An easy and formal method for generating structural
high leverage policy in system dynamics models
Showing H. Y oung, Associate Professor
Chia Ping Chen, Doctoral Student
Department of Business Management
National Sun Y at-Sen University, Kaohsiung, Taiwan
Fax: 886-7-5252367; E-mail: syoung@ mail.nsysu.edu.tw
Abstract
Designing high leverage policy is a very crucial and challenging step in system
dynamics approach. However, very limited formal methods were developed in this
area. Literature showed three kinds of these methods: the algorithm method, the
mathematical method, and the guideline method. The algorithm method is easy to use
and suitable for nonlinear models, but can only obtain “parameter policy”, not
“structural policy”. The mathematical method can obtain “structural policy”, but is
not easy to use and not suitable for nonlinear models. The guideline method uses
guidelines induced from some special cases to design policy; it is easy to use, but its
generality is very weak. The objective of this research is to develop an easy and
formal method for generating structural high leverage policy in system dynamics
models. The idea of the method came from our experimental studies of microworlds.
In which we observed that if the subjects repeatedly play a microworld by trial and
error, they could often implicitly lean how to control the microworld even when they
did not know the underlying structure. This kind of cognitive behavior is useful for
controlling system dynamics models. So we imitate it to develop a conceptual
framework for generating structural high leverage policy. And then we follow the
conceptual framework to direct the development of our method. In short, this kind of
cognitive behavior has two major activities: selecting information and organizing
information. We adopt the genetic algorithm as the mechanism for selecting
information. This algorithm is suitable for searching huge solution domain;
probabilisticly searching with natural selection, but without blind search; being able to
obtain global satisfactory, not local optimal solution. We adopt the back-propagation
algorithm as the mechanism for organizing information. This algorithm is a kind of
artificial neural network with learning capability, and has showed its effectiveness in
system control. The actual operation of this method is as follows. First, we use the
conventional optimization algorithm to obtain the optimal trajectory of decision
output, so called the open-loop solution. Then we use a hybrid software of genetic
algorithm and back-propagation algorithm developed in this research to find out a
decision function that produces approximation to the optimal trajectory, what is called
close-loop solution. In the hybrid software, genetic algorithm is used to find out the
independent variables of the decision function from the observable level variables in
the system dynamics model, and back-propagation algorithm is used to generate the
functional relationship among the independent variables. We have applied the method
to the model developed by Forrester in his paper “Market Growth as Influenced by
Capital Investment’. The result showed that the performance of the policy obtained
by our method is better than that of Forrester’s policy.
Introduction
It seems crucial and challenging to design a high leverage policy for improving
the system behavior in system dynamics area. However, very limited formal methods
were developed in this area. This study developed a formal and easy method
combining the genetic algorithms and the artificial neural network to obtain a high
leverage closed loop policy.
Table 1 shows the position of the literature about the formal methods obtaining
the high leverage solution in system dynamics. We positioned these according to two
dimensions: one is the solution level, the other is the approach.
Table1: The category of the formal methods about designing the high leverage policy
Approach|Optimal algorithm Mathematical method —|Guideline from
Solution level from the control theory _|simulation experiments
Open loop Burns & Malone(1974)
»Bradford Group(e.g.
Coyle, 1985)
Closed loop |Wire *Talavage(1980)
Mohapatra &
Sharma(1985)
This study s method Macedo(1989)
+Ozveren &
Sterman(1989)
Wire and »Graham(1977)
flow #ranco(1990)
System's boundary
The three solution levels are open loop solution, closed loop solution and
system’s boundary. The open loop solution means that the solution function has not
the variables from the system. If the system is fluctuated by some little impact, the
open loop solution without information feedback can not adjust itself to the new state.
So it’s not robust. The closed loop solution means that the solution function has the
variables from the system. The system’s boundary solution means that the solution
may lie outside the structure, or be generated from reconstructing the structure.
There are three kinds of approaches: the algorithm methods, the mathematical
methods and the guideline methods. The algorithm methods are most friendly for
users, and they can deal with the nonlinear system, but they can not obtain the closed
loop solution. The mathematical methods almost originated from the control theory,
they are difficult for the users without advanced mathematical background. They can
not directly deal with the nonlinear system, however they can obtain the closed loop
solution. The guideline methods induce guidelines from some special cases; they are
easy to use, but their generality is very weak.
