Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
Industrial Engineering Department
Bogazici University
Bebek — Istanbul 34342 — Turkey
sema_mhmt@hotmail.com; hakan.yasarcan@boun.edu.tr
Abstract
In system dynamics methodology, a formal mathematical model of a dynamic system
consists of a stock-flow diagram and a set of equations. It is possible to simplify and
express a system dynamics model as a set of differential equations, which can then be used
to obtain the corresponding block diagram for that system dynamics model. In the paper,
we obtain simplified differential equations for two system dynamics models and based on
the differential equations, we construct two block diagrams. Differential equations serve as
a bridge between the two systems modeling perspectives, system dynamics and control
theory. In addition, we also show other mathematical forms that can be used to express a
dynamic model such as approximate integral equations, difference equations, and integral
equations. In Appendix A, a summary of Laplace transforms, transfer functions, and block
diagrams are provided as a quick reference. In Appendix B, 18 generic system dynamics
models, their simplified differential equations, and their corresponding block diagrams are
presented. We carefully formulated SD models and their corresponding block diagrams
and verified their behavior by simulating them and by observing the same exact behavior
from the SD model and its block diagram. Similar to “differential equations”, this paper
aims to construct a bridge between control theory and system dynamics.
Keywords: approximate integral equations; block diagram; control theory; differential
equations; frequency domain; Laplace transform; stock-flow diagram; system dynamics
model.
Block Diagrams of Generic System Dynamics Models -1-
Sema Mehmet and Hakan Yasarcan
Introduction
Laplace transform is widely used in control theory, which is a method of converting
a set of ordinary differential equations to a set of algebraic equations that can be easily
solved. A transfer function is the ratio of a system’s output to its input in the Laplace
domain, which is also known as the frequency domain (Olivi, 2006). Block diagrams are
often used to represent dynamic systems in control theory. Each block in a block diagram
has at least two Laplace domain signals connected to it, one input signal and an output
signal, and an associated transfer function that transforms the input signal into the output
signal. Blocks are connected via their signals (i.e. the output signal generated by a block
can be the input to another block). Thus, a complete block diagram represents the dynamic
relationship between one input or many inputs to a system and one output or many outputs
of that same system (Bequette, 2007; Seborg, 2004).
In system dynamics (SD) methodology, a formal mathematical model of a dynamic
system consists of a stock-flow diagram and a set of equations, which together correspond
to a set of approximate integral equations. It is also possible to express these models as a
set of differential equations (Barlas, 2002; Forrester, 1961 and 1971; IE 533, Unpublished
Lecture Notes; IE 550, Unpublished Lecture Notes; Sterman, 2000). As mentioned before,
a block diagram represents a set of differential equations in frequency domain. Therefore,
it is natural that a block diagram of an SD model can be obtained. Jay Wright Forrester, the
founder of SD, developed the field adapting servomechanistic ideas (Forrester, 2007; Lane,
2007). Today, servomechanism theory is known as classical control theory. This paper
aims to build a bridge between SD and its roots (i.e. control theory). For this purpose, we
constructed block diagrams of well known generic SD models providing details about SD
modeling concepts. Such a link between the two fields of dynamic systems will help
control theorists to understand SD models and will assist system dynamicists in
representing their models using block diagrams, which will hopefully enable them use the
analysis methods of control theory. Another aim of this paper is to show different
mathematical representations of an SD model. Therefore, after giving the stock-flow
diagram and equations of two example models, we also provide their approximate integral
equations, difference equations, differential equations, and integral equations.
Block Diagrams of Generic System Dynamics Models -2-
Sema Mehmet and Hakan Yasarcan
The first example given in the paper is a basic population model and the second
example is a stock management model with three different delay structures; a supply line
delay, a decision delay, and a perception delay. In Appendix B, we give SD model,
corresponding differential equation(s), and block diagrams of 18 commonly used
structures: compounding, draining, first-order linear, production, goal seeking (stock
adjustment), capacitated growth, growth with overshoot, a first order and a third order
continuous material delay, a first order and a third order continuous information delay,
discrete material delay, discrete information delay, oscillating, simple goal setting,
epidemic, stock management with a first order and a third supply line delay. Block
diagrams that we present are not only exact replicas of their corresponding SD models, but
they also include all the details present in the SD models. In addition, we present a
summary of Laplace transforms, transfer functions, and block diagrams as a quick
reference. We carefully formulated SD models and their corresponding block diagrams and
verified their behavior by simulating them and by observing the same exact behavior from
the SD model and its block diagram.
A basic population model
Stock-flow diagram of a basic population model is given in Figure 1.
O i n >
Biris LT en
Birth fraction Death fraction
Figure 1. Stock-flow diagram of a basic population model
In the stock-flow diagram given in Figure 1, “Population” is a stock variable, which
is an accumulation formed over time. “Population” (p) can only change via “Births” and
“Deaths”, which are flow variables. There can be one, two, or more than two flows
Block Diagrams of Generic System Dynamics Models -3-
Sema Mehmet and Hakan Yasarcan
attached to a stock variable. In this simple example, there are only two flows attached to p,
where “Births” is the inflow and “Deaths” is the outflow. Therefore, “Births” fill in and
“Deaths” drain out p. “Birth fraction” (bf) and “Death fraction” (df) are the parameters of
the population model, which consists of the stock-flow diagram given in Figure | and the
equations | and 2.
Births = bf x p (1)
Deaths = df x p Q)
To be able to simulate the model, numerical values must be assigned to bf, df; and
simulation-time-step (D7). bf and df'can assume non-negative values and DT can assume a
value between zero and one. If the value of DT is strictly between zero and one, the model
corresponds to an approximate integral equation. If the value of D7 is one, the model
corresponds to a difference equation. DT cannot be equal to zero.
The approximate integral equation of the basic population model
The relationship between the stock variable, which is p, and the flow variables
attached to it, which are “Births” and “Deaths”, imply Equation 3 (see Figure 1).
Pupr = P, + (Births — Deaths) x DT (3)
Inserting equations | and 2 into Equation 3 and simplifying the equation result in
Equation 4, which is the corresponding approximate integral equation of the model (IE
533, Unpublished Lecture Notes).
Pur = B, + (bf —df)x p, x DT (4)
In continuous time simulation, an approximate integral equation or a set of
approximate integral equations are used; the value assigned to DT must strictly be less than
Block Diagrams of Generic System Dynamics Models -4-
Sema Mehmet and Hakan Yasarcan
one and greater than zero (IE 533, Unpublished Lecture Notes; IE 550, Unpublished
Lecture Notes).
The difference equation of the basic population model
In discrete time simulation, a difference equation or a set of difference equations are
used. Assigning one to DT in Equation 4 and simplifying the equation result in Equation 5,
which is the corresponding difference equation of the model (IE 533, Unpublished Lecture
Notes).
Prvr = (1+ bf —af)x p, (6)
The differential equation of the basic population model
Equation 4 can be re-written as Equation 6.
