THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 159.
IDENTIFYING AND DISPLAYING IMPORTANT FEEDBACK PATHS
Robert L. Eberlein
Pugh-Roberts Associates Inc.
Abstract. Fundamental to the practice of system dynamics is the
identification of feedback. The theory of linear model analysis
and model simplification provide tools for doing this in the
setting of linear state space models. The application of these
tools in the field of system dynamics has been very limited
primarily because the tools are inaccessible and difficult to
use. Many of the difficulties can be overcome by linking the
analysis more closely with the original nonlinear model. We do
this first by using time plots of model variables to describe
behavior and second by deriving a nonlinear feedback model that
can be used to exhibit the important feedback structure. The
theory for doing this is heuristic, but allows the techniques to
be automatically applied with interaction only in the domain of
the original nonlinear model and its simulation.
INTRODUCTION
The identification and understanding of feedback is fundamental
to the practice of system dynamics. Though there is a good deal
of theory dealing with this problem, the theory has had only very
limited applications. The primary reason for this is the
inaccessible nature of the work that has been done. In this
paper we will develop some rules of thumb that can be used to
perform the process of identifying feedback. This combined with
an easily.understood presentation of the problem and results
allows widespread access to some very useful tools.
The process of system dynamics involves the analysis of models.
At the core of this analysis is the determination of what
feedback paths in a model are responsible for the behavior of
interest. The tools we discuss in this paper are intended to aid
in this analysis. These tools require that the definition of
behavior be restricted and carefully specified.
In most cases behavior of interest is a composite of less
interesting but more basic behavior. For example, the behavior
of interest might be a lack of innovation, which results from
instability in the ratio of experienced to inexperienced staff.
The instability in the staff is of less interest, but can more
easily be cast into framework of behavior we will discuss in this
paper. The process of casting something of interest into
something that can be analyzed is often tortuous and not always
successful. To aid in this we will derive an approximate
representation of things that can be understood, in the hope that
160 = THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, ULTUBER, 1900
connections can be made to those that need be understood.
Nonetheless, successful application of tools being described here
still requires a solid understanding of the model under
consideration.
Richardson (1986) has outlined a number of different dimensions
along which the analysis of feedback structure occurs. The three
basic frameworks identified are: the traditional methods of
repeated simulation with experimentation, consideration of how
the model output is determined by different time patterns of
inputs, and analysis of the eigenvalues of a model. We deal only
with the last of these in this paper, but tie the work that is
done to the more traditional methods involving time plots.
Although the theory is based on the assumption of linearity, we
will discuss applications to nonlinear models.
BEHAVIOR
Behavior is a very general concept which must be made specific,
and thus quantifiable, in order to use it as a basis of analysis.
The concept of behavior we use has its foundation in the theory
of linear analysis of dynamic models. In order to explain this
concept. we will first show how we go from a nonlinear feedback
model to the linear form that we need. Then we will discuss what
the behavior we are considering means in the context of the
linear model and extend this back to the nonlinear model.
A feedback model consists of levels, which accumulate over time,
and rates and auxiliaries that depend on the levels, and
determine how the levels change. The linear model we base our
analysis on has only levels, and the rate of change in a level is
specified as a linear combination of all the levels. We use a
linearization that is an approximation of the nonlinear model of
interest. The method used to arrive at this approximation is
quite intuitive. In a linear model changing state 1 by 6 will
cause a proportionate change in the rate that integrates into
state two. That proportion is used as the coefficient for state
1 in the equation determining the rate at which state 2 changes.
For example, suppose that the equation for LEV1 is given by
L LEV1.K=LEV1,J+DT*R1.JK (1)
RRL. KL=.3*LEV1.K*LEV2.K
with LEV1 having a value of 10 and LEV2 a value of 100. If we
change LEV1 by § we see that R1 changes by .3*5*100, this gives a
proportionate change of 30. Similarly if we change LEV2 by &
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 161
then R1 changes by .3*10*5 giving a proportionate change of 7%
Thus the linear equation for LEV1 would be
L LEV1.k=LEV1. J+DT* (30*LEV1.J+3*LEV2 .K) (2)
The actual linear equation clearly depends on the values of
levels and will be different at different times during the
simulation.
