To Main Proceedings Document
Coflow Structures: Some Problems and Solutions In
Representing Psychological Characteristics and Processes
Ralph L. Levine
Department of Psychology and
Department of Resource Development
Michigan State University
East Lansing, MI 48824, USA
1-517-332-2317
leviner@ pilot.msu.edu
And
Hoa X. Nguyen
Department of Psychology
Michigan State University
East Lansing, MI 48824, USA
1- 734-995-0971.
nguyenho@pilot.msu.edu
ABSTRACT
The coflow structure is useful when changes in one level simultaneously drive
another set of changes in a second level. Coflow structures can represent how
psychological variables, such as average attitudes, are carried over as a group of people
change status or designation. In modeling a problem of an agency, we had difficulties
using the classic form of the coflow process to represent average attitudes towards the
organization. Total Attitude, which is defined as a state variable, is hard to interpret.
This is particularly true when attitudes are also influenced by organizational experience
and may erode over time. This paper compares an isolated coflow structure with the
coflow structure combined with an eroding process. Adding an eroding process to the
classical coflow structure generated interesting behavior modes, such as logarithmic
decline to a minimum and growth toward a higher equilibrium point.
We suggest directly using Attitude as a stock rather than using Total Attitude as
the level. Changes in (Group) Attitude would depended on the influence of those coming
in from the previous stage, experiences happening at that stage, and perhaps upon a
naturally occurring eroding process. The influence of new people coming into the group
affected Attitude in proportion to their numbers. If only a few entered the organization,
their attitudes would have little effect on the group’s Attitude. We suggest two
alternatives to the classical coflow formulation. We also show how each formulation
behaves when we allow attitudes to erode.
INTRODUCTION
The coflow structure becomes very useful when some change or action in a
fundamental level concomitantly causes a change in another level. For example, in
modeling the production of computer code, one could also represent the generation of
programming bugs by directly connecting the production of lines of code with the rate of
bug generation. In addition, the coflow structure is frequently used to deal with the
change in psychological, demographic, and social characteristics of people who are
changing status, such as the change in the level of sophistication with new technology of
new hires. In this case, the second level, which deals with the characteristic, is the total
accumulation of the rate of change in the characteristic variable. To make sense of this,
one has to divide the level by the number of people in the organization to obtain the
average value of the psychological characteristic.
GOALS OF THIS PAPER
There is very little published about the coflow process. This paper will first point
out some of the problems with using the classical approach to the coflow structure when
the coflow dynamics are combined with other processes. These problems deal with the
definition of the psychological variable being modeled. We shall introduce two
alternative model formulations, both of which directly quantify the psychological
variable as an average or mean value. Since the coflow process can frequently be
combined with other processes, the second goal of this paper is to show some interesting
behavior modes, which emerges when the coflow process is combined with other
fundamental atoms of structure, such as a draining process. In some circumstances, one
observes a parabolic growth process, followed by a collapse of the value of the
characteristic variable associated with the original driving level.
BACKGROUND
(Figure 1 at end of paper)
It is best to describe the background concerning the problems we ran into in using
the coflow structure in our modeling efforts. We were interested in modeling some of the
problems encountered by an organization, which represented our university's effort in
dealing with urban problems within the community. The work was being done by
graduate students majoring in a number of different areas of social science and natural
resource management. Our modeling effort focused on the conflict the workers were
experienced when they devoted so much time to helping people in the community at the
expense of their own graduate careers. As part of the modeling process, we wanted to
represent the students’ attitudes toward the Center as they came into the organization and
also spent time working for the Center. At first, before actually working for the Center,
they came into employment with a much more positive attitude than after spending some
time working on some project in an urban setting. Figure 1 shows a simplified version of
part of the model, which deals with the number of students recruited and various ways in
which their attitudes might be influenced in this situation.
As one can see, the coflow process in our model is also supplemented with other
structures which capture (1) the external influences on attitudes from the experience in
the Center and (2) the spontaneous erosion of attitudes over time. To amplify a bit,
attitudes can be influenced in many ways. The coflow process in our model focused on
how attitudes were modified by newcomers who had come in with a strong positive
feeling about the Center before they actually began to work for the organization. On the
other hand, we felt that experience on the job would also have a profound effect on
people over time. Finally, we assumed that extreme values were relatively hard to
maintain, and thus we put in a draining process to represent spontaneous decreases in the
worker s attitude.
SOME BASIC PROBLEMS WITH THE CLASSIC COFLOW STRUCTURE
It can be seen from Figure 1 that the level representing the psychological
characteristic, which in this case are the attitudes of the trainees and workers, is in total
attitude units. This number is relatively meaningless, until one uses an additional
converter or auxiliary variable, Av Trainee Att, which can easily be quantified on a
scale ranging, for example, between -100 and + 100 points. In terms of units,
Av_Trainee Att is equal to Total Trainee Att units/ People. Changes in that variable
would describe mean behavior over time.
