System Dynamics '95 — Volume II
A new approach for finding dominant feedback loops:
Loop by loop simulations for tracking feedback loop gains
Dong-Hwan Kim
(Electronics and Telecommunications Research Institute, Seoul, Korea)
Electronics and Telecommunications Research Institute (ETRD,
6th-floor Leaders office Bldg 1599-11, Seocho-Dong, Seocho-Ku, Seoul, Korea.
TEL: 82-2-587-7472 (Office)
82-2-573-2192 (Home)
FAX: 82-2-588-7337 (Office)
Traditionally, feedback loops have been analyzed in two ways. First, as in causal loop
analysis, the positive or negative relationships between variables are summed up to judge the
polarity of feedback loops. This approach can be said as a qualitative method. Second approach
for analyzing feedback loops are analytic methods mainly developed for dealing with linear
models.
For the problem of understanding the behavior of feedback loops, the qualitative methods
and analytic approaches give little help to modellers. In this paper, third approach for
understanding the behavior of feedback loops are suggested. That is a loop by loop simulation
method for tracing the feedback loop gains.
First parts of this paper explain the concept of a feedback loop gain and the loop simulation
method. Second parts of this paper experiment the loop simulation method with two S.D.
models; the commodity cycle model which shows equilibrium forces and the two shower model
which shows fluctuating system behaviors without external shocks. Last parts of this paper
discuss about the danger of understanding S.D. model with qualitative analysis of causal loops
and raise a question on the way of interpreting cyclic or chaotic behavior as shifts in dominant
feedback loops.
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A new approach for finding dominant feedback loops:
Loop by loop simulations for tracking feedback loop gains
Dong-Hwan Kim
(Electronics and Telecommunication Research Institute, Seoul, Korea)
From the beginning of the system dynamics, feedback loops have been regarded as a starting
point to understand dynamic and complex systems. In general, causal loops and the polarity of
feedback loops have been used as a major tool for explaining model behavior. However, system
dynamists have found that it is not an easy work to find and understand all feedback loops in a
model. Furthermore, as George Richardson pointed, system dynamists are really in need of a
general tool for finding dominant feedback loops (Richardson 1986).
Traditionally, feedback loops have been analyzed in two ways. First, in causal loop analysis,
the positive and negative relationships between variables are summed up to judge the polarity of
feedback loops (Weick 1979; Hall 1994). This approach can be said as a qualitative method.
However, the qualitative method has following defects.
1) It cannot deal with the conditional or complex relationships, polarities of which are
determined by variables located out of the feedback loop.
2) The strength of a feedback loop cannot be determined.
3) As a result, it cannot find which loops are dominant in system behavior.
Second approach for analyzing feedback loops are analytic methods mainly developed for
linear models. Although the analytic methods are useful in some area, to my knowledge and to
the knowledge of George Richardson in 1986, its applications in analyzing nonlinear feedback
loops are severely limited (Richardson 1986).
For understanding the behavior of feedback loops, the qualitative methods and analytic
approaches give little help to modellers. As a general rule, one should simulate his problems for
which qualitative insights and analytic formula are of no use. In this paper, third approach for
understanding the behavior of feedback loops is suggested. That is a loop by loop simulation
method for calculating feedback loop gains.
First parts of this paper explain the concept of a feedback loop gain and the loop simulation
method. Second parts of this paper experiment the loop simulation method with two S.D.
models; the commodity cycle model which shows equilibrium forces (Meadow 1970) and the
two shower model which shows fluctuating system behaviors without external shocks (Morecroft
et al 1994). Last parts of this paper discuss on the danger of understanding S.D. model with
qualitative analysis of causal loops and raise a question on the way of interpreting cyclic or
chaotic behavior as shifts in dominant feedback loops.
1. Definition of feedback loop gain
SD models have many feedback loops. One can conceive a feedback loop as an independent
entity or structure with its own property emerged from individual variables. In this perspective, a
feedback loop is regarded as having its own attributes which are protected from the change of
individual variables. Within this perspective on the feedback loop, its numerical property cannot
be observed, because only individual variables have a real numerical value.
