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A METHOD POR INITIAL FORMULATION .
OF SYSTEN DYNAMICS MODELS" ca
. roa
Dr. R.G. Coyle
Director
System Dynamics Research Group
University of Bradford | m1.
England . 1.
vw
vit.
Even the experienced practitioner of eystem dynamics can encounter
serious conceptual problems in getting started on a model, and
to add more and more to his model. A technique ~ ‘list extension’ ~ is
described which, from the purpose of the project and the importance of
fecdback loops, guides the evolution of the simplest adequate model.
This model is expressed as an influence, or causal loop, diagram,
The influence diagram should be tested to ensure that its
structure contains the necessary elements of a dynamic model. If it
fails the test attention is directed to the area of the system where
further elucidation is needed.
The techniques heve been applied in many practical cases and have
been found to give useful results and to increase the efficiency of the
modelling process.
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TABLE OF CONTENTS
Introduction
Initial Phases of List Extension
A. The Closure Test
B. An Example of List Extension
Type Assignment Procedures
Structural Incoherence in Influence Diagrams
Structurally Resolvable Incoherence
Double Coherence
Application of List Extension, Type Assignment
and Cotierence Testing in Practical Modell ing
References
sho
Bos “WIC tiie
bal a
+2 Introduc on
The firet stage in the modelling project is the definition of its
purpose. The second stage is the construction of a diagram showing the
causal relationships and the model boundary, Those diagrams are called
‘influence’ or ‘causal loop’ diagrams.
An experienced modeller often scems to find no difficulty in writing
down the influence diagram directly from inspection of the system, The
iar situation, often
novice, or the experienced modeller in an unfam
encounters two difficulties; knowing how to start the influence diagram
and knowing how to stop. The resule if often a diagram which includes
every conceivable variable. This is not good practice; it reflects a poor
understanding of how the system operates and an even poorer one of what the
study is supposed to be for.
The practice of writing down the influence diagram directly suggests
that the model boundary can, in some way, be realised intuitively. It is
not, however, clear how this realisation comes about, it is difficult to argue
that the model boundary is ‘correct’ in the sense that it contains enough
of the system to generate the dynamic behaviour which has been observed
in the system and which is sufficiently important to justify doing the
“modelling at all, and it is not easy to:conmunicate that boundary~identification
skill to the newcomer to dynamic modelling, This paper therefore suggests
@ procedure which is flexible and easy to apply and which meets the following
eriteria:~
a) _ it focusses attention on the purpose of the model
b) starting from a subset of variables which reflect the purpose
of the model it leads through successive steps and simple testa to
4 model containing the minimum number of variables and feedback loops
which can be deduced from the stated purpose of the model,
c) it verifies that this minimal model contains only feedback loops,
inputs, and outputs, and that there are no loose ends, If the
minimal model proves to be inadequate then further detail or
sophistication can be added in the appropriate areas so that the
model will guide its own evolution,
4)
e)
f)
LIST
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it is easy to learn and practical to apply
it is flexible enough to be a servant to, and not a master
of, real human modelling skill
it leads to tests which verify that no errors have been made
in the structural modelling
We now discuss a technique which meets these criteria, This is the
EXTENSION METHOD.
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WN Lnitial Phases Of List Extension
The list extension method starts from a series of six to cight
columas on a piece of paper. From right to left, the columns are labelled
the Supplementary List, the Model List, Pirst Extension, Second Extension,
and 60 on.
The Supplementary List contained ‘artificial'variables which the
modeller has created as indicators, to him, of model performance but which are
not part of the system itself,
‘The Model List contains the names of the variables whose behaviour
of the model should explain, or the control of which is aimed at. There
should not he more than three to five such variables, and one or two
is better at the commencement of a new study.
For each variable in the Model List one writes, in the First Extension
colum, the names of the variables which most immediately affect it, draving
the influence lines. Variables in the Model List may affect other variables
in the same list, as may be the case for any of the lists, and variables
in an earlier list may affect those in a later list. The lists must,
therefore, be scanned for these connections and the influence lines drawn
in, An attempt should be made to add polarity signs to the links,
After the First rxrension ‘column has been completed, the ‘closure
test’ is applied to see if a dynamic model has been produced, or if further
links are needed to create the requisite feedback loops.
The Closure Test
This is a simple procedure for verifying that the influence diagram
contains only feedback loops and input structures i.e. that there are no
loose ends. If the diagram passes this test then it has the makings of a
dynamic model, If not, further detail is not only justifiable, but necessary.
The property of CLOSURE means that a model must contain at least one
feedback loop, and that all its variables lie on a loop, have been defined
as exogenecus inputs to a loop, or provide supplementary output from a
loop.