When the procedure we have discussed is used to compute a model
linearization the result is in the form
IK
1
Ax F (3)
with A the matrix with coefficients determined by the technique
discussed above, Exogenous variables have been left out of
equation 3 for simplicity. The matrix A will be referred to as
the dynamics matrix. In this equation x represents the vector of
levels in the model, x will also be weferred to as the state
vector.
Given a linear dynamic model in the form of equation 3 it is
difficult to determine in advance what behavior may result. To
aid in this, the dynamic system of equation 3 can be transformed
to a series of scalar differential equations. These scalar
equations have easily determined dynamics and can be used to
determine what the dynamics of the original system are. The
transformation of the model given in equation 3 involves both a
transformation of the A matrix and of the state vector x.
The matrix A can be transformed into a diagonal matrix having the
eigenvalues of A along the diagonal. These eigenvalues represent
the behavior generated by the (linear) model in the following
sense. If 4 is an eigenvalue of A then, in the absence of any
external influence, there is a scalar linear combination of the
states x (that we will call &) having the property
bane a)
That is, & changes over time in a way that is proportional to
itself. The eigenvalue } determines the behavior of the variable
€ over time. If A is positive then there will be exponential
growth, if negative exponential adjustment and if complex there
will be oscillations. For a model with N states there will be N
equations in the form of equation 4.
Because the levels in the model can be written as a linear
combination of the different €’s the dynamics of the system given
in equation 3 can be described by the different eigenvalues. How
TRE TRE SCS NT SrNEL IAL LUNG GP EE as CEE Ee SUE CT SEER he Eee 1
the dynamics defined in this manner manifest themselves in the
original states depend on what linear combination of the €’s
yields the state variable. This linear combination is given by
the appropriate entries of the right eigenvectors of the matrix A
(Chen 1970 and Porter and Crossley 1972 give a detailed
description of this decomposition) .
The eigenvalues and eigenvectors of a matrix provide a useful
method for breaking down the dynamics of a complex feedback
system. The basic building block of this analysis is the linear
transformation of the states to the eigenstates (the &’s), and
back again. This transformation can be made at any time, so that
the time path of the eigenstates can be used to determine the
time path of the model states. We use the idea of the eigen-
states in order to break down the time paths of the states into
different components. This approach allows us to easily present
the contribution of a behavior mode to a states overall behavior.
Showing a Mode
The discussion of modes as being represented by the eigenvalues
of the A matrix does not lend itself to easy inspection of what
the eigenvalues really mean. Eigenvalues do imply certain
patterns of behavior (damping times and periodicities), but modes
can combine in the model in a variety of ways. We base our
presentation of behavior modes on the time paths of the model
variables. The presentation of behavior modes we describe has a
strict theoretical basis only in the case of the linear system
discussed above, but can also be applied to nonlinear systems.
We apply a decomposition to the time paths of all the variables
of the original nonlinear model. This decomposition is based on
a linearization of the model as discussed above. The values of
the states from a nonlinear model simulation are used to generate
values for the eigenstates (the &’s) by pre multiplying by the
matrix of left eigenvectors. That is,
be = 7x (5)
where 1 is the left eigenvector written as a column vector. The
contribution of a mode to the current value of a state is given
by multiplying the eigenstate by the right eigenvalue as in
(25 by (6)
for state j in the k’th mode. The above calculation assumes that
the right and left eigenvectors are normalized to have an inner
product of dne. When the eigenvalue is complex twice the real
part of the associated calculation is used. Because the trans~
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 163
formations involved are invertible adding up all of the
contributions thus calculated will yield the original behavior.!
The above manipulations allow us to break down the behavior of a
state into components attributable to different modes. For the
linear model of equation 3 in the absence of any exogenous inputs
those components would be determined by equation 4. These
components would be exponential growth, exponential adjustment,
damped oscillation or expanding oscillation. When exogenous
variables are impacting the model (as is normally the case) the
behavior attributable to modes will not be quite so
straightforward. The exogenous variables might, for example,
introduce a changing goal for an exponentially adjusting mode.
