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System Dynamica Generalized Modeling
for Forecasting Multiproduct Substitution
Zhao Chong Jie
Zhejiang University PRC
ABSTRACT
The System Dynamics Generalized Substitution Modeling is presented.
This modeling considered the influence factors of circumstance by
introducting action function. The methodelogy is based on the System
Dynamics with econometrics,combining three postulates in product
substitution and decomposing multi-product into several two~product
substitution. Parameter estimation, which existed in all System
Dynamics Modelings,is one important but still unsolved problem.
Now this problem has been solved in our paper by orthogonal simulation,
it is based on the orthogonal theory and generalized least squares(GLS)
INTRODUCTION
The history of technical development tell us:in nature there is no
technique that is forever advanced,because advance and backwardness
are relative. It is the dialectical relation between advance and back-
wardness that a series of technological substitution can be produced.
So man can speak that the history of technical development actually
is the history of technological substitution. Moreover,the substitution
in technology must be showed by products. The substitution of products
is as a resuit of technological substitution.
That people demand products is only for their function not for the
products themselves. Different forms of products may have same func—
tion. Based on this principle, product substitution can be produced
and continued.
Appearence of large amount of substitution products makes the fore~
casting for single product lose sense, because forecasting is related
not to products but to their function. So the study for multi-products
substitution appear.
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BACKWARD
1) Logistic Curve
Although introduction of Pearl curve was only for approximating the
growth curve in biological phenomana . Because of the similarity in
their outside properties,it"s equation is used as startpoint of pro-
ducts substitution modél.
It's form is:
yea/(1+bee7° *) (1,1)
2) Mansfield model
Mansfield took substitution of multi-technology in multi-department
as analysis object and tried to explain the nature of technological
substitution from causality.
His conclusion was that :
m(i,j) = nla, g)/(1te7P (45) BG) ty (2,1)
b(i,j) = ali,1)+a(i,2). pli, j)+a(i,3)es(i, J) (2,2)
here :
p: win capability s : infest scale
m/n : user share a,l : constant
i :departmen: j : technology
3) Fisher model :
The form is :
dy/(y dt) =k (1-y) (3,2)
or: ye 1/ (1497 (t-4(1/2)) (3,2)
4) Sharif model for multi-produtt substitution
The model for forecasting two-product substitution in Sharif model
was still used. The difference was that two-product in odler models
were transformed into two-group product. The principles of substitution
can be stated as following :
a) The oldest product in the market Py being technologically least
advanced will lose it's market share to all other products: (p, to p,)-
b) The newest product in the market Py being advanced will obtain
it's market share to all other products (p, to Phi)
c) The intermediate product (pp top,_,) obtain its m rket share to
older products and lose its market share to newer products.
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In order to determine the share of intermediate products (px), follow-
ing analytical procedure was used.
a) Group all older products (p, to Py4) aB Pet + considering the
substitution between Pot and others :
af(pg1)/at =e, f(pg1) (1-£(pg1)) (4,1)
b) Group all newer products (p. to Py ) as Pon considered the sub-
stitution between p and sumer e
Af (DgnJ/at = cy £(Pgn) (1-f(py,)) (4,2)
From f(pg4) and £(Pgn) we can obtains
£(py) = 1-f(p,4)-f(p,,,) (4,3)
for £(p,) and f(p_) following result can be directly obtained;
af/at =c f (1-f) (4,4)
Only f is transformed into £(p,) and f(p,).
Discuss
These models contain several supposes}
1) all products have same properties:
In Sharif's model products are divided into two groups and then one
group products are considered as one product. It means that products
in one group have same properties. But that isn't true. It is because
these products have different character that we study the problem
of multi-products substitution.
2) Unchangeableness of factors
All models showed above consicer influences of circumstance only
statically. It is showed that in equation constant are used as propers..
ties factors. Suoh as b(i,j) (in equation (2,1)), K (in equation(3,1))
and cy ey (in equation (4,1) (4,2)) ete. In fact circumstance factors
are not always constant. There are not only statical but also dyna-
mical factors. All static factors can be combined together and repre-
sented by constant, but dynamic factors must be alon considered.