From the users’ viewpoint, the algorithms methods are easy to use, so this
study’s method will base on these, but can obtain the closed loop solution, that is,
positioning on the gray cell in Table 1.
Method
Imitating human brain
According to our past research about human beings making decision in the
microworld, we found the subjects’ performance increased through many times
practice in the microworld experiments, although they may not know the underlying
structure of the microworld (e.g., Y oung et al., 1991; Wang and Y oung, 1992; Y oung
et al., 1993; Young et al, 1994; Young and Wang, 1995; 1996). The finding shows
that human brain seems to be an effective controller for the microworld with the
characteristics of dynamics complexity. So this study’s method will imitate the human
brain.
Working hypothesis of human brain
We now construct a working hypothesis about the human brain how to operate in
the microworld according to our observation in experiments and some literature
(Forrester, 1961; 1964; 1968; Sterman, 1989; Morecroft, 1988). Then we will develop
the study‘s method based on the working hypothesis.
Fig. 1-(a) shows the working hypothesis of human brain. There are two layers in
human brain: one is the superficial cognitive activity; the other is the deep cognitive
mechanism. Both layers are influenced by the whole system objectives, for example,
the growth of system, the stable of system, or making a profit, and so on. We can
identify and observe at least three superficial cognitive activities in the microworld
experiments. They are screening, interpreting, and organizing in sequence. When
human being interacted with the microworld, his brain screened, interpreted, and
organized the information of the system state of the microworld, and then made
decision, which led to changes of the system state of the microworld. The changed
system state produced new information and make the interaction continue.
The underlying layer of the superficial cognitive activities is the deep cognitive
mechanism. The behavior that the subjects adopted to improve performance in the
microworld is a kind of implicit leaming. The operation of neural network of human
brain may account for the implicit learning by some research (e.g. Cleeremans, 1993).
We take this hypothesis to develop this study’s method.
The conceptual framework of the method
Fig. 1-(b) shows the conceptual framework of the method, which is based on the
working hypothesis of human brain. In the conceptual framework, we use the genetic
algorithm and the artificial neural network to serve as the underlying mechanism of
the screening activity and the organizing activity respectively.
The genetic algorithm is a kind of the optimal search algorithms. It imitates the
natural evolution and hereditary to search the potential solution space and obtain the
optimal solution. The potential solution space is analogous to the population of some
being. The objective function is analogous to the natural environment. According to
the principle of evolution- survival of the fittest, the survival individuals can produce
their child generation through mating, crossover, and mutating. After some number of
generations, the process converges. The best individual hopefully represents the
optimal solution. The advantage of the genetic algorithm is that it is suitable for
searching huge solution domain; probability searching with natural selection, but
without blind search; being able to obtain global satisfactory, not local optimal
solution (the reader is referred to Goldberg (1989) for further details of the genetic
algorithm).
The artificial neural network imitating the neural network of the human brain
consists of numbers of the processing elements. The processing element imitating the
neuron is the most basic operator in the artificial neural network. Its operation is to
sum up all inputs value weighting by each connecting weight, and then transfer the
sum value to an output value. This study adopted the back-propagation algorithm--a
kind of artificial neural network. The network architecture of this algorithm is that the
network consists of numbers of layers, and the layer consists of numbers of the
processing elements. The layers between input and output of the network are called
the hidden layers. Every processing elements of the same layer can not connect each
other. The stimulus information propagates forward element by element through the
network, and emerges at the output layer of the network as an output value. The error
information is the gap between the output vale and the desired value, and propagates
backward layer by layer through the network, and adjusts every connecting weight.
The advantage of the back-propagation algorithm is that it can serve as a substitute of
human brain; its effectiveness for system control is supported by some evidences; and
it can approximate any continuous function to map input value to desired output value
(the reader is referred to Haykin (1994) for further details of the artificial neural
network).