Thus, much of the apparent behavior due to the mode would arise
because of the sensitivity of the mode to external inputs.
We can break down the behavior of variables into components due
to modes, but the key requirement for an understandable
presentation is that each variable have a short list of
influential modes. Most variables in a model will have a large
number of eigenstates influencing their behavior. This large
number is reduced by only considering those that have the
potential to cause a substantial portion of the variation in the
variable of interest. A mode will influence behavior strongly
only if the eigenstate associated with the mode is changing, and
the component of the right eigenvector corresponding to the state
is large. To quantify this we take the variance of the
eigenstate (&,) over the simulation and multiply by the absolute
value of (z,); (the entry for state j in the eigenvalue for mode
k). If the result of this computation is larger than some
fraction (say 30%) of the variance of the state then the mode can
be considered influential enough to display.
Mechanically, the above decomposition gives first a list of modes
important in determining a variables behavior, then a decom-
position of the variables behavior into components attributable
to the different modes. Thus if there is interest in determining
why a variable displays a particular pattern of behavior one can
associate that behavior with a mode or modes. This association
puts us into a framework of analysis that is more exact, and thus
programmable, than the standard analysis of time plots. The
approach described above requires approximation and is not very
1, Note that this is only true for the states themselves since
other variables will be approximate by virtue of
linearization. Model variables that are not themselves states
can have the appropriate right eigenvector determined by
taking a linear combination of right eigenvectors for the
model states.
164 = THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
robust, but it bridges the gap between the traditional time plot
and the less well known eigenvalue.
LINEAR MODEL SIMPLIFICATION
The intention in this paper is to outline techniques for ident-
ifying important feedback paths. We develop these techniques
based on the theory of model simplification. In the context we
have been discussing a simplified model has fewer states than the
original model. Because there are fewer states there are fewer
potential feedback paths. What remains within the simplified
model are the important states, and hence the important feedback
paths.
The process of simplifying a model involves removing such
important feedback paths from the framework of the large model
and imbedding them in a smaller model. The small model can,
thus, be used as a means of displaying the important feedback
paths. The only substantive difference between a full
simplification and the process we are discussing is the
adjustment of the parameters of the simplified model. We shall
not pursue this final stage of simplification but in other
regards the language and literature we use is that of
simplification.
We have outlined what we mean by a behavior mode, and now seek a
structure that generates this behavior. In the language of
simplification we would like a model with a smaller number of
states than the original model that still generates this behavior
mode. This process of simplification requires approximation, not
every variable in the original model will be in the simplified
model. Variables with little importance to the behavior modes of
interest will be omitted. A model is simplified by keeping
important variables and discarding unimportant variables. This
process is inherently judgemental, and the rules we discuss have
this judgment built into them.
The simplification of dynamic models has a long history stemming
largely from an early desire to ease computation and allow
_aggregation. There are a variety of approaches to simplification
depending on purpose and Genesio and Milanese (1976) and Sandell
et al (1978) give extensive reviews of the literature in the
area. The early work on simplifications intended to retain
eigenvalues was done by Davison (1966) and Marshall (1966). In
this work the emphasis was on dominant modes, modes that seemed
to be the most. important to determining behavior. The selection
of dominant modes varies depending on purpose (Gopal and Mehta
1982, Mahmoud and Singh 1982). For the analysis done here we
replace dominant modes by modes of interest, where these can be
selected in the above discussed framework.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 165
Model simplification based on selected eigenvalues was addressed
by Perez Arriaga (1981) who developed rules for selecting what
states to include in a simplified model. The concentration in
this theory is on retaining interpretability, that is, on having
a simplified model that sense can be made of. Forrester (1982)
has applied these techniques in order to identify important
feedback paths. A more formal analysis of the problem of model
simplification is contained in Eberlein (1984). In all these
works the step to actual application in a system dynamics model
is long and difficult. The theory is developed for state space
models in the form of equation 3 and the transformation of a
system dynamics model to this form loses much of the original
model meaning. We will show ways to tie the analysis back to the
original nonlinear model structure, but first we review the
existing analysis.
