180 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
NODEL RESTRUCTURING
Robert L. Eberlein
dames H. Hines
Pugh-Roberts Associates
ABSTRACT
Models are continually changed and altered, and it is clear that more effective
techniques for accomplishing this can make modeling easier and faster. One
common way that models are changed is to remove selected pieces of structure from
the model. This type of model alteration can be done for debugging a model, for
tuning a model, and for performing policy analysis. In this paper we present
techniques that automate the processes involved in removing components of model
structure. By taking out some of the drudgery of changing a model, and
guaranteeing consistency in the resulting altered model, the techniques greatly
enhance modeling fluency and speed. These techniques are implemented in a
software package that is part: of Professional DYNAMO.
INTRODUCTION
Often in the process of model development, analysis, and presentation it is
desirable to alter the feedback structure of a model, Such alterations can range
from the addition or deletion of a few measurement equations, to the complete
rebuilding of a model's feedback structure, Modification of structure is time
consuming and error prone, ‘In this paper we outline a procedure that allows more
automated structural alteration when the goal is to remove or inactivate selected
model components. The automated procedures save the modeler time in the initial
alteration of structure, and allow easy replication of the restructuring process.
We begin the paper by reviewing typically encountered situations in which
structural alteration can be useful, and the common features of these situations.
We use these common features to characterize the information necessary to
restructure a model, and the nature of the restructured model. Information is
recast into the form necessary to use the restructuring program that is available
aS a supplement to Professional DYNAMO Plus. Following this, we present the
theory of model restructuring more formally, discussing both the required
information, and the algorithmic approach used to accomplishing restructuring.
Finally, we present an example illustrating how the restructuring program
described can be applied to an actual model.
THE STRUCTURE OF MODELS
Tt is true that structure determines behavior, and clearly modelers determine
structure. Still, it is often the case that modelers are faced with the task of
finding out why the structure they have created is behaving the way it is.
Forrester (1941 Appendix N) describes a situation in which a number of decision
makers, working ta accommodate seasonality, in fact induce seasonality.
Forrester (1971) describes why things often work counter-intuitively, and the
importance of understanding surprising model behavior is widely recognized.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 181
Homer (1983) describes techniques for dealing with the behavior of specific
components or substructures in a model as a means of validation.
There are two important points that are common to the above issues. First, the
behavior of individual structures within a model is not sufficient to determine
the overall behavior of the model. Second, it is much easier to evaluate the
behavior of an individual piece of structure than that of the full model. If
surprising or irregular behavior can be traced to a specific component of the
model structure, its validity can be judged in the context of that component of
model structure,
There are three phases in the model development and analysis process where
delineating key structures driving behavior is of great importance. The first,
is at the initial model development stage, when it is necessary to just get the
model working without numerical overflow. The second, is in the validation
stage, where it is necessary to have the model produce good, or at least
sensible, results. And the third phase, is in the understanding of what policies
are likely to be useful and why. The issues are somewhat different, but there is
a great deal of overlap.
Problems arising in the early stages of model construction are often very coarse
in nature. A model may “blow up," with exponentially increasing variables
_ Causing simulations to abort early on. The sources of such behavior may be
obvious (for example sign reversals in accounting identities) but they can still
take time ta track down. Problems cannot always be isolated when the bad output
from a sector inflyences the rest of the model, because the input to the sector
is also likely to show the problem behavior.
Finding the source of odd and inexplicable behavior can be accomplished by the
successive removal of pieces of model structure, until the structure, or linking
of structures, responsible for the problem is found. Structures and components
within a model can be removed by making the key variables that feed back into the
rest of the model exogenous, or in some cases constants. When this is done the
output of different model sectors can be reviewed; if some outputs of a sector
display problematic behavior, while the inputs to the sector seem okay, there is
a problem in the sector.
To actually decouple elements of structure, variables that switch between
endogenously generated values and exogenously specified values are often inserted
into a model. The resulting equations tend to complicate the model and make
maintenance more difficult. There is also a tendency to clutter the model with
all the names of the exogenous variables and the associated switches that
together simply act a5 a substitutes for endogenous variables. In writing the
new equations all locations where switches and alternative variables are
necessary must be identified. This is very time consuming and tends to interrupt
the flow of tracking down behavior.