The fact that the level itself is not interpretable is problematic. Although there are
many examples of good SD models in which levels have to be modified through
converters, in general, levels should be defined to be a interpretable and meaningful as
possible. State variables can be defined as those variables which are monitored by the
actors or stakeholders in any given situation. The problem of defining attitudes as total
attitudes leads to some difficulties. For example, from Figure 1, we see that attitudes are
influenced by the workers experience in the Center. The change in Tot_Wkr Att is
accounted for partly from rate variable, Wkr Experiences. The units of this rate variable
have to be in total attitude units per unit of time. It is extremely difficult to calibrate and
work with these units. It would be better to define the level as an index of average
behavior for the group. Thus, instead of having the average attitude defined as a
converter, the average attitude would be defined directly as a state variable or level
concept. Eventually, we shall do this later in the paper.
THE BEHAVIOR OF THE CLASSICAL COFLOW STRUCTURE
(Figure 2 at end of paper)
The Simple Case First
To simplify matters somewhat, Figure 2 shows an even more simplified version of the
classical coflow structure. In this case, there is only one set of workers in which
increases through recruiting and decreases through quitting or retiring. The characteristic
is the Total Wkr Att, which is the summation of the workers’ attitudes towards the
organization. In the classical version of the coflow process, the average worker attitude
is an auxiliary variable, which in this case is Av_Wkr Attitude. Note from Figure 2, we
have modeled the worker sector as a first order material delay by specifying the rate of
leaving the Center is a function of the number of workers. This is a drainage process.
(Figure 3 at end of paper)
Basic Results
Run #1. For the entire series of simulation runs, we assume one can quantify average
attitudes toward the Center on an attitude scale which runs from - 100 to +100, zero being
neutral. In our base run, we started the Center with a single worker and recruited people
per month. The average stay in the Center was for one year or twelve months. The initial
total attitude level was 40, and with only one worker at the start of the run, the average
attitude was 40 as well. Finally, we initially assume that incoming people are much more
positive about the Center than those who have been there for some time. Thus, we set the
initial average attitude value of the incoming recruits (IncomingA tt) at a value of 75. We
would interpret this as being very positive.
The initial run, as seen in Figure 3, indicates that the Workers show logarithmic growth
pattern, a result which is well known, because we have modeled the level as a first order
material delay. In this particular run, when the input rate equals 4 people per month and
the time constant associated with Outrate equals 12 months, Workers will come to an
equilibrium point at 48 people. Note also that the mean attitude, ie, Av_Wkr Attitude,
shows the same pattern as material delay representing the number of people working at
the Center.
Adding a Drainage Process to Total Attitudes
(Place Figure 4 at end of paper)
Run #2. Thus far the behavior of the model displays fairly well known behavior. For
example, if the initial attitudes of the workers were above 75, say 83, the average attitude
would decrease over time to the value of 75. Frequently, however, other influences on
attitudes besides from the effect of new people entering the organization also have to
included in the model. One such process deals with the gradual decay or decline of
attitudes. Satisfaction and positive attitudes have to constantly be maintained and fostered
to prevent erosion of these psychological variables. Let’s this additional process into the
model in addition to the original coflow process. Figure 4 shows the augmented model
which includes a draining process to represent the erosion attitudes.
(Place Figure 5 at end of paper)
In run #2, we open the draining process and set the TimeToErode Att, the constant
associated with draining process to four months, implying that workers’ positive attitudes
decay relatively quickly. Figure 5 shows what happens to the three major variables in this
situation. Again Workers show the characteristic logarithmic pattern over time. This is
to be expected because it is not affected at all by the attitudes. Note also that
Total_Wkr Att has the same general pattern. However, looking at the third variable,
Av_Wkr Att , which is the average attitude toward the Center, one sees a new pattem
emerging, namely logarithmic or parabolic growth, followed by overshoot and collapse to
an asymptote. In this run, average attitudes fell below the initial value, name 40 attitude
units. Note that this pattern is not the same as the usual s-shaped growth, overshoot, and
collapse mode, frequently found and modeled by system dynamicists in a number of
other situations .
This behavior pattern is relatively different from what might be expected. In many
situations, a draining process would yield a simple and gradual steady state error. In this
situation, there is overshoot. Note also that only the average attitude value has this
pattern. The components of the mean curve show a different pattern, unlike the case
where there is no spontaneous draining process, run #1.
These results seemed somewhat surprising to us. Our first thought was that somehow
the pattern observed was an artifact of the computer integration process. We re-ran the
simulation, with an extremely small dt, and Powersim’s Runge-Kutta 4 (Variable)
algorithm. The results were exactly the same. Indeed, all simulation runs reported here
use the RK4 (Variable) approach to deal with a potential stiff system.
