase the damping of oscillatory behavior modes bymaking the real part d more negative (the settling time criterion) or, when adjustment costs are significant, decrease the (damped) fre-quency of oscillation x (the frequency criterion). More general measures are based upon more sophisticated objective functions relatingto the ability of the system to absorb exogenous disturbances. Examples include the variance of a specific system variable or the fre-quency response of the variable. A summary of criteria is provided inTable 1. Since all of these measures are ultimately related tothe system eigenvalues and the DDW, we have chosen in this work to focus on these directly and relegate other measures to subsequentwork. [ E λ  , E w ] ←  DDW  ( G, p , b , SimData , t  s , t  e , τ  , δ   ) t ← t  s initialize time tracker to start-time while t ≤ t  e while time tracker less or equal than end-time x ← SimData tes}t{ x to state of system at time t[ λ * , W * ] ← Weights ( G ( x , p ), b ) function call to obtain base eigenvalues and weights for j=1: length ( p ) for every parameter in ppr ← p initialize pr to ppr {j} ← (1+ δ   ) p {j} modify parameter j by δ [ λ , W ] ← Weights ( G ( x , pr ), b ) function call to obtain eigenvalues and weights E λ {j, t} ← δ   -1 ( λ  - λ  * )./  λ  * store eigenvalue elasticity to parameter j at time t E w {j, t} ← δ   -1 ( W - W * )./  W * store weight elasticity to parameter j at time t end )rof (dne t ← t + τ  increment time tracker by τ end end (while) end [ λ , W ] ← Weights (J, b ) λ ← Eigenvalue ( J etaluclac)eigenvalues R ← Eigenvector  ( J ) calculate right eigenvector matrix a ← R -1 b calculate projection on the right eigenvectors W ← ( a ./  λ )* R calculate dynamic weight matrix end where G ( x , p ) is a symbolic representation of the system’s Jacobian in terms of state variables x and parameters p ; b is a constant vector associated with the linearized model; SimData is a matrix containing the simulated values of the n state variables across time; and δ   is the perturbation parameter. λ and λ * are vectors of length n, and W and W * are matrices of dimension n x n . Elasticity matrices E λ and E w have two additional dimensions to storeelasticities by parameter and time. * denotes the base case values; the symbol ./ indicates element-by-elementdivision. Fig. 2. Pseudo-code for DDWA algorithm (see onlineAppendixfor details).676 M. Saleh et al./European Journal of Operational Research 203 (2010) 673–683  The LEEA can aid in finding the desired changes to d and x , and explain why the effect occurs in terms of the changes in feedback loopgains they imply. Correspondingly, the DDWA can help find changes that reduce the weights w of the undesired reference modes in a par-ticular system variable, i.e., reduce the amplitude of the variable’s oscillations. Both methods are necessary in order to address more gen-eral measures of the degree to which external disturbances can be absorbed and dampened by the system.In order to address these measures, we must consider the elasticities of the real and imaginary parts separately, as the real numbers e d ¼ ð @  d =@   g  Þð  g  = d Þ ; e x ¼ ð @  x =@   g  Þð  g  = x Þ , respectively. Note that it is not  the case that Re f e g ¼ e d or Im f e g ¼ e x .Kampmann and Oliva(2006)found that it is easier to work with the influence measure instead, defined as l ¼ ð @  k =@   g  Þ  g  . For the influence measures , l ¼ ð @  k =@   g  Þ  g  ; l d ¼ ð @  d =@   g  Þ  g  ; l x ¼ ð @  x =@   g  Þ  g  , it is indeed the case that Re f l g ¼ l d ; Im f l g ¼ l x . In addition to simplifying interpretation,theinfluencemeasuresalsoremovetechnicaldifficultiesinvolvedwheneigenvaluesareclosetozero.Whiletheeigenvalueelasticitymea-sure is scale invariant, the influence measure is not invariant to time scaling, i.e., the measure depends upon the choice of time unit in themodel. Since the measure is still invariant to all other scaling (choice of variable units), we consider this a minor drawback.  3.2. The model and its behavior  To illustrate the above concepts, we apply them to a simple linear model; a simplified version of the inventory-workforce model de-scribedinchapter19,inSterman(2000). Stermanusesthismodeltomaketheargumentthatinteractionsbetweeninventorymanagementpolicies and labor adjustments cause a dampened oscillation with frequency and amplitude similar to the business-cycle. The stock andflow diagram of the simplified linear model is portrayed inFig. 3.Themodel consistsof aninventorysectorandalaborsector. Thetwosectorsarelinkedviaproduction,whichisafunctionof labor, andhiring, which is influenced by the desired inventory adjustment and expected demand. The model contains four state variables: Inventory  Table 1 Stabilization policy criteria and corresponding effects on eigenvalues and decomposition weights w of a policy change in a system element g  . Policy criterion Description Change in eigenvalue k ¼ d Æ i x ; x > 0Change inBDW w AppropriatemeasureDamping Increases the rate of decay of oscillation (or decreases the rate of expansion) @  d @   g  g  d < 0N/A x ð t  þ T  Þ  x ð t  Þ Frequency Decreases the frequency of oscillation (lengthens the period T  ) @  x @   g  g  x < 0N/A T  Variance Reduces the variance of a targetvariable (or the weighted average variancesof several variables)No simple relation @  w @   g  g w < 0 R  x ð t  Þ 2 dt  Auto-spectrum Reduces variance of targetvariable(s) within a target frequency rangeNo simple relation @  w @   g  g w < 0Filter infrequencydomainFrequency responsegainReduces the gain (amplification) inthe target frequency range for a particularcombination of disturbance exogenous and output variablesBased upon transfer function G ð i x Þ Work inprocess InvProd StartRateProdRateInvShipmentRateProductivityAdjustmentWIPDesired ProdStart RateDesiredProdCustomer Order RateDesiredInvDesired InvCoverageMin Order ProcessingTimeSafety StockCoverageManf CycleTimeDesiredWIPWIP AdjustTimeInv AdjustTimeProd Adjustfrom InvStandardWorkWeek28910VacLabor Vac CreationRateVac ClosureRateHiringRateQuitRateAvg Duration of EmploymentAdjust For Labor Labor AdjustTimeDesired HiringRateDesiredLabor Adjust For VacVac AdjustTimeDesiredVacAvg TimeFill Vac<StandardWorkWeek><Productivity>134567 Fig. 3. Model structure and feedback loops. M. Saleh et al./European Journal of Operational Research 203 (2010) 673–683 677
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