Closely related to this process of tracking down problematic model behavior is
that of adjusting model parameters to attain acceptable historical fit. In the
early stages of this process it is often desirable to reduce the amount of
feedback to make the problem more tractable (Homer 1983). The difficulty in
adjusting model parameters, with all model feedback active, arises for the same
reason that problematic behavior is hard to trace. Historical inaccuracies in
any variable are likely both caused by and causing inaccuracies in other
variables. Adjusting a set of parameters appropriate to one or more variables
182THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
may simply be compensating for the fact that there are other variables in the
model that are not right. In such situations finding parameters that
simultaneously make a large number of variables historically accurate can be very
time consuming,
The solution to this problem of parameter adjustment is precisely that used for
problematic behavior. Feedback can be cut out using switches that substitute
exogenous values, usually historical values, for different variables. With these
switches in place the modeler can work on individual components of model
structure, As the components become acceptable the parameter values obtained can
be incorporated into the full model, and feedback restored. The full model can
then be used as the basis for final parameter adjustment which, normally, will be
slight.
After problematic behavior has been removed from a mode] and the parameters have
heen adjusted to the satisfaction of the modeler the model should be an effective
tool for analyzing and explaining problems and policies. Much of this
understanding and explanation is in terms of fundamental feedback paths.
Understanding comes about through working with the model and conducting a variety
of different experiments. Typically, this experimentation involves the isolation
of structures that are key to causing the problems, and making policies work.
This can again be accomplished by deactivating selected structures with switches,
where such switches are likely to be more selective than in the earlier modeling
phases, :
There are also more analytical ways of arriving.at the fundamental elements of
structure responsible for determining behavior. Perez~Arriaga (1981) outlines
techniques for identifying important model levels, Forrester (1982) outlines
techniques for determining important model causal loops. In this work there is-
ay possible implicit, removal of structure. The removal is not completely the
Same as that we are describing, though Eberlein (1986) applies our approach to
restructuring in this context.
The common element in the above discussion is the alteration and selective
removal of feedback paths. In small models this can often be easily accomplished
by the addition of switches or the explicit alteration of structure. As models
become larger, however, the difficulty of selectively removing structure
increases very rapidly; the number of potentially altered variables as well as
the number of times those variables are used becomes very burdensome. The
partial removal of feedback through, for example, omission can also cause
difficulties because the model will be different from what it was thought to be.
For all sizes of models techniques that automate the alteration of model
structure and enforce consistency have great potential benefits.
The cutting of model feedback links by the creation of switches does not
necessarily take complete advantage of the less complicated feedback structure
that results. If the full model is still retained for simulation purposes, even
though many variables are being ignored, the computational requirements of
simulation will be as high as for the original model. It is possible to make a
model smaller by pulling out selected pieces entirely, however, the difficulty of
fixing up loose ends and later putting the pieces back in, tends to make this
impractical. Even when large quantities of structure have been made inactive,
simulation can. still be as time consuming as it was for the original model.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 183
AUTOMATED RESTRUCTURING
The approaches used in altering the feedback structure of a model typically
involve two steps: first, the identification of a set of structures or variables
of particular interest, and second, the severing of feedback links. In
automating restructuring we emulate this, working with groups of model variables
to be retained, and groups to be made exogenous. In this section we specify the
form and content of the information required for our automatic restructuring
algorithm to work. The software we have developed then creates a restructured
model based.on this information. The discussion in this section is heuristic and
explanatory, the formal theory behind the algorithms used to accomplish
restructuring is presented in the next section. For a more detailed discussion
of the actual procedures involved in performing a restructuring the reader is
referred to the program documentation (Professional DYNAMO 1987).
In restructuring a model attention is focused on those elements causing trouble,
or being compared to data, or of policy interest, that are closely related.