Though it is clear that the linear state space representation of
a dynamic model is removed somewhat from the original model the
variables of the state space representation are all contained
within the original model. It is essential that the variables of
the simplified model also make sense relative to those in the
original model. The key requirement of a simplification intended
to display structure is that the structure retained in the
simplified model make sense. It is always possible to derive a
simplified model containing a behavior mode, we did so in
presenting the one variables eigenstate model of equation 4.
Such a model does not have any useful interpretation in terms of
variables of interest. The one variable in the eigenstate model
is a complicated linear combination of the states of the original
model.
This process of simplification normally requires approximation
and this approximation can have two faces. Either the variables
of the simplified model may be transformed so as to differ from
those of the original model, or the model may produce different
dynamics. In both of these cases displaying the feedback causing
a behavior mode is an approximation in the sense that some
variables contributing to a mode are likely to be ignored. The
actual modes generated by a simplified model may be different
because feedback having a small effect on the mode is ignored.
Alternatively, if the modes are the same, the variables may have
a different interpretation because they have incorporated the
effects of other variables in their internal feedback. Faced
with this required approximation it is necessary to identify the
"best." simplification, thereby the most important feedback paths.
In order to identify the best way to simplify a model we use the
concept of an exact simplification. A model is said to be
exactly simplifiable if a separate block of equations with no
feedback to the rest of the model can be identified and it
166 = THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
produced the modes of interest. Rewriting equation 3 by
partitioning the state vector as x
x.
1 Air Al2 *y (7)
% Bai Baz %2
exact simplification requires that A, = 0 or A, = 0. If the
upper left hand block generates the nodes of interest then the
model
(8)
By By As
can be used as the exact simplification. Most models are not
exactly simplifiable, but the characteristics of an exactly
simplifiable model identify the desirable features of a simpli-
fication. Importantly they identify which variables should be
retained in the simplification.
The logic behind the theory of identifying important states is
somewhat backward. We first consider what the properties of the
simplification based on the selected states would be. Given
these properties we can measure how good a simplification it is
relative to an exact simplification. This will tell us how good
a choice the included states were. If they are to be prac-
ticable, the actual rules for selecting states need to be
applicable before the state selection is made. This will be the
case for the rules we consider, though there are some issues that
do not break out this easily (see Eberlein 1984 chapter 3). Our
goal is to break down the states x into two components, x, which
we shall refer to as the first block variables and % which we
shall refer to as the second block variables.
The eigenvector matrices for an exactly simplifiable model have
characteristics analogous to those of the A matrix. If the block
of equations generating the modes of interest is not influenced
by a second block of equations (A,. = 0) the values of the
eigenstates associated with the modes of interest are independent
of variables in the second block. Thus, when we decompose the
behavior of the variables, the components due to the modes of
interest will be independent of the behavior of variables in the
second block. This means that we could give the second block
variables any values and still get the modes. On the other hand,
if the first block of equations does not influence the second
block (A, = 0) none of the modes of interest will be apparent
in the second block. The contribution of the modes of interest
to the behavior of variables in the second block will be zero.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 167
The above characterization of an exactly simplifiable model can
be rephrased in terms of characteristics of the right and leit
eigenvectors. The right eigenvectors determine how much a
variables displays a mode, the left eigenvectors how much a
variable excites a mode, This interpretation has its basis in
aguations § and 6 which show what a variables does to an
eigenstate and what an eigenstate does to a variable. Variables
nat are important to the generation of a mode must beth
influence, and be influenced by, the eigenstate; they must have.
feedback through the mode. Eecause of this it
fice to look at either the right. or left e nvect
in asolation; both must be taken into account when judging the
importance of a state.
For an exactly simplifiable model the second b
not have feedback through the modes of interest.
will have only zero components in the left eigenvectors of the
mcdes of interest or only zero components in the right. These
zero components mean, respectively, that the existence of a
able does net affect the modes or that the modes do not
ect the vaciable. in eather case it is ar that the
ciable is not key to the mode.
it is important to note that a variable that is not key to
generating modes may still display them very clearly. The
relevant issue in the generation of a mode is what happens when
the variable is taken away. For example, consider a variable
that does not feed back into the rest of the model. The variable
may display a mode very clearly, but its removal changes nothing.
