Newton’s Laws of System Dynamics
John Hayward
School of Computing & Mathematics
University of South Wales
Pontypridd, CF37 1DL, UK
john. hayward @southwales.ac.uk
July 27, 2015
Abstract
This paper proposes an understanding of system dynamics models by analogy
with Newtonian mechanics. By considering the second derivative form of a model,
and extending the concept of feedback loop impact, it is shown that Newton’s three
laws of motion have their equivalent in system dynamics, and that the impacts of
the forces on the stocks are the measure of the force that determines stock behav-
ior. The concepts of mass, inertia, momentum, and friction are explored as to
their usefulness in understanding model beh . The Newtonian understand-
ing is applied to two standard system dynamics models, inventory-workforce and
economic long-wave, where their behavior is analyzed using force dominance on
the stocks and the laws of motion. The method, and conceptual understanding,
is commended for further exploration.
Key Words: System dynamics, force, Newton’s laws of motion, loop impact, feedback loop,
loop dominance
1 Introduction
A fundamental principle of system dynamics is that the structure of a system is responsible
for its behavior (Sterman, 2000, p. 28). Such structure is expressed in the stock/flow rela-
tionships and the feedback loops between the model variables. The latter are particularly
important as they represent endogeneity in the system, and help explain complex behavior
in the variables through the change in dominance of different types of feedback.
In order to quantify feedback loop dominance, numerous methods have been developed
(Kampmann & Oliva, 2009; Duggan & Oliva, 2013). Broadly, the methods can be divided
into two categories: those that relate loop gains to system behavior, expressed in eigenvalues
and eigenvectors, through elasticity analysis (e.g. Forrester, 1982; Goncalves, 2009; Kamp-
mann, 2012); and those that relate pathway connections to variable behavior expressed
in their change and graphical curvature over time (e.g. Ford, 1999; Mojtahedzadeh et.al.,
2004; Hayward, 2012; Hayward & Boswell, 2014). In each method the main structural el-
ement under consideration is the feedback loop, whose changes in dominance become the
chief way of explaining variable behavior. The two categories of methods are related: the
loop gain is the product of the link, or pathway, gains between adjacent elements in the loop
(Kampmann, 2012); also the loop gain is the product of the impacts between adjacent stocks
in the loop (Hayward & Boswell, 2014), that is, the pathway participations (Mojtahedzadeh
et.al., 2004). Whatever method is used, a loop with n stocks has n degrees of freedom, and
thus requires n numbers to fully describe its effect, whether they are metrics, eigenvalues, or
impacts. If n > 1 the loop gain is insufficient on its own to capture the loop dynamics.
System dynamics is not the only modeling methodology that uses structure to explain be-
havior; such linkage between the expression of a model and its results is the essence of all
modeling whether it is, for example, agent-based, chaos theory, or Newtonian mechanics.
Where such methodologies differ from system dynamics is the varied way in which “struc-
ture” is expressed in, for example, transition diagrams, abstract mathematical relationships,
or physical laws. One of the attractions of system dynamics is that structure is refined to
visual expressions, causal loop diagrams and stock/flow diagrams, which cut through the
model’s complexity to provide an intuitive framework from which model behavior can be
explained. Nevertheless it may be possible to provide additional understanding of a system
dynamics model by applying a different approach to model structure. This paper explores
such an approach.
The starting point for such an alternative understanding of structure is to note that the
link between structure and behavior in the pathway participation metric (PPM) method
(Mojtahedzadeh et.al., 2004), and the loop impact method (Hayward & Boswell, 2014), is
an equation for the second derivative of stock variables in terms of other variables. In each
method the model equations are differentiated, placing them in second derivative form, thus
focusing on the curvature in variable behavior, whilst retaining the causal structure of the
model in each term that contributes to that curvature. In essence the understanding is close
to that of Newtonian mechanics where acceleration is determined by various forces, here
identified with different loops. However it is clear from the loop impact method that the
“forces” on a given stock are less connected with a loop, but more with the effect of adjacent
stocks which may be part of a loop, or may be exogenous. Indeed the loop impact method,
which may be better termed the stock impact method, has made steps towards interpreting
model behavior in Newtonian mechanics terms, even if this was not explicitly made clear
(Hayward & Boswell, 2014).
The purpose of this present paper is to use concepts from Newtonian mechanics to understand
system dynamics models!. Firstly, the concept of loop impact, as introduced by Hayward
& Boswell (2014), is investigated further by discussing the notion of the “impact” of a
cause on motion generally. The concept is reinterpreted in as the impact of a force on a
system dynamics to illustrate Newtonian m«
r this paper attempts the oppo:
Newtonian terms.
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tem dynamics models and behavior in
to understand
motion, with the concepts of force, momentum, mass, inertia and friction explored as to
their usefulness in understanding the behavior of any system dynamics model. Finally the
ideas presented are applied to two existing system dynamics models to evaluate their use.
Put informally this paper addresses the question: If Sir Isaac Newton had tried to understand
a system dynamics model, without the benefit of Prof. Jay Forrester’s insight, how would
he have explained its behaviour?
2 The Concept of “Impact”
Initial Considerations
The proposal is to define the concept of the impact of the force exerted by one stock on
the motion of another stock, in particular its acceleration. In order to help understand the
notion of impact on motion, an example is given of how the word “impact” is used with
regard to motion generally.
The following situation happened to the author at a recent International System Dynamics
Conference. For readability the narrative is given in the first person.
During the conference lunch break I wished to attend the presentation by one
of the software developers. Realizing I was late for the meeting I rushed to the
lecture theater, which turned out to be a massive tiered room, seating hundreds.
The presentation had already started and the presenter was explaining his wares
to the solitary person in the audience, who sat somewhere in the middle of the
room. My entrance was greeted with a “Wow! A hundred percent increase. Do
come in”. It is clear my entrance had impact on the relieved presenter. I took a
seat, and the presentation carried on.
Two minutes later another person arrived. The presenter again noted his ap-
pearance and asked him to sit down, but he said it with a little less emotion than
my entrance. Two minutes later yet another person arrived, now an audience of
three became four. The presenter’s comments were briefer still. A new arrival
continued roughly every two minutes until we were up to seven people, by which
stage the presenter had stopped noting their appearance. It struck me at the time
that although the addition in numbers in the room was uniform, the impact of the
change on the presenter became smaller the larger the audience became.
What the presenter appeared to be doing was reacting to the change in numbers in the room
by comparison with the number of people already in the room. Had there been a hundred
people at the beginning it is doubtful if any of the new arrivals would have been noticed.
Although the arrival rate would have been the same, one every two minutes, the impact of
the change would be far less as the constant change is proportionally less. Thus the impact
of change can be defined as the ratio of the change to the initial base value, that is:
dx/dt
impact of change = de /dt
@
where x is the variable being changed. This is a ratio measure of change, effectively the
derivative of the logarithm of the variable d(In x) /dt.
Impact of Force on Motion
The previous definition of impact concerned the way a cause affects a change in the variable,
impact on change, that is “velocity”. Of more relevance to the loop impact method is the
impact a force has on the “acceleration” of the variable. Thus the following definition is
given for the impact of a force F on the changes in a variable x:
d?x/dt?
Iz, 4 —_ 1
Pe “da /dt (1)
that is, the ratio of the acceleration to the velocity of x, the derivative of the logarithm of
the velocity. Consider the following illustration:
A ball is thrown upwards into the air with initial velocity u, subject to constant gravita-
tional force of acceleration —g. Air resistance is assumed negligible. Let 2 be the vertical
displacement of the ball from the ground, then d?x/dt? = —g. Thus dx/dt = u — gt, where
t is time, and x = ut — it”, assuming the ball starts at z = 0. Thus, from (1), the impact
of gravity on the ball’s motion is given by:
g
Lge gt—u
Although gravity is constant, its impact on the motion of the ball is not constant. The
impact is greatest as the ball is slowing down, near the top of its motion, because gravity is
inducing greater curvature in the graph of x against time, figure 1. The impact is infinite
while the ball is temporarily at rest, t = u/g, and then changes polarity as the ball starts
falling. The sign of the impact indicates whether the force is reinforcing the motion, positive,
or resisting it, negative.
