574 THE 1987 INTERNATIONAL CONEERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
THE PARAMETER QUASI-OPTIMIZATION
FOR SYSTEM DYNAMICS MODELS
Qifan Wang Guangle Yan
System Dynamics Group
Shanghai Institute of Mechanical Engineering
Shanghai, China
ABSTRACT
With the sustained development in science and technology, the
method of system. optimization has more and more widely been
applied to the area of science, technique, engineering, economy,
etc, The optimization theory is strongly supported by the birth
and the dévelopment of computers. System dynamics models are good
at understanding complex systems with the characterisics of
high-order, multi-loops and nonlinear (say socio-economic
systems), The purpose of this paper is to combine. the modeling
process of a system dynamics model with the optimization method
so as to make the system synthesis more perfect. Because of the
specific properties of large scale systems, there are some
serious difficulties in completely optimizing systems. In many
cases, it is impossible to find an overall optimization for
complex systems. However the quasi-optimization for dynamic
systems is still available. This paper develops some ideas of the
parameter quasi-optimization for system dynamics models and
presents a practical method. Its advantages include that the goal
of the parameter quasi-optimization is clear, the precision is
controllable, the quasi-optimal indices are conveniently
regulated, the whole process can be automatically completed by a
computer, repeating computations and simulations are not needed.
Besides system synthesis, this method can also be applied to
system analysis such as the parameter quasi-optimization after
decoupling a system, selecting the dominant loops, separating the
interest substructures, etc. The final goal of this paper is to
make a solid fundation for a common used modeling software
package.
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS 'SOCITY. CHINA 575
PARAMETER QUASI-OPTIMIZATION
FOR SYSTEM DYNAMICS MODELS
INTRODUCTION
The system simulation technique has been more and more widely
applied to analyzing and synthesizing. complex systems, While
there are still much to do in the combination of system
simulation and system optimization. The complete process of
system analysis and system synthesis mainly contains three
phases: modeling, simulating and optimizing, simplely note M-S-O.
Here exist two relative links. One is the combination of M and S,
called "Decision Support System Based on Simulation." This link
has been paid much attention to in modeling and analyzing system
dynamics models. The other is the combination of S and 0, called
"Simulation Optimization Techniques." Usually the system
optimization of system dynamics models relies on subjective
experience. First estimate some sets of system parameters,
compare all the the results, then select the best one. Obviously
this "trial and error" method is time-consuming and costly. It is
actually difficult to get the most optimal results. Therefor the
problem of deeply studying the simulation optimization techniques
has laid before us.
SYSTEM OPTIMIZATION AND QUASI-OPTIMIZATION
System optimization is actually to seek a optimal or
quasi-optimal one from all the results in a system by the help of
some mathematical algorithem for evaluating extreme values. There
are some characteristics in system optimization. First, system
optimization is governed by the objective functi:as. Different
objective functions lead to different optitimal scheme and
produce different results, A scheme is optimal for some objective
functions and is not optimal, even is bad, for some other
objective functions. Second, system optimization is governed by
the time period system working. Any optimal scheme always relates
to some relative time period. Long term optimal schemes generally
contradict short term optimal schemes. In addition, system
optimization is governed by system boundaries. An optimal scheme
for subsystems ‘is not always optimal for their whole system.
There exist various complicated connections, exist the exchanges
of imformation and energy between subsystems or subsystems and
the whole system, It can be proved in theory that the local
optimization of a system is global optimal if and only if the
system is definited in a convex set, otherwise the system has no
such a property.
System optimization technique is actually a kind of mathematical
method, Some optimal schemes generally exist in a system in
theory for some special time periods, special space and special
676 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
objective functions, but practically systems do not always work
by the theoretical optimal schemes. The main reasons include:
1) Various indices in a system usually restrict, sometimes
contradict one another. It is impossible to make all the
indices arrive at ideal optimization.
2) The optimal results in a system for a long term period
generally go against those for a short term period.
3) Actual complex systems are difficult to be satisfied with
convex conditions,
4) Under the permission of technical indices, an experienced
modeler would rather take the advantages of simple
mathematical calculations and convenient practical
applications than make the model compeletely accurate with
a higher cost.