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System Dynamics '95 — Volume II
In order to catch the numerical property of the feedback loop, one should focus on linking
points between the feedback loop and individual variables. By definition, a variable in the
feedback loop produce effects on itself. This feedback effects may be strong/weak and
positive/negative according to the nature of the feedback loop. From the standpoint of a
particular variable, one can calculate the feedback effects numerically. In this paper, a feedback
loop gain is defined as 'a magnitude of feedback effects from the viewpoint of a particular
variable’. It can be described as following mathematical form.
G=AXf/ AX wevereceneeteennnneeennnnnnenenens (1)
G_:a feedback loop gain
AX : a magnitude of change in the original variable
AXf : a magnitude of feedback effect to the original variable
In this equation X is an original variable in the loop from which we start to calculate the
feedback loop gain. The magnitude of feedback effect to the- original variable (AXf) can be
defined as
AXf= Xf - Xm
~- (2)
In this equation, Xf is an updated value of the original variable which reflects a feedback
loop effect resulting after the introduction of change (AX) into the original variable. Xm is an
another updated value of the original variable which reflects the feedback loop effect without the
change of an original variable.
In a closed feedback loop, each variable is updated every time by the feedback loop effects.
Xm reflects this momentum of feedback loops which is generated by the previous value of the
original variable. In order to obtain a pure loop gain of AX, this momentum should be subtracted
from the feedback loop effect of AX. In a word, AXf can be interpreted as a net feedback effect
which is generated by the change of an original variable (AX).
From the equation (1) and (2), we can define a feedback loop gain as follows.
G= (Xf - Xm) / AX nna
This is an operational definition of a feedback loop gain. Equation (3) can be programmed
straightforwardly into computer languages.
The logic behind the Equation 3 can be demonstrated by figure 1. Figure 1 shows the
calculation process of the feedback loop gain which is composed of three variables; X, Y and Z.
In figure 1, an inner circle represents a momentum of a feedback loop. Xm is generated from the
momentum of a feedback loop.
For calculating the feedback loop gain, AX is inserted into the loop. That is, a small fraction
of X is added to the previous value of X. By simulating variables of the loop after the
introduction of AX, we can get Yf , Zf and Xf sequentially. However, Xf contains the momentum
of the feedback loop. In order to calculate the net feedback effect of AX, the momentum (Xm)
should be subtracted from the Xf. As a result, a feedback loop gain (G) of AX is calculated from
the Xm and Xf.
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[Figure 1] Calculation process of a feedback loop gain
2. Computer implementation of feedback loop gains
Note that a feedback loop gain is not static, but dynamically changes with the time. The
amount of Xf and Xm changes as external variables affecting the feedback loop do change. For
this reason, we should not assume the feedback loop gain (G) as a static value. We should trace
the feedback loop gain as the system evolves. We can trace several feedback loop gains
simultaneously by simulating model equations, and then we can select a dominant feedback loop
which has the largest gain in the specified periods.
The amount of Xf and Xm can be calculated by a loop by loop simulation. The procedure of
loop simulation is somewhat different with that of a system dynamics simulation in which level
variables and auxiliary variables are updated separately. In the loop simulation, each value of
variables in the loop is updated sequentially in order of causalities. This is for excluding external
effects which come into the feedback loop during the simulation. With the loop simulation, one
can get the feedback loop gain which is not contaminated from other effects.
In this research, the algorithm for finding feedback loops and their gains is implemented in
EGO (Equations as Graphic Objects) which is an object-oriented simulation environment for
system dynamics (D.H. Kim 1995). Figure 2 shows a simplified flow chart for calculating a gain
for a given feedback loop.
First six steps in figure 2 are for calculating Xf of equation 3. At first step the amount of delta
X (AX) is calculated as 10 % of the value of original variable (X) and is added to the original
variable. From second step to fourth step, all variables in the given feedback loop are updated
one by one. In this process, updated variables are recovered to their original value as soon as next
variables of them are updated.
From seventh step to twelfth step, the momentum of the feedback loop (Xm) is calculated.
Calculation processes for Xf and Xm are identical. Both are different only in whether or not the
original variable is increased.
At last step in figure 2, one can calculate the feedback loop gain according to the equation 3.
Its polarity represents the polarity of the feedback loop. Moreover, the amount of the gain means
the strength of the feedback loop, because it indicates how much feedback the original variable
receives.
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System Dynamics '95 — Volume II
(0! Increase original variable X by deliaX. (deltaX =X*0.1) \
2) Update next variable in the loop
3) Recover the variable updated previously
No
Next variable
the original
variable (X) ?