Height x
nooe Impact ftorce J
Impact of force 1
Time t
Fig. 1: Impact of force of gravity on vertical motion of a ball.
Physically the impact of a force is measuring the extent to which the force can change the
motion of the object (indicated by its position variable), given that it is already in motion.
2Care is needed with the mathematical sign in the derivative of the logarithm of a negative velocity.
For the thrown ball, when it is moving very fast, the constant gravitational force is inducing
only a small change in the motion of the ball, whereas when the ball is moving slowly the
change in its motion due to gravity is much larger, in percentage terms.
For example, if the ball is moving at 50 m/s, then the change of velocity in a tenth of a
second is 2%. For a velocity of 10 m/s, the change is 10%, and for 2 m/s the change is 50%.
Thus a constant force has more impact on a slow moving object compared with a fast one,
as the change in the position of the object is more noticeable.
The impact of a force on motion can be viewed as a measure of curvature, similar to the
radius of curvature used in analytical geometry. However, whereas the latter is constant when
the curve is circular, the impact is constant when the curve is exponential. Consider a force
that induces exponential acceleration, d?x/dt? = e”, for a constant a. Then da/dt = e/a,
and the impact of the force on x is Ip,., a, a constant. The impact is positive if
a > 0 reinforcing the motion, and negative if a < 0, resisting the motion. Impact is a
more natural method of measuring curvature for dynamical motion, where linear processes
generate exponential behavior.
Thus the impact of a force, in the sense defined here, is the impact on the motion of an
object, units [T~+]. It should not be confused with the impact an object makes when it
collides with another object, which is a force applied over a very short period of time, units
{MLT~?], also called a shock.
Impact of a Stock on a Stock
Consider a stock y influencing a stock x, figure 2, equation (2):
dx/at = fly) @)
Fig. 2: One stock, x, influenced by another, y.
Following Hayward & Boswell (2014), and (1), then the impact of y on x is defined by:
Pa/dt? y
ne Se = rw (3)
where the underline subscript indicates the causal pathway of the impact. This definition is
referred to as the pathway participation between y and x by Mojtahedzadeh et.al. (2004),
where f’(y) is the link gain (Kampmann, 2012). Although (3) is called loop impact in
Hayward & Boswell (2014), it can be seen that the concept of impact is independent of the
whether the link between the stocks is part of a feedback loop. Thus (3) will be called Stock
Impact as it represents the effect on the acceleration of x due to changes in y.
Of course if a system has feedback loops then the stock impact will represent the impact
of the loop on the stocks. Consider the general linear system with two stocks, figure 3,
equations (4-5):
& = ax+by (4)
cx + dy (5)
ra
|
flow = ax
flow = ox flow = dy
Fig. 3: Generic linear 2 stock system.
with constants a,b,c,d. There are two first order feedback loops, L;, Ly with gains G,; = a
and G2 = d; and one second order loop, L3 with gain G; = bc. Using definition (3), the
stock impacts are given by:
laa(Ls) = a (6)
Iy(la) = d (
Iyella) = “EMD (8)
Layla) = Soot) (9)
where the loop identifier has been added to the notation for impacts in order to indicate
the loop in which the pathway is embedded*. The stock impacts I,,(L1) and I,,(L2) are
the direct impact of a stock on itself, via first order feedback loops, L;, L2, and are equal
to the loop gains. Thus it is natural to refer to them as loop impacts. Because the stock
impacts [,,(Z3) and I,,(L3) are part of a loop, L3, they could also be called loop impacts,
provided it is clear that they are the impacts of the loop on each stock, which in general
will not be equal. Although neither are equal to the loop gain, their product does equal the
gain, I,,(L3)Iy2(L3) = G3 = be; a special case of the loop impact product result (Hayward
& Boswell, 2014, appendix C), which shows that gain is the product of impacts in the loop
regardless of model complexity and non-linearity. Note the polarity of the impacts (8-9)
may flip sign, but do so together such that the polarity of the loop gain G3 = bc is preserved.
However the concept of impact is more general than that of feedback loops. Let ¢ = 0 in
(4-5) figure 3, which breaks the loop L3 with I, = 0, where the loop indentify, L3 must now
3Further examples of computing impacts analytically are found in Hayward & Boswell (2014) and Hayward
(2012).
be dropped. y is now an exogenous influence on x, with impact I, = bdy/(ax + by). This
can no longer be referred to as loop impact as there is no loop, thus the term stock impact
is preferred in this case. This impact can also flip polarity, although this time there is no
loop gain to preserve.
With the concept of stock impact established, and with its definition, (3), the same as that
of force impact, (1), the question is now asked as to whether the influence of one stock
on another, as in figure 2, constitutes a force in the Newtonian sense. Differentiating (2)
to obtain the acceleration of x gives: d?x/dt? = f’(y)y. Thus it is clear that y does not
quantify a “force” affecting x. Rather it is the time derivative of the right hand side that
is analogous to the Newtonian concept of force. Nevertheless stock impact is playing the
same role as the impact of a force introduced earlier. Thus it can be stated that if a stock
y affects a stock x (through the latter’s flows), then y exerts a force on x, whose impact is
measured the same way as a force, even though the force is not quantified by the value of y.
These concepts will be used to draw out the Newtonian analogy in system dynamics in the
next section.
It is noted that one significant difference between system dynamics, and Newtonian me-
chanics, is that in system dynamics stock variables can be in completely different units. In
mechanics the state variables are position coordinates. always in the same units. By contrast
in the second order model of figure 3, equations (4-5), there is no reason why x and y should
be in the same units, which makes it difficult to compare the effects of forces on x with
forces on y. However the impact of a force always has units of “per unit time”, and thus are
independent of the units of the stock imparting the force and the stock effected by the force.
Thus forces from variables with any units can be compared through their impacts.
Before progressing it should be noted that the concept of measuring the impact of a variable
by comparing the ratio of its second and first derivatives is not new, or confined to system
dynamics. In functional data analysis this ratio is used as a growth factor in the analysis of
time series data, where it is called the relative acceleration (Ramsay & Silverman, 2002). The
ratio also occurs in the concept of financial risk aversion, where it is measures the curvature
of a utility in an way that is invariant under certain transformations.(Pratt, 1964).
3 “Newton’s” Laws of System Dynamics
In order investigate a Newtonian interpretation of system dynamics, some conventions will
be assumed. Firstly, if a variable is stated as a stock then it is assumed to have flows, even if
those flows do not explicitly appear in a model diagram. Although it is possible to include a
stock with no flows in system dynamics software, in a finished model such a stock is constant,
effectively a parameter/converter. The use of a stock will always indicate a variable which
can be potentially changed through its flows.
Secondly, a flow with no connecting element is assumed to be constant in time, explicitly
and implicitly, even though some software packages allow it to be a variable. If the intention
is for a flow to vary over time then a connecting element should be used to indicate this.
Any such variable element connected to a flow is deemed to originate in a force.
In what follows the stocks labeled by z, y etc. represent any system dynamics stock, such as
population number, debt, inventory, motivation, burnout, etc. The three laws which Newton
developed for mechanics are now widened to apply to any type of stock variable, and could
be named: The three laws of stock dynamics. The section is developed abstractly with no
particular application in mind.
Law 1 — Uniform Motion
A stock will remain level or change uniformly unless acted upon by a force.
This is the equivalent of Newton’s first law of motion, applied to a single stock 2, and
represents the system in figure 4, with equation « = k, where k is the net flow. The law
applies regardless of the number of flows. The stock either stays at “rest” or in motion at
the same “speed”, unaffected by any force to either slow it down or speed it up. The graph
of x against time will be linear, for example figure 5; the lack of curvature indicating no
force. At this stage no concept of mass or momentum is required.
oS ¢
L* J
Fig. 4: Model of stock with no applied forces.
Fig. 5: Behavior of stock with no applied forces and four different net flows.
For the model in figure 4 to give unique behavior, both its initial value, xo, and its initial
net flow, that is its initial “velocity” io, are required, as in Newtonian mechanics.