(Place Figure 6 at end of paper)
Run #3. The last run showed that the average attitude first went up and then came
down below its starting point. The time constant, TimeToErode Att, was set at four
months. In the next run, we changed the constant to 60 months to slow down the process.
Figure 6 shows the results for the mean attitude only since the same logarithmic pattern
appeared again for the other two variables. In this situation, the general pattern of log
growth, overshoot and then decrease to an asymptote occurred again, but less vigorously.
The negative loop was less strong, as would be expected when the constant was 60
months instead of four months. Nevertheless, the overall pattern persisted.
The Effect of Excess Workers
Run #4. In the previous runs, we started with a very few workers, far below the
equilibrium point, 48 people. In the next run, we kept the total attitude value at 40 units,
but changed the number of workers from 1 to 100 people working for the organization.
The average attitude value equaled .4 for this run, indicating that, as a whole, the workers
were almost neutral about the organization at the beginning of the run..
(Place Figure 7 at end of paper)
As one can see from Figure 7, workers can down to an asymptote of approximately
48 people. In this run, Av Wkr Attitude (Figure 7c) showed the same logarithmic
pattern as the classical case without the draining process. The only difference is that
Figure 7C shows a steady state error. The average attitude starts at a low point and
grows logarithmically to a steady state error. There is no evidence of overshoot under
these conditions. At this point, we have two distinct behavior modes.
When Initial Workers and Average Attitude are Above their Equilibrium Points
(Please place Figure 8 at end of paper)
Run #5. Thus far, we have seen that average attitudes either reach their asymptotes
smoothly or overshoot first before reaching an equilibrium point. In this run, we again
set the initial value of Workers at 100 people, but this time we set the Total Wkr Att at
4000 units, making the average attitude equal to 40 instead of .4 as in our last run. The
results of this run, found in Figure 8, indicate that, although both the Total Wkr Att
variable and the number of workers came down smoothly, the average attitude decreased
to a minimum value which was less than the final equilibrium point. This is the mirror
image of the pattern of run #2, which showed the pattern of log growth, overshoot, and
collapse down to an asymptote.
When Workers Are Below Their Asymptote and Attitude Starts Above its
Asymptote
(Please place Figure 9 at end of paper)
Run #6. In the next run, we set the initial number of workers at 20 and the
Total Wkr Att value at 800, giving us an initial value for the average attitude of 40
attitude units. Setting parameters in this manner insured that the initial number of
workers would be below its asymptote of 48 and the average attitude would be above its
final asymptote. The results are shown in Figure 9. One can see from Figure 9C that the
average attitude went down to its final equilibrium point in a standard manner. Unlike
Run #, there was no evidence of a “dip” before reaching its asymptote.
A Four-fold Summary of Findings
(Please place Figure 10 at end of paper)
Thus far we have shown a variety of behavior modes of the mean worker attitude,
generated when studying the effects of both the classical coflow structure and another
process which represents the spontaneous loss of positive worker attitudes. There are
some rules of thumb which help us know under what conditions one would obtain simple
logarithmic behavior and when one would predict either a pattern of growth, overshoot,
and collapse or a pattern of dropping, dipping , and then increase to a higher asymptote.
Setting initial levels and estimating final equilibrium points. The variables of
interest in our analysis are (1) the number of workers and (2) average worker attitude.
This model containing the draining process consists of only two state variables, and is
quite easy to estimate, given the value of the parameters, where the variables will be at
equilibrium. We can use those estimates to develop our categorization scheme, which
describes the conditions which generate specific behavior patterns.
Figure 10 summarizes these rules of thumb by means of a four-fold table. The rows
represent the whether initially, the number of workers were above or below the
equilibrium point, namely 48 workers in this situation. The columns represent whether
the initial average attitude was below the maximum or minimum value. Runs #2 and 3,
started with workers being set at one person, far below the equilibrium point for workers,
namely 48 persons. The Av Wkr Attitude in these runs was below its maximum. This
generates a growth, overshoot, and collapse trajectory.
On the other hand, if again the initial number of workers is below its maximum, 48,
but this time the average attitude starts above its minimum value, then we get the pattern
in Panel 10-B.
If the system was started with a large number of workers, above the equilibrium
point, which in this case was 48 persons, and the initial average worker attitude was
lower than its maximum, then the logarithmic pattern emerged over time. This can be
seen in Panel 10-C.
Finally, if both the initial number of workers and their attitudes were above their
respective minimums, then the trend was downward, “undershoot,” and up to an
equilibrium point. This can be seen in Panel 10-D.
Thus, from panels 10A and 10D, one can see that, when the initial values of the
variables are below their maximums, transient overshoot or undershoot occurs.
