On Reliability Improvement of S.N. Model
Li Zhouwei
Xinjiang Institute of Scie-Tech Information, China.
Abstract
This article expounds the necessity of improving
reliability of the S.D.model and based on the concepts of
"BIG SYSTEM", "“STRICTNESS” and “PARAMETER ACCURACY" when
developing a model puts forward some tentative methods to
improve reliability of the S.D. model.
Preface
When we are developing a new mode!, we mean to study
a certain system and to learn its performing mechanism, we
mean to work out the necessary counter-measures and
relevant policies so that we may achieve our set goal and
realize effective control over the system. Then to what
extent will a $.D. model reflect the regular performing
activities of an actual system and to what. extent can the
primary resolusions on the system be taken as reliable?
These will, more than often, influence greatly or even
determine the whole process of decision-making. A
seriously distorted model and the resuits worked out with
it will undoubtedly lead the decision-makers to astray
and this in turn will‘ endanger the realization of the set
goals when developing the model. It is in the nature of
things the reliability of a model to become the utmost
concern for all decision-makers. This article is to
expound some tentative methods for improving the
reliability of the S.D.model.
System Dynamics ‘90
1. Introduction of "Big System" Concept
into Development of S.D. Model
A system, in general, is a setof different yet
interacted parts combined closely to perform a commmon
function for the sane goa!” A system may be big as well
as small depending on comparison®
Any objective we are
studying can be regarded as a smaller system (A), which is
ineluded in a bigger system (B), that is A C€ B,
therefore, the funcrion and behavior of system (A) is
necessarily restricted by system (B). In this case he who
wants to obtain a comprehensive and accurate understanding
of system (A) must not confine his study in system (A)
only, instead ,he should make considerable efforts to
study system (B), especially factors closely related to
system (A) and other interaction and interrelation between
the two systems. Only in this way can he get to know the
mechanism of system (A) by nature and the behavior of
system (A) on the whole; can he avoid being one-sided and
ignoring the relevant key factors when developing a model
hence the great improvement of reliability in developing a
model. For example, once we are set to study the tendency
of cotton textile industrial development ina certain area
then work out the effective counter measures, we shoule be
engaged not only in the research into the development
pattern of the cotton textile industry, into the raw
material supply, into the marketing, and technological
development trend, but also in the research into the
development trend of the cntire industry including woolen
tixtile, flax textile, silk weaving and artificial fabrics,
and into their effects on the cotton textile industry. For
another example, the study of population. In order to gras
666 System Dynamics '90
the core of the population growth tendency accurately and
put forward counter-measures for population control right
to the point, one must not only study the development
pattern and tendency of the population system proper, but
also study the supporting abilities and shifting trends of
those systems as economics, society and national environment
that the population system falls within, and study their
influences upon the population growth
Generally speaking, when we are making a research into
problems concerning regional economic development
strategics and counter-measures, it is altogether
necessary for us to perform a careful study of the
economic system, which includes all cconomic sectors in
the region. Yet it is far from enough. We must take all
those factors following into our consideration as well:
the regional scie-tech system and the economic society
system that closely related to the economic system; the
State and even international scie-tech system and the
economic society system; and their ever shifting trends;
the post-effects on regional economic development; and
interrelationship changes triggered off by the State and
international systems. It is boosting to say that we have
had a thorough understanding of the regional economic
development trends before we have all abovementioned
researches done.
2. Introduction of "Strictness" Concept
into Development of S.D. Model
The simulation of a S.D.model is of a structural and
fuctional kind. When a S.D. model is being developed,
System Dynamics '90 667
special] attentionshould be paid to the internal micro-
structures of the system and the information feedback
mechanism of the system, in which the behavior pattern of
the system is deeply rooted. Then with the help of the
simuiation technique of a computer, one may analyse the
relationship within the systematic structural funetion
and the dynamic behavior, and work out the countermeasures
to solve the problems. Therefore, the most important things
before using a S.D. mode] to do some research work
methodologically are to study and analyse the system
proper thoroughly and comprehensively so as to grasp its
nature, to know cleariy the interactions of the main
feedback loops and influences upon the systematic
behavior. The author believes that the behavior of any
comparatively independent system would fall into a pattern
of its own, otherwise the system would not be an
independent. one; and the only way to precisely illustrate
the behavior pattern of a system is to use mathematic
expressions, which would. depict. accurately the operating
characteristics of the system. It is evident that a model
such developed would be significantly improved in its
reliability and practicability, and the emulation program
for countermeasures, optimum seeking and countermeasure
suggestions worked out would, no doubt, be of great
importance and reliability to those decision-making
bodies as well as persons. That éxplains why we insist
that much attention be paid to the strictness especiaily
when developing a model.
Let's take population growth for example and study
the natura! birth - death process in a region. If a stands
for the age, + for time, Aa for a considerably small
668 System Dynamics ‘90
span of the age, Aa>0, then the total population between
ages [a,e+Ac] at the certain time t will be p(a,t)Aa,
( p(a,t) : function of the age distribution density of a
population). As time Ar passes on to time ++ Ar. sone
poeple will have died for some reasons, and the death
number will be #(a,1)p(a,t)Aaht ( pla, 4) =: funetion
of a comparative death rate), and the rest keeping alive
by the time s+Ar will be at the age of [e+ Aad,a+Aa
+ Aa’']., since both a and t have the same dimension,
dafdr=1 , thereforth Aa’=Ar , and by the time
t+Ar , the total number of people at the age of
[a + Ao’, a+Aa+Aa’] will be
pla + Aa’, t+ ArthAa, hence the equation:
pla, Aa — pla + Aa’, t+ ArAa= pla, pla, DAcAI® ....... (1)
Considering the factor of immigration, then:
pla + Aa’, t P Ar)Aa — pla, t)Ac= —u(a, rt) pla, Aad?