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The test for closure 1s very simple:
starting from any point in the influence
diagram it must be possible to return to
that point by following the influence
‘lines, in the direction of causation, in
such @ way as not to cross one's track.
This test applies to all points in the diagram, with certain exceptions
which are noted below,
The test can be simplified somewhat by exploiting the property itself.
This means that, having chosen an arbitrary starting point and traced a path
vhich returns to that point then a number of intermediate points will have
been passed and these, of course, lie on the feedback loop just traced out.
Since they lie on this feedback loop they lie on a loop and can be dropped
from further consideration, One must still apply the closure test to any
remaining paths in the diagram to see whether they pass it, whether their
variables are covered by one of the exceptions or whether, indecd, the system
is not totally closed.
If the influence diagram is not closed, attention moves to the Second
Extension list. This contains, for each variable in the First Extension
which is not part of a feedback loop, which is not one of the exceptions Lo
the closure rule, or which the modeller can justify representing in wore
detail, the names of the variables which most immediately affect it. The
necessary causal Links are then drawn between the variables in the Second
Exten
links between variables in the Second Extension list, and between variables
already entered in the First Extension and the Model List, and the new
variables entering the Second Extension. When the Second Extension has beea
ion and the variables in the First Extension, together with causal
completed the closure test is again applied, and the process either terminates
or continues.
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OEE Ae ben
> 519 -
B._An Example of List Fxtension
The list extension technique really comes into ite om In complicated
systems, When the system consists of a controller and ite complement,” the
practical need to satisfy management that a credible model has been built
usually forces one to include more detail than is really needed for an
adequate model, Even in such cases, it te a good plan to have a discipline
for developing the early versions of the model, and the list extension method
provides this.
Consider the problem of improving control of profitability and
Production in.a mining enterprise which produces a metal, the market price
of which is markedly unstable, Traditional policy has been to produce at
@ constant rate, but it seems plausible to management that gearing production
to price might improve profitability. Management nced a design for a
Production policy which will enable the company to do as well as possible in
the face of fluctuating prices. Since the price is so unstable, forecasting
seems impossible, so the production is to be tied in to average price. World
production is large compared with the mine's output so that this is unlikely
to affect the price.
In a mine, preparatory tunnelling, or development, precedes production.
Since only Limited
jounts of production machinery can be deployed in a given
developed area the size of the developed reserves affects produccion.
Investment in developed reserves affects profitability, ‘and profitability
affects the level of reserves which can be supported financially, thus
affecting target reserves, Development produces waste rock which competes for
shaft hoisting capacity with the ore from mining.
This is a very complicated problem and one's first reaction might be to
produce a very detailed model, Such a model could well be ideal for short-
range tactical production planning but that is not what is required. Figure 1
shows how list extension, as it might be applied to this problem, leads to a
fairly simple model boundary thus creating the framework for further modelling
if the initial model proves to be inadequate. This framework, or, model
boundary, shows the facets of the system which have to be considered in order
to find a closed set of feedback loops. Some of these areas might require
modelling in more detail or with greater subtelty but the modeller now has a
* Briefly, the ‘controller’ in a system is the part whose workings can be
ascertained exactly, e.g. the firm itself, The 'complement' interaccs with
the contrallar hur cannot he modelled uith earrainty 0 @ tha Firms marker
- 520 ~
modelling guide related to management's problem and should be able to
produce more useful results than would be.the case of the simply built, say,
a very detailed production model.
To return to list extension, in this case, the model contains six
feedback loops. The first, A, appears at the Second Extension but the
diagram is not closed by its emergence because profitability has not been
declared to be supplementary variable. A second loop, B, appears at the
Third Extension but does not close the model.
The Third, fourth and fifth loops, C and its two unlettered parallel
branches, are found at the Fourth Extension, The diagram is, however,
unclosed until the detection cf loop D at the Fifth and Sixth Extension,
at which point it becomes the simplest model which can be built of the
system, Whether it is the most adequate model is entirely another matter.
It is unlikely that it is, but the modelling process has now startcd and
the model itself, and its output will guide its own elaboration.
ing the test of closure, will raise
The influence diagram, while p:
many questions in the analyst's mind about how the system might be changed
from its present form to a better one, For example, should the model
parameters be fixed or should they be converted into true variables dependent
on other parts of the system, perhaps including quantities not yet in the
diagram?
From the building of influence diagrams ve turn to two procedures
which abstract importent information from them and check that they are
free from certain types of error. These two procedures - Type Assignment
and Coherence Testing - will firet be described from a theoretical stand-
point, Later we consider their use in practical modelling situations.