‘Sy Let X¥= updated value of X
6) Recover the variable updated previously |
7) Do not change orignial variable X |
LS
8) Update next variable in the loop
ieee
9) Recover the variable updated previously |
0) Next variable =
the original
variable (X) 2
(11) Let Xm = updated value of X
12) Recover the variable updated previously
13) Calculate the feedback loop gain
G= (Xf- Xm) /delta X
[ Figure 2] Simplified flow chart of calculating a feedback loop gain
This loop by loop simulation is performed for every specified feedback loops at every time
interval of recording model behaviors. Even in a small model, a variable may have hundreds of
feedback loops. Accordingly, the loop by loop simulation method requires a lot of times.
However, it can help modellers to find out dominant feedback loops and can provide many
insights for understanding fundamental structures of system behaviors.
3. Experimenting with the commodity cycle model
For experimenting the performances of feedback loop gains algorithm, the commodity cycle
model was selected, because it shows the forces toward equilibrium state. Generally speaking,
systems characterized by the equilibrium states are expected to have the negative feedback loops
as their dominant feedback loops. Figure 3 shows the time behavior of feedback loop gains about
inventory (Inv) variable.
In commodity cycle model, EGO found out four feedback loops containing ‘Inv’ variable.
One can inspect contents of the feedback loops in a text window as figure 4. EGO automatically
makes names for each loops. A name of a feedback loop is constructed as ‘variable name + Loop
+ sequential number’. For example, 'InvLoop2' is a name for the feedback loop on 'Inv' variable
which EGO found at second time.
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in
1.00 634 Inv
0.80 570:
0.60 50754
0.39 44404
0.19 38064
InvLoop! = InvLoop2
0.00 3172.
019 2374 f$\—™
“0.40 19034
InvLoop3
InvLoop4
-0.60 12634
0.80 6344
15.0 30.0 450 60.0 75.0 90.0 1050 120.0 135.0 150.
[Figure 3] Time behavior of feedback loop gains in commodity cycle model
*** Feedback Loops of Inv
InvLoop! :: Cov => Rcov => Price => Ep => Dpcap => Rdac => Cuf => Inr => Pr => Inv
InvLoop2 :: Cov => Rcov => Price => Ep => Dpcap => Ctir => Ctcr => Peap => Rdac => Cuf => Inr
=> Pr=> Inv
InvLoop3 :: Cov => Rcov => Price => Ep => Dpcap => Ctir => Cter => Peap => Inr => Pr => Inv
InvLoop4 :: Cov => Rcov => Price => Eppe => Peer => CR => Inv
[Figure 4] Feedback loops in commodity cycle model
In figure 3, one can observe three important points about the dynamic nature of feedback
loop gains.
1) Some feedback loop gains are not stable, but dynamically change with time.
2) As the commodity system goes to the equilibrium state, all feedback loop gains tends to
be stabilized.
3) The cyclic behavior of 'Inv' variable in the initial period can be explained by the
fluctuation of the InvLoop4.
Comparing the time behavior of ‘Inv' and that of 'InvLoop4', one can conjecture that
"InvLoop4' is a dominant feedback loops. In figure 3, 'InvLoop3' and 'InvLoop4' tell us that the
feedback loop gain changes as the system evolves. From these observations, one can conclude
that the dynamics of feedback loops can be used as an explaining tool for dynamic behavior of
system variables.
InvLoop4 was experimented for inspecting the power of a dominant feedback loop. As can
be seen in figure 3, InvLoop4 is a dominant negative feedback loop. In order to stabilize the 'Inv’
variable, one must add more strength to InvLoop4. In this experiment, ‘consumption requirement
adjustment delay (Crad)' was reduced from 3 months to 2 months. The results are displayed in
figure 5.
The feedback loop gain of InvLoop4 was enforced from -0.2 to -0.3. However other feedback
loop gains are identical with those of figure 3. While 'Inv' was stabilized after 135 months in
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System Dynamics '95 — Volume II
figure 3, the stabilization of 'Inv' starts from 45 months in figure 5. From these results, one can
conclude that the dominant feedback loop really has the power of changing system behaviors.
Gain Inv
4.00 6321 Inv
0.80 wsl/
0.60 5087.