Such a variable is not generating the mode in the sense we are
discussing.
Given the importance of both the left and right eigenvectors it
does not suffice to look at either in isolation; both must be
taken into account when judging the importance of a state. The
general concept in measuring the importance of a state is based
on the matrix formed by summing the outer products of the right
and left eigenvectors over the modes of interest. More
precisely, we define the generalized participation matrix G as
Mm
ise
pe EU) Gd)» (9)
k
where i and j are indices corresponding to the states and the
summation on k is over the mcdes of interest. For an exactly
simplifiable mcdel G has the structure
G=it H or = § (10)
' * O01
oO
Oo}
I
oO
168 — THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986.
with the * denoting some potentially nonzero matrix and I the
identity matrix.
It is worth noting that in the case’in which there is only one
mode of interest the matrix G is the matrix of derivatives of the
eigenvalue with respect to the corresponding entry in the A
matrix (Porter and Crossely 1972). Also, in this case, the
diagonal elements of G are what Perez Arriaga (1981) termed the
participation factors. We base our selection of states on the
matrix G, with three attendant measures of a states importance.
First we consider the values of the diagonal elements. These will
sum to the number of modes selected, and large. values indicate
importance. The diagonal elements have the advantage of being
dimensionless and thus independent of the choice of units in the
original model.
The other measures considered involve off diagonal elements and
some adjustment for the units of measurement is required. We do
this by multiplying each element by the corresponding element of
the A matrix. This is a measure closely related to the eigen-
value elasticity and when this is done for an exactly simplifi-
able model only the upper left hand block of G will have nonzero
elements. Any nonzero elements indicate an important feedback
link involving two states. To get a summary of the important
states, we take the sum of the absolute values across a row and
compare this to the sum of the absolute values of all the entries
in the matrix.? When the ratio is large it indicates the state
corresponding to the row to be important. An exactly analogous
calculation is made relative to columns.
The three calculations outlined above will yield the same results
for an exactly simplifiable model. When the simplification is an
approximation as it normally will be there may be some differ-
ences, To deal with these we take the union of all indicated
variables.. The sensitivity for cutting off variables is not
strongly indicated by any theory. We have used a critical value
of between .25 and .5 when comparing absolute values as outlined
above. The result of all of this is a list of what variables to
include in the simplified model. These variables are indicated
because the feedback paths among them are important for
generating the modes. Thus we have identified important paths.
2. It is straightforward to show that the row sum of elements of
the weighted G matrix will equal the product of the right and
left eigenvector entries of the eigenvectors corresponding to
the row times the eigenvalue, all summed over all included
eigenvalues. Thus, the sum of all entries will simply be the
sum of the eigenvalues, The analogous result holds starting
with column sums.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 169
DISPLAYING FEEDBACK
The display of feedback requires that the analysis move back to
the domain of the original nonlinear model. Thus far the
identification of feedback structure has been for the linear
state space representation of the model. In this section we
outline a practical approach to going from the identification of
structure in state space to the identification of the structure
in the model. The amount of structure retained by the techniques
we discuss is likely to be somewhat excessive, but the techniques
have some valuable attributes. Foremost among these is the
ability to get the same model simulations by adding exogenous
values for the excluded model variables.
The selection process we outlined in the previous section will
cause some states to be retained and others to be discarded. An
important fact about feedback paths is they all pass through
states. Thus the removal of a state is essentially the same as
the cutting of feedback links. Given that we have removed
states, it is at the states that the breaking of feedback links
begins. We replace each of the removed states by an exogenous
variable, then step back to look at the results. Variables that
are not used are simply removed, variables that are used, but not
themselves in feedback paths can be made exogenous since they
depend only on exogenous variables. Making such variables
exogenous will likely leave other variables unused and, thus,
allow their removal.