3) One direction substitution
In these models influ ence factors are constant, so man cann't change
the symble of coefficient (positive or negative). It means that it is
new or old of product that determine the substitution of products.
But in fact , only the characters of products can determine the subs-
titution, although several new products at first are very good. after a
period of timm because of the change of circumstance factors,their
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advantage may be not advantage but shortcoming. So at this time newer
products may be substitutea. by olders.
BULLLinG OF GENERaLIaby iiuvel
1) Substitotion of multi-product
Comparing with two-product substitution, because of intrdduction of
intermediate products, multi-product substitution is more complicated.
Usually intermediate products in one side substitute older products,
but in the other side they can be substituted by newer products. in
fact the substitution between different products can be determined
by product character and their adopted degree to circumstance. Owing
to the change of circumstance, in a group of products intermediate
products can substitute several products and at same time can be subs-
tituted by others. Now we have known that it is unilateral that man
only pay attention to new or old of products.
In the other side each product in one group of products react to the
other products. The amount of substitution not only based on circums-
tance factors, but also the absolute amount of their market shares.
It means that for example substitution between product i and product |
j influence the substitution between product i and product k.1... (ex-
cept i.j ).
2) Suppose of model
a) Decomposability :
In multi-product substitution in one unit time the amount of substitu-
tion for one product can be decompose as the amounts of substitution
of the product and other products.
b) Timelessness ;
In any unit time substitu+ion between two products only based on the
stale of two-produéts and circumstance factors at that time, not based
on past process.
c) Independence :
At any time the amount of substitution between product i and product
j only depend on influence factors and market shares of product i and
j (f; and f, ).
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3) Restrain condition
We use market share to represent the level of product substitution.
Specialization of market share producé following restrain condition;
oa f(i) =1 (5,1)
i=1
here f(i): market share of product i
ni: product number
4) Diagram
For the restrain condition " the sum of all market share equal to 1 "
iceg? f(i) =1. This condition certainly can be replaeedinto following
i=1
condition: s.. af(i)/dt =0 = f(a) tetgh (5,2)
ort 5” ar(a)/at =¥ F ae(i,5)/at = 0 (5,3)
ist i=ij=1
je#i
here df(i,j) : the substitution amount that product 553) obtain
from product ic i)
af(i) : the substitution amount that product i obtain
from the other ( in time..interval dt )
Because the substitution amount that pi obtain from pj is also the
amount that pj (through pi ) lose.
Hence: af(i,j)/at + df(j,i)/at = 0 (6,1)
S af(i)/at = 0
dependenes™ and
We use this condition to diagram, the restrain
can be satisfied. After introducting suppose "
"Decomposability" , we obtain:
(taking three products as example)
Sa Ye atin Seiate A cate
Pr P35
7
1
R25 i
/
=—— ae
here x,? influence factor i=1.2.-'m
Riot Ray =df(i,j)
Pi:Pg,P3 : product 1.2.3.
5) Building of model
After dete:mining diagyam of the system, mvl+i-products Substitution
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Has been divided into several two-products substitution. Now our
task is to determine their action form. Considering the analysis
of Mansfield and Fisher (mentioned above ), We notice that re-
place only m(i,j)/n(i,j) given by Mansfield's mog@#l into f(i,j)
(the meaning is same ) b(i,j) (in equation(1,3)) introduced by
Mansfield actually ia coeffiences k (in equation (1,4)) in the
Fisher's.model. It means that for two-products substitution the
proportion factor k comprehend fumction acted by all influnence
factors to products substitution, but in Mansfield's model only
contain ststistic “win capability " and “infest scale". Now we -
represent these influence factors by one function , their variables
are X, X52+° Xp given by diagram mentioned above. these not only
camtain statistic influence factors, but also dynamic factors.
Using the suppose "timelessness", the variables in the function
don’t contain time varible.