Pupr — Pr “ae
= (bf —df )x 6
or Of axe ©
Equation 7, which is the corresponding differential equation of the model, is obtained
from Equation 6 by taking the limit of D7 to zero (IE 533, Unpublished Lecture Notes).
dp , Pipe P, : .
“= jim | #22 “* |=(of —df)x 7.
7 sim op TO axe @)
Block Diagrams of Generic System Dynamics Models ~5-
Sema Mehmet and Hakan Yasarcan
The integral equation of the basic population model
Equation 7 can be re-written as Equation 8.
tap =| (of —df)x pxdt (8)
Ja =J
Po
Equation 9, which is the corresponding integral equation of the model, is obtained
from Equation 8 (IE 533, Unpublished Lecture Notes).
P, = Py + | (of -af)x pxat (9)
Block diagram of the basic population model
Block diagram of the basic population model is given in Figure 2. More information
on Laplace transforms and block diagrams is provided in Appendix A.
Population
Integrator Birth fraction
Deaths
Death fraction
Figure 2. Block diagram of a basic population model
Block Diagrams of Generic System Dynamics Models -6-
Sema Mehmet and Hakan Yasarcan
Stock management with three different delay structures: a supply line
delay, a decision delay, and perception delay
Stock-flow diagram of a stock management structure with a three different delay
structures is given in Figure 3.
Sy Li
Stock
< Control Flow > Acquisition Flow, Loss Flow
Decision Making
Acquisition Delay
Delay Time te
FO Goes Percenved
% Flow Stock
Desired Supply
Line
x Perception
Lescaera \_Bomation
Formation x
Stock Adjustment péisapila Detiy
Supply Line aa
Indicated Control Adjustment
Flow 4 ere
Desired Stock
Stock Adjustment
Weight of Supply Time
Line
Figure 3. Stock-flow diagram of a stock management structure with three different delay
structures
In the stock-flow diagram given in Figure 3, “Supply Line” (SL), “Stock” (S),
“Perceived Stock” (PS) and “Control Flow” (CF) are stock variables, which are
accumulations formed over time. CF is, at the same time, a flow variable. The other flow
variables are “Acquisition Flow” (AF), “Loss Flow” (LF), “Perception Formation” (PF),
and “Decision Formation” (DF). SL can only change via CF and AF, S can only change via
AF and LF, PS can only change via PF, CF can only change via DF. CF is the inflow of
SL, AF is the outflow of SL, simultaneously, AF is the inflow of S, LF is the outflow of S,
PF is the inflow of PS, DF is the inflow of CF. Therefore, CF fill in and AF drain out SZ,
AF fill in and LF drain out S, PF fill in PS and DF fill in CF. “Indicated Control Flow”
(UICF), Desired Supply Line (SZ*), “Supply Line Adjustment” (SZA), and “Stock
Adjustment” (SA) are intermediate calculation variables (i.e. auxiliary variables) of the
model. “Decision Making Delay Time” (dmdt), “Weight of Supply Line” (ws),
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
“Acquisition Delay Time” (adr), “Stock Adjustment Time” (sat), “Desired Stock” (S*) and
“Perception Delay Time” (pdt) are the parameters of the stock management model that
consists of the stock-flow diagram given in Figure 3 and the equations 10-16.
ICF = LF + SA+ SLA
*_
sA= S*-PS
sat
ws.
SLA = ws! x SEE=SE
sat
ap = SE
adt
SL* = adt x LF
DF= ICF —CF
dmdt
PF = S-PS
pdt
(10)
ay
(12)
(13)
(14)
(15)
(16)
To be able to simulate the model, numerical values must be assigned to dmdt, wsl,
adt, sat, S*, pdt, and simulation-time-step (DT). dmdt, wsl, adt, sat, S*, and pdt can assume
non-negative values and DT can assume a value between zero and one. If the value of DT
is strictly between zero and one, the model corresponds to a set of approximate integral
equations. If the value of DT is one, the model corresponds to a set of difference equations.
DT cannot be equal to zero.
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
The set of approximate integral equations of the stock management model with three
different delay structures
The relationship between the stock variable, which are S, SL, CF, PS, and the flow
variables attached to it, which are AF, LF, CF, DF, and PF, imply equations 17-20 (see
Figure 3).
Sysop = S, +(AF —LF)x DT (17)
SL,, pp = SL, +(CF — AF)x DT (18)
CF, yp = CF, +(DF)x DT (19)
PS,, pr = PS, + (PF)x DT (20)
Inserting equations 10-16 into equations 17-20 and simplifying the equations result
in equations 21-24, which are the corresponding set of approximate integral equations of
the model (IE 533, Unpublished Lecture Notes).
Srey -5,+(S-uF )xor (21)
adt
SLi, pr = SL, (cr )xor (22)
adt
*_ —
UF 4 SPS 4 gf x QU XLF = SL) _ op
CF yp = CF, + sat sat XxDT (23)
dmdt
PS..p1 -3,+(S25 or (24)
pdt
Block Diagrams of Generic System Dynamics Models -9-
Sema Mehmet and Hakan Yasarcan
The set of difference equations of the stock management model with three different
delay structures
Assigning one to DT in equations 21-24 and simplifying the equations result in
equations 25-28, which are the corresponding set of difference equations of the model (IE
533, Unpublished Lecture Notes).
SL
Spr -5,+( 5-17] (25)
SLy.pp = SL, + G = =) (26)
adt
*_ —
ir+5 PS ysl OLE St) _ Cg
CF. pp = CF, + Sat Ga sae (27)
PS j.pr = PS, (2 5) (28)
pdt
The set of differential equations of the stock management model with three different
delay structures
Equations 21-24 can be re-written as equations 29-32.
(29)
(30)
Block Diagrams of Generic System Dynamics Models -10-
Sema Mehmet and Hakan Yasarcan
* -_
LF+ chin) +wsl x (adt LF SL) —CF
CF or — CF, sat sat Gl)
DT dmdt
(32)
Equations 33-36, which are the corresponding set of differential equations of the
model, are obtained from equations 29-32 by taking the limit of DT to zero (IE 533,
Unpublished Lecture Notes).
ds lim Sur —S, = SL op (33)
dt = bro DT adt
ASL tim ( Shsor = SL.) _( op SE (4)
dt = pro DT adt
S*-PS (adt x LF - SL)
5 LF + +wsl -CF
dCF = lim CF,.or — CF, = sat wes sat (35)
dt — PT>0" T dmdt
aPS _ lim PS or ~PS, \_{ S=PS (36)
dt — br" DT pdt
The set of integral equations of the stock model with three different delay
structures
Equations 33-36 can be re-written as equations 37-40
Ss, L
fas =|(S-uF xa (37)
i »\adt
Block Diagrams of Generic System Dynamics Models -ll-
Sema Mehmet and Hakan Yasarcan
‘fast = ier - =) x dt
0
& adt
*_ —
CR, 1 r+ PS sop y (ade x LP SL) og
facr =f sa ime xdt
(38)
(39)
(40)
Equations 41-44, which are the corresponding set of integral equations of the model,
are obtained from equations 37-40 (IE 533, Unpublished Lecture Notes).
t
S,=S, + {S-ur ea
» \adt
SL, SSE, + [(cr- xa
‘ adt
*_ —
t Lr+5 PS ys] (edt x LF SL) _¢p
CF, = CF, +] Si ime xdt
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
(4)
(42)
(43)
(44)
« Tho
Block diagram of the stock management model with three different delay structures
Block diagram of the stock management structure with third order delay is given in
Figure 4. More information on Laplace transforms and block diagrams is provided in
Appendix A.