From Equilibrium Conditions: Changing TimeT oL eave
In all the previous simulation runs, we started out of equilibrium. We next performed
a series of simulation runs starting in equilibrium. We either changed the parameter on
the drainage process associated with Workers or changed the parameter of the drainage
process associated with Total-Wkr Att.
Run #7 (Baseline Equilibrium). We first will look at the classical coflow structure
when the drainage process representing spontaneous decreases in average attitude is
turned off. We set the number of workers equal to 48, the equilibrium value for Workers.
In addition, we set the equilibrium value of the second level, Total Attitude equal to the
value of the IncomingAtt, which equaled 75 on the attitude scale. In this run, we wanted
to see what would happen if we changed the time constant, TimeToGo on the material
delay from 12 months to 18 months. This should take both levels out of equilibrium to a
new Set of values. What would happen to the average attitude, Av_Wkr Attitude, which
is the ratio of the two levels?
(Place Figure 11 at end of paper)
Panels 11-A and 11-B indicate that, when the time constant associated with first order
material delay is changed from 12 to 18 months, the levels responded as they should,
namely to increase in both cases. However, note that once the average attitude was in
equilibrium, it never changed its value. This was because both the numerator and the
denominator of the ratio defining the mean attitude changed by the same proportion. The
transfer rate of the total attitude level, is a common function of the Outrate variable.
(Please Place Figure 12 at end of paper)
Run #8 (equilibrium with draining process). In this run, we started the simulation
under the previous conditions, which were set for the classical coflow process without the
draining actively engaged. At time 10, two things happened. First the switch for starting
the drain process was turned on. Second, the time constant for the material delay,
Workers, was changed again from 12 to 18 months. Figure 12 shows that, unlike the
previous run, Average Attitude responded to the drain process and went smoothly to a
new low equilibrium point.
From Equilibrium Conditions: Changing TimeT oErode_Att
(Place Figure 13 at end of paper)
Runs #9 and #10 (equilibrium with draining process). The next two runs were
conducted in a way which, after, starting both levels in equilibrium, the time constant
associated with the draining process, TimeToErode_Att was changed to either a larger or
smaller value at Time 10. Initially the value of the parameter was 10 months. Then, at
Time 10, the value of the time constant was changed either up or down by six units.
The results of these two equilibrium runs can be seen in Figure 13. One observes
that, when the changes in the constant associated with the negative, draining loop are
initiated, the average attitude goes smoothly to a new asymptote. There was no sign of
overshoot or going below the final asymptote, which was found under some conditions
seen in Figure 10.
DEFINING AVERAGE ATTITUDE AS A STOCK VARIABLE
One of the initial concerns of this paper dealt with the problems of working with
the classical coflow process by defining the level as the total attitude, when actually it is
the average attitude that is most meaningful. The average attitude is what one wants to
monitor in this situation. In the classical case, one has to define the average as a auxiliary
variable, which would be the ratio of the total variable and the number of cases or people
in the organization. We would like to suggest two related solutions to this problem. We
first developed a solution in working on a model of dynamic problems of the Center For
Urban Affairs (Levine, et al, 1998). Although it worked, we were modified it slightly to
be more consistent with respect to dimensional analysis. In the meantime, we recently
became familiar with another solution to this problem, worked out independently by
Professor James Hines at M.I.T.! His solution was simple and elegant. In this paper, we
shall refer to his version as the Hines Coflow structure. In addition, we will also describe
our solution to defining average attitude directly as a level. The two solutions are very
similar to each other. Ours is a bit more disaggregated and consequently less simple the
the Hines coflow process.
(Please Place Figure 14 at end of paper)
The Hines Coflow Model
Figure 14 shows a version of the Hines coflow process with an additional draining
process to represent spontaneous loss of the intensity of the workers attitude toward the
organization. Note that the stock is principally modeled as a stock adjustment process or
first order information delay. Second, in this approach, one forms the ratio of the number
of workers to the RecruitRt. This ratio, dimensionally is in terms of time units, so that it
can become a key input into the stock adjustment process. Mathematically it is
equivalent to the classical version of the coflow process.
Results of Runs Without Draining Process
(Please Place Figure 15 at end of paper)
Runs #11 to #14 (Hines coflow model with draining process). Figure 15a and 15b
show the results of runs, where initially, the average attitude value was above and below
the goal. There is no surprise behavior here. The Hines coflow model gives identical
results as the classical case.
The next set of runs were conducted with a combination of the coflow and draining
process, by setting the switch, SW1 to a value of 1. Figure 15-C to 15-F show the
identical results in each of those which resulted in previous runs associated with the
classical case when we added a spontaneous drain on the attitudes (see Figure10).
In summary, the Hines solution to the problem seems strictly comparable to classical
case. The overshoot and undershoot observed previously also were observed in approach
as well.