The operational framework of the method
The learning of back-propagation algorithm is a kind of supervised learning. The
supervised learning must provide the neural network with a set of desired sample for
adjusting output error. However, the conceptual framework does not have the
mechanism to produce the target decision value. So we add the optimal algorithm into
the conceptual framework to resolve above problem. Fig. 1-(c) shows the operational
framework of the method, which is modified from the conceptual framework.
human brain microworld
whole system
objectives se
+
/ ad oe
ation
Le \
deep cognitive mechanism
superficial cognitive activities
Fig.1-(a): The working hypothesis about human brain interacting with microworld
whole system
objectives
output
3
2
Z
i=]
S
computer
model
information
Fig.1-(b): the conceptual framework of the method
Faia
whole system
objectives —
decision__|
i 5 — open oral open computer
cs Fi yo error fond algorithm loop aioel
4h er Lore .
Pi solutio solution
jae a
reg" a pt
CGA see
Ler sere ‘<
Le information
Fig.1-(c): the operational framework of the method
The operational process--two stage
According to the operational framework, we can further develop the operational
process. There are two stages in the process. The first stage is to obtain the open loop
solution of the system by the optimal algorithm. The second stage is to obtain the
closed loop solution based on the open loop solution by GNN (Genetic Neural
Network software, combining genetic algorithm and artificial neural network. We will
discuss more about it later).
How to obtain the open loop solution
There are two kind of the open loop solution: the trajectory solution, and the
constant value. In order to obtain the trajectory open loop solution, we adopt Fourier
series to substitute original policy acting on the sensitive variable. The Fourier series
can approximate any function. We use Powell algorithm (Vensim provides the
function) to estimate the optimal parameter value of the Fourier series.
As to the objective function, we use L»-gain as the objective function of the
stable oriented models. L»-gain is defined as Eq.(3), in which the numerator represents
the norm of output variable, and the denominator represents the norm of input
variable. If L2-gain is less than 1, then the system between input and output dissipates
energy (Schaft, 1992). According to Lyapunov’s direct method, if a system dissipates
energy, then the system will approach stable. We use conventional norm H) as the
objective function of the growing oriented models. Eq.(1) is the objective function,
the first term of the right-hand side represents the profit, the second represents the
cost, and the third represents the growth potentiality.
How to obtain closed loop solution
We use GNN based on the open loop solution to obtain the closed loop solution,
which mathematical implication is shown in Fig.2. Given the desired trajectory of the
decision point and the optimal trajectory of the other variables in the system, the
problem is to obtain some variables and the functional relationship between these
variables and the decision variable. So we develop GNN to solve the problem.
The illustration of GNN
We combined the genetic algorithm and the artificial neural network to develop
GNN software. Fig.3 shows the calculating process. Firstly, a group of sets of
variables are randomly produced to be the variables of potential closed loop solution
function by the genetic algorithm. Then the artificial neural network adjusts the
functional relationship among the variables until the error between the function output
and the desired decision value can not be decreased. Last, the genetic algorithm
selects some sets of variables from original group according to the final error of each
function. Meanwhile, the genetic algorithm uses these sets of variables to produce
another sets of variables through mating, crossover, and mutating. Then GNN
continues to run next cycle until the genetic algorithm converged.
We can use an imaginary experiment to illustrate the calculating process of
GNN. Let us imagine that a group of subjects play the same microworld, but they are
restricted to look at different sets of information. These subjects play repeatedly until
their performance can not be improved. The researcher selects some sets of
information based on the subject’s performance, and changes these to new sets of
information through the operations of mating, crossover, and mutating. Then the
experiment uses new subjects and new sets of information to run next iteration.
When we obtain the variables of the closed loop solution function by GNN, we
can base on the variables and the open loop solution to obtain the connecting weights
of the variables by any artificial neural network software. With the variables, the
connecting weights, and the network architecture, we can construct a high leverage
policy.
Fig.2: The mathematical implication of the method about the second stage
(4) |
2
GA ANN £(x,*())—u(t)
(5) (3)
error u*(t)
Fig.3: The calculating process of GNN
Result
In the following we apply the method to two different types of model: the
growing oriented model--Forrester’s market growth model (Forrester, 1968), and the
stable oriented model--Forrester’s customer-producer-employment model (Forrester,
1961). The operational process and every step results are summarized in Table 2.
Table 2: Operational process and each step results
Stage | General operational Growth Model: Stable Model:
process Forrester’s market growth model | Forrester’ s customer-producer-
employment model
Open
Loop
Select one sensitive
decision variable.