Figure 4. Block diagram of the stock management structure with third order delay
Conclusion
In this paper, block diagrams of well known generic SD models are constructed.
Such a link between system dynamics and control theory will help control theorists to
understand SD models and will assist system dynamicists in representing their models
using block diagrams. This paper presents the preliminary work of an ongoing master
thesis, which mainly focuses on modeling and analyzing inventory control systems. The
plan is to use both system dynamics and control theory as methodological approaches.
In the paper including its Appendix B, we present twenty different SD models and
their corresponding block diagrams. Block diagrams that we present are not only the exact
replicas of their corresponding SD models, but they also include all the details present in
the SD models. We carefully formulated SD models and their corresponding block
Block Diagrams of Generic System Dynamics Models -1B-
Sema Mehmet and Hakan Yasarcan
diagrams. We simulated the SD models using Vensim and their corresponding block
diagrams by using Matlab’s Simulink and observe the same exact behavior from each one
of the SD models and their corresponding block diagrams.
Acknowledgements
This research is supported by a Marie Curie International Reintegration Grant
within the 7th European Community Framework Programme (grant agreement number:
PIRGO7-GA-2010-268272) and also by Bogazici University Research Fund (grant no:
6924-13A03P1).
This paper was also published by Bogazici University (Mehmet and Yasarcan, 2015).
References
Barlas Y. 2002. System dynamics: systemic feedback modeling for policy analysis. In
Knowledge for Sustainable Develop
Systems. UNESCO/EOLSS: Paris and Oxford; 1131-1175.
: An Insight into the Encyclopedia of Life Support
Barlas Y. 2013. Unpublished Lecture Notes, JE 550 Dynamics of Socio-Economic Systems.
Bogazici University.
Bequette B.W. 2007. Process Control: Modeling, Design and Simulation. Prentice Hall,
New Jersey.
Forrester J.W. 1961. Industrial Dynamics. Pegasus Communications, Waltham, MA.
Forrester J.W. 1971. Principles of Systems. Pegasus Communications, Waltham, MA.
Block Diagrams of Generic System Dynamics Models -14-
Sema Mehmet and Hakan Yasarcan
Forrester J.W. 2007. System dynamics-a personal view of the first fifty years. System
Dynamics Review. 23: 345-358.
Lane D.C. 2007. The power of the bond between cause and effect: Jay Wright Forrester
and the field of system dynamics. System Dynamics Review. 23: 95-118.
Olivi M. 2006. The Laplace transform in control theory. Lecture Notes in Control and
Information Science. 327: 193-209.
Seborg D.E., Edgar T.F., and Mellichamp D.A. 2004. Process Dynamics and Control. John
Wiley&Sons Inc.
Mehmet S. and Yasarcan H. 2015. Block Diagrams of Generic System Dynamics Models.
Research Paper Series No: FBE-IE-01/2015-01. Bogazici University, Istanbul.
Sterman J.D. 2000. Business Dynamics: Systems Thinking and Modeling for a Complex
World. \rwin/McGraw-Hill: Boston, MA.
Yasarcan H. 2013. Unpublished Lecture Notes, JE 533 Systems Theory. Bogazici
University.
Block Diagrams of Generic System Dynamics Models -15-
Sema Mehmet and Hakan Yasarcan
Appendix A: Laplace transforms, transfer functions, and block diagrams
In this appendix, we present a summary of Laplace transforms, transfer functions,
and block diagrams.
Laplace transform method
Laplace transform of a time domain function, f(t), is given in Equation A.1
(Bequette, 2007).
F(s)=L[f(]= Jf@xe xdt (A.1)
where s is a complex variable.
An example of Laplace transformation: exponential function
In this section, the Laplace transform of an exponential function is obtained as an
example (Equation A.6).
Le~*']= few xe“ xdt (A.2)
where a is a constant. Equation A.2 can be re-written as Equation A.3.
b
~<at) 43 -(s+a)t
Le J=tim fe xdt (A3)
Assay JP
Le~*"]= tim| £ (AA)
| —(e+a) |,
Block Diagrams of Generic System Dynamics Models -16-
Sema Mehmet and Hakan Yasarcan
Le~*]= lim
bo]
-(sta)eb
E a (A5)
—(sta) sta
Finally, Equation A.6, which is the Laplace transform of an exponential function, is
obtained from Equation A.5.
Lew ]=— (A6)
sta
The Generic form of the Laplace sform of a time delayed fi ion (pure delay)
If f(t) represents the value of a particular function at time ¢, its time delayed version,
f(t — 9), represents the value of that function at time t — 6, where 0, which is a positive
constant, is the duration of the delay. In this section, the generic Laplace transform of a
delayed function is derived (Equation A.12).
L[fe-0)] =f f@-0)xe~" xdt (A.7)
0
Equation A.7 can be re-written as Equation A.8.
L[fe-0)]= | f@-0) xe xdt (A.8)
0
L[fe-0)]= | fe-0) xe" xe” x dt (A.9)
0
L[f@-a)J=e"" xf re -0)xe°"'™ xd(t — 0) (A.10)
0
Block Diagrams of Generic System Dynamics Models -17-
Sema Mehmet and Hakan Yasarcan
If a change of variables, t =t— 0 is used to integrate the function (Equation A.10),
Equation A.11 is obtained.
Lf -O]=e" xfs ye" xa 2 (AD)
Finally, Equation A.12, which is the generic form of the Laplace transform of a time-
delayed function, is obtained from Equation A.11.
LIf(t-0)| =e x F(s) (A.12)
According to Equation A.12, Laplace transform of the delayed version of a function
equals to e* times the Laplace transform of that function (Bequette, 2007; Seborg et al.,
2004).
The generic form of the Laplace transform of a first order derivative
The generic form of the Laplace transform of a first order derivative of a function is
obtained by using integration by parts technique (Equation A.17).
{#0 ja
iG
xdt (A.13)
Equation A.13 can be re-written as Equation A.14.
. i ;
420) =lim (en x FO nat (A.14)
dt | tod dt
| 2o|- inf Foxe tox [foxes a (A.15)
t
Block Diagrams of Generic System Dynamics Models -18-
Sema Mehmet and Hakan Yasarcan
f2o|- inl ose 1 +94) Oe "x a| (A.16)
Finally, Equation A.17, which is the generic form of the Laplace transform of a first
order derivative of a function, is obtained from Equation A.16.
{xo ©). sxI[fO]-F) (Al)
Laplace sforms of ly used time d in functions
Laplace transforms are used for solving most dynamic problems and, in solving such
a problem, Laplace transform tables are usually used to save time. Accordingly, a Laplace
transform table for some of the common functions is also provided in this Appendix (Table
A.1). For a more comprehensive Laplace transform table, see, for example, Bequette
(2007) or Seborg et al. (2004).