A Disaggregated C oflow Model
We were unaware of the Hines alternative formulation of the coflow process when we
solved the problem of how people take their beliefs and attitude with them entering and
leaving organizations. In the past, the senior author had modeled attitude and belief
systems using stock adjustment processes (Hunter et al, 1984). As indicated earlier in
this paper, the coflow process can represent the influence of new people coming into an
organization. Originally, we too worked with the ratio of workers to the input rate.
However, since we were focusing on attitude change, where other independent sources of
change have an effect on attitudes, we felt the time factor should represent the individual
difference parameter dealing with how perusable people who were coming into the
organization might be over time. We wanted to treat the coflow process independently of
persusibility.
In order to accomplish this, we thought about the relationship between the number of
new recruits and their influence on the mean worker attitude value over time. We
assumed that if only a few new people came into the organization, they would not have
too much influence on the group as a whole. On the other hand, a large number of new
people might have a very big effect on attitudes, beliefs, and behaviors over time. We
accomplished this by disaggregating workers into two groups, “trainees” and “older
workers” respectively. Although we could have disaggregated attitudes into trainee
attitudes and old worker attitudes, to be comparable with the two previous models, we
decided to present the influence of new people on a single stock variable, namely average
Worker Att.
(Place Figure 16 at end of paper).
The model is represented in Figure 16. The attitude sector of the model is almost
identical to Hines Coflow process. The worker sector is represented by a second order
material delay process, composed of trainees and workers. In order to make things a bit
comparable from a parameter perspective, we set the time constant for training equal to 4
months and for the time to serve as a worker at a value of 8 months, make the total time
equal to one year.
The primary change in worker attitudes was modeled as a stock adjustment process,
which is similar to the Hines coflow approach. This change is proportional to the relative
number of newcomers coming into the organization. We assume that, if the agency were
to bring in a large number on new people, relative to the number of existing workers, they
would have an important effect on worker attitudes. On the other hand, if just a few
people come into the organization, they would not affect average attitudes very much.
Note that we have included a draining process in the model as before to see if one
gets the same set of patterns as was found in the first two models.
(Place Figure 17 at end of paper)
Run #15 (disaggregated model, no draining process). In this situation, we wanted
to see how this alternative to the coflow model behaved when there was no drain process
active. We set the parameters as close as possible to those used in the other two coflow
models. Thus, the constant, TimeToChng, was set at Two months. Figure 17 shows the
results of this baseline run. The behavioral patterns for both the Worker and
Av_Wkr Att Variable were comparable to the patterns found for both the classical and
Hines coflow models. For example, the total number of employees, i.e., trainees and
workers, equaled 48 and the average attitude reached 75 in about the same time as the
previous simulation runs.
Setting Trainees At Equilibrium.
(Please Place Figure 18 at end of paper)
Runs #16 to #19 (Disagregated model with draining process). The next series of
Tuns were performed by first setting the initial value of the number of Trainees at 16,
which is the equilibrium value under these conditions. We next undertook four
simulation runs, where Workers were either above or below the equilibrium point, 48
people and Worker Att was either above or below a maximum or minimum as before.
The results are seen in Figure 18. One can see that these patterns match the classical and
Hines coflow models very well. The three models appear to generating identical modes
of behavior.
Setting Workers At Equilibrium.
(Place Figure 19 at end of paper)
Runs #20 to #23 (disaggregated model with draining process). The next series of
four runs set Workers at 32, the equilibrium point, and allowed the Trainees and
Worker Att variables to vary in the same way as before. Figure 19 shows these results.
Again, the patterns found in each of the panels is similar to what was found before. Cells
A and D show either overshooting and collapse toward a equilibrium point or in Cell D,
Worker-Att first drops and then hits a minimum before reversing direction and rising to
an equilibrium point.
MOVING TOWARD INSIGHTS
(Place Figure 20 at end of paper)
There were two goals to the paper, namely (1) describing and dealing with the
problem of indirectly defining the average characteristic in the coflow process and (2)
seeing the implications of combining the coflow structure with other mechanisms, such
as a draining process. With respect to developing an alternative to the classical coflow
structure, both the Hines approach and our disaggregated model appear to directly focus
on an average psychological characteristic, such as the mean attitude toward an
organization held by new people coming into the company. The difference between the
two alternative formulations appears to be minimal. The Hines model is simple and to
the point. Our formulation adds to the complexity of the modeling process, which at
times is not conducive to understanding. On the other hand, if the problem being
modeled calls for more detail, so that one should disaggregate the total workers into
subgroups, then our formulation would be appropriate.
Illustrating the Structural Aspects of These Behavioral Patterns.