0.33 4424)
0.19 3792
InvLoop! = InvLoop2
0.00 3160
-o19 25281 TnvLoop3
0.40 ash! TnvLoop4
060 1264]
080 6324
150 300 45.0 60.0 750 900 105.0 1200 136.0 150
[Figure 5] Results of enforcing InvLoop4
When Meadow said that "a change in the consumption loop has a greater impact on the
stability of the system", he was aware that InvLoop4 is a dominant feedback loop (Meadow
1970, p.70). However, he has not used a systematic and convenient method to find this fact. With
the analysis of the feedback loop gains, one can find the dominant feedback loops without time-
consuming trials and errors and furthermore one can understand how the dominant feedback
loops can be exploited to achieve policy goals.
4. Experimenting with the two-shower model
Previous experiments was performed to investigate the feedback loops in a stable system. In
this section, two-shower model is experimented to investigate the time behavior of feedback loop
gains in the system which shows fluctuating behaviors.
As Morecroft et al presented, system behaviors of the two-shower model depend on whether
or not two person have the same desire for temperature of water (Morecroft et al 1994). First, the
model with different desire for temperature at shower head (27 and 25 respectively) is
experimented. It shows fluctuating behaviors. Results of experiments are presented in figure 6.
In figure 6, all feedback loop gains fluctuate with the time. Moreover, the polarities of
feedback loops change with regularities. All feedback loop gains graphed in figure 6 fluctuate
between the negative polarity and the positive polarity. In this situation, the polarities of
feedback loops should be understood as dynamically.
As graphed in figure 6, fifth feedback loop (temperatureAtShowerHead1LoopS) is dominant.
Note that a shift of dominant feedback loops did not happened during entire simulation period! It
seems that one dominant feedback loop generates the cyclic behavior of 'temperatureAt-
ShowerHead1". In this example, the fluctuating system behavior should not be explained by the
shifts in dominant feedback loops, but by the oscillation of one dominant feedback loop from
negative polarity to positive polarity and vice versa.
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Gain temperatureAtShowerHead|
7.00 32.00
0.80 28.804 temperatureAtShowerHead|
0,60 25.60) temperatureAtShower
0.39 22.39 HeadiLoopS
temperatureAtShower
Head! Loop6
0.1919.19f
0.00 16.004 temperatureAtShower
Head i Loop7
0.191280}
040 953]
060 6.40
080 2204
15.0 30.0 450 600 750 900 1050 1200 1360 150.
[Figure 6] Time behavior of feedback loop gains in two-shower model
*** Feedback Loops of TemperatureAtShowerHead!
TemperatureAtShowerHead1LoopS :: TemperatureGap1 => FractionalAdjustment1 => RequiredAdjustment] =>
ChangelnTapSetting1 => TapSetting1 => TemperatureAtTapl => TemperatureAtShowerHead1
TemperatureAtShowerHead1Loop6 :: TemperatureGap1 => FractionalAdjustmentl => RequiredAdjustmentl =>
ChangelnTapSetting] => TapSetting] => FracHotWaterAvaiableTol => MaxFlowOfHotWaterl =>
TemperatureAtTap] => TemperatureAtShowerHead1
TemperatureAtShowerHead1Loop7 :: TemperatureGap| => FractionalAdjustmentl => RequiredAdjustment] =>
ChangeInTapSetting] => TapSetting] => FracHotWaterAvailableTo2 => MaxFlowOfHotWater2 =>
TemperatureAtTap2 => TemperatureAtShowerHead2 =>TemperatureGap2 => RequiredAdjustment2 =>
ChangelnTapSetting2 => TapSetting2 => FracHotWaterAvaiableTol => MaxFlowOfHotWaterl =>
TemperatureAtTap1 => TemperatureAtShowerHead1
[Figure 7] Feedback loops in two-shower model
The dynamics of feedback loops found in figure 6 is somewhat different with the explanation
provided by George Richardson (1986, 1991). The explanation for the cyclic or chaotic behavior
by the shifts in dominant feedback loops seems to be not applicable to this case. With regard to
the systems like this example, one should explain the cyclic or chaotic behavior by the
fluctuation of some dominant feedback loops.
As a second experiment with the two-shower model, a model on two persons with identical
desire for temperature at shower head was experimented. The experiment was performed to find
out what differences are there between the stability of a commodity model and the stability of
two-shower model.