Block Diagrams of Generic System Dynamics Models -19-
Sema Mehmet and Hakan Yasarcan
Table A.1. Laplace Transforms of Common Time Domain Functions
Time domain
Laplace domain function
function
f(t) F(s)
O(t) (Equation A.18) | 1
S(t) (Equation A.20) ds
Ss
a
a =
Ss
f(t-0) ee” x F(s)
1
t az
Ss”
n!
t" sm
at 1
e sta
at 1
txe (s + ay
sin (at) = =
cos (at) 3 <
oo sxL[fo]-f0)
7 8" xF(s)—s"" x fO)~ 8"? x f'O)— 8" x ["O)~ ~~ LO)
It
Unit impulse (d(t)) is defined by Equation A.18 and its integral from negative
infinity to positive infinity, which is equal to one, is given in Equation A.19.
Block Diagrams of Generic System Dynamics Models
1
6) = lim €
"10 for
for (A.18)
t>é
«260
Sema Mehmet and Hakan Yasarcan
[ow xdr = fo( x ar = tim [Exar =1 (A.19)
Bs 0 erg €
Unit step (S(t) ) is defined by Equation A.20.
s(t)= 0 for t<0 (A.20)
lL for 120 .
It is also possible to use a Laplace transform table (Table A.1) to obtain the inverse
Laplace transform of a Laplace domain function, which is defined by Equation A.21.
L'FF(]= £0 (A21)
Note that the inverse Laplace transform of the Laplace transform of a time domain
function is itself (Equation A.22).
fO=L'LfOo]] (A.22)
Solving linear differential equations using Laplace transforms: an example first
order equation
To solve a differential equation with Laplace transform, Laplace transform of both
sides of the differential equation must be taken. Then, the resulting algebraic equation must
be solved for Lf]. Finally, the inverse transform must be taken by using Laplace
transform table.
An example first order differential equation is given below (Equation A.23):
dx(t
BO ay ae! (A.23)
dt
Block Diagrams of Generic System Dynamics Models -21-
Sema Mehmet and Hakan Yasarcan
Initial condition is x(0) = a, where a is a constant value.
Taking the Laplace of Equation A.23 gives Equation A.24.
dx '
{20 = wi] =1e'| (A.24)
[20] Ux] = Le'] (A.25)
Equation A.26 is obtained from Equation A.25 using Table A.1.
(sx L[x(O]-x(0))-L[x()]= = (A.26)
Equation A.26 can be re-written as Equation A.27.
(s—1)x Ufs(e)]-a=— (A.27)
Solving for L[x(4)] gives Equation A.28.
1 a
L[x@]= Gay +
(A.28)
Equation A.29 is obtained by inverting Equation A.28 to the time domain using
Laplace transform table (Table A.1).
x(t)=txe' +axe' (A.29)
Block Diagrams of Generic System Dynamics Models -22-
Sema Mehmet and Hakan Yasarcan
Solving linear differential equations using Laplace transforms: an example set of first
order equations
An example of a set of first order differential equations is given below (equations
A.30 and A.31):
an +x%,Q=1 (A.30)
os wf) )
x()- =0 (A31)
Initial conditions are x,(0) =a, and x,(0) = a,, where a; and az are constants.
Taking the Laplace of equations A.30 and A.31 gives equations A.32 and A.33.
[2a ol=z) (A.32)
Lb,(o]- 22). 10] (A.33)
Equations A.34 and A.35 are obtained from equations A.32 and A.33 using Table
All.
5xX,(s)—x,(0)+.X,(s) =4 (A.34)
Ss
X,(s)—sxX,(s)+x,(0)=0 (A.35)
Equations A.34 and A.35 can be re-written as equations A.36 and A.37.
Block Diagrams of Generic System Dynamics Models = 23
Sema Mehmet and Hakan Yasarcan
sx X(s)+X,(s)=b44,
Ss
X,(s)—sx X,(s) =—a,
Equation A.38 is obtained from Equation A.37.
X,(s) =-a, +sx X,(s)
Equation A.39 is obtained by inserting Equation A.38 into Equation A.36.
AH XERG REND AKO EVs a
Ss
Solving for X2(s) gives Equation A.40.
a, xs
X,(s)=— 1} 4 +
* sx(lts?} s?+1 5? +1
Equation A.41 is obtained from Equation A.40.
x,q-1-eaks a,
Equation A.42 is obtained by inserting Equation A.41 into Equation A.38.
X,(s)=-a, tft ears, 4 |
Ss
sv +l 4]
Equation A.42 is simplified to Equation A.43.
X,(s)=—a, ¥1=(1-a)x{-
2
S
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
(A.36)
(A.37)
(A.38)
(A.39)
(A.40)
(A4l)
(A.42)
(A.43)
-24-
(A.44)
(1-a,) a,Xs
X(s)= 274
ifs) stl s? 41
Using Table A.1, the two Laplace domain functions (equations A.41 and A.44) is
inverted to the time domain, which are given below:
x, (t) = (I-a, )xsin (¢) + a, xcos(t) (A.45)
x,(t)=1-(l—a,)xcos(t)+a, xsin(t) (A.46)
Solving linear differential equations using Laplace transforms: an example second order
equation
An example second order linear differential equation is given below (Equation
AAT):
d°x(t dx(t ant
TAO ax BOO ax x()= e (A.47)
Initial conditions are x(0) =a, and x(0) =a,, where a; and a2 are constants.
Taking the Laplace of Equation A.47 gives Equation A.48.
[0)- 4x x 0) +4xL[x(o]= Le] (A.48)
Equation A.49 is obtained from Equation A.48 using Table A.1.
(s? x X(s)—sx.x(0)—(0))—4x(sx X(s)—x(0))+4x X(s) = sy (A.49)
S
Block Diagrams of Generic System Dynamics Models -25-
Sema Mehmet and Hakan Yasarcan
Equation A.49 can be re-written as Equation A.50.
(c2-4x544)x X(s)—a, xs —a, +40, = (A.50)
st+2
Solving for X(s) gives Equation A.51.
1 a a,xs {a= deen) (A351)
—4x5+4)x(s+2) s’-4xs+4 s°-4xs+4
Equation A.52 is obtained from Equation A.51.
X(@)=-—-—~+ a, = ae © Dice wa, x u = (A.52)
16x(s+2) 16) s-2 \4 (s-2y
Equation A.53 is obtained by inverting Equation A.52 to the time domain using
Laplace transform table (Table A.1).
ee 1 oe {1 oy
xD Exe” +(a-Z)xe (4(F 20,44, ree ‘ (A.53)
Transfer functions and block diagrams
Transfer function of a dynamic system is the ratio of the output variable to the input
variable in the Laplace domain. In general, g(s) is used to represent a transfer function that
is defined in Equation A.54 (Bequette, 2007; Seborg et al., 2004).
Ky
g(s) ae ) (A.54)
where u(s) is the input variable and y(s) is the output variable in the Laplace domain.