5). Sometimes mathematical calculations are difficult for complex
systems, therefor some approximate calculations have to be
adopted.
6) The realizations of theoretical optimal schemes are sometimes
difficult because of the limitations of facilities
conditions.
Because of all the reasons above, real complex systems can only
arrive at an area near some optimal operative point in stead of
at the completely optimal operating point. Naturally the final
results can only be quasi-optimal.
However the limitations to the operation of real systems are not
extremely strict. Some schemes work very well in practice, but
they are not theoretically optimal. Therefor if we say studing
and exploring system optimization has theoretical meaning in
guidenees and elicitation, studying and exploring system
quasi-optimization has practical meaning.
PARAMETER OPTIMIZATION ALGORIYHM
System dynamics models are good at understanding large scare
complex systems with the characteristics of high order, multiple
loops and nonlinear. The more complex a system is, the farer the
distance. between its "cause" and "effect" either in time or in
space. A behaviour change in a complex sysyem may be caused by
some trivial factors, The present behaviour of a complex system
may be caused by the changes of various factors long before. This
counterintuitive behaviour intrinsically existing in complex
systems make the realization of system optimization greatly
difficult,
The characteristics of a complex. system are that it contains
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 577
complicated nonlinear feedback loops and a lot of variables and
parameters. Under the influence of nonlinear characteristics, on
one hand, systems are insensitive to the changes in many system
parameters. They stubbornly resist the policy changes. On the
other hand, systems are specially sensitive to some "leverage
points." This two-flod properties of a complex system enlighten
us that we can avoid the resistive actions of insensitive
parameters and effectively quasi-optimize systems only by
carefully refining a few sensitive parameters,
Because of the nonlinear properties, the gradient information in
a complex system can not be easily obtained. Algorithms without
calculating derivatives can be only used either to optimize or to
quasi-optimize parameters. Here we adopt the variable polyhedron
algorithm,
The basic ideas of the variable polyhedron algorithm are like
this: In an n-dimensional space E", construct a polyhedron with
n+l apexes, in which any n apexes do not lie in an
(n-1)-dimensional sub-space of E" . Calculate the indices value
for each apex. Select an apex which produces a maximum indices
value, and reflect it through the core of other apexes with the
hope to find another apex which can produce a smaller indices
value so as to construct a new polyhedron. Step by step-we can
get some better results. If we do not get satisfied apexes after
some reflection, the polyhedron should be condensed or expanded
to form a new polyhedron. Repeating this process, the apexes of
the polyhedron would convergent to the local minimum value of the
objective functions. The flow chart of the variable polyhedron
optimization algorithm is shown in Figure l.
PARAMETER QUASI-OPTIMIZATION OF A TYPICAL MODEL
According to the basic ideas of system parameter
quasi-optimization, we take the famous Prey-Predator model for
example to test the simulation quasi-optimization program
The problem of prey and predator was first put forward by Italian
biogist Umberto D'Ancona in 1920's, At that time he observed the
ten years' data of the fishing ammount at the port of Fiume from t
1914 to 1923. He found the ratio of the fishing ammount of
predator fish (e.g. shark) during the war period is much higher
than in peace time. To make it clear, Italian mathematician Vito
Volterra did a lot of research work, Finally he developed a
mathematical model to discribe this problem:.
k=ax-by-ex=(a-e)x=bxy
ya-cy+dxy-ey=-(c+e)y#dxy
where x is the state of prey, y is the state of predator, e is
the rate of fishing, a, b, c and d are constent factors, Its
system dynamics flow chart is shown in Figure 2.
578 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
This is an extremely typical system. A special model of this kind
of systems is Rabbit-Lynx model. Now we take it for example to do
the simulation optimization tests. The simulation structure of
this model is shown in Figure 3, where T are TABLE functions,
This system has strong nonlinear properties. It contains prey
RABBIT and predator LYNX two level variables. The outputs of the
two level variables in the ideal case are taken as optimization
indices. Two factors CC (Carrying Capacity for Rabbits) and RRPLS
(Required Rabbits per Lynx for Subsistence) are taken as
controllable parameters. With the help of the variable polyhedron
algorithm, we simulated and quasi-optimized the model by
regulating the controllable parameters for some different initial
parameter values. Four tests had been done. All the results are
shown in Table 1. The outputs of level variables RABBIT and LYNX
under ideal conditions (A) and under the 3rd test conditions (B)
are separately shown in Figure 4 and Figure 5, We can see from,
all the results of the tests, the model after parameter
quasi-optimization has been very near to the ideal system.