This process of removing variables by making other variables
exogenous continues until all endogenous variables are part of
the feedback structure, and all exogenous variables are used
directly in defining these. An example of this process is given
in the simple two state predator prey model of Figure 1. If we
choose to remove the wolf population and retain the deer
population we first cut the links going into the wolf population.
This leaves the rodent population unused and we therefore remove
this. Further, it is not the wolf population per say, but the
wolf roaming that determines contacts. Thus we remove the wolf
population altogether, The deer-wolf contacts depend on the deer
population and cannot therefore be removed. The resulting model
contains deer population, available food, deer kills and deer
wolf contacts. Carrying capacity and wolf roaming enter as
exogenous variables.
The model that results from the simplification just described
contains the feedback essential to the modes of interest. It
will not necessarily produce these modes however. To get the
model to generate behavior modes adjustments would have to be
made to the model parameters. We will not pursue this issue
since our primary interest is just in having the essential
170 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
x
\
a {su
-
Figure 1. A Feedback in a simple predator prey model.
feedback structure. However, if the variables that we have now
made exogenous are given the same values as they had in the full
model simulation the simplified model will reproduce the original
model behavior.
While the simplified model can reproduce the original behavior
the relationship between making changes in the original and
simplified model is not as clear. The goal of a simplification
is to arrive at a model where changing parameters has the same
effect as in the original model. To do this, however, it is
necessary to alter the parameters of the simplified model. We
again do not pursue this, but feel it important to recognize this
restriction.
APPLICATION TO A MODEL OF WORKER BURNOUT
In order to illustrate the techniques described we will apply
them to a model of burnout as presented in Homer (1985). This
model describes how the process of working at an activity can
easily lead to cycles of high and low productivity and effort.
The model contains only four states in the active feedback
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 171
structure: energy level, hours worked per week, perceived
accomplishments per week and expected accomplishments per week.
Because the model is small to start, a dramatic reduction in size
cannot be expected, but the model does serve to illustrate the
techniques. The model listing and a complete description and
behavior analysis are contained in Homer (1985).
Because the model is nonlinear the linearization yields different
results at different times. These distinct linearizations can be
viewed as reflecting the different phases that the model goes
through. The application of the linear analysis is valid during
such a phase, but cannot be used to bridge the gap between phases
(discussion of an approach to bridging this gap is contained in
Eberlein 1985). A time plot of key model variables is shown in
Figure 2. The model shows a definite transient stage and then
goes to a steady state in which hours worked, energy level and
effectiveness show continual swings. During these swings there
also appear to be some shorter term oscillation that is most
readily apparent in hours worked per week.
FLO. 1 ———— paid. ,80.)
1,— —— HuWc40. 80.)
0. =
80, Mf
Qs,
75) ps
20. “f ‘
60. fi
ft
yy
6a, f-
4a, :
2}
50,
29.| 7
f
a./
40.
a. o. 1s. 30. 45. 60. 75.
TIME
Figure 2. Energy Level (EL), Hours Worked per Week (HWW) and
Perceived accomplishments per Week (PAW) in the Worker
Burnout Model
In. the linearization we focus on the period during which this
shorter term oscillation is active. We perform a linearization
at time 42. The eigenvalues of this linearization imply two
oscillatory modes, one of period about 12 and one of period about
172 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
6. The one mode of these that might correspond to the behavior
we are interested in getting a handle on is the 6 month
oscillation}. This shorter term mode is important in determining
the behavior of hours worked per week. In Figure 3 we show the
component of hours worked per week that is attributable to this
mode. Two things are to be noticed: first, thought the component
goes through the major shifts that hours worked does, the timing
is very different, and second, the component shows the shorter
term oscillation and has timing consistent with hours worked.
HWW in mode number 1.
100... oo
40,
20. eT
8 a. 18. 27. 36. 45. 54. 63. 795.
eee SIMULATION VALUE COMPONENT DUE TO MODE
Figure 3. Behavior due to a mode for Hours Worked per Week
These findings are consistent with a mode that is causing the
short term oscillations. Because the model is nonlinear the
transformation of the state vectors does not end with something
showing only the short term oscillation. The nonlinearity causes
an environment in which the linearization we have is valid only
during certain periods. During those periods the component of
hours worked due to the mode of interest shows a short term
oscillation. During the other periods the other dynamics
dominate and the transformation of the states essentially follows
3. Analysis of the longer oscillation does not reduce appreciably
the complexity of the model, though it does suggest that
expected accomplishments per week might be removed
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 173
suit. The difference in timing reflects the fact that the
transformation has components from other states in the model.