The four-fold table of patterns showed some somewhat surprising modes of behavior
which appear to be replicable with all three models. Given the time restraints and the
length of this paper, we shall only begin to move in the direction of analyzing the
influence of loop structure on the patterns observed in this study. Figure 20 shows the
loop structure of the classical coflow and the draining process. It can be seen that there
are three negative loops involved in this situation. Again, as was mentioned previously,
the worker’s sector of the model drives the attitude sector of the model. There is no
influence in the opposite direction. Loops B1 and B3 are associated with the draining
processes related to a decrease in Total Wkr Att and Workers respectively. B2
represents another draining process, which is coupled with the coflow relationship
between the OutRate variable and the TransferO fA tt rate variable.
In the model represented in Figure 20, one can find only two stocks and only three
loops, yet the model exhibits a variety of behavior modes. System dynamicists have
always been interested in gaining understanding about the dynamics of the problems
being modeled by understanding the specific role of the loop structure in underlying the
behavior of the model (Coyle, 1977; Richardson, 1995; Mojtahedzadeh, 1996, Ford,
1999).
An Example of Searching For Loop Dominance
To gain some insight concerning dynamics of combining the coflow and draining
process, we are going apply a relatively new type of loop analysis, recently developed by
Ford (1999), to one of the most interesting patterns described previously, name the
pattern where the Av_Wkr AA variable made a logarithmic drop down to a minimum,
and then increased to a higher equilibrium point. This is the variable of interest in the
analysis The the search for dominance begins by partitioning the total time curve into one
of three general patterns or trends, namely an exponential trend, a logarithmic trend, or a
linear trend. The first derivative of this variable describes the net rate of change during a
given period. In Ford’s procedure, one first takes the absolute value of the first derivative
and then calculates the movement of this rate of change to obtain the second derivative.
Tf the second derivative is positive in an time interval, then the shape of the curve is
exponential. If the second derivative is negative, the shape of the curve is logarithmic.
Finally, if the second derivative exactly equals zero, the one is dealing with an linear
trend.
(Figure 21 at end of paper)
Using the POWERSIM language we found it extremely easy to obtain the second
derivative described above. Figure 21 shows the system when Workers equals 100,
Av_Wkr Att equals 40, and the drainage process was turn on. The second derivative is
plotted in 21-B. We found that in the range of time 0.0 to time 11, the second derivative
was negative, indicating a logarithmic trend. From around time 11 to time 18, the second
derivative tumed positive, indication exponential trend. Finally from time 18 to the end
of the simulation run, the second derivative turned negative again.
Thus, we have partitioned the total time curve into three intervals. The next step
in Ford’s procedure is to start with the first interval and systematically turn off each of
the loop singally as well as in combinations to see if a given loop or set of loops affects
the pattern for that interval. In this paper we are only going to report the analysis of the
first interval for illustrative purposes. We first deactivated loop #1 (see Figure 20), which
controls the draining process. After running the simulation with loop #1 deactivated, we
found that the second derivate, between time 0.0 and 11, was negative. Since this
matched the original run with Loop #1 intact, we concluded that this loop was not
dominating the behavior of the system in this interval.
After activating Loop 1 again, we next deactivated Loop #2 by severing the
connection between the Av_Wkr Att and, associated with the rate variable
TransferOfAtt. The second derivative again was negative during this interval, which
matches the intact results. Thus, we concluded that Loop #2 was not dominating the
behavior of Av_Wkr Att during this interval.
The next candidate was Loop #3. Av_Wkr Att is not in this loop. We
deactivated the loop, ran the simulation and again found that the second derivative was
always negative during this time interval.
After controlling each loop singly, we next undertook deactivating pairs of loops
to look for an interaction effect. When we deactivate Loops #1 and #3, we found that the
second derivative during this initial interval was always negative in value. The same was
true of simultaneously deactivating Loops #2 and #3. Again the second derivative was
negative, indicating that this combination of loops, acting together, did not determine the
logarithmic behavior observed in the variable of interest.
(Figure 22 at end of paper)
Hitting some pay dirt. Finally, we simultaneously deactivate Loops #1 and #2
and ran the simulation again. Figure 22 shows the results of this analysis. The original
behavior pattern can be seen in 22-A, while the pattern obtained by deactivation is shown
in 22-B. Finally, the variable, Derv2AvAtt, the second derivative, can be seen in 22-C.
This time the second derivative is positive, indicating an exponential trend over this
initial interval.
We interpret this outcome as evidence that only in combination can the loops
dominate the behavior of the variable of interest. They comprise a “shadow pair.” Neither
dominates singly. This is what we conclude from this analysis. At this moment we are in
the process of applying Ford’s procedure to the next two periods demarcated in Figure
21. These will be reported at the conference in August. We have found some interesting
multiple loop dominance. Moreover, although initially we thought the second and third
interval were unitary in what could dominate the behavior during those times, we found
that we had to disaggregate both the second and third time periods into subintervals. For
example, in the second period, when deactivating Loop#3, we found that its second
derivative was positive from time 11.28 to 17.25, which matches the baseline during this
interval, and went negative from 17.25 to time18.125. Since the baseline derivative was
positive, Loop #8 became briefly dominant during this second subinterval.