Block Diagrams of Generic System Dynamics Models -26-
Sema Mehmet and Hakan Yasarcan
Block diagrams
Transfer functions are often used in block diagrams. The relationship between input
and output defined by Equation A.54 is depicted in Figure A.1.
u(s) y(s)
——> a(s) >_>
Figure A.1 The most basic block diagram representing an input-output relationship in the
Laplace domain
Block diagrams have three main types of elements which are signals, unary operator
blocks, and m-ary (i.e., many-ary) operator blocks given in Figure A.2.
r(s) u(s) yi(s)
8:1(s)
ya(s)
82(s) it
Figure A.2 A block diagram with two blocks
In the block diagram representation given in Figure A.2, r(s), u(s), vi(s), and y2(s) are
signals. u(s) is the input signal and y\(s) is the output signal of g\(s). y(s) is the input
signal and y»(s) is the output signal of go(s). gi(s) and g»(s) are unary operator blocks which
operates on the input signals connected to them with the transfer functions to form the
output signals. r(s) and y2(s) are the input variables and u(s) is the output variable of the
summation block which is an m-ary operator block. An m-ary operator block is shown as a
circle and has two or more input variables and a single output variable (Wescott, 2006).
Block Diagrams of Generic System Dynamics Models =e
Sema Mehmet and Hakan Yasarcan
Reduced block diagrams
A generic example is given in Figure A.3.
Figure A.3 A generic example of block diagram
The relationship between inputs and outputs of the system are given in equations
A.55 and A.56, A.57 and A.58.
_ ys)
sans (A355)
g,(s) = 2 (A.56)
¥4(5)
¥3(8) = y3(s)x Gain (A.57)
u(s)—y, (5) =14(5) (A358)
Combining gi(s) and g»(s) into a single transfer function, g3(s) is obtained which is
given in Equation A.59.
yils) , Val) _ yal) sks
w(s) ¥(8) 63)
83(8) = 8,(8)xg2(9)=
Block Diagrams of Generic System Dynamics Models -28-
Sema Mehmet and Hakan Yasarcan
Equation A.58 can be rewritten as Equation A.60.
Yo(s)
u,(s)— y(s) x Gain = ——>—_ A.60
(8) yo (s) FOKa® (A.60)
u,(s)=—22©)_4. y, (s)x Gain (A61)
&(s)xg,(s)
i(o)= yon] Sais ges (A.62)
&(5)x 8, (s)
Overall transfer function of the process, g,(s) is given in Equation A.63.
gZy(s)= Yols)_ 8S) * 828) (A.63)
u,(s) 1+Gainx g,(s)x g,(s)
The reduced form of the block diagram given in Figure A.3 is depicted in Figure A.4,
which is also a block diagram and an equivalent of the diagram given in Figure A.3.
ux(s) ya(s)
——>__ als) }3=[>-———>
Figure A.4 Reduced form of the example block diagram
Block Diagrams of Generic System Dynamics Models -29-
Sema Mehmet and Hakan Yasarcan
Appendix B: The corresponding block diagrams of basic system
dynamics models
We first convert generic system dynamics model structures to differential equations
and, later, we obtain corresponding block diagrams of these structures from the differential
equations. To save space, the derivation process is not provided, but only the resulting
differential equations and block diagrams are given. If the reader is interested in the
derivation process, she can read the paper and Appendix A and carry out derivations
herself. Note that a comprehensive model usually contains one or many of these generic
structures. Moreover, a constant of a simpler structure may turn into a variable, even into a
state variable, in a more complex model.
Compounding structure
Stock-flow diagram of a compounding structure is given in Figure B.1.
Featon 2
Figure B.1 Stock-flow diagram of the compounding structure
The inflow equation of the model is given in Equation B.1.
Inflow = Fraction x Stock (B.1)
where “Fraction” is a nonnegative constant value.
The diagram in Figure B.1 and Equation B.1 define a compounding structure. The
simplified differential equation that corresponds to this structure is given in Equation B.2.
Block Diagrams of Generic System Dynamics Models -30-
Sema Mehmet and Hakan Yasarcan
ae = Inflow = Fraction x Stock (B.2)
It
Block diagram of the compounding structure is given in Figure B.2.
Stock Ee Inflow
Fraction
Figure B.2 Block diagram of the compounding structure
Draining structure
Stock-flow diagram of a draining structure is given in Figure B.3.
Fraction
Figure B.3 Stock-flow diagram of the draining structure
The outflow equation of the model is given in Equation B.3.
Outflow = Fraction x Stock (B.3)
where “Fraction” is a nonnegative constant value.
The diagram in Figure B.3 and Equation B.3 define a draining structure. The
simplified differential equation that corresponds to this structure is given in Equation B.4.
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
«B10
ae = —Outflow = —Fraction x Stock
It
Block diagram of the draining structure is given in Figure B.4.
(B.4)
Outflow
Fraction
Figure B.4 Block diagram of the draining structure
First order linear structure
Stock-flow diagram of a first order linear structure is given in Figure B.5.
Outflow
at ee 2.
Fraction |
Figure B.5 Stock-flow diagram of the first order linear structure
The inflow and outflow equations of the model are given in equations B.5 and B.6.
Inflow = Fraction 1x Stock
Outflow = Fraction 2x Stock
where “Fraction 1” and “Fraction 2” are nonnegative constant values.
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
(B.5)
(B.6)
«Boe
The diagram in Figure B.5 and equations B.5 and B.6 define a first order linear
structure. The simplified differential equation that corresponds to this structure is given in
Equation B.7.
Bick = Inflow — Outflow = (Fraction 1— Fraction 2)x Stock (B.7)
Block diagram of the first order linear structure is given in Figure B.6.
Fraction 1
Outflow
Fraction 2
Figure B.6 Block diagram of the first order linear structure
Production structure
Stock-flow diagram of a production structure is given in Figure B.7.
ZZ
Zs
Production rate
Productivity.
Figure B.7 Stock-flow diagram of the production structure
The inflow equation of the model is given in Equation B.8.
Block Diagrams of Generic System Dynamics Models -33-
Sema Mehmet and Hakan Yasarcan
Production rate = Productivity x Stock 2 (B.8)
where “Productivity” is a nonnegative constant value.
The diagram in Figure B.7 and Equation B.8 define a production structure. The
simplified differential equation that corresponds to this structure is given in Equation B.9.
——— = Production rate = Productivity x Stock 2 (B.9)
dStock 1
dt
Flows are not connected to the state variable named “Stock 2” because the focus of
this structure is on representing “Production rate” flow. Similar to other simple model
structures, the production structure usually is a part of a more comprehensive model.
Block diagram of the production structure is given in Figure B.8.
Stock 2 Production rate Stock 1
Productivity
Figure B.8 Block diagram of the production structure
Block Diagrams of Generic System Dynamics Models -34-
Sema Mehmet and Hakan Yasarcan
Goal seeking structure
Stock-flow diagram of a goal seeking structure, which is also known as stock
adjustment structure, is given in Figure B.9.