Normally, variables that have short causal paths connecting them would be grouped
together. In such a grouping a few key variables will be the most important to
consider. Such variables are often those for which data are available, or
generally important concepts for the system being modeled. In the approach to
restructuring we use, these key variables will be the ones that need to be
identified by the modeler. Subsidiary variables, closely related to the
variables of interest, need not be specified; they will be kept automatically if
needed,
In severing feedback links we work with the model variates, just as switches
that move a variable from endogenous to exogenous do, Often, making a variable
exogenous entails cutting more than one causal link, nonetheless, the conceptual
process is still one of cutting links. The selection of variables to be made
exogenous is, in some sense, the opposite of the selection af variables to focus
on. Key variables outside of the area of focus, that influence the area of focus
are clear candidates to be made exogenous.
The modeler needs to take a more active role in deciding what is to be made
exogenous. While what is important to retain is usually quite clear, the
feedback links existing between the unimportant elements and the important
elements of a model can be subtle, and difficult to find. To effectively exclude.
elements of structure all of these feedback links need to be identified. The
amount of structure to be retained is made larger by feedback. This is an
advantage, in that important things will be kept, but a disadvantage in that
unimportant things will also tend to be kept. It will often be necessary to make
more than just a few important key variables exogenous.
Following the above logic, the inputs into a model restructuring are two lists,
the first of variables to be retained in the restructured model and the second of
variables to be set exogenous. The restructuring algorithms then determine the
restructured model, We outline the steps involved.
First, and optionally, the modeler may wish to trace through the causal structure
to find the largest possible list of exogenous variables. A variable is
effectively exogenous if its value can be determined based solely on other
exogenous variables. That is, a variable can be made exogenous if it is not
involved in any feedback. When a modeler sets a variable to be exogenous, it is
possible that other variables are effectively made exogenous. Depending on the
184THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
model and purpose, setting all possible variables as exogenous may be desirable.
This practice can reduce the size of a model by eliminating intermediate
computations on exogenous variables, On the other hand, if a small number of
exogenous variables drive many more, the required number of exogenous variable
inputs to a model can be driven up. Under these circumstances, if the mechanism
for automatically determining additional exogenous variables is invoked it can
result in a larger list of variables to be set exogenous.
Given the lists of variables to be retained and variables to be set exogenous a
new simpler restructured mode! can be assembled that contains only the variables,
endogenous and exogenous, required to compute the retained variables. The
restructuring starts with the list of variables to be retained and identifies all
of the model variables involved in feedback loops with the retained variables.
Tn’ addition, all exogenous variables necessary to compute the retained variables,
and the variables on which they depend, are identified. Feedback links involving
exogenous variables are not used in use in identifying the variables to retain.
Rather, when an exogenous variable is ericountered no further dependencies are
considered. The resulting model contains only what is essential to computing the
retained variable given values fer the exogenous variables,?
The equations for all the kept endogenous variables are the same as those in the
original model. The equations for the exogenous variables are no longer needed,
since the variables will be input exogenously. Any other variables and their
equations are simply dropped. The resulting model, with more limited feedback,
serves the requirements of isolating the sources of dynamic behavior, and,
because extraneous variables are simply dropped, the model size will normally
decrease substantially. The model that results is a working DYNAMO model (in
either compiled form or as a model listing) that can be used as any other model,
While the resulting model is definitely smaller, it is likely to have a number of
exogenous variables that were not in the original model. In order to simulate
the model values are required for these exogenous variables. The preferable
source of these values will depend on the purpose of the restructuring. For the
purposes of building and validating models the most appropriate source of
exogenous variables is the available historical data. When historical data are
not available, or it is not desirable to use them, the original model can be used
to generate the exogenous variables. This can be done by simulating the original
model and saving values for the needed exogenous variables. Finally, it will
often be appropriate to set exogenous variables to some constant value when
working with the restructured model.
There are a couple of points relative to arrays and macros that should be
mentioned. The discussion in this paper is all stated in terms of variables, and
if there are neither macros nor. arrays in a model the meaning of this is. glear.