Law 2 ~— Change of Motion Due to Force
The acceleration of a stock produced by a net force is in proportion to that force
and the inverse “mass” of the stock.
This is similar to one of the ways of Newton’s second law is expressed, where the acceleration
is proportional to the net force, and inversely proportional to the mass (Physics Classroom,
2015). In the system dynamics case the equivalent of the mass is inverse of the sensitivity
of the stock to the force, the converter a = 1/m, in figure 6. In the system dynamics model,
the influence of the force on the stock appears as a converter, y, giving « = ay. The measure
of the force is dy/dt as the equation for the acceleration is # = ay = y/m, with the impact
of the force on the stock 2 the same as the impact of y, as noted in section 2. This equation
is the equivalent of the mathematical expression of Newton’s law of motion:
dy 1)\ x Pr
PS = (=) = =n 1
dt (+) de ae (10)
a pemass/inertia of x m= I/a
reaction of = gy
xtoy a wrt y
: pp force measure F = dy/dt
y ofy
Fig. 6: Stock with applied force.
Consider the effect of a step change in y on x, at t = 2, for three different values of sensitivity
a. The lower values of a means the force has less effect on the stock x, figure 7 (a), which
can be interpreted as a stock with greater inertial resistance to the force. That is, a force
has less effect on a higher mass stock. The measure of the force, y, is a pulse, figure 7 (b),
as is its impact %/+. Thus a converter with a step change represents an impulsive force’.
60
14 bb]
*a=03 LowMass
12 a=02 Converter y 40
1 a=0. Hignwass ef Pn Foe F
5 | 20
5 o
en ne 8
2 os
8
oa
o2F
o
o 2 4 6 8 10 0 2 4 6 8 10
Time t Time t
Fig. 7: (a) Effect of pulse force on stock x showing the inertial effect of high mas
a. (b) Step value in converter y is equivalent to pulse force y = dy/dt
It should be noted that the converter y is playing the part of momentum. The three runs for
x in figure 7 (a) represent a system with the same momentum change with differing masses.
Thus an alternative version of the second law can be given, that is the rate of change of
stock momentum is proportional to the applied force. This interpretation will become more
helpful when the applied force is due to an external stock/flow system.
‘The force is a delta function but appears as a finite spike in figure 7 (b) due to the fixed step length
used in the numerical integration.
Strategic Decision Support for Startup Company Using System
Dynamics: an online startup company’s case
Yutaka TAKAHASHI and Nobuhide TANAKA*
School of Commerce, Senshu University
Faculty of Economics, Gakushuin University*
2-1-1, Higashimita, Tama, Kawasaki, Kanagawa, Japan
+81-44-900-7988
takahasi@isc.senshu-u.ac.jp
Abstract
Startup companies need to decide when, what, and how to utilize their resources in
particular business activities. The necessity of these kinds of decisions is similar to that
of long-standing companies. However, there is a significant difference; most startup
companies have very limited resources including money, compared to long standing
companies. In this situation, a computer simulation reflecting each individual company
is valuable. Examining simulation results can be helpful to real company owners and
managers. This paper explains a process of introducing system dynamics simulation into
startup businesses decision-making and shows a case of use of system dynamics model in
a startup company selling original cosmetics online.
Keyword: startup, strategy, decision support, simulation, a cosmetic company.
1. Introduction
Many startup companies have limited financial and other resources. Naturally, they need
to decide when, what, and how to utilize their resources in particular business activities.
The necessity of these kinds of decisions is similar to that of long standing companies.
However, there is a significant difference. The fact that most of the startup companies
have very limited resources means that they cannot use their resources for multiple
activities simultaneously. This naturally causes a situation in which startup companies
choose only one choice that they really do. This study’s purpose is to show how system
dynamics models and simulations help decision makers in startup companies with limited
Forces Due to Stocks
The most common way the concept of force will be used in a system dynamics model is when
changes in a stock affect other stock/flow systems. For example, let a stock y > 0 exert a
constant force on two other stocks 21,22 with different sensitivities to y, that is different
masses with respect to y, figure 8. The equations are #; = ayy for i = 1,2 and y = k, where
k is a constant. Let a; = 0.2 and a2 = 0.4, making x the least sensitive of the two stocks
to the common force. For a constant force k = —4, and yo = 20, both stocks are brought to
rest by t = 5, with wy reaching the higher value, figure 9. The common force due to y has
had more effect on 22 due to it having less inertia than x, thus less resistance to change. For
x2 to have reached 10, like x1, before stopping, a force of twice the magnitude would have
been required.
i
reaction of x2 toy =a2y
A jo yoedwi,
Time t
Fig. 9: Effect of stock y, representing a constant force, on other stocks 1, x2
The impact of the force due to y on each of the stocks #1, #2 is the same, figure 9, as it is a
ratio measure, #;/%; = y/y = —k/(yo — kt) for t < 5. The impact of y is independent of the
sensitivities of 2,72 and increases in magnitude as the force has more effect on the motion
of the stocks. For t > 5 the impact is zero as both stocks are at rest, due to y having reached
zero. y plays the role of the momentum of the stocks 2;.
10
Thus the interpretation of the the second law of motion in system dynamics is that a stock S
that affects other stocks in the system represents the common momentum of the other stocks,
with its flow being their common force. The differing effects S has on the other stocks is due
to those stocks having a different inertial response, their “masses”, the inverse of the rate
multipliers; the “lighter” stocks have more response. However the impact of S, as defined
by Hayward & Boswell (2014) is the same on each stock, as for each stock its acceleration
is measured next to its own rate of change. Thus the impact of one stock on another can
be seen as a scale-free measure of the force of one stock on another. This interpretation is
independent of the stock connections being part of a loop, thus independent of the concept
of feedback.
Forces Due to Feedback Loops
When feedback is involved then a stock in a loop will be affected by a force, and be a force
itself. Consider the second order linear system (4-5), figure 3. In the second order loop Ls,
stock « exerts a force on y and y exerts a force back on x. Set a = d = 0 so that Ls is
the only loop, with gain G3 = bc. Following the Newtonian analogy, the parameters b,c are
the inverse of the masses of each stock with respect to the other. The gain is therefore the
inverse of the product of the masses and thus represents the inertial resistance of the loop.
If Gs < 0 the system oscillates indefinitely with angular frequency equal to the square root
of the |G3|. Thus a higher “mass” loop will be more sluggish and oscillate slower.
In general the linear model has two first order loops, one on each stock. Set b = 0 in (4)
to decouple the stock from y, & = ax. If a < 0 the loop is balancing, a draining process,
which can also be seen as a form of frictional resistance # = az. Thus a first order loop
can be interpreted as the force of a stock on itself. If a > 0 the loop is reinforcing, a
compounding process, whose nearest mechanical analogy is the snowball gathering material
and thus accelerating itself.
Thus each stock in the linear system is subject to two forces, whose balance will determine
its behavior, as measured by their impacts. Because all the forces are part of loops these
impacts can be called loop impacts on the understanding they are scale-free measures of
the forces on each stock in those loops. The condition for stability depends on the gains:
G, + Gy < 0 and G,G_ > G3 (see appendix). Thus for stability at least one the first order
loops must be balancing.
Consider a system where Ly is reinforcing and the other two loops balancing, such that
is stable. The second order loop balancing, L3, could be seen as an attempt to
stabilize the system, for a given set of parameters. The loop impacts on each stock (6-9) are
computed in the model and their dominance determined, figure 10(a).
Stock 2 starts with the “frictional” force, due to L, dominating, transitioning to L3 at
t = 12.5 and back to L; at t = 25.5, figure 10(a). The frictional force is a factor in
bringing x to equilibrium, whereas in the middle period the impact of L3 is destabilizing as
it has changed polarity to positive (reinforcing) effect on x, figure 10(a), causing x to start
accelerating until its impact is less than the magnitude L, again>.