QO 4 > [stock]
2s
Adjustment flow
i]
Goal
Adj me pill \_/
fraction
Figure B.9 Stock-flow diagram of the goal seeking structure
The model equations are B.10 and B.11.
Adj flow = Adj fraction x Discrepancy (B.10)
Discrepancy = Goal — Stock (B.11)
where “Goal” is a constant value and “Adjustment fraction” is a nonnegative
constant value.
The diagram in Figure B.9 and equations B.10 and B.11 define a goal seeking
structure. The simplified differential equation that corresponds to this structure is given in
Equation B.12.
Set = Adjustment flow = Adjustment fraction x (Goal = Stock) (B.12)
Block diagram of the goal seeking structure is given in Figure B.10.
Block Diagrams of Generic System Dynamics Models -35-
Sema Mehmet and Hakan Yasarcan
Adjustment
flow
Discrepancy Stock
Adjustment fraction
Figure B.10 Block diagram of the goal seeking structure
Capacitated growth structure (S-shaped growth caused by a capacity limit)
Stock-flow diagram of a capacitated growth structure is given in Figure B.11.
Zs
Outflow
Fraction 1
Ratio
Effect of Ratio on “__ Capacity
Fraction 1
Standard
Fraction 1
Ma Fraction 2
Figure B.11 Stock-flow diagram of the capacitated growth structure
The model equations are B.13-B.17.
Inflow = Fraction 1x Stock
Fraction 1 = Effect of Ratio on Fraction 1x Standard Fraction 1
Effect of Ratio on Fraction 1 = f (Ratio)
Stock
Ratio = ———_
Capacity
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
(B.13)
(B.14)
(B.15)
(B.16)
«3B 6
Outflow = Fraction 2x Stock (B.17)
The diagram in Figure B.11 and equations B.13-B.17 define a capacitated growth
structure. The simplified differential equation that corresponds to this structure is given in
Equation B.18.
dStock
= Inflow — Outflow
dt
(B.18)
_ ,{ Stock
“ \ Capacity
| x Standard Fraction 1x Stock — Fraction 2x Stock
Block diagram of the capacitated growth structure is given in Figure B.12.
Effect of Ratio
on Fraction 1
Ratio
| fRtio|
Outflow
Standard Fraction 1
Fraction 2
Figure B.12 Block diagram of the capacitated growth structure
As an example, assume that f(Ratio) is given by Equation B.19.
Effect of Ratio on Fraction 1 = f (Ratio)= 1-0.75x Ratio (B.19)
The corresponding part of the block diagram representing f(Ratio), which is obtained
under this assumption (Equation B.19), is given in Figure B.13.
Block Diagrams of Generic System Dynamics Models -37-
Sema Mehmet and Hakan Yasarcan
Effect of Ratio
on Fraction 1
Figure B.13 Block diagram of an example f(Ratio)
Growth with overshoot structure (caused by a delayed effect of capacity limit)
Stock-flow diagram of a growth with overshoot structure is given in Figure B.14.
2S
Inflow
Fraction we
Fraction 2
Ratio
rf Effect of Effective Capacity
Standard Ratio on Fraction |
Fraction |
Effective
Kate Adjustment flow
_ time
Figure B.14 Stock-flow diagram of the growth with overshoot structure
The model equations are B.20-B.25.
Inflow = Fraction 1x Stock (B.20)
Block Diagrams of Generic System Dynamics Models -38-
Sema Mehmet and Hakan Yasarcan
Fraction 1 = Standard Fraction 1x Effect of Effective Ratio on Fraction! (B.21)
Effect of Effective Ratio on Fraction 1 = f (Effective Ratio) (B.22)
_ Ratio — Effective Ratio
Adj flow B.23
f Delay time 6 )
Rages (B.24)
Capacity
Outflow = Fraction 2x Stock (B.25)
The diagram in Figure B.14 and equations B.20-B.25 define a growth with overshoot
structure. The simplified set of differential equations that corresponds to this structure is
given in equation B.26 and B.27.
HBG, = Inflow— Outflow
dt (B.26)
= Standard Fraction 1x f (Effective Ratio)x Stock — Fraction 2x Stock
APR eciive Raft a Effective Ratio
iffective Ratio = Adi flow = ‘apacity (B27)
dt . Delay time
Block diagram of the growth with overshoot structure is given in Figure B.15.
Effesive Effect of fective
[Ratio [Ratio on Fraction 1
Ley
LJ
WEtfedive Ratio)
Adjustment
flow
1VDelay time
capacity
Standard Fraction + Fraction 2
Figure B.15 Block diagram of the growth with overshoot structure
Block Diagrams of Generic System Dynamics Models -39-
Sema Mehmet and Hakan Yasarcan
Continuous material delay structures
Stock-flow diagram of a first order continuous material delay structure is given in
Figure B.16.
ioe: _ Stock PO)
Inflow J iit
Delay time
Figure B.16 Stock-flow diagram of the first order continuous material delay structure
Outflow equation of the model is given in Equation B.28.
oupow-— 0% (B.28)
Delay time
The diagram in Figure B.16 and Equation B.28 define a first order continuous
material delay structure. The simplified differential equation that corresponds to this
structure is given in Equation B.29.
= Inflow — Outflow = Inflow -—————_ (B.29)
Delay time
dStock Stock
dt
where “Delay time” is a nonnegative constant value.
Block diagram of the first order continuous material delay structure is given in
Figure B.17.
Block Diagrams of Generic System Dynamics Models -40-
Sema Mehmet and Hakan Yasarcan
1/Delay time
Figure B.17 Block diagram of the first order continuous material delay structure
Stock-flow diagram of a third order continuous material delay structure is given in
Figure B.18.
Oo Stock | BP Stock 2
Input Flow 1 [| Flow 2
Delay time for
each stage
Order’
‘Delay time
Stock 3
Output
Figure B.18 Stock-flow diagram of the third order continuous material delay structure
The model equations are B.30-B.33.
Flow1= ——__Stockt
Delay time for each stage
Flow? =—_Stock2_
Delay time for each stage
Gurput= Stock 3
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
Delay time for each stage
(B.30)
(B.31)
(B.32)
-41-
Delay time
Delay time for each stage =
™ f a Order
(B.33)
The diagram in Figure B.18 and equations B.30-B.33 define a third order continuous
material delay structure. The simplified set of differential equations that corresponds to this
structure is given in equations B.34, B.35, and B.36.
GSOC pie Fowl Typat-— °°? _ (B.34)
dt Delay time/Order
Atock? _ rigw 1 Flow2=——S0ck!__Stock 2 (B35)
dt Delay time/Order Delay time/Order
dStock 3 = Flow2—Output = Stock 2 _ Stock 3 (B36)
dt Delay time/Order Delay time/Order
where “Delay time” and “Order” are nonnegative constant values and “Order”
corresponds to the number of state variables (i.e., stocks) in the material structure.
Block diagram of the third order continuous material delay structure is given in
Figure B.19.