When there are arrays the entire array is treated as a single variables, Thus if
one element of an array is kept, or made exogenous all are. Second, the nature
Sf macros is to increase the number of variables that are hidden - those
evariables that exist and are used for computation but are never seen by the
modeler. To be consistent with this hiding, macros, like arrays, are all or
Mone. If something within a macro is kept, the whole macro is kept.
+>. The determination of initial conditions may involve the inclusion of
additional variables as well as the alteration of some active equations to
initial value equations. :
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA185
The model resulting from the restructuring we have outlined has a minimal number
of variables, For the purposes of computation this is preferable to a model that
removes feedback but retains all variables, However, sometimes it is preferable
to cut the feedback in a model while, at the sane time, retaining all of the
mocel variables. This permits the inspection of what the model would have
produced for a variable’s values. Most commonly, this would be useful when
parameters are being adjusted on the basis of a variable, but that variables
value is not desired to influence any other variables in the model.
To accommodate situations in which computed values are also
desired we define an intermediate class of variables which we refer to as mixed
variables. For the purposes of restructuring mixed variables are treated as
endogenous variables, For the purposes of simulation all equations using a
mixed variable use an exogenously supplied value, but the simulated value is
created for review. Mixed variables have the further advantage that they gan be
applied selectively to elements of an array. This allows mixing of endogenous
and exogenous impacts for a given subscripted variable. .
The approach to restructuring we have outlined is straightforward and intuitive.
The resulting models are consistent with Professional DYNAMO Plus and can be
created in either compiled form or as a new set of model equations. The use of
an algorithm in the creation of the aodels guarantees consistency, so that the
models are always well formulated. Such an automated technique can substantially
reduce the time needed to alter a models structure,
FORMAL THEORY
Intuitively the ideas behind restructuring a model are very straightforward.
First determine all the variables that need to be kept, second determine what
variables can be made exogenous and third find atl the feedback loops that go
through the variables to be kept but not the exogenous variables. The feedback
loops found in this way are what need to be kept. If one goes ta a stock and
flow diagram and cuts and traces in the above manner the result will be a
restructured model, However, shart of actually going to a causal loop diagram,
some formalism is required to state fully the techniques of restructuring... We
present in this section a brief description of the algorithms used to ae camplish
restructuring. The discussion essentially repeats that of the previous section
in a more formal manner.
In the presentation of the formalism we use x,y and z to represent any variables
in a model. A name enclosed in curly braces (as in (RETAIN}) is used to denote a
set. When the name of a set.is in uppér case it indicates the set is defined
globally, lower case indicates a definition local to a function or operator, The
symbol € is used to indicate that a variable is an element of a set, while ¢
indicates that a variale is not an elenent of a set. Ly indicates the union of
two sets and f the intersection. {a} - {b} indicates the intersection of {a}
with the complement of (b), that is, any elements of (b} that are in, (a) are
removed,
Prior to beginning the restructuring it is assumed the two sets {RETAIN} and
{SET} have been defined: {RETAIN} consists of all variables that are to be
retained, {SET} of all variables that are to be set exogenous,
186 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY,- CHINA
We first. define -> (arrow) as a boolean operator such that x -> y is true if
there is a direct link from x to y and false otherwise, Note that x -> y does
not imply y ~> x, and will, in fact, rule this possibility out in many cases.
The arrow operator could equivalently have been defined as true if x appears in
the equation defining y, and as false otherwise.
The goal of the restructuring is to arrive at a new set
Bay Zr 72 Za vee Bont 7“) Ey Za 7) My
x € (RETAIN, 2ayeeeq2n € (SETH a)
of variables that are kept in the restructured model and a second set
CEXOG} = {KEEP} 91 (SET) , . (2)
of variables required as exogenous inputs into the model.
The set {KEEP} is clearly a finite set, but @ practical way of arriving at it
needs to be. specified. We use the arrow operator in order to define the set
valued function
(define({list3)} = {2 | 2 -> x, x € {list} }. ()
(define(<list})} gives the set of ali variables that directly enter into the
determination of every element of {list}. The {define()? function can, in turn,
be used to set up the recursive function (include({add},{list}} with the
following algorithm
Step 0: If {add} is empty go to step 5
Step ts {current}? = (define ((add))? U (ist?