Cancel the policy acting
on the decision variable.
PCR: production capacity
ordering
LDF: labor desired at factory
Set up the objective
function.
Eq.(1)
Eq.(3) or Eq.(4)
Use Fourier series to
replace original policy
acting on the variable.
Eq.(2) and T=150
Eq.(2) and T=350
Use Powell algorithm to
estimate the optimal
parameter value of the
Powell and
Multiple_start =V ector
Ag, -, A7=0 0, 1e+5]
Powell
|A 9=750 4400, 1200]
Ay, .. Az, By, «.. B7=0 4-20, 20]
Fourier series. By, ... B7=0-1e+5, 0]
Obtain some open loop |Fig.4 Table 4
solutions by different
methods or different
objective functions.
Select a satisfactory [The simulation output of the The simulation output of the
open loop solution.
heuristic mathematical method
(Young and Chen, 1998) is best.
objective function: “min
gain(IA FPC)” is best.
Preprocess the data of
the selected variables
from open loop solution
for GNN.
Selecting 24 variables
Variable
transformation:y;(t)=In(x;(t))
2;(t)=(yj(t)-yjmean)/yjs.d.
Iwy(t)=1/(1 +e 21)
Selecting 47 variables
Variable transformation:
2; (t)=(x;(t)-xjmean)/xjs.d.
wj(t)= 0.1+ [(2;(t)-zjmin)/ (zjmax-
zjmin)](0.9-0.1)
Closed ithe training data set: 73 patterns _|the training data set: 176 patterns
Loop ithe testing data set: 145 patterns _|the testing data set: 351 patterns
delete former 5 patterns
Set up the parameter [Table 5 Table 5
value of GNN.
Run GNN.
Select the set of GNN converged at the 11™ GNN converged at the 8"
variables with best generation. generation, and again diverged at
performance from the |From the 11" to 80", total 5600 [the 21".
result of GNN. sets, we selected the best set of |From the 8" to 20", total 1222
variables: BL, DRA, DRC, sets, we selected the best set of
SEDM, and revenue. variables: DFOF, EDPC, FRFIF,
ILLF, and RMPAF.
Obtain the connecting Fig. 5 Fig. 6
weights of the variables
by any ANN software.
Construct a potential
high leverage policy
with the variables, the
connecting weights, and
network architecture.
Test Reproducing test Fig. 7: good Table 6: good
Robustness test Fig. 8: good Table 7: not good
Table 3: Equations indicated in Table 2
max J = [ (DR4x100) dt — i. (Pco x1000) at — i (seM —sE) at
t x2 xm t x4 xm
) +A, xCOs( )
tx6 Xn, +A, xcos(POX a4. xcos(t1on
Fourier function =A0 +A, xCOS(
+A, xCOS( )
t x12 xn t x14 xn tx2 x1
+A, xCOS( )+A, xCOs( ) +B, XSIN( )
tx4xn t x6 xm t x8 xm
+B, xSIN( ) +B, xSIN( ) +B, xSIN( )
txt) 4B, xsiv(L2 +B, xsiv(ten
+B, xSIN( )
f (x(¢) 100)? at
min gain(x) = | —________
f (INPPC(t) -100)? ae
x(t)=BLCPC(t) or CASPC(t) or DQDFPC(t) or IAFPC(t) or MENPC(t)
or
max NPTDF = fPNPRF oat
(1)
Comparison between two Methods
3.091 M BN | |
348.18 se 4
1.547 M
342.74 se aE
3,480
337.31 se
0 20 40 60
Time
-M: the heuristic mathematical method; -A: this study’s method
Fig.4: Comparison of the growth model’s open loop solutions between this study’s
method and the heuristic mathematical method developed by Y oung and Chen (1998).