If we apply the rules for state selection discussed above with a
40% cut off rule then two states, the energy level and the hours
worked per week are identified as being important.‘ .Based on the
selection of these two states we generate a simplified model by
the method outlined. The simplified model contains 10 active
variables and 2 exogenous variables, both of these coming from
the perceived adequacy of work which has been removed. A listing
of the simplified model is contained in the appendix.
The basic feedback in the simplified model is clear in Figure 4.
Although there are feedback loops, involving the energy level and
its renewal and depletion directly, the fundamental loop is a
Major negative loop through both levels. As the energy level
rises so the hours worked per week rise which begins to deplete
energy. This loop, with its attendant delays has the potential
to cause oscillation. When the simplified model is simulated
using constant values for the two exogenous inputs the results
4. Because the mode identified is a complex mode it will
necessarily require at least two states for its generation.
174 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
are a very high frequency oscillation that as is shown in Figure
5. Because we have not tried to adjust the parameters of the
simplified model the frequency of oscillation does not match that
of the original model, but the causes and character of the
oscillation are similar.
QT EL 4d — —_Hwuc40. ,80.9
“THA AN I
ill vival vw W
76. v
/\ NY LAG A,
MA DADAAKINS
5. 75 15. 22.5 38, 37.5 45, 52.5 60, 67.5 75.
TIME
Figure 5. Simulation of Hours Worked per Week (HWW) and Energy
Level (EL) in the simplified worker burnout model.
In this example we have started with a nonlinear model and shown
how we can get at the structure underlying some of its behavior.
This same approach will not necessarily work for all visible
patterns of behavior. The longer term oscillation of the model
is a more nonlinear phenomenon. Although we have seen one root
implying oscillation, at most times the roots for the model are
explosive (eigenvalues greater than zero), implying that the
model would send things off in one direction if no nonlinearities
were encountered. The linear analysis can be used to get at the
causes of the explosive roots, which are important in the
explanation of the longer term oscillation.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 175
CONCLUSIONS
In this paper we have outlined practical tools for understanding
model dynamics. The tools are based on the theory of linear
model analysis, but can be usefully applied to nonlinear models
as the example has made clear. The key feature that
distinguishes the tools described is the ease with which they can
be applied. Because information is presented in the same time
domain that most modelers are comfortable working with the costs
of applying the tools are quite low. The tools work from a
nonlinear feedback model to another smaller nonlinear feedback
model. Though the second nonlinear model will not necessarily
generate the original behavior modes, it does contain the
essential feedback structure for doing this.
Work in linear model analysis frequently ends with the reminder
that most models are nonlinear. Theoretical tools for dealing
with nonlinear models are neither common nor powerful and,
clearly, all additions are welcome. The emphasis on the need for
a better theory of nonlinear model can, however, cause the linear
model théory to be ignored. Equally as important as developing
new theory is finding out when the old theory does and does not
work, It is hoped that the kind of tools outlined in this paper
can make that determination a reality.
176 — THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
REFERENCES
Chen, C.T. (1970). Introduction to Linear System Theory. Winston
Inc., New York.
Davison, E.J. (1966). A Method for Simplifying Linear Dynamic
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93-101
Eberlein, R.L. (1984) Simplifying Dynamic Models by Retaining
Selected Behavior Modes. PhD Thesis, Alfred P. Sloan School
of Management, Massachusetts Institute of Technology,
Cambridge, Massachusetts
Eberlein, R.L. (1985) Linear Analysis and Model Simplification.