5A second order balancing loop will always have one of its impacts positive, with the other negative to
maintain a negative gain. Thus one stock in the loop is always being ac ted and must have
11
1 T T 2 7
: H [al : [b]
oa “3 } ay La mona: uo} l2 eames |
gems pot —s 15 : x -
y a --
o6}/“ Poo. wee
5 0:
13
oO 10 20 30 40 co) 10 20 30 40
Time t Time t
Fig. 10: Second Order Linear System (4-5) with a = —0.25,b = —0.1,d = 0.03, showing the change
of force/loop dominance on each stock for the same loop structure: (a) Stable, with c = 0.15; (b)
unstable, with c = 0.01.
Initially stock y has L3 dominating, and with a change of polarity changes the direction of y
and causes it to accelerate downwards, t = 7.2. There is brief period where the reinforcing
loop Ly dominates, causing y to accelerate more, from t = 13.4. However L3 ultimately
dominates, from ¢ = 17.7, and because it has changed polarity again to have negative
(balancing) effect on y, and y is brought to stability.
That the loop L3 is balancing and of constant gain is not a good indication of stock behavior
as the impact of its force on each of the stocks is variable and changes polarity. There is
a brief period where L3 is dominant on both stocks, t = 17.7 — 25.5, and could be said to
dominate the system; but it is the balance of forces on each stock that is key to their behavior.
Stability comes from the final dominance being a force with negative impact. Thus stability
has been achieved through sufficient friction being applied to x, which is slowed enough to
control y.
The system can be destabilized by reducing c to 0.01. The loop structure remains identical,
but now G,Gy < G3 and the system is unstable with « + —oo, y — oo. For y, the force
due to x, via L3 is unable to regain dominance over that of L2, the destabilizing force. For
x, the loop L; is unable to regain dominance over Ls, as is there is insufficient friction to
counteract the force from y. The reduction in the absolute value of the gain G3 can be
interpreted as the loop L3, and thus the system, having more inertia, that is more mass, and
thus less effective in controlling the destabilizing influence of Lo.
If instead the the gain of L3 were increased, the system remains stable, but will oscillate
first as (Gy — Gy)? < —4G3 (see appendix). This could be interpreted as a less massive
system being over responsive to the corrective effects of L3 as the frictional dissipation L, is
relatively less effective.
Thus the loop impact method of Hayward & Boswell (2014) can also be understood in terms
of force dominance on each stock, with the use of Newtonian terms such as mass, inertia and
friction providing an alternative explanation of behavior.
balancing force of greater magnitude if stability is to be achieved.
Law 3 — Equal and Opposite Forces
The force on a stock through a flow has an equal and opposite force on a stock
at the other end of the flow.
For example, consider a stock x with a draining process flowing into stock y, figure 11 with
equations ¢ = —ax, y = ax. Not only is the flow conserved, but the forces are equal and
opposite in effect, # = —y. Their impacts are identical Ip2(B) = Iy(B) = —a, as the sign
of the force is also determined by the sign of the flow on each stock. As such the behavior
of x and y mirror each other, both decelerating to stability, assuming x,y > 0, figure 12.
Examples of this mirroring effect of Newton’s third law can be seen in the epidemic SIR
model (Hayward, 2012).
Fig. 11: The force due to loop B has an equal and opposite effect on each stock.
> / =
xe / any
a NM
O4r 7
/
/
/
|
V
10
Time t
Fig. 12: Equal and opposite forces on stocks in a conserved flow, with same impact on behavior.
4 Application to Inventory-Workforce Model
To help understand how Newtonian concepts can be applied to a system dynamics model,
consider a standard two state inventory-workforce model subject to an exogenous demand on
sales, figure 13 (Ventana, 2011). The equations are given on the system dynamics diagram.
There are two forces on the inventory, one from demand, and one directly from workforce
connected with loop B2. There are three forces on the workforce: the frictional self force
connected with B1; one from the inventory via the inventory adjustment control, connected
with B2; and one from demand via sales which is not connected with a feedback loop; an
external force in the Newtonian sense.
Let the system start in equilibrium with the inventory at 100, the workforce at 20, and a
demand of 10. Let demand rise steadily from 10 to 12 between t = 5 and t = 10. The
13
ff a(t)
S Inventory laveinert |
production sales rate s = d(t)
rate r= pW
¢ desired inventory 10 “inventory
shortfall i
production required
for sales fis)
fis) isa
smoothing
function
adjustment time tl
* seiiapitnn sequined to
productivity per person adjust inventosyra
units per person month p
total production units
3 DS arose workforce <— per month u=atfis)
staff needed to adjust ore
inventory S=Wo-w <- — WO= wp
1)
hire fire rate
=Si2 workforce
adjustment time t2
Fig. 13: Two state inventory-workforce model. (Ventana, 2011)
force exerted by the demand only applies during this period, and is the slope of the line,
figure 14(a). Thus demand has an impact on the inventory from ¢t = 5, however at t = 9.1
the restoring force from the workforce, part of loop B2, exceeds the demand impact and
dominates the rest of the inventory’s return to equilibrium.
30
= Reduce workforce adjustment ime [b]
Reduce inventory adjustmenttime
25 —— Base run
B20
8 :
5
8 20 g
8 = Equibrium
5
515 3
z 4
iS os %
grr 8
—— Worrtorce (person) & Be
5 ~ Demand (units per month)
oe Lerf—e2 Los
uilbrium
0 0 =
0 10 20 30 40 0 10 20 30 40
Time t Time t
Fig. 14: Results of inventory-workforce model. D indicates dominance of exogeneous demand d(t).
(a) Base run for workforce with transitions of force/loop impacts, t) = t2 = 2.5, p = 0.5, Ip = 100.
(b) Force/loop dominance on workforce, base run compared with reducing inventory and workforce
adjustment times, t; = 1 and tz = 1 respectively.
The more involved dynamics takes place on the workforce. The impacts of the three forces
on the workforce can be computed numerically, using the method of Hayward & Boswell
(2014), and the transitions indicated on the graph, figure 14(a). Initially, from t = 5, B2
is dominant as the workforce increases to match the production required by the inventory.
However there is also a small but increasing effect from the exogenous demand. By t = 9.2 it
takes both B2 and the demand forces, indicated by D, to dominate the behavior of workforce,
thus showing the corrective action from sales is helping to adjust the the desired workforce.
This continues until t = 10.9 when the frictional force of the workforce, due to the stock
14
adjustment loop B1 starts to slow its growth. That the demand continues to have an impact
on the workforce after it has stop changing, t = 10, is due to the delays in production required
for sales.
At t = 13.5 the corrective force B2 again dominates causing the workforce to peak, and
accelerate downwards. The impacts of B2 on the two stocks has changed polarity with the
change in direction. The remainder of the motion is repeated change between B2 and B1.
The latter is a dissipative force damping the oscillations caused by B2.
Let the scenario in figure 14(a) be the base run, and consider the differing effects of reducing
the inventory adjustment time t; and workforce adjustment time t2. Using the Newtonian
analogy, tz controls the friction, and t; effects the inertia of the second order loop Bp. Figure
14(b), compares the effects of adjustment time reduction on the force/loop impacts on the
stock Workforce using the loop picker algorithm of Hayward & Boswell (2014).
Reducing inventory adjustment time brings in the first occurrence of B1 dominance earlier,
compared with the base run. However its appearance is too brief to give sufficient correction
as the impact of B2 has increased. This has happened because the inventory adjustment
time t, contributes to the “mass” associated with the loop B2. It now has less inertia and is
thus harder to control. The resulting effect is a greater number of oscillations compared with
the base run, figure 14(b) middle run, even though it reaches equilibrium slightly faster®.
Reducing the workforce adjustment time, figure 14(b) top run, also has the first occurrence
of B1 starting earlier than the base run. However this has been achieved by reducing the
duration of B2 dominance, as B1 is now stronger compared with B2. The weakness of B2
can be seen in the first period of its dominance, which needs a longer period of assistance from
the demand, B2 D, to overcome B1. This results in fewer oscillations and thus equilibrium
is reached faster. In Newtonian terms this can be interpreted as increasing friction at the
expense of the corrective and exogenous forces.