Model
List
- pal - - 522 -
WI Type Assignment Procedures
It is necessary to decide, for each variable, whether it 1s to be a
level, rate or auxiliary, This step is often taken for granted and it is,
indeed, often obvious that a particular varinble should be @ level. We
Profitability
y
First
Extension
shall exploit this property in what follows. In practice however, the
intervening substructure between levels and rates is often treated in a
cavalier fashion and toose modelling can’ often be covered up by progranming
dodges. We therefore need a procedure which will determine uniquely the type
of each variable and direct our attention to the underlying problem area
MN
hy:
= production]
= Rate
Costs
4£ a unique type assignment fails to energe.
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H 3 Ss gE 2 7 g We start from the necessary relationships between the variable types,
gg 33 S .
a8 gé 3 3 i Z as shown in'Table 1.
“ t ODT | For most of this section we shall denote variables by letters rather
Gob FA YY
a | o | y a than by names, so as to show the technique of type-assignment unencumbered
8 4 g
z 3 A oy ; z ZI 3 by preconceptions about a variable's type which might’ be generated by its
4 a
ca oe 3 8 é & § name, In practice, a diagram with names should be used.
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Table 1.
‘The Table shows the relationships which must hold between the
- 523 -
Relationships between Variable Types -
types of successive variables in an influence diagram.
Types for varies Types for
Preceding Tyee Succeeding
Variable Variables
R L A or R
Lor A A A or R
i) if a delay i) if a delay
in the in the
influence influence
Link link
R R
R
ii) if no delay
in the link
Loor A
ii) if no delay
in the link
L
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Consider a simple model involving only three variables, X, Y, and 2,
in Fig. 2a).
To start the type-assignment process suppose that we have some reason
for choosing variable X to be a level, perhaps because it is a stock of
completed product. Write the variable type near the name, and enclose it
in a box to show that X is the chosen starting point. We can now work
either forvards to ¥ or backwards” because a level can only be preceded by
a rate, Thus Z must be a rate, The dotted lines and their directing arrows
indicate the sequence of derivation of the variable types and have nothing
to dé with the direction of causation in the feedback loops.
To type ¥ we first work forvards from X and then backwards from Z,
noting that neither the ¥ - Z link nor the X - Y link contains a delay, so
that the conditions i) for a rate in Table 1 do not apply.
From the X - Y link, ¥ can be either a rate of an auxiliary, and, from
the Z - Y link, Y can be a level or an auxiliary,
We write these conclusions onto the diagram and it is fairly obvious
that only if Y is an auxiliary will the forward and backward derivations
of its type be consistent.
‘The type-assignment for thie model must, therefore, be
X- Level, Z- Rate, Y - Auxiliary
In this case, the assumption about X led to COHERENT conclusions about *
Zand Y, The absence of coherence indicates that something is seriously
amiss with the model, but its presence only means that the model is ‘correct’
in a very limited and technical sense.
« Forwards means in the direction of the influence link and vice-versa.
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Fig. 2 A Simple Type Assignment
x
Pig, 2a) 17 ‘The given influence diagram
Fig. 2b: Initial assumption that Xx
is: 2b) a) is a level
7\
ze
Fig. 2c)
Deduction that 2 must be
a@ rate
Determination of the
Fig. 2d) erEaet
mere
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In solving assignment problems we have found the following empirical
rules to be useful:~
i)
ii)
ili)
iv)
v)
vi)
if a variable is a level, work backwards first to find the
preceding rate, .
if a variable is a rate, work forwards first to find the
succeeding level,
apply these two steps as often as possible before attempting
to assign types to those variavles for which there is more
than one possibility,
even where a variable's type appears to be unequivocally
determined by rule i) or ii), always check if possible by
following another path to the variable to make sure that the
type assignment is consistent, This will reveal any errors
in the influence diagram,
In practical modelling, the types of several variables are
often obvious and one. therefore has several simultancous starting
points. ‘Type assignment then needs to be applied only to
the interconnecting structure and its main value is in showing
whether ‘or not that part of the model has been worked out in
sufficient detail,
Where two variables are connected only by a delay they must be
rates which provide further starting points,
The notation includes numbers, written at the side of the dotted lincs,
to indicate the approximate sequence of the derivation. The numbers are
purely explanatory and are not part of the technique of type-assignment.
Usually, ‘at any one time, there are alternative steps in the type-assignment
Process,
The numbers do not, therefore, mean that step n must precede step
n4L but, generally, the lover the number the earlier in the type-assignment
Process that step should occur.
“ECM Liban,
= 527 -
For a real system, the starting assumption is not made arbitrarily,
but on the basis of what is known about the character of the particular
variable. A variable which appears, from ite name, to have the character-
istic of accumulation is probably a level. One which seems to be a flow
may well, be a rate, but it may be an average or level so in general it is
better to start from a variable vhich is known to be a level, only starting
from rates when they are connected