Otherwise, because of the difference between substitution of multi-
products and two-products (for latter, two-product's sum of market
share equal to one constantly) the factor (1-f) represent latent
capability of whole market in one side, also represent the other
product's market share in the other side. For multi-product substitu-
tion, the first explanation wasn't suitable but we can use "inde-
pendence" suppose and combine with second explanation. So we can
obtain:
af(i,j)/dtex 2£(i) £(5) (7,1)
If action function of influence factors will be considered, the
equation become:
a£(4,5)/At =E( 5 5) (xyxQ--m,) £C4) £(5) (8,1)
ist.2.0' ij
here 6 (5,5) (xyxQ7 Xp) } gemeral action function form of in-
fluence factors 1%, Xa.
(for df(i,j) the meaning are the same as above)
For n kindg of product substitution, use "decomposability"suppose
its general equation should be writen as:
att )/at B05 y(t a) t(4) £(3)
je
(9,1)
B55 leee xy) =F 5 5) (ym) fet,Qe0n
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Taking three products as example:
ag(1)/at Gia ay £(1)° £(2) + 4 5) £(1)- (3)
at(2)/at = - G4 5) £(2)-£(1) + Gy 5)-£(2)-£(3) (942)
at(3)/at = - G(4,3)-£(3)-£(1) - G5 5)-£(3)° £(2)
The Relation Between General Model and Others Model
1) When the amount of product equal to two (n=2)
a) If let (4 0) (Ey aq) =k, it will become the Fisher's model.
b) If let 84 2) ey) sa(i,1)+a(i,2)-r(i,j) ta(i,3)-s(i,j) it
will become Mansfield's model.
2) When the amount of product larger than two (n>2)
Let G4 5) yo xy) =constant, now the general equation become the
Sharif's multi-product substitution model.
Take threeproducts as example, the model become:
af(1)/dat =c-£(1)°(£(2) + £(3))
af(2)/dt = -c:£(1)-£(2)4e8-£(2)-£(3) (10,1)
at(3)/at = of(3)-(£(1) +£(2))
It satisfies obviously the equation:
af(2)/dt = -af(1)/dt -at(3)/dt
By use the initial condition:
= 2) beato = 1
We can obtain £(2)= 1-f£(1)-£(3) immediatly.
The equations are the represents of Sharif's suppose of "All Pro-
ducts Have Same Properties",but Sharif represented these only by
cy and e) Coefficient c, obtained by equation (4,1) usually don't
equal to -cy obtained by equation (4,2). In fact, they satisfy re-
frain condition Ef(i)=1 only by f(2)=1-f(1)-£(3),but they don't
ensure to satisfy refrain condition in subsystem:
af(i,j)/at =+ df(j,i)/at
In the former it can satisfy the conditions only through taking
errors into f£(2).
Now we can see that because of introducing acté&n function of in-
fluence factors,suppose “decomposability" "independence" and "time-
lessness" our model not only contains advantages of former models,
but also overcome shortcomings of three sides (All products have
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same properties,unchangeablenss of factors and one direction subs-
titution) and make the model more reasonable.
SMPLUNENCE FACTOR ANALYSIS
The action function general form of influence factors is mentioned
above. Now let's discuss these factors and their effections in de-
tail further.
For product substitution,related factors usually follewing:
a) profitability for the manufacturer of the newer product
b)size of investment necessary for economic production,
c) stage of perfection in production technology due to time and ex-
perience,
d)overall growth of industrial production (expansion of economy),
e)utility adjusted price ratio between competing producs,
f)sales and promotional efforts for the newer product,
g)useful life of older product and capital rquipments for manufac-
turing (durability and obsolescence),
h)quality characteristics of the newer product
i)stage of diffusion characterized by adopted population and poten—
tial adopters.
THese factors are generally divided into three kinds. The first
acts on producer the second acts on user, the third acts on the
both. Obviously,all factors act on either producer or user.
Based on this character and combine with further analysis, we can
get these following conclusion:
Influence factors are subjective and product substitution isd!t
introduced by them. It is the producer and user that determine the
product substitu tion. All factors make action only through producer
and user.
Because producer provide and user realize the posibility of product
substitution,so the factors influencing both are acted at the same
time. Also, as the amount of products substitution has only one re-
sult,it shows that there are interaction between producer and user.