5 Notational Refinement
Further insight into the Newtonian understanding of the previous model can be obtained
by computing the impacts of the three forces analytically, following the method of Hayward
& Boswell (2014). This method requires the system dynamics model to be reduced to
differential equation form and the definition of stock impact (3), to be applied to each stock
on the right hand side of the each equation according to causal pathway. Unfortunately
the differential equation form of the system dynamics model does not retain the pathway
information.
For example the differential equations for the inventory-workforce model, figure 13, are:
I = pW-d(t) (11)
There is no indication in this model that the W in equation (13) proceeds via a different
5A threshold is required to stop registering forces when they get very small. This defines the numerical
approximation to equilibrium.
pathway to the W in equation (14). (Likewise the pathway from the exogenous d(t).) In this
model the issue is resolved by the Ws being in different equations, i.e. the different target
stocks, but in models where there are multiple pathways between the same stocks there will
be no such clear distinction.
Thus a notation is proposed so that the differential equations are written in a form as to
retain the causal network information. Using the inventory-workforce model as an example
consider the causals link from Inventory stock I to the auxiliary variable production required
to adjust inventory a, via i, the inventory shortfall, figure 13. The equation can enhanced
with the subscript of any intermediary variables, As i = Jo—J and a = i/t1, then substituting,
a = (Io—Ji)/t1, where the underline on the subscript indicates it is used for a causal pathway.
Thus the formula shows J connects to a via i.
The next auxiliary variable in the pathway is total production units per month, u = a+ f(s).
Using the pathway subscript notation this is written uw = (Ip — Jia)/ti + f(s), where both
intermediary variables have been subscripted on the causal stock J. Thus the formula now
shows J connects to u via i and a. The general form of this process is given in appendix B.
Continuing this process to the flow h of the target stock W, will give
1 (4 = vats)
ha — (2s
top th
with the underlined subscript on J preserving the entire causal pathway from cause I to
target W through its flow h. Thus using the equations of figure 13 the differential equations
in network form of the inventory-workforce model are:
I = pW,-d,(t) (13)
. 1 [1 (to — Liauwosn
we = EE (At + rapanasa(t))) ~ Wa (14)
to |p ty ——
The subscripts of J, d(t) and W in equation (14) clarify which sections of a causal chain
they have in common, and where they differ. Thus these “networked” differential equations
preserve the causal topology of the system dynamics model figure 13.
The impacts of the stocks, and exogenous demand, on a target stock, can be derived by
differentiating along the appropriate pathway. For example, following (3), the impact of I
on W though the inventory shortfall pathway is defined as the partial derivative connected
with the W of that network.
yia weit
W — Oliauwosh W
ow
a
TiauWoShW = Dy
(15)
iauWoSh
The double vertical line in (15) indicates the causal pathway derivative. A general definition
is given in appendix B.
Thus the three impacts on stock W are:
pW = alt)
Tiauwosnw(B2) me Ih —T+tif(@ — tipw (16)
tif’ (dd(t)
Tas fuWoShw ho +hf(d —hpWw (17)
1
Twsnw(Bl)) = s (18)
by
16
Because the paths are not shared by loops the subscripts can be omitted without confusion,
and loop identifiers added, I(B2), Iu, 1(B1) respectively’.
Returning to the analysis of the inventory-workforce model, both I(B2) and Iy are indepen-
dent of the workforce adjustment time, t2, thus it is possible to increase the frictional force
without any direct effect on these impacts, just the indirect effect through W; hence the
success of that policy on reducing oscillations. Reducing t, increases I(B2), the reduction
in inertia in this loop referred to earlier. I4/I(B2) x t; showing that reducing inventory
adjustment time, weakens the effect of the target of sales compared with production. The
resulting higher relative impact of B2 on W increases the number of oscillations.
6 Application: Economic Long-Wave Model
For a more challenging application of the Newtonian view of system dynamics consider
Sterman’s (1985) economic long-wave model. The equations and parameter values are taken
from Kampmann (2012) and it is assumed the reader has some familiarity with the model,
which has become one of the benchmarks for analytical methods. The analysis given here is
just an initial exploration as to whether there are Newtonian aspects to the model’s behavior,
and as such, the investigation concentrates on the mathematical interpretation alone, rather
than the conceptual meaning of the model.
The stock/flow diagram of Kampmann (2012) is given in figure 15, where his equations of
table 2 are embedded in the diagram®. Although the model has only 3 stock variables, the
connections are complex with 36 loops, of which 16 are independent (Kampmann, 2012).
More than one independent loop set can be chosen because many of the loops share edges
in parts of their structure.
TEven if the subscripts had been retained some notational simplification would be possible by combining
adjacent auxiliaries ranching or merging of pathways.
8A typographi preciation equation of Kampmann’s (2012) paper is corrected using his
accompanying Vensim model, d = K/r; denominator is 7 rather than 4.
17
oS
a= Gp SS
\
\
Os = gd |
-_* iy
%
= Kappa \
Z oa Ste \
SE Stere) ‘
oO
(k*-K)/tauk +
(o*S)taus \
a |
tauk
ik |
Ob = Os
7a .
= mao
ay t pat
delia
aa
‘st = dBhx
~~
readability.
nee
Fig. 15: Economic long-wave model, with model equations given on each stock, flow and auxiliary.
f and g are graphical converters, expressed in functional form. Some connectors are dashed for
Following the method of the previous section the model is written as three networked differ-
ential equations, using figure 15 and (Kampmann, 2012, table 2):
dk SaK ca f [
SS a Mea 19
dt KBa T (19)
r Kds* o* go, oO S
——_TrBawjesrorgo] 7 9etae
> KT Bet KX o* go—OT Kot go Kezst o* go S|
dS Kay | Kaorgo + SS +
dt ~~ 7 3 Kago
7 KBys ~
SaKexa fl :
- 20
KBa (20)
ete ali
Z eT Bye prot go-oT Kot go. Kezstotao!| SR apaanaego” ~~
dB Kao | Kaorgo + SSE +
dt o> Kago
(21)
From a Newtonian point of view, the number of forces on each stock due to other stocks
is unique. For example K has 6 forces, indicated by the 6 different pathway subscripted
variables in (19). The use of subscripted variables prevents any algebraic reduction taking
place through factoring or cancellation of stocks, unless they have come via the same pathway,
Thus the networked differential equations (19-21) preserve the network topology of the
system dynamics model, as explicitly expressed in the stock/flow diagram, and implicit in
the model equations (Kampmann, 2012, table 2). Stocks are denoted by upper case letters
to make them easier to identify. Note that the forces from Kas*o-go aNd Kers*o*go identically
oppose each other, as the K’s would cancel in the fraction had they not been annotated by
the path, (equations 20-21, last term of numerator in g’s argument).
Thus equation (19) shows that K has 6 forces. 3 are self forces: via the flow d, (Ka); via x
directly, (era); and via c through the graphical function f, (K.y2a). These are first order
loops. There is one force from S$ via the flow a, which is the equal and opposite reaction
(Newton’s third law) from the draining loop on S. Finally, there are 2 forces from B, one
directly via a and one through the function f.
The forces from $ and B on K are all part of a number of higher order loops. Although there
is some choice of the loop with which they can be identified, the forces of S and B on K are
unique. Thus 6 unique stock impacts on K can be derived representing all the forces, and
their relative strengths compared. The impacts can either be computed numerically in the
simulation model, following the method of Hayward & Boswell (2014), or they can be derived
analytically by differentiating the differential equations (19-21) by the appropriate variable,
and ensuring the differentiation only occurs along the path of the force whose impact is being
19
resources.
This strategic decision is not easy even if companies’ owners and staffs are well
experienced in another company. One of the reasons is that startup companies’ products
and services are often very new; therefore, similar products’ market information and
knowledge are not applicable to a particular startup company.
Another reason is that startup companies are relatively vulnerable. Longstanding
companies would be able to survive the effects of an unsuccessful decision because their
relatively rich resources can cover costs of failures. However, startup companies are often
likely to fail because of one inappropriate decision.
Bianchi and Bivona (2002) investigate the whole business of startup companies and
show “how managing processes of accumulation and depletion of strategic assets,
detecting inertial effects of decisions made in the past, and selectively acting on policy
levers are likely to help entrepreneurs in understanding opportunities and pitfalls related
to e-commerce strategies.” Indeed, these help to improve business people’s thinking in
middle or long range, but real site business people also need more decision supports to
improve the current situation.