All the tests mentioned above were finished by the use of a
simple micro-computer Apple-II. Each test spent about five
minuts., The process of simulation quasi-optimization was
programed in the FORTRAN language. The computer program is
common-used. It can deal with various nonlinear units. It is an
economical and practical tool in the realization of parameter
quasi-optimization of system dynamics models.
Conclusion
Parameter optimization of system dynamics models is an important
part of system analysis and system synthesis. System optimization
improves system qualities and precision so as greatly to meet the
gap between dynamics models and real. systems. Compared with the
"trial and error" method in parameter adjestment, parameter
optimization (or quasi-optimization) has many advantages which
include that the goal of the parameter quasi-optimization is
clear, the precision is controllable, the quasi-optimization
indices are conveniently regulatable, the whole process can be
automatically completed by a computer,repeating computations and
simulations are not need. In addition; parameter optimization can
also be applied to system analysis, such as the parameter
quasi-optimization after decoupling a system, selecting the
dominant loops, separating the interest substructures, etc. The
computer program for simulation and parameter quasi-optimization
developed in this paper is flexible. It can simulate and optimize
many kinds of complex systems by the help of a very simple
micro-computer. It is an economical and practical tool in
simulation and parameter quasi-optimization of system dynamics
models,
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY, CHINA'579
TABLE 1.
a a: a Toomeece nana | I
I test I controllable I initial I optimized I ideal I
I I parameters I values I values I values I
I------ [-------------- I-----+---- I+----------I-------- I
Io I cc I 2500 I 2399.89 I 2400 I
T I RRPLS I 180 I 199.70 TI 200 I
[------ J-------------- [--------- Tenennesean [-------- I
T 2 I cc I 2200 I 2400.01 I 2400 I
1 I RRPLS I 220 He 197.24 1 200 I
I------ Ieonenceannnete io I--e------2- Tannen nan I
I3 cc I 3000 I 2400.04 I 2400 I
rT I RRPLS I 250 I 198.87 I 200 1
I------ I---------+---- I--------- I----------- I-------- I
I4 1 cc I 2000 I 2399.86 I 2400 I
I I RRPLS I 150 I 198.56 I 200 I
I------ J-------------- I--------- I----------- I-------- I
Construct a
polyhedron
Construct a
new polyhedron
Calculate the
objective values
for each apex
Reflect the apex for
Max. £(x) through
the core of. other apexes
Conden or
expand the
polyhedron
End
Figure 1.
580 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
Preggtor
Prey
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Figure 2
Te
REPL LM
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Jisvar
Ts Tq,
[nar _[RNBF
us Rupe ~ LX
6 ‘700
£ + IRDEN
inee *] S frye Ian 2S “S fRapprr LZ 1oC.
Figure 3
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THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 581
wee eteneeteceeLeseetoce
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1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
OP.DIV= «1162
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THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 583
REFERENCES
1. Michaol, G. Ketcham, John W, Fowier, Ton T, Phillips, New
Directions for the Design of Advanced Simulation Systems,
WSC'84,
2. Charles R. Standridge, Integrated Simulation Support System
Concept and Example,America Computer Society, 17th annual,
Symposium on Simulation.
3. Alberts Garcia-Diaz, Hogg, G. L., Combined Simulation and |
Network Optimization Analysis of a Production / Distribution
System, Simulation, No.2, 1983,
4. M. Braun, Differential equations and their Applications, 2nd
Edition, New York: Springer Verlag, 1978.
5. Qifan Wang, On the Statistical and Estimative Problems of the
Parameters. of System Dynamics, Proceedings of the 1986 China
National System Dynamics Conference.