Step 2: {new) = {define({add})} - C{list} U {SETI3)
Step 3: {include} = (include (new), (current?)
Step 4: finished
Step 5: Cinclude}) = {list}
Step 6: finished
The set {include) returned by this function is given in either step 3 cr step 5.
This set contains all of the variables that are needed to calculate {add}, pads:
all variables in {list}.
To see that (include()} will terminate after a finite number of recurs first note
that if any variable » is an element of {define(add?} then x will be an element
of {list on successive recursions. Since there are only a finite number of
variables and recursion continues only as long as (define(add)} is non-empty it
follows that there can be only a finite number of recursions.
With the function (include? thus defined we can set
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA187
{KEEP} = {include ({RETAIN}, CRETAIN}) > (4)
To show that this generates the right model we note that if y -> 2 and z € (KEEP)
either y € (RETAIN? or y € {SET} or because y € define(z) at some recursion y €
{current} and therefore y € {KEEP}. In particular, if y--> 21, 21 -> Zey vee
@na¥ 7) Zmy En —> Ky xX € CRETAIN}, and zi4..¢y2m € (SET}} then y € {RETAIN}.
In discussing the procedures for restructuring a model we considered the
possibility of making an exhaustive list of potentially exogenous variables. We
do this using the recursive boolean function trace({check},{hold}) intended to
determine whether a set of variables is completely exagenous.
trace({check},{hold}) is defined with the fullowing algorithm
Step 1: (potential? = (check? U {hold}
Step 2: {new} = (define((check})}} - (SET)
Step 3: if {new} is empty go ta Step 8
Step 4: {current} = (new) fl fhold}
Step 5: if (current? is empty go to step to
Step é: trace = FALSE
Step 7: finished
Step 8: trace = TRUE
Step 9: finished
Step 10: trace = trace({new),{potential})
Step 11: finished
To see that trace() will terminate after a finite time note that {hold} isa
Growing set that is bounded above by the total number of variables which is
finite. If, in any recursion, the set does not actually increase in size the
algorithm terminates because (new) is empty. Since {hold} is bounded in size and
the recursion terminates as soon as it stops growing it follows that the
recursion must terminate.
If there is a causal path from x to y Step 2 of the algorithm guarantees that the
path will be followed to y. Feedback from x to y similarly would return to the
point where x € {define{z}) so that x € {mew}, but since x € {hold} the routine
will return FALSE, Conversely, if x is exogenous, it ultimately depends only on
{SET} (and constants and possibly time) so that {new} will eventually be empty
and the procedure will return true.
The function trace is inclusive in the sense that if any element of {check} is
determined not to be exogenous all will. Thus it is necessary to invoke trace
for every variable in a model individually in order to determine if the variable
can be made exogenous,
188 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY, CHINA
We have set out a procedure for arriving at a restructured model given an initial
collection of variables that can be made exogenous ({SET}) and variables that are
important ({RETAIN}). So far we have only dealt with the actual variables to
keep of discard, but the equations for the variables follow directly. if x €
{KEEP} and x is not exogenous then {define(x)} € {KEEP? so that the original
equation for x can be retained. If x has been made exogenous the equation for x
tan be discarded. Note that (define()} will be different if used for
initialization equations then if used for active @quations. Thus the active and
initial condition inclusions can be determined separately by the execution of
{include()} with two distinct {define()} functions. Mixed variables, as
discussed in the previous section, do not require any alteration in feedback
structure but only a rewriting of equations.
The above discussion outlines formally what is required in order to restructure a
model. The algorithms, because they require little numerical computation, are
quite fast. By construction the resulting model will be complete and well
formulated, No simultaneous equations or indeterminate initial conditions will
arise when the model is restructured. The resulting model can stand on its own
as a model of part of a process. The existing exogenous variables only represent
a redefinition of the model boundary.
APPLYING RESTRUCTURING
Automated restructuring is proving its worth in the modeling effort ar Pugh-
Roberts Associates. A recent example serves to illustrate the power of the
techniques. In late 19786 and early 1987 we developed a model for one of the
principle pension fund managers in Chile's privatized social security system.