In this situation, a computer simulation reflecting each certain company’s real situation
and stakeholders’ real understanding and question is valuable. Examining simulation
results are helpful to real company owners and managers. This paper explains a process
of introducing system dynamics simulation into startup companies and shows a real case
of a startup company selling original cosmetics online. The authors are directly
communicating with this company and sharing real business data. The purpose of
modeling and simulation is to investigate outcomes of strategic decisions and to help
decision making for companies with limited resources.
2. Model and Simulation
The model in this study is a startup company, called here “LP,” which sells original
cosmetics.
LP’s main product is an original skincare cream produced with cranberries. It is unique,
and their philosophy (high quality, ecological production, contribution to society)
particularly attracts middle-aged women.
They offer a “tester,” a small portion of their products. Prospective customers try a tester,
and then they decide to continue to use the products. Offering the tester is not only to
keep out of troubles with customers but also to have an important means to reach potential
customers.
determined’.
For example, using the stock impact definition (3, 15), the impact of B on K via pathway
By» fra is given by:
aK
Ip,» freak — OB
ea)
Bf OK \B_ BSB
ja) ® \OBwta) KKB OK K OBER
(22)
where all variables are held constant except for B along the B,+ fra pathway. Once the dif-
ferentiation is performed the subscripts on the right hand side can be dropped and algebraic
manipulation can take place. However the subscript on the impact symbol I, on the left
hand side, is essential in order to precisely locate the path from the stock causing the force.
The use of a loop identifier in this model is less helpful as some paths are shared between
loops.
A simulation is performed and the variables examined from ¢ = 128, once the limit cycle
behavior is established. The change of dominance of stock impacts on K are given in figure
16(a). Growth is dominated by Isax, that is the reaction of K to the outflow of S. The only
exception is a brief period from K itself via the KcraKk pathway, enhancing the accelerated
growth. Thus it can be said that the growth in K is largely a reaction to the outflow of
S, that is a result of Newton’s third law of motion. The same reaction force governs the
change to decline of K and the acceleration that immediately follows. This cause of behavior
is standard in chain models (Hayward & Boswell, 2014). The remainder of the decline is
caused by the frictional dissipation force, impact Lrax.
14 [al
IKdK 1SaK 1 KexaK 1 Sak 1KdK
Stock K
°
Force Impacts on K
{
{
{
i
730 740 150 160 170
Time t Time t
Fig. 16: Economic long-wave model with 7 = Ts = 6 = 1.5, 7 = 20, = 3 and y = 1. (a) Stock
K with regions of force (stock impact) dominance. (b) Stock impacts for 5 of the 6 forces on K.
Ikcfrak is too small to show; all bar Lax are zero during K’s decline.
Comparing the impacts of the forces, figure 16(b), shows the frictional force Ixax = —1/T,
is a small constant, but that all the other forces collapse to near zero throughout K’s decline
allowing Iax to dominate. The two forces via the graphical function f are very small, except
for Iga*yeax near turning points only!®, even then it is swamped by the other forces. Other
°l the subsequent analysis the numerical method was used with the analytical formulae providing a
check.
l0The impact is not smooth due to f being based on a look-up table. Replacing f with a smoothed
function would eliminate the sudden changes.
20
forces, such as Igax, are non-zero and do affect the motion, but they are not strong enough
to explain the type of curvature, the acceleration and deceleration; they only influence its
extent. Thus K’s behavior can generally be explained by the relative effects of a reaction to
a force on S (Newton’s third law) and friction on K.
For the stock S' there are 16 forces: 4 direct forces; 2 via f; and 10 via g. To simplify the
understanding, the 2 forces via f are ignored as these pathways have already been shown
to be small, figure 16(b), and the ones via g are treated as a single combined force. Thus 5
impacts on S can be compared, figure 17(a). Essentially there are four pha: a short period
of growth, due to a short impulse from g, about t = 130; a period of steady growth, where a
number of forces dominate in combination (given by the arrow on the figure); a short impulse
from g to cause S to decelerate, after t = 140; and its subsequent decline through Igas, the
frictional foree on S. The force dominance in the second period, the steady growth in S,
is largely spurious as the forces through g during this period are zero, and the remaining
four forces, Ixdos; Isas; Ixeras; Leas, though not zero, nevertheless balance to almost zero,
figure 17(b) (labelled Sum of other forces on the figure). Instead this second phase of steady
growth is best explained by Newton’s first law of motion, with S increasing under its own
momentum after the impulse caused through g in the first phase. g only has effect in two
short phases because all forces through it have their impacts proportional to g, and most of
the time g is either zero, or at saturation, thus horizontal with no gradient, figure 17(b).
Tas Tks 1 a
Keres 1BaS Kaas
Fi sus 1goS ~ LowLevelotForces I goS 1 SaS
re » .
i 5 §
o4 i S
si 3 3
8 i & &
Ber | A A
| 2 2
ar | c 2
1h I
0 + H H ii
130 135 140 145 150
Time t Time t
Fig. 17: Economic long-wave model. (a) Stock S' with regions of force dominance, negligible force
on region marked by arrow. (b) Force impacts for those through g, (jos), compared with the sum
of the other 4 forces on $, (Licdos + Isas + Ixcras + Ipas), showing a period of negligible force,
t = 132..139. Graph g against time, showing it has no force (gradient) except at critical turning
points of S.
Examining the impacts of the 10 forces through g shows that most are negligible. The rapid
acceleration of S about t = 130 is dominated by Ipg-o*gos, the target setting for K. The
change from growth to decline of $ about t = 140 is dominated by Iico-gos (initially assisted
by Ixezas), With the following short period of deceleration again dominated by Igy+o*gos. Both
Tgp+o-gos ANd L¢o+gos are connected with the capital A adjustment process. Thus the dramatic
changes in S are caused by two brief, but intense, periods of acceleration and deceleration
caused by the capital adjustment process, with the remainder of the behavior either following
Newton’s first law, uniform growth, or frictional dissipation, giving exponential decline. That
g is effectively acting like a switch, goes some way to explaining the severe non-linearity of
the limit cycle.
21
The impacts on B, the backlog can be examined, though these are not included for lack of
space. It is hoped that sufficient insight has been given to show the value of using Newtonian
concepts, particularly that of force, to explain behavior from structure in a complex system
dynamics model, with numerous loops.
7 Conclusion
The paper developed the concept of loop impact, proposed by Hayward & Boswell (2014),
using the analogy with Newtonian mechanics. It was possible to identify the force exerted
by one stock on another as the net rate of change of the stock representing the force. Key
to the analogy is the concept of impact as a ratio measure of acceleration, which is the same
for a mechanical force and the effect of one stock on another. These concepts correspond
with the definition of loop impact, but are also applicable to exogenous influences. Newton’s
three laws of motion have their analogy in system dynamics and, together with the concept
of mass, inertia and momentum, have a natural interpretation which can assist with under-
standing model behavior. The ideas were applied to an inventory-workforce model, and the
economic long-wave model, where Newtonian interpretations were given to model behavior
by examining the dominant forces on individual stocks. However it is not the value of the
forces that determines dominance, but their impact as a ratio measure of the force compared
with the net flow of the stock being changed. A notation was developed for the model differ-
ential equations that allowed the network topology of the system dynamics model to remain
intact, enabling easier analytical computation of the impacts on stocks.
The word “force” has a precise meaning in physics which does not always transfer to social
systems. Consider the following definition of “social force”:
A social force is an element of society which has the capability of causing cultural
change or influences people (Business Directory, 2015).