Figure 8 shows the time behavior of feedback loop gains in two-shower model with identical
desire for water temperature. As demonstrated by Morecroft et al (1994), 'temperature-
AtShowerHead1' comes to an equilibrium state. However, this equilibrium state is not robust. It
is very weak to external shocks.
This point can be explained with the feedback loop gains. In figure 8, feedback loops are
fluctuating in spite of the stable behavior of 'temperatureAtShowerHead1'. Only seventh loop
remains stable after initial fluctuations. The oscillating behavior of feedback loops implies that
the temporary equilibrium state is weak. If there is any change in fifth or sixth feedback loops, its
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System Dynamics '95 — Volume II
effect will be fluctuating with that of feedback loop gains.
Gain temperatureAtShowertlead!
1.00 32.004
0.80 22.80] pa NeuperdturestShowerHead|, _ temperatureAtShover
¥ i] f [| f] Head iLoopS
neo 25604 \ AN
0.33 22.35] M HT TL I TTT TV AT Miiteseersturessstoner
aig iaral Head|Loop6
0100 16.0044 temperatureAtShower
5 CI CI [HeadiLoop7
-01912.804 LILILILILILI LI LI L
040 9594
280 6.40]
00 320
16.0 300 45.0 600 750 90.0 1050 120.0 1950 150.
[Figure 8] Feedback loop gains in two-shower model with identical desire for temperature
How can one achieve stability in the face of different desires for shower temperature? From
figure 6, one can observe that the unstable system behavior comes from the oscillating nature of
feedback loops. If one can reduce the power of the dominant feedback loops, one can increase
the stability of shower system. The power of dominant feedback loops can be reduced by
decreasing 'judgementalCalbration' from 0.1 to 0.01. The results are displayed in figure 9.
Gain temperatureAtShowerHead!
0.10 32.004
0.07 28.804
0.06 25.604
0.04 22.394
0.02 19.194
0.00 16.00.
temperatureAtShoverHead!
temperatureAtShover
HeadILoop7
temperatureatShower
0.02 12.804
Head |Loop6
06.03 9.59]
0.06 6.404
“0.08 3.204
temperatureAtShower
Head|LoopS
150 30.0 45.0 600 750 900 105.0 1200 135.0 150.
[Figure 9] Results of reducing 'judgementalCalbration' from 0.1 to 0.01
In figure 9, all feedback loops are stabilized to the negative side so that the shower
temperature comes to stable stage. Note that polarities of feedback loops can be changed by
adjusting one parameter! These results raise a new research question on the relationship between
parameters and the polarities
of feedback loops.
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5. Discussions and Conclusion
In this paper, a new method for finding dominant feedback loops was introduced and their
practical implications in system dynamics models were presented with two examples. I think that
a loop by loop simulation method for tracing feedback loop gains may provide a new tool for
analyzing and understanding S.D. models.
Through this research, some questions have been raised. First question is on the qualitative
analysis of causal loops. If feedback loops fluctuate from positive to negative polarities with the
time, qualitative analysis of causal loops may be incorrect in interpreting model behaviors and
even might suggest wrong policy recommendations. Second question is on the notion of shifts in
dominant feedback loops. If unstable systems can be explained by the fluctuation of one
dominant feedback loop, we should pay more attentions to the parameter adjustment than the
system restructuring. Third question is on the dynamic relationships between parameters of
variables and polarities of feedback loops. Studies on their relationships might give much more
understandings on the dynamic interactions between system structures and parameters in
determining system behaviors. I think that more experimentations and technical developments on
using dominant feedback loops to find policy levers should be performed in the future.
[References]
Hall, R. I. 1994. Causal policy maps of managers: formal methods for elicitation and analysis.
System Dynamics Review, 10(4).
Kim, D.H. 1995. Introduction to EGO. Unpublished personal memo.
Meadows D.L. 1970. Dynamics of Commodity Production Cycles. Wright-Allen Press, Inc.
Morecroft J.D.W., E. R. Larsen and A. Lomi, 1994. To Shower or No to Shower: A Behavioural
Model of Competition for Shared Resources. Proceedings of 1994 S.D. conference.
Richardson G.P. 1986. Dominant structure. System Dynamics Review, 2(1).
Richardson G.P. 1991. Feedback Thought in Social Science and Systems Theory, University of
Pennsylvania Press.
Weick K.E. 1979. The Social Psychology of Organizing, Addison-Wesley Publishing Company.
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