Input Flowt Flow 2
Output
Delay time
Delay time
for each stage
Order
Figure B.19 Block diagram of the third order continuous material delay structure
Block Diagrams of Generic System Dynamics Models -42-
Sema Mehmet and Hakan Yasarcan
Note that every material delay structure is an application of aforementioned draining
structure. Hence, a material delay structure contains one or many draining structures.
Continuous information delay structures
Stock-flow diagram of a first order continuous information delay structure is given in
Figure B.20.
Zs
Adjustment flow
Input
Delay time Discrepancy
Figure B.20 Stock-flow diagram of the first order continuous information delay structure
Equations of the model are given in equations B.37 and B.38.
An flow = Discrepancy (B37)
Delay time
Discrepancy = Input — Stock (B.38)
The diagram in Figure B.20 and equations B.37 and B.38 define a first order
continuous information delay structure. The simplified differential equation that
corresponds to this structure is given in Equation B.39.
dStock . Input — Stock
BlOOk = Adj flow =P» (B.39)
dt Delay time
where “Delay time” is a nonnegative constant value.
Block Diagrams of Generic System Dynamics Models -43-
Sema Mehmet and Hakan Yasarcan
Block diagram of the first order continuous information delay structure is given in
Figure B.21.
Adjustment
flow
1/Delsy time
Figure B.21 Block diagram of the first order continuous information delay structure
Stock-flow diagram of a third order continuous information delay structure is given
in Figure B.22.
Stock 1 | / Stock 2 \
Discrepancy 1 Discrepancy 2 Discrepancy 3
ctl
Adjustment flow 1 Adjustment flow 2 px Adjustment flow 3
N Q
Delay time for
each stage
Order Delay time
Figure B.22 Stock-flow diagram of the third order continuous information delay structure
The model equations are B.40-B.46.
Discrepancy 1
Adj flowl= (B.40)
Delay time for each stage
Block Diagrams of Generic System Dynamics Models -44-
Sema Mehmet and Hakan Yasarcan
Discrepancy 2
Adj flow 2 =
fi Delay time for each stage
Discrepancy 3
Adji low 3 =
: fi Delay time for each stage
Discrepancy 1 = Input — Stock I
Discrepancy 2 = Stock 1— Stock 2
Discrepancy 3 = Stock 2 — Output
Delay time
Delay time for each stage =
sd f 2 Order
(B.41)
(B.42)
(B.43)
(B.44)
(B.45)
(B.46)
The diagram in Figure B.22 and equations B.40-B.46 define a third order continuous
information delay structure. The simplified differential equations that correspond to this
structure are given in equations B.47, B.48, and B.49.
dStock1 _ , ,. fowr= Input — Stock 1
dt “ Delay time/Order
dStock 2 _ Ads flew2'= Stock 1 — Stock 2
dt . - Delay time/Order
dOutput _ NeW = Stock 2 — Output
dt . . Delay time/Order
where “Delay time” and “Order” are nonnegative constant values.
(B.47)
(B.48)
(B.49)
Block diagram of the third order continuous information delay structure is given in
Figure B.23.
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
-45-
Delay time
Figure B.23 Block diagram of the third order continuous information delay structure
Note that every information delay structure is an application of aforementioned goal
seeking structure. Hence, an information delay structure contains one or many goal seeking
structures.
Discrete material delay structure (pure delay)
Stock-flow diagram of a discrete material delay structure is given in Figure B.24.
Delay time
Figure B.24 Stock-flow diagram of the discrete material delay structure
Output equation of the model is given in Equation B.50.
SEED) for 0<t< Delay time
Output(t) = 4 Delay time (B.50)
Input(t — Delay time) for +t 2 Delay time
Block Diagrams of Generic System Dynamics Models - 46 -
Sema Mehmet and Hakan Yasarcan
where “Delay time” is a nonnegative constant value.
The diagram in Figure B.24 and Equation B.50 define an infinite order (i.e., discrete)
material delay structure. The simplified differential equation that corresponds to this
structure is given in Equation B.51.
dStock
——— = Input — Output
ae Ip Ip!
Stock(0
raputt)- LEO _ for 0<t< Delay time @5))
= Delay time
Input(t)— Input(t — Delay time) for t2 Delay time
Block diagram of the discrete material delay structure is given in Figure B.25.
Input . (a Stock
ate mS
Step
Output
Discrete
Delay
Figure B.25 Block diagram of the discrete material delay structure
Block Diagrams of Generic System Dynamics Models -47-
Sema Mehmet and Hakan Yasarcan
Discrete information delay structure (pure delay)
Stock-flow diagram of a discrete information delay structure is given in Figure B.26.
Initial value
\
Input Output
SS a
Delay time
Figure B.26 Stock-flow diagram of the discrete information delay structure
Output equation of the model is given in Equation B.52.
(B.52)
Initial value for 0<t< Delay time
Output(t) =
Input(t — Delay time) for t2 Delay time
where “Delay time” is a nonnegative constant value.
The diagram in Figure B.24 and Equation B.50 define an infinite order (i.e., discrete)
information delay structure. Note that there is no simplified differential equation that
corresponds to this structure as there is no stock in Figure B.24.
Block diagram of the discrete information delay structure is given in Figure B.27.
LT] Input =| ay Output
sas 4 >
Step Discrete
Delay
Figure B.27 Block diagram of the discrete information delay structure
Block Diagrams of Generic System Dynamics Models - 48 -
Sema Mehmet and Hakan Yasarcan
Oscillating structure
Stock-flow diagram of an oscillating structure is given in Figure B.28.
p| Stock | PO)
Inflow 1 Outflow 1
Productivity
Consumption
Fraction multiplier
Figure B.28 Stock-flow diagram of the oscillating structure
The model equations are B.53, B.54, and B.55.
Inflow I = Productivity x Stock 2
Inflow 2 = Fraction x Stock 2
Outflow 2 = Consumptio n multiplier x Stock 1
(B.53)
(B.54)
(B.55)
The diagram in Figure B.28 and equations B.53, B.54, and B.55 define an oscillating
structure. The simplified differential equations that correspond to this structure are given in
equations B.56 and B.57.
dStock 1
. = Inflow 1 — Outflow 1 = Productivity x Stock 2 — Outflow 1
t
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
(B.56)
-49-
dStock 2
= Inflow 2 — Outflow 2
dt
= Fraction x Stock 2— Consumption multiplier x Stock 1
(B.57)
where “Productivity” and “Consumption multiplier” are nonnegative constant values
and “Fraction” is a constant value.
Block diagram of the oscillating structure is given in Figure B.29.
Outflow 2 a le
ol
Inflow 1
Consumption
multiplier
Outflow 1
Fraction
Figure B.29 Block diagram of the oscillating structure
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
« 50~
Simple goal setting structure
Stock-flow diagram of the simple goal setting structure is given in Figure B.30.
x Goal adjustment
Control flow i
Stock adjustment A
time
ZS
Goal adjustment
flow
Figure B.30 Stock-flow diagram of the simple goal setting structure
The model equations are B.58 and B.59.
Control flow=— 200! stock _ (B.58)
Stock adjustment time
Goal adj flow= Stock — Goal
Goal adjustment time
(B.59)
The diagram in Figure B.30 and equations B.58 and B.59 define the simple goal
setting structure. The simplified set of differential equations that correspond to this
structure are given in equations B.60 and B.61.