The definition gives the impression that it takes a change in the force to give a change
in the culture, that is if the force remains constant then the social variable in the culture
affected by it remains fixed. If the social force were removed, the social condition would fall
back to its original value prior to the force being acted, rather than continue under its own
momentum. This is an almost Archimedean, or pre-Newtonian idea of force. The concept is
about balance rather than acceleration. However, this can be replicated in system dynamics
using the Newtonian understanding if both the stock representing the force, and the one
it accelerates, have natural frictional dissipative forces, figure 18. In this case a change in
the level of the force k will cause y to increase to a new limit, and thus «x will rise, initially
accelerating, and then slowing to reach a higher equilibrium. In this sense such a force
could be said to cause a cultural change, whilst still imparting acceleration. Such natural
dissipative forces have been propsed for psychological and soft variables by Levine (2000)
and Hayward, Jeffs, et. al. (2014).
The work in this paper is an initial attempt to provide another approach to understanding
the connection between system structure and behavior. It is seen as an enhancement to the
feedback understanding of a model rather than a replacement. In many models the stock
impacts are a measure of the effect of feedback on a stock, and give an indication of loop
22
—# gf" dissipation time
x 202)
4 natural loss of x
reactionofxtoy “aya
Noe’ dissipation time
y
force of y = k L* | natural loss of y
Fig. 18: Social force y acting on social variable x, including internal dis
pative forces.
dominance where loops can be easily be identified, as in the inventory-workforce model. In
a model as complex as the economic long-wave, there may be a number of loops that can be
associated with a given stock impact. Thus, provided a suitable independent loop set can
found, the impacts can be applied to the force effects of feedback loops on stocks.
The proposed method, and its Newtonian interpretation, needs to be applied to a wide range
of models in order to investigate the extent to which the analogy can help understand model
behavior. It is hoped that the work presented in this paper gives encouragement for further
research into a Newtonian understanding of model behavior and its connection with system
structure. An additional benefit of a Newtonian approach to system dynamics, and the use of
the mathematical formalism presented, is that it may encourage researchers in mathematical
modeling, and social physics, to consider system dynamics as a serious analytical approach
to modeling that would enhance traditional non-linear anal
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24
Appendix A: Second Order Linear System
For a second order linear system the criteria for stability of the equilibrium can be expressed
in terms of loop gains. The Jacobian J of the linear system, (4-5) figure 3, can be written
as J = : v , which has eigenvalues A = [a+ d+ \/(a + d)? — 4(ad — be)|/2. The system
is stable if both eigenvalues are negative, that is trace p = a+d < 0 and determinant
q = ad — be > 0 (Drazin, 1992, ch.6). Using the gains of the loops G; = a, Gz = d and
G3 = be, then stability can be determined by the sum of the first order loop gains being
negative, G; + Gy < 0, and their product being bigger than the second order loop gain,
G,G2 > G3. That is, there is sufficient dissipation in the system to counteract the effects of
the second order loop. One corollary of these conditions is that the system is stable only if:
either L3 is reinforcing and both L, and Ly» are balancing; or L3 is balancing and at least
one of the first order loops is also balancing.
The system oscillates (a + d)? < 4(ad — be). This can be expressed in gains as (Gy — G2)? +
4G3 < 0. Thus G3 < 0 is a necessary condition for oscillation, that is the second order loop
L3 must be balancing.
Classification criteria for stability, saddle behavior, oscillation etc. can be expressed on the
the standard p — q plane (Drazin, 1992, p.176) using the loop gains, figure 19.
Appendix B: Pathway Notation for System Dynamics
Models
Causal Pathway Notation
Causal pathways can be indicated by subscripts for the intermediate variables in a causal
chain. Let x be a cause of y, write y = y(x), and y be a cause of z, z = z(y). Then
2(y) = 2y(2)) = 2(y(ay)) = 2"(2y)
where the subscript on the argument of the combined function z' = y o z indicates the
pathway from x to z is via y. Thus in a function combination the name of the intermediary
variable becomes a subscript.
For example let y(x) = 32? and z(y) = 2—4y, then z = 2— 1282. The subscript makes clear
the pathway from x to z via y. -
Let w = w(z) be a cause of z then
w= w(2(yz)) = w(2(Y@y)z)) = w(E(Y @ye))) = w'(@yz)
where w’ = (yozow). Thus, in general, for path collections a, b:
f(%a)u = f(a)
a
a 2
io) g coin
7 5 8 eX/o
iH a 4 58| 4
o 3 ay
a ° 3 S33| xr
8 a S$ 8359}
g x
2. 8
*
° 8
3
8
« a
E 7 é
3 is 3
H 2 3
2 § 2 Fy °
° s 5 x
nu 5 H -
oO 8 3 3
SO og 8 g » 3
7 2 S
oO ala = < Se
k—° 4 #2 VG
se % 0 89
s 2 2 &
5 q °
t = v
3 3 &
2 5 +
3 kd a
£ co 3 &
% a
é 8 3
z z
°
a >
y 3
: 8B
& ra
boagek
2 4 BS
a aage
o go
oo
Fig. 19: Stability of second order linear system and loop gain.
Consider an example with three causal pathways from x to w. Let y(x) = 32°, 2(2,y) =
2a — 4y and w(z, z) = xz. Then
w= frz= ye (2x, —4yz) = 4/x (20, - 1222,)
where the three different pathways are indicated by the different subscripts. The direct
pathway from x to w has no subscript.
If such a pathway were part of a system dynamics model then, for numerical simulation,
equilibrium analysis etc., the subscripts can be removed and algebraic simplification and
numerical computations performed, w = \/2x?(1—6x?). However for the computation of
stock and exogenous impacts, or other network related calculations, the subscripts should
be retained.
26
Stock Impact Notation
Consider a first order system dynamics model with stock x, with net flows f, and with 7
causal pathways to itself, i.e. m first order loops. Let a, be the name for the collection of
intermediary auxiliary variables in pathway yu. The system dynamics model in networked
equation form is:
= f(s = f (Gay) Lag) -+ + Bays ++ Zag) (23)
q, is the variable x along pathway a,,.
Equation (23) can be written in a more concise form, using the conventions of multivariate
calculus and differential geometry, as:
b= f(t) w=... (24)
The stock impacts on x are derived by differentiating (24) by time:
where the pathway derivative is defined by:
af} a
dz||,
a Of
Oxa,
the derivative along one pathway in the first order model. The stock impacts on x are:
a af
tay.e — a
ay
By definition all these stock impacts are first order loop impacts.
Consider an nth order system dynamics model with stocks z;, 7 = 1...n, with net flows
fi, and with 7; causal pathways from x; to xj. Let aj; be the names of the list of causal
pathways from a; to 2;. aj; is a matrix of lists of possibly differing lengths 7,;. An individual
causal pathway in the list is indexed by ju;; drawn from the range 1...7;;, thus giving the
matrix of lists ij ,,;, Which can be abbreviated to a;;,, without confusion. Each element of
each list is a collection of intermediary auxiliary variables in pathway jij. The nth order
system dynamics model in a concise networked equation form is:
i= Fil2jaj4) i,g=l...m, py Hl... ayy (25)
Zja;,, 18 the variable xj along pathway aji, = aji,,, connected to x;. There are 7; pathways
connecting these variables.
The stock impacts on x; are derived by differentiating (25) by time:
(26)
Ox; 2;
j=l
27
where the pathway derivative is defined by:
(27)
the derivative along one pathway a;;, in the nth order model. Note in xj,,,,, the underlined
subscripts a,;,, themselves subscripted, define the pathway, whereas the non-underlined sub-
script j indicates the variable name. The use of the underline subscript notation should
avoid confusion for the many other uses of subscripts in dynamical models.
The stock impacts of 2; on 2; are:
a Oh
Ejajenti ~ By
ay
ay
a a;
Qin
When i = j these are first order loop impacts. When i # j these impacts may be part
of higher order loops, however this is not guaranteed, as some stocks may not be part of
feedback loops. Thus the term stock impact is preferred. If impacts are part of higher order
loops they may be referred to as loop impacts, but as in the case of the economic long-wave
model, they may be part of more than one loop.
The notation is easily extended to include multiple pathways from any number of exogenous
forces.
The double vertical line notation for the pathway derivative has been chosen for two reasons.
Firstly to avoid confusion with the use of the single line in the evaluation of integrals.
Secondly to avoid confusion with different index notations for differentiation.