Early in the validation process the model was tuned to historical data.
Subsequent tests of the dynamic responses of the model, however, revealed that
the formulation for salesmen was flawed. After rewriting the salesman equations,
we were faced with the problem of retuning the model.
The relevant feedback structure of the model altered by the new equations is
sketched in Figure {. We use the fictional name "AFP* for our client in order to
preserve confidentiality. Among the factors affecting the decision to hire
salesmen is the market share of affiliates (workers affiliated with a particular
pension fund manager) and the strength of competitors’ sales forces. Sales
force, in turn, affects the number of affiliates as well as salesmen hiring.
Affiliates and salesmen form a tight complex of feedback loops. Model output
after the alteration of the sale force structure indicated substantial
differences between the simulated and historical values for AFP's sales-force,
the sales-force of competitors and the number of AFP affiliates. We felt that
retuning to achieve a better fit would like yield more accurate parameter values.
@ better fit would also bolster confidence in both the structure and the new
parameters.
To retune we used the restructuring program now available in Professional DYNAMO
Plus to break the structure into the three simpler ones illustrated in Figure 2.
the first structure takes affiliates and AFP's salesmen as exogenous, making it
straightforward to discover the parameters governing competitors’ hiring
decision. The second structure takes affiliates and competitors’ salesmen as
exogenous, permitting the easy uncovering of the parameters affecting AFP’s
hiving. Finally, AFP’s and competitors’ salesmen are taken as exogenous,
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA189
Figure 1
Relevant Structure for New Salesmen Equations
J90THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
Figure 3.
Structure 1: AEP Saleen, AFP Afiiais and Compesior ‘Structure 2: AFP Affililiates, Competitor Affiliates, and
Affiliates are exogenous in order to tune ‘Competitor Salesmen are exegonous in order
the Competitor Salesmen decision. to tune the AFP Salesmen decision.
Structure 3: AFP Salesmen and Competitor Salesmen.
are made exogenous in order to tune
the response of Affiliates to Salesmen.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA191
permitting us to get a fix pn the parameters governing the affiliates’ response
to the activities of the salesmen,
Having decided how to break up the complex of feedback loops, we next chose
sources for the exogenous inputs. We used historical data on affiliates and
AFP‘s salesmen as the exogenous input to the first structure, achieving a very
good fit for competitors’ salesmen. Because the data on competitors’ salesmen
are yearly, and available only from 1984, we used the continuous output of the
tuned first structure (beginning in 1982) as the exogenous input to the second
structure. Finally, we used the continuous sales-force output from the first two
structures as the exogenous input to the third structure in order to find values
for paraneters governing affiliates’ response to salesmen.
Putting the parameters resulting from tuning the three individual structures
resulted in a good fit far the full model, differing little from the fit for the
individual structures. If we had tried to tune the full model all at once things
would not have gone as smoothly. For example, putting in the correct parameter
values for the competitors’ salesmen decision without altering other parameters
in the original model does not improve the fit over the untuned model. The set
of parameters that would improve the fit using the full model would be different,
and on tuning the other components of the model would need to be altered. Full
model tuning would have been a repetitive and time consuming process of tuning
and retuning the same parameters.
Cutting feedback loops and using exogenous data for tuning purposes got us a
better fit faster. The restructuring program provided a quick easy means for
breaking the model up into smaller pieces and, in combination with Professional
DYNAMO’s translation options, provided a quick means of putting in the proper
exogenous data were they were néeded.
CONCLUSIONS
Part of modeling is changing modeis, In this paper we have outlined a technique
for accomplishing many common changes in an efficient and straightforward manner.
The alteration of model structure can be accomplished automatically by
selectively making variables exogenous, thus cutting feedback links. Because the
approach guarantees the resulting model to be consistent, the potential for error
is substantially reduced over less automated techniques, The techniques can be
applied in the model building and validation stage as well as the policy analysis
phase. In all cases the techniques serve to streamline many of the cumbersome
alterations that might otherwise be employed. Through the use of an example we
have illustrated some of the strengths of these techniques.
192THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
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