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Forrester, N.B. (1982) A Dynamic Synthesis of Basic Macroeconomic
Theory: Implications for Stabilization Policy Analysis. PhD
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Institute of Technology, Cambridge, Massachusetts
Genesio, R. and M. Milanese. (1976) A Note on the Derivation and
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Gopal, M. and S.I. Mehta (1982). On the Selection of Eigenvalues
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Transactions on Automatic Control AC-27, pp. 688-690
Mahmoud, M.S. and M.G. Singh (1981) Large Scale System Modeling.
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Marshall, S.A. (1966) An Approximate Method for Reducing the
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Porter, B. and R. Crossley (1972) Modal Control: Theory and
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Perez Arriaga, J.I. (1981) Selective Modal Analysis with
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Department of Computer Science and Electrical Engineering,
Massachusetts Institute of Technology, Cambridge,
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Richardson, G.P. (1986). Dominant Structure. forthcoming in
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108-128
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 177
PQ paaat
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APPENDIX: SIMPLIFIED WORKER BURNOUT MODEL
HWW. K=HWW. J+ (DT/TAHWW) * (THWW.J-HWW. J)
HWW=HWWI HOURS WORKED PER WEEK (HOURS/WEEK)
HWWI=40 HOURS WORKED PER WEEK INITIAL (HOURS/WEEK)
TAHWW=1 TIME TO ADJUST HOURS PER WEEK (WEEKS)
THWW. K=MIN (LHWW, HWW. K*EELHW. K*EPAHW. K)
INDICATED HOURS WORKED PER WEEK (HOURS/WEEK)
LHWW=80 LIMIT ON HOURS WORKED PER WEEK (HOURS/WEEK)
EELHW. K=TABLE (TEELHW, EL.K, 0,1, .2)
EFFECT OF ENERGY LEVEL ON HOURS WORKED
(DIMENSIONLESS)
TEELHW=0/.4/.7/.9/1/1
TABLE FOR EFFECT OF ENERGY LEVELS ON HOURS WORKED
EPAHW.K=1.2 EFFECT OF PERCEIVED ADEQUASCY OF HOURS WORKED
(DIMENSIONLESS)
EL. K=EL.J+DT* (ER. JK-ED.JK)
EL=ELI ENERGY LEVEL (UMPH)
ELI=1 ENERGY LEVEL INITIAL (UMPH)
ER. KL=ERN*EHWER .K*EHEFR.K
ENERGY RECOVERY (UMPH/WEEK)
ERN=.3 ENERGY RECOVERY NORMAL (UMPH/WEEK)
EHWER . K=TABLE (TEHWER, HWW.K, 0,120, 20)
EFFECT OF HOURS WORKED ON ENERGY RECOVERY
(DIMENSIONLESS)
TEHWER=1.3/1.2/1/.7/.5/.35/.25
TABLE FOR EFFECT OF HOURS WORKED ON ENERGY LEVEL
EHEFR.K=TABHL (TEHEFR, EL.K, .8,1, .05)
EFFECT OF HIGH ENERGY ON FURTHER RECOVERY
(DIMENSIONLESS)
TEHEFR=1/.9/.7/.4/0
TABLE FOR EFFECT OF HIGH ENERGY ON FURTHER
RECOVERY
ED. KL=EDN*EPAED . K*EHWED .K*ELEFD .K
ENERGY DRAIN (UMPH/WEEK)
EDN=.06 ENERGY DRAIN NORMAL (UMPHH/WEEK)
EPAED .K=2 EFFECT OF PERCIEVED ADEQUACY ON ENERGY DRAIN
(DIMENSIONLESS)
EHWED . K=TABLE (TEHWED, HWW.K, 0,120, 20)
EFFECT OF HOURS WORKED ON ENERGY DRAIN
(DIMENSIONLESS) z
TEHWED=.3/.6/1/.15/2/2.5/3
TABLE FOR EFFECT OF HOURS WORKED ON ENERGY DRAIN
ELEFD .K=TABHL (TELEFD,EL.K,0, .2, .05)
EFFECT OF LOW ENERGY ON FURTHER DEPLETION
(DIMENSIONLESS)
TELEFD=0/.4/.7/.9/1
TABLE FOR EFFECT OF LOW ENERGY ON FURTHER
DEPLETION