Table 4: Comparison of the stable model’s open loop solutions with
different objective functions
result}! NPTDF gain gain gain gain gain
jective BLCPC_| CASPC |DQDFPC| IAFPC_| MENPC
max NPTDF 4710010.5| 4.5281 1112.3) 2.5858] 567.34] 11.633
min gainBLCPC 35085315] 0.2828) 3.1322] 0.2506) 0.6646) 0.4694
min gainCA SPC 3526868.5| 1.1696] 0.9483 0.428) 1.1432) 0.5872
min gainDQDFPC 3546074| 0.8504) 1.9827, 0.259] 0.7031) (0.5127
min gainIA FPC
3521606.3| 0.7356| —1.9707| 0.2096] 0.5277| 0.4281
min gainMENP'
Cc 3560326.5] 9.2369] 14.112} 5.1947} 6.8034] 0.0036
Table 5: The parameter value of GNN
models} The growth | The stable
GNN parameter model model
the parameter of genetic algorithms
population size 80 94
chromosome length 24 AT
generation number 80 25
mating probability *crossover probability 0.7*0.6 0.9*0.9
mutation probability 0.01 0.03
chromosome [fixed or not? fixed fixed
type fixed number (fixed) Ng 7
penalty multiplier (fixed) 1.2 12
reproduction |Proportional: convergent pressure 3 1
methods _|rank-based: no.1 number
the parameter of neural network
learning rate 0.05 0.1
momentum parameter 0.01 0
noise factor 0 0
stop criteria |error tolerance 0.001 0.001
maximum learning cycles 100 500
initialized number 3 1
network architecture-layer number 3 3
neurons number of input layer 24 AT
neurons number of hidden layers 3 3
neurons num!
ber of output layer A.
<BL> <DRA> <revenue> <DDRC> <SEDM>
PCO
PCO =EXP(LN(OUT/(1-OUT))*1.71877+9.27733)
OUT = 1/(1+EX P(-1*(H1*3.98928+H2*-4.50871+H3*-3.01871)))
H1 = 1/(1+EX P(-1*(W 1*0.305184-+W 2*0.543068+W 3* 0.363065 +W 4*1.00581+ W5*0.998152)))
H2 = 1/(1+EX P(-1*(W 1*-1.65855+W 2*-0.0559+W 3*-1.5359-+W 4* 1.50866+ W5*2.67165)))
H3 = 1/(1+EX P(-1*(W1*0.105585-+W 2*-0.83289+W 3*-0.85289-+W 4*-0.33741 +W5*-0.95505)))
W1 =1/(1+EX P(-1*(LN(BL)-11.9895)/1.72565))
W2 = 1/(1+EX P(-1*(LN(DRA)-11.2542)/1.72596))
W3 = 1/(1+EX P(-1*(LN(revenue)-15.8594)/1.72596))
W4 = 1/(1+EX P(-1*(LN(DDRC)-0.7363)/0.003451))
W5= 1/(1+EX P(-1*SEDM))
price = 100
revenue = DRA*price
Fig.5: Diagram and equations of the growth model's high leverage policy
<DFOF> <EDPC> <FRFIF> <LLF> <RMPAF>
H1 = 1/(1+EX P(-1*(X 1*w11+X 2*w124X 3*w134X 4*w144X 5*w15)))
H2 = 1/(1+EX P(-1*(X 1*w21+4X 2*w224X 3*w23+X 4*w244X 5*w25 )))
H3 = 1/(1+EX P(-1*(X 1*w31 +X 2*w32+X 3*w33+X 4*w344X 5*w35)))
OUT = 1/(1+EX P(-1*(H1*wh1+H2*wh2+H3*wh3)))
X1 =0.8*((DFOF-x1m)/x1s-y1min)/(ylmm)+0.1
X2 =0.8*((EDPC-x2m)/x2s-y2min)/(y2mm) +0.1
X3 =0.8*((FRFIF-x3m)/x3s-y3min)/(y3mm)+0.1
X4 =0.8*((LLF-x4m)/x4s-y4min)/(y4mm) +0.1
X5 =0.8*((RMPAF-x5m)/x5s-y5min)/(ySmm)+0.1
x1m=4.71389282, x2m=30093.22159, x3m=0.698456352, x4m=1.116144058, x5m=3005.067463;
x1s=0.070924391, x2s=1068.491835, x3s=0.007880488, x4s=1.428607458, x5s=113.762688;
ylmin=-1.92222158, y2min=-1.524071469, y3min=-1.936169338, y4min=-0.781281136,
ySmin= 1.807770392;
ylmm=3.858390322, y2mm=3.101063413, y3mm=3.858390113, y4mm=3.732034352,
y5mm=3.495289642;
wl1= 0.171587, w21=4.985481, w31= 1.218813
w12= 2.758465, w22=-0.780286, w32= 4.973371
w13=-0.724769, w23=5.785361, w33= 1.145750
wl4=-4.233764, w24=3.310918, w34= 1.346264
w15=1.145976, w25= 6.089832, w35=-8.820408
whl= 4.745557
wh2=-6.000803
wh3=-3.972231
Fig.6: Diagram and equations of the stable model’s high leverage policy
Reproducing test
1.726 M
350 se
OB=C10Se
859,436 ‘i
345 se E-closed =
-7,176
340 se :
733
Time
Fig.7: Result of reproducing test of the growth model
Robustness- 1
5M _ ob-1
83.27M
2.501M_ ob-1
ob-1
months
Robustness-2
SE:RS=0.12
SE:RS=0.24
SE:RS=0.36
SE:RS=0.48 -
20 40 60 80 100 120 140
months
Fig.8: Result of robust test of the growth model
Table 6: Result of the reproducing test of the stable model
scenario: 2-year sine curve