In summary, we have first considered the behavior of the coflow structure in
isolation. However, usually one models other processes along with the coflow structure.
In this paper, we only added a draining process, but this complication lead to some
surprising and interesting behavior patterns. The use of Ford’s procedure for assessing
the dominance of the loop structure reinforces the conclusion that the dominance
relationships are indeed complex in this situation. For example, when we deactivated
Loop #1, we in effect moved from the complete model to the simpler coflow situation.
Y et we found that the second derivative did not change during that initial time interval.
This indicated that, under those circumstances, the drainage loop was not dominant by
itself.
REFERENCES
Coyle, R. G. (1977). Management System Dynamics. Chichester, United Kingdom, John
Wiley.
Ford, D. (1999). A behavioral approach to feedback loop dominance analysis. System
Dynamics Review 15, 3-36.
Hunter, J. E., Levine, R. L., & Sayers, S. (1984). Attitude change in hierarchies. In
Hunter, J., Danes J, and Cohen, J (eds.), Mathematical Models Of Attitude Change
and Group Dynamics, Academic Press, New Y ork, pp. 231-259.
Levine, R., Hovmand, P., Lounsbury, D. (1998). Squeezed between two masters: The
dynamics of an organization which interfaces both community and the academy.
Proceedings of the Sixteenth International Conference of the System Dynamics Society,
Quebec City, Canada, July, 1998.
Mojtahedzadeh, M T. (1996). A path taken: Computer-assisted heuristics for
understanding dynamic systems. Ph.D. thesis. Rockefeller College of Public A ffairs
and Policy , University at Albany, State University of New Y ork, Albany, NY
Richardson, G. P. (1995). Loop polarity, loop dominance, and the concept of dominant
polarity (1984). System Dynamics Review 11(1), 67-88.
FOOTNOTES
| We want to thank Professor Hines for send us a copy of his modification of the classical
coflow process
Perception_of_OutRt
Ch_In_OutgRt
TimeToP erceive
TrainingTime
Experienced_Wkrs Trainees
Induction_Rt
ransferRt
TimeToGo Av_Trainee_Att initialAVR ecruitAtt
o,
oc Tot what Total rainee_A
Att_OutRT TransferAT .
~— ErotleWkrA: EffodeTr_Att]| IncreaselnTraineeAtt
Wkr_#xperiences rainee_E
OS
eriences_in_Org
Figure 1. Typical example of how a coflow process is combined with other processes.
TimeToGo
xO)
Workers
Recruit\Rt
Av_Wkr_Attitude
ChinRecruitAttRt TransferOfAtt
IncominglAtt
Figure 2. Simplified version of the classical coflow process.
8
<t
&
s
g
°
a
Workers
Now s
Av_Wkr_ Attitude
=-
TraineelnitialAtt
~2-
Figure 3. The basic output of the classical coflow structure.
Workers
Recruit\Rt
TimeToGo
v_Wkr_Attitude
Total_Wkr_Att
Oo
ChinRecruitAttRt TransferOfAtt
IncomingAtt sw2
Spontan_DecinAtt
TimeToE rodeAtt
L—_7)
Figure 4. Simplified version of the classical coflow process and a draining
process on worker attitudes.
Total_Wkr_Att
¥
G
xs
5
=
= —Av_Wkt_Attitude
—) -2- RecruitsInitialAtt
Figure 5. Average attitudes of workers when attitudes
erode relatively quickly: TimeT oErodeA tt = 4 months.
1005
80+
2
———
1
604
——Av_Wkr_Attitude
404 — RecruitsinitialAtt
2
Figure 6. Av_Wkr_Att when TimeToErode_Att = 60 months.
Total_Wkr_Att
¥
¢
8
Ss
——Av_Wkr_ Attitude
-2- RecruitsinitialAtt
0 20 40 60 80 100
Time
Figure 7. Attitude change when initially Workers
are above equilibrium and Av_Wkr_ Att is below
equilibrium.
w
ry
s
Ss
Total_Wkr_Att
Workers
Av_Wkr Attitude
=
-2- RecruitsInitialAtt
0 20 40 60 80 100
Time
Figure 8. Results of a run where both initial Workers
and the Av_ Wkr- Att were above their respective
equilibrium points.
rey
S$
s
S
S
Total_Wkr_Att
——Av_WEr_Attitude
—7— Recruitsinitialatt
111-1
20 40 60 80 100
Time
Figure 9. Simulation run where initial Workers are below their asymptote
and Av_Wkr_ Att is above its Asymptote.