6. Qifan Wang, A Study of the Complex Systems and Their
Characteristics from the View of System Dynamics, Journal of
Shanghai Institute of Mechanical Engineering, pp. 47-54,
Vol.9, No.l, 1983.
584 THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA
APPENDIX
* RABBIT-LYNX MODEL
NOTE a
NOTE NAME ~- RABBIT-LYNX MODEL
NOTE AUTHOR - WILLIAM SHAFFER
NOTE THIS MODEL REPRESENTS THE PREDATOR-PREY RELATIONSHIP
NOTE BETWEEN RABBITS AND LYNX. THIS RELATIONSHIP PRODUCES THE
NOTE OSCILLATIONS IN POPULATIONS OF THESE ANIMALS.
NOTE
NOTE #--2 22 - nnn nnn nee en en nnn enn nnn nen
NOTE
NOTE RABBIT SECTOR
NOTE
L RABBIT. K=RABBIT. J+(DT) (RNBR.JK-RKL.JK-RTRAP.JK)
N RABBIT=INRAB
NOTE RABBITS (RABBITS)
c INRAB=700
NOTE INTITAL RABBITS (RABBITS)
R RNBR.KL=(RABBIT.K)(RNBF.K)
NOTE RABBIT NET BIRTHS (RABBITS/YEAR)
A RNBF,.K=TABLE(TRNBF,RDEN.K,0,1.25,0.25)
NOTE RABBIT NET BIRTH FACTOR (1/YEAR)
T TRNBF=1.50/2.40/2.20/1.10/0.00/-1.00
NOTE TABLE FOR RABBIT NET BIRTH FACTOR
R RKL.KL=(LYNX.K)(RKPL.K)
NOTE RABBITS KILLED BY LYNX (RABBITS/YEAR)
A RKPL.K=TABLE(TRKPL,RDEN.K,0,1.25,0.25)
NOTE RABBITS KILLED PER LYNX (RABBITS/LYNX-YEAR)
T TRKPL=000/150/250/325/375/400
NOTE TABLE FOR RABBITS KILLED PER LYNX
A RDEN.K=RABBIT.K/CC
NOTE RABBIT DENSITY (DIMENSIONLESS)
Cc CC=2400
NOTE CARRYING CAPACITY FOR RABBITS (RABBITS)
R RTRAP.KL=(RABBIT.K)(FRABTR)
NOTE RABBITS TRAPPED (RABBITS/YEAR)
Cc FRABTR=0.0
NOTE FRACTION OF RABBITS TRAPPED (1/YEAR)
NOTE
NOTE LYNX SECTOR
NOTE
L
=LYNX,J+(DT)(LNBR.JK-LTRAP.JK)
YNX
N J
NOTE LYNX (LYNX)
THE 1987 INTERNATIONAL CONFERENCE OF THE SYSTEM DYNAMICS SOCITY. CHINA 585
c INLYNX=6
NOTE INITIAL LYNX (LYNX)
R LNBR.KL=(LYNX.K)(LNBF.K)
NOTE LYNX NET BIRTHS (LYNX)
A LNBF.K=TABLE(TLNBF,LSUBR.K,0,2,0.5)
NOTE LYNX NET BIRTH FACTOR (1/YEAR)
Tr TLNBF=-4.0/-0.6/0.0/0.3/0.5
NOTE TABLE FOR LYNX NET BIRTH FACTOR
A LSUBR.K=RKPL.K/RRPLS
NOTE LYNX SUBSISTENCE RATIO (DIMENSIONLESS)
c RRPLS=200
NOTE REQUIRED RABBITS PER LYNX FOR SUBSISTENCE
NOTE (RABBITS/LYNX-YEAR)
R LTRAP.KL=(LYNX.K)(FLNXTR)
NOTE LYNX TRAPPED (LYNX/YEAR)
c FLNXTR=0.0 5
NOTE FRACTION OF LYNX TRAPPED (1/YEAR)
NOTE
NOTE CONTROL STATEMENTS
NOTE
SPEC DT=0.125/LENGTH=30/PLTPER=0.5
PLOT RABBIT=R/LYNX=L(5,25)
OPT TXI=20
RUN
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