NPTDF /gainB |gainC |gainD gainl |gainM
original model baserun _—_[ 3345853.5| 10.579) 7.7413] 3.13] 2.674} 2.0087
original model new policy |3501079.8| 1.7019) 2.5091) 0.7566} 1.8962] 0.5125
open-loop solution 3521606.3} 0.7356] 1.9707| 0.2096] 0.5277} 0.4281
closed-loop policy 3552401.3] 1.2381) 3.3604) 0.5904; 1.419] 0.5208
Table 7: the result of the robust test of the stable model
scenario: 4-year sine curve
NPTDF /gainB |gainC |gainD gainl |gainM
original model baserun__[ 3616565.3| 4.4418) 3.9684) 1.2668/ 1.2496] 1.0473
original model new policy |3622099.3| 3.2196] 2.726] 1.3318| 3.0075] 1.2072
closed-loop policy 3618647.3} 4.9186] 10.134) 2.4356] 4.3962] 0.5423
Discussions
In the simulation experiments, we found some interesting phenomena worth
discussing.
I.
Besides the policies of the system, the system performance is influenced by the
objective function. Table 3 shows that if we take gain (IA FPC) as objective
function, then the system performance is best. However, if we take NPTDF as
objective function, then the system performance is worst.
In the second stage, we use the best set of variables by GNN to construct a high
leverage policy. We found that the best sets of variables by GNN are different
every time. However, these policies from different sets of variables made similar
good performance. From above simulation experiments, we infer that there exist
more than one high leverage policy in nonlinear dynamic system.
It is hard to explain the result of the method, because the method includes two
nonlinear algorithms. The reason for the difference of the robustness between the
two the model simulation is that, we guess, the artificial neural network has the
capability of pattern recognition. It is very similar among the growth model’s
pattern of behaviors in different situations, but it is very different among the
stable model’ s. So the growth model is robust, but the stable model isn’t.
The policy constructed by the best set of variables from GNN may not pass the
testing stage. In this time, we suggest to add one cycle loop process between the
second stage and the testing stage, that is to select next best set of variables from
the output of GNN to construct another policy, and then test it.
Summary and further study
This study’s method imitates human brain. According to the working hypothesis
of human brain, we construct the conceptual framework of the method, and then base
on it to develop the operational framework and process. There are two stages in the
operational process: the first is to obtain open loop solution, the second is to obtain
closed loop solution by our GNN software.
The result of the method's test shows that the reproducing is good, and the
robustness of is half good and half bad.
In the users’ view, this study’s method is friendly and easy. Users without
enough mathematical background can use the method by following the procedure. We
will replace the routine of the procedure with a computer program in the future. So the
method will be easier for users.
However, the method has two flaws. One is that it is hard to explain the result of
the method. The reason results from two nonlinear algorithms constituting the
method. Maybe the flaw can be resolved until the two algorithms have well-developed
theorem. The other flaw is that GNN takes much time to run in PC. If we use higher
speed computer, the flaw will be resolved.
As to the management implication of the method, in addition to derive high
leverage policies, there are two themes: how to screening decision relevant
information, and how to construct the objective hierarchy of organization. Now we are
studying these themes.
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