For example, using the comma notation for partial differentiation, f;,;, the pathway derivative
(27) can be written:
Fidllasin = Fidllasin
the partial derivative of variable i by variable j along the pathway a;;,, from j to i.
jin
In differential geometry covariant derivatives are used as a coordinate free form of differen-
tiation, with semicolons and single vertical lines as notation (Rund, 2012). Thus a covariant
derivative along a pathway would have no confusion between the derivative notation and the
pathway notation. For example Filglasen is the horizontal covariant derivative of variable i
by variable j along pathway aj;,. Although there is no immediate use for this notation it is
noted that equations (26) may be geodesics in a curved space for certain system dynamics
models, suggesting a geometrical approach could be useful as a future line of research in this
area.
28
LP reaches consumers by using existing mail order catalogs (paper media), existing
Internet shopping sites, and their original website. Paper media is relatively powerful in
cultivating new customers. However, it is very expensive compared to the Internet media.
Therefore, LP’s normal marketing is on the Internet.
Based on the above considerations and interviews with LP owner, the model in this
study focuses on transitions of customers shown in figure 1.
__ total regular
4 customer
Tester Satisfied
winning tester | Buyers | new regular eet having
customers
with
complaint | losing regular
buyers customers complaint
27 customers
Figure 1. Basic structure of the LP model focusing on customer transitions
LP’s main product that occupies almost all sales is skincare cream. It is consumed at a
steady pace. Therefore, the sales are considered as in proportional to the customer size.
Winning customers depends on two advertisement media: Internet advertisement and
paper (mail-order magazine) advertisement. These advertisement media have contrasting
characteristics shown Table 1. LP has limited financial resources for advertisement;
therefore, their usual promotion medium is the Internet.
Table 1. Characteristics of advertisement media.
Cost Consumer Exposure | Possibility of gaining new customers
WWW | cheap frequent low
Paper expensive | limited high
Based on ZINTX (2015) and interview with the LP owner
LP understood their total advertising costs. Consequently, the cost of gaining one
customer and effect of each advertisement medium can be calculated based on the existing
LP’s data by the authors. As mentioned above, LP’s usual advertisement medium is the
Internet. When the marginal effect of Internet advertising is constant, the fluctuation away
from the sales from Internet advertising alone is the effect of paper advertisements. Thus,
the marginal effect of paper advertisements can be derived.
Customer starts from as “tester buyers.” LP’s sales data show the fact that almost all
customers use the tester at least once. The number of new customers buying a regular size
without a tester was negligible.
After the first trial, tester buyers decide whether they would buy the same product or
not. In the case of LP, a tester is not free. Therefore, most tester buyers shift to “regular
customers” if they value the quality of the products. This rate is stable irrespective of
implementation of paper media advertising.
The terms to continue “regular customer” status vary among customers. The majority
of customers continue “regular customer” status for three and four months. One and two-
month customers are relatively smaller in number than three month customers.
The continuation term distribution suggests customer transition includes a structure of
lower order delay. Indeed, the calculation using real data shows that the number which is
a square of the average stay time, in which customers keep their regular status, divided
by variance of stay time is approximately two (see Sterman, 2000, p. 465).
Transition time, or the time to stay in one stock, can depend on customer volume; more
customers, less satisfaction. Therefore, this model has a transition rate (rate of
dissatisfaction and average time to cancel) as functions of customer volumes. Each of
these functions is defined as a horizontal line when the customer size is small and as a
diagonally right down function when it increases. The customers stop buying the product
when they perceive a problem, but “loyal core fans” would stay irrespective of other
people’s behavior. The change from horizontal to right-down is when “satisfied regular
customers” and “regular customers” reach “threshold” and “average time to cancel
threshold,” respectively.
Figures 2 and 3 show LP’s real situation (reference mode) and the results of the
simulation. Customer numbers and costs values are standardized as one hundred for the
time 0 value. They are derived from parameters based on interviews and arithmetically
calculated values based on existing data. The LP owner and authors examined each
variable’s definition. The simulation values sufficiently trace real history. MAPE of
“Total Regular Customers” is 8.41%, and MAPE of “Tester Buyers” is 19.69%. See
Sterman (1984).
tester buyers
200
150
100
50
0 1 2 3 4 5 6 7 8 9 10 1 12
Time (Month)
tester buyers
tester buyers : base run
Figure 2. Base simulation result of Tester Buyers and actual data (Reference mode)
total regular customers
200
150
100
50
0 1 2 3 4 5 6 q 8 9 10 ll 12
Time (Month)
total regular customers
total regular customers : base run
Figure 3. Base simulation result of Regular Customers and actual data (Reference mode)
3. Discussions
After the review of simulation model and a base run, the LP owner agreed to test some
scenarios including interventions, which would work effective. One of those scenarios
was to employ the popular effective paper advertisement more frequently. It means, the
owner hoped to test his advertisement strategy. The price of popular magazines’ paper
advertisements is much higher than the price of ordinary paper media. Therefore, LP
cannot afford the same volume with more frequency. The LP owner thought if these
advertisements were conducted more time, the volume should be smaller. The simulation
result shows this intervention can retain customers (figure 4).
200
150
100
50
o 1 2 3 4 5 6 7 8&8 9 10 I 12
Time (Month)
tester buyers : popular magazine ads
total regular customers : popular ine ads
Figure 4. Tester Buyers and total regular customer with twice popular magazine
advertisements and halving effect size
This scenario is slightly better than the base run and reference mode (real past data).
However, the popular magazine advertisements are expensive. In reality, LP uses it only
when the magazine offered the discount advertisement price. Therefore, this scenario
would be preferable, but LP cannot drive this situation by itself. The LP owner understand
the result, possibility of implement of his scenario, and challenges.
Following this examination, we conducted several examinations that reflect the LP
owner’s ideas and questions, sometimes with additional structures. These examinations
assisted his decision-makings. Usual qualitative dialogs between LP insiders and
advertisement magazine representatives often showed prospective without solid basis.
However, simulations can show results with evidence and assumptions. Besides, the
system dynamics model is relatively small and shows its details in easy way to
stakeholders. Company owners and stakeholders can decide their next strategy with
confidence. Thus, computer simulation using system dynamics improve quality of
decision-makings.
4. Conclusion
Real company owners are aware of many possibilities of various strategies. At the same
time, it is not feasible to distinguish the result or outcome of each strategy. Computer
simulations can show the differences and characteristics of each strategy’s outcome.
Besides, system dynamics models can explain why these outcomes occur, how we
manage it, and which strategy we should choose. This research shows the process to give
real site stockholders insight and a case study. In particular, the stakeholders’ needs are to
solve current problems with limited data and experience, rather than to draw a future
design. This research succeeds that system dynamics simulations are able to assist
relatively short-term decision-makings as well as long-term policy makings.
Acknowledgement
This research is funded by The MEXT (Ministry of Education, Culture, Sports, Science
and Technology, Japan) Programme for Strategic Research Bases at Private Universities
(2012-16) project "Organisational Information Ethics" $1291006. E-Grant, Co., Ltd.
helped the authors to collect real data on startup companies. The authors appreciate LP’s
understanding and agreement of publishing under the fictitious name.
References
Bianchi, C. and Bivona, E. 2002. “Opportunities and pitfalls related to e-commerce
strategies in small—medium firms: a system dynamics approach.” System Dynamics
Review Vol. 18, No. 3, pp. 403-429
Geoffrey, J. and Tong, V. "TATCHA: Marketing the Beauty Secrets of Japanese Geisha."
Harvard Business School Case 313-149, June 2013.
Sterman, JD., (1984) Appropriate Summary Statistics for Evaluating the Historical Fit of
System Dynamics Models. Dynamica, 10 (Winter), 51-66.
Sterman JD. 2000. Business Dynamics: Systems Thinking and Modeling for a Complex
World. Irwin/McGraw-Hill, Boston.
ZINTX . 2015. “Media Cost Comparison (using standard industry CPM measurement).”
http://www.zintx.com/wp-content/uploads/2015/02/MEDIA5-NZ-Media-Price-
Comparison-Chart.pdf
WWW resource was retrieved on 1*' February, 2016.