So the principle on which we consider is that: for product substitu-
tion only these factors that have influence to producer and user
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should and must be considered.
When thinking their action,wecshould consider from statistic and
dynwmic aspects.Some factors(such as function of product)have been
fixed,wher products jgst enter into market,these factors can be com—
bined together and represented mit one constant.But dynamic factors
(such as price and cost),which aren@t fixed over whole living period
of product,must. be representai alone.At same time we must consider
their influence form to product substitution.
For example,for textile product substitution we consider average
income as dynamic factor.The result is:
G(x)=a, (y/x-k(n-y) Jax
here xtaverage income/man.
yitextile product/man.
k,n,a,:constant.
(1 have discussed in detail at another paper)
PaRAMETER ESTIMaTIUN IN TH# GENERAL MODEL
After giving the general model of n procucts substitution,we cansi-
dered how to estimate parameter in action function in actual model.
For this kinds of model, there are several difficulties in parameters
estimation. They are:
a) The model consists of differential equations.
b) There are same parameters in different equations.
ec) Crossequation are interrelzted.
As a matter of fact, as estimating parameters in system dynamtc
models,we often meet difficulties above. ( df(i,j)/dt =-df(j,i)/dt)
In the past,parameters of system dynamic model are determined by
subjective to do qualitative analysis. After setting his polynomial
model, Sharif considered that because of difficulties in determining
parameters, parameters in polynomial model usually can be deter-
mined by subjective.So advatages of the model are that they can
help us to do qualitative analysis.
For these problems our ideal showed following:
1) differential equations are represented by difference equation.
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2) Estimating all equation's parameters at same time and using gene-
ral least square to eliminate the interrelation in equations.
a) When parameters are linear,we can solve large equation. (I have
related it in detail at another paper.)
b) When parameters have general forms (linear or nonlinear) we can
solve these by orthogonal simulation combining with general least
square. Computer orthogonal simulation actually is that: choosing
optimum parameters combination using orthogonal test, simulating
object value in each group -of parameters and then choosing optimum
parameter group in given region, based on the theory of orthogonal
design to search for the best way.
For our general model, simulation model: are:
z ~
Min. 4 Ff (m)-1 (%
=, FB Y FB)
a.
s.t (i,k) = a Gp. ay(xere x) Fliyket) F(5,k-1)+
= (i,j)01 m
Jai a
+£(i,k-1) (11,1)
let f(i,o) =f(i,o) Lea. oR
here F: #(4,k)
x ¥ = |Ple.)
ko : actual value matrix of f(i)
F(n,k) at time k,
. a
RF F4,x)
a ”
A * £(2,k) ecalcilated value matrix
~- of f(i) at time k.
f(n,k)
Ht covariance matrix.
The computer simulation process is : (please to see next page)
This method has widely suitibility and flexibility :
1) The problem of determining nonliear parameters can be solved.
2) Choosing parameters region can be controtéd qualitatly, it ensure
the reasonable of model parameters. ;
3) When all observation data are not obtained, it has not mugh in-
fluence.
4) Crossequations are considered , so it increases the efficience
of parameters estimation.
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(Gem)
Giving small interval of
each parameter
Selecting the best parameters combination
in givyen.;small interval according to orth»
ogonai searching methode
Any parameter reaches
gmall interval boundg
This: paramete’
reaches large
interval bound
parameters are §
ble in a determinate
removing its
a interval
List stage > obtain H i _ ol
Fig.2 computer diagram
REFERENCES
1)Halold A. Linstone & Devendra Sahal "Technological Substitution"
1976 by American Elsevier
2)M. Nawaz Sharif & Chowdhury Kabir "System Dynamics Modeling for
Forecasting Multi-level Technological Substitution " Tech. Fore
Socia Chang Vo.8 no.4 1976. :
3)M. nawaz Sharif & A.K.M.Abdul Haq. "Determinants for Forecas-
ting Technological Substitution " Tech. Fore Socia Chang No.13
1979.
#)M.N.Sharif & K.Ramanathan "Polynoméial Innovation Di#Pusion
Models " Tech. Fore Socia Chang No.2! 1982.
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