= Control flow =————!__—_ (B.60)
dStock Goal — Stock
dt Stock adjustment time
dGoal Stock — Goal
= Goal adj flow= : (B.61)
dt Goal adjustment time
Block Diagrams of Generic System Dynamics Models -51-
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where “Stock adjustment time” and “Goal adjustment time” are nonnegative constant
values.
Block diagram of the simple goal setting structure is given in Figure B.31.
Goal
adjustment flow
Control flow Stock
Stock
adjustment timet adjustment time
Figure B.31 Block diagram of the simple goal setting structure
Epidemic model structure
Stock-flow diagram of the epidemic model structure is given in Figure B.32.
Infection fraction
Infected
In Infection rate
Removal fraction
Contacts
Contact fraction ad
Figure B.32 Stock-flow diagram of the epidemic model structure
Model equations are B.62, B.63, and B.64.
Infection rate = if x Contacts (B.62)
Block Diagrams of Generic System Dynamics Models -52-
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Contacts = cf x SxI (B.63)
Removal = rf xI (B.64)
where if stands for “Infection fraction”, cf stands for “Contact fraction”, and rf stands
for “Removal fraction”. S and / stand, respectively, for “Susceptible” and “Infected”.
The diagram in Figure B.32 and equations B.62, B.63, and B.64 define the epidemic
model structure. The simplified set of differential equations that correspond to this
structure are given in equations B.65 and B.66.
© = in— Infection rate = In—if xcf xSxI (B.65)
< = Infection rate — Removal = if x cf x Sx I —rf x I (B.66)
t
where if, cf, and 7f are nonnegative constant values.
Block diagram of the epidemic model structure is given in Figure B.33.
Infected
Removal
Removsl frection
Infection rate
le
—
Infection fractiom Contact fraction
Figure B.33 Block diagram of the epidemic model structure
Block Diagrams of Generic System Dynamics Models -53-
Sema Mehmet and Hakan Yasarcan
Stock management with a first order supply line delay structure
Stock-flow diagram of a stock management structure with a first order supply line
delay is given in Figure B.34.
Stock
Acquisition Flow Loss Flow
Desired Stock
Acquisition Delay
Time
Stock Adjustment
a
Supply Line Desired Supply
Adjustment Line Stock Adjustment
Weight of | supply 7 a Time
Line
Figure B.34 Stock-flow diagram of the stock management structure with a first order
supply line delay
The model equations are B.67-B.71.
CF = LF +SA+SLA (B.67)
*_
sa-5"-8 (B.68)
sat
*_
SLA = wsl x SEAS (B.69)
sat
ap = 5 (B.70)
adt
Block Diagrams of Generic System Dynamics Models -54-
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SL* = adt x LF (B.71)
where CF stands for “Control Flow”, LF stands for “Loss Flow”, SA stands for
“Stock Adjustment”, SLA stands for “Supply Line Adjustment”, S* stands for “Desired
Stock”, S stands for “Stock”, sat stands for “Stock Adjustment Time”, ws/ stands for
“Weight of Supply Line”, SL* stands for “Desired Supply Line”, SL stands for “Supply
Line”, AF stands for “Acquisition Flow”, adt stands for “Acquisition Delay Time”.
The diagram in B.34 and equations B.67-B.71 define a stock management structure
with a first order supply line delay. The simplified set of differential equation that
corresponds to this structure is given in equations B.72 and B.73.
WS _ apie =" re (B.72)
dt adt
*_ =
QSL _ Cp aR = ty 8S eye MRE OSE SE (B.73)
dt sat sat adt
Block diagram of the stock management structure with a first order supply line delay
is given in Figure B.35.
Block Diagrams of Generic System Dynamics Models -55-
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Desired Supply Line
Acguisition Delay Time
Loss Flow
Supply Line Adjustment
‘Supply Line
Step
Stock Adjustment
Stock Adjustment
Time
Desired Stock
‘Acquisition Delay
Time
Figure B.35 Block diagram of the stock management structure with a first order supply
Block Diagrams of Generic System Dynamics Models
Sema Mehmet and Hakan Yasarcan
line delay
« 56 <
Stock management with a third order supply line delay structure
Stock-flow diagram of a stock management structure with a third order supply line
delay is given in Figure B.36.
a ar
[Sappiy Lie] ay [Suppiy Lin] soa |
1 2
l F xcquisiton Flow 3 |
Control Flow J Acquisition Flow 1! F Acquistéon Flow 2 L
Desired Stock
Stock Adjustment
Acquistion Delay
Supply Line Desited Supply Time
‘Adjustment Line _—
Stock Adjustment
a Time
Weight of Supply
Figure B.36 Stock-flow diagram of the stock management structure with a third order
supply line delay
The model equations are B.74-B.81.
CF = LF +SA+SLA (B.74)
*
SA= = (B.75)
sat
*
SLA = wsl x SEAS (B.76)
sat
SL* = adt x LF (B.77)
SL = SLI + SL2 + SL3 (B.78)
Block Diagrams of Generic System Dynamics Models -57-
Sema Mehmet and Hakan Yasarcan
SLI
r= SEt (B.79)
adt/ Order
Arg =_—_S!? _ (B.80)
adt/Order
4p3 = —S3__ (B81)
adt/Order
where CF stands for “Control Flow”, LF stands for “Loss Flow”, SA stands for
“Stock Adjustment”, SLA stands for “Supply Line Adjustment”, S* stands for “Desired
Stock”, S stands for “Stock”, sat stands for “Stock Adjustment Time”, ws/ stands for
“Weight of Supply Line”, SL* stands for “Desired Supply Line”, SL stands for “Supply
Line”, adt stands for “Acquisition Delay Time”, SZ/ stands for “Supply Line 1”, SZ2
stands for “Supply Line 2”, SL3 stands for “Supply Line 3”, AF/ stands for “Acquisition
Flow 1”, AF2 stands for “Acquisition Flow 2”, AF3 stands for “Acquisition Flow 3”.
The diagram in B.36 and equations B.74-B.81 define a stock management structure
with a third order supply line delay. The simplified set of differential equation that
corresponds to this structure is given in equations B.82, B.83, B.84, and B.85.
@ _yp3-pp-—53___ yp (B.82)
dt adt/ Order
*_ _
ASLI _ op Api = LP +55 4 ws xP SE__ SLI. g3
sat sat adt/Order
ASL2 _ ype qpp = SE! SLD (B.84)
dt adt/Order — adt/Order
@SI3 _ 4p9— p32 52 ____SI3__ (B.85)
dt adt/Order adt/Order
Block Diagrams of Generic System Dynamics Models -58-
Sema Mehmet and Hakan Yasarcan
Block diagram of the stock management structure with a third order supply line delay
is given in Figure B.37.
Figure B.37 Stock-flow diagram of the stock management structure with a third order
supply line delay
Block Diagrams of Generic System Dynamics Models -59-
Sema Mehmet and Hakan Yasarcan