Belo
Initial
Number
Of
Workers
Abov
Initial Average Worker Attitude
Start Below Start Above
Figure 10. Patterns of average worker attitudes as a function
of initially being below or above maximums or minimums.
Draining Process Is Off
Workers
8
<
&
s
g
°
ec
5 am se |
=~ Av_Wkr_Attitude
40 —7— Recruitsinitialatt
Figure 11. Equilibrium run for the classical coflow process when
the constant for the material delay, Workers, was increased.
Draining Process Turned On At Time 10
8
qt
5 3
2
604
= Av_WEr_Attitude
40+ ,
—— RecruitsInitialatt
ii
ii,
0+ +_ +_ +
0 20 40 60 80 100
Time
Figure 12. Equilibrium run for the combined coflow and draining
structures when the constant for the material delay, Workers, was
increased at time 10.
= Av_Wkr_Attitude
—7~ TraineelnitialAtt
0 10 20 30 40 50 60
Time
=, —Av_Wekt_Attitude
=7—TraineelnitialAtt
0 10 20 30 40 50 60
Time
Figure 13. Equilibrium run for the combined coflow and draining
structures when TimeT oErode_Att is moved up or down by six
i oy’)
Workers
RecruitRt OutRate
TimeToChng TimeToGo
oO 1)
Av_Wrk_Att
Ch_|n_att Decay_of_Att
IncomingAtt TimeToE rode Swl
Figure 14. The Hines coflow model, where Av-Wrk_Att
is treated as a stock variable.
Below
Above
ree
1 Worker Attitude
= RecruitsAtt
Below
Ins
pleasent.
= Worker_Attitude
=- RecruitsAtt
Above
= Worker_Attitude
=> RecruitsAtt
he ay ee ee 2
=\- Worker_Attitude
—Recruitsatt
=i
== Worker_Attitude
-2- RecruitsAtt
1-11-11
1 Worker_Attitude
-2- RecruitsAtt
Hines Coflow With A Draining Process On Attitudes
Figure 15. Results of runs with and without draining
TimeToGo
TmeToTran§fr
O XO
rainee Cc) Workars
Input_of_Trainees ransferRt ecrease_In_Wkrs
Total/Workers
Prop_Nw_Trainees
TimeToChni
oO
IncgmingAtt
C) ofan
Ch_In_A
TimeToE rode
Figure 16. Disaggregated model. Alternative formulation of the classical
coflow process, where the influence of new people on worker attitudes is
proportional to their relative number of recruits.
=7—Trainees
4+ Total_Workers
y
G
o
£
ig
5
s
=
fo
2
a
p2—12—$12-
=> Worker_Att
—7~IncomingAtt
Figure 17. Baseline run for a disaggregated coflow model with no
draining process active.
Below
Set at 20
Initial
Numb.
Of
Workers
Above
Set at 55.
Initial Worker Attitude
Below Set at0 Above Set at 75
——Worker_Att = —Worker_Att
aa 171 == RecruitsAtt ~2- RecruitsAtt
0 5 1015 2025 0 5 1015 2025
Time Time
= Worker_Att => Worker_Att
2- RecruitsAtt II 1- RecruitsAtt
0 10 20 30 0 10 20 30
Time Time
Figure 18. The aggregated model output, when Trainees
are set at equilibrium and Workers and Worker Attitudes
are set either above or below maximum or minimum
Below Set atO
Initial Worker Attitude
Above Set at 75
22-222
Below
Set at 0
Initial
No. Of
Worke
Sa a aa
—1- Worker_att 1 Worker_Att
_ RecruitsAtt _ RecruitsAtt
“2: “2
Above
Set at 30 persinici
_Worker_Att Mpstieied
—— Worker_Att
5
—5—RecruitsAtt
2 —»—RecruitsAtt
2
Figure 19. The aggregated model output, when Workers
are set at equilibrium and Trainees and Worker Attitudes
Set either above or below maximum or minimum.
aa B3
Recruit Rt. ————> Workers OutRate
Av_Wkr_ Att
+
4
. . + i) B2 i *
ChinRecruitA ttRt
Total Wkr_ Att TransferO fA tt
B1 Nee
+
Spontan_DecInAt
Figure 20. Causal loop representation of the classical coflow structure,
combined with a draining process.
jas
S
ey
s
S
6
Av_Wkr_Attitude
2
s
Derv2AvAtt
Figure 21. Partition of the total time curve into intervals
containing either an exponential, logarithmic, or linear trend
according to the second derivative.
=
S
ey
s
S
&
Av_Wkr_Attitude
s
Av_Wkr_Attitude
Derv2AvAtt
Figure 22. The response of the system when a combination of
Loop #1 and #2 are simultaneously deactivated.