Pigldy AGA, = saw new (2)
WithAe being taken off, equation (2) will be:
pCa + Aa’, ¢-+ At) — plas 1) = —p(a, 1) pCa, NAL
telarar, lee (3)
In order that. the population growth equation can be also
available for computer processing, a, t in the equation
are, as usual, the integers. x,(¢) for the totall number
of people at the age of i (not exceeding i+1) in the year
of t, i will be an integer (0, 1, 2, 8, 4, 5 ..... 2
i.e.
i+
=f “plastda, F=0,1,2,000m (4)
System Dynamics ‘90
Integrating a on the both sides in equation (3) from i to
{it+1J, and assuming Ar=1, then based on (4):
ith ze
SC) eee) = = f, ala, Sples Dea
eg
+(' g(a, 12a, F=0, 1,577, mol, ... (5)
Applying the intermediate value theorem to the first right
term, then:
en), F€li,i + 1]5
(ion ia4
\, Ea, pCa, Oda = le, of Plas tda= pl
Provided #(f, 1) meet the following:
jak elas 1) SeG. 0) S
igacitt bas 2
w(t) =p2(E,1) is tne death rate of those who age
{#1 in the year of [2], then equation (5) will be:
waa( 1) = xi) — wi Cexs€e) + 2,C2),
F=0,1,2,°%¢,m—1,
it
In the equation 2c) ={ S(a,t)da, da, this indicates the
i
total number of people at the agr of [i] yet not exceeding
{¢+t1] who immigrate or migrate during the time from year
{2#] to year [#41].
Let's assume that g(t) is the total number of babies
born in the whoie year of [ft] by women of child bearing
ages [a@, @2 ], {t] is the average birth rate of women in a
year, &() the female proportional function, 4(t) the
birth pattern, then:
4
de) = plas = Bc) [kas aCe, Noles Dee
os ith
= 8) yh, ACas t)ACas t)pCas dee (8)
Applying the integral intermediate value theorem to
the right side of the above equation:
670 System Dynamics '90
7
i
att +1
{/ Aa, DACa, Nelastde = RE, OME, ‘ade FeLi +1:
given 4G) =A(8,1), AC) = ACS, 2), then:
ian
AE WE D[ pe, Neem BOONE) FEL + 1s
Iy we substitute the equation above instead of equation
(8), then:
6) = 60) SALE,
BF a gg ew ge (9)
in the equation above, 4A,t) can meet the normalized
2
conditions > AG) =1
baa,
If x(e) “is the total unmber of babies that live up
to the year of [t],and. #,(s) . is the deathrate of babies
during years from [#-1] to [#]. Then:
n= pb. eee (10)
Combining equations (6), (9) and (16), we will get a
set of difference equations, in another word, a complete
set of equations of the population growth:
6) = 7) SOO
2) = C= gl) 6%),
ae 1) = (= we) le) + gol),
n@ +1) = (1 2G))aG) + 2.0),
emt 1) = CL = pyar t) tma€e) + SniG).
The difference equations above can also be converted
into a set of matrics which would not be illustrated here,
vet. its S.D. flow diagram’ is as foliows:
System Dynamics '90 671
PRO) Bay PaO)
ee
G En)
Van
i i !
‘@ pose) nash) finns)
’
2(¢+ 1)
tm-iCt-+ 1) = Xm(t +1) rae 2
DK)
It is easy encugh to deduce the PYNAMO equations from
the difference equations and the flow diagram. The author
believe every system has its own nature, only by means of
areful study, can one get to know its nature, can one
describe the mechegnism and behavior pattern of the system
accurately in the form of mathematic expressions. In
short, we should pay much attension to the “strictness”
concept when developing a model.
3. Introduction of Parameter Accuracy Concept
into Development of S.D. Model
Since.a S.D. model is specifically developed to show
the feedback relationship, therefore its behavior pattern
will be inert to parameter changes. Yet this is not
necessarily to mean that parameters for a §.D. model can
be carelessly selected, and only the contrary is true. It
is so simpiy because that. no matter how strict in
developing a model, and how precise in describing the
nature of system behavior in mathematic expressions, the
emulation result. value would be definitely in correct if
the parameters selected are not correct. And no matter
what efforts in improving the mode] hence made will not
compensate the losses resulted by parameter errors. Let's
take above discussed population growth for example. If the
equations are correct, but the parameters put into the
672 System Dynamics '90
equations are incorrect, how can we expect the emulation
results worked out to be correct! From above, it may be
clear that we should not only pay much attension to study
of the basic structure of a system and various feedback
relations when manipulating S.D. method in our reseach,
but also emphasize the correctness of parameters. The
author thinks that historic parameters should be selected
out of good sources; abnormal Parameters should be
carefully verified; Parameters concerning future
development tendency should be evaluated by adopting
various kinds of scientific methods; and additional care
should be given to avoid one-sided conjecture in selecting
the table functions
Conclusion
The reliability of a S.B. model will be greatiy improved
if a careful study is made and special attention is paid to
the abovementioned three aspects when developing the model.
It can be taken as granted since it has already be proved
successful as the author applied the three concepts te the
researches into Rational Development of Water Resources in
Xingiang and Its Economic Development.
Reference:
1. On Principle of Syster
Jaw Forrester Ginhua Press (1986)
2. S.D.
Wang Qifan Qinhua Press (1988)
3. Cn Poputation Contro]
Song Jian Science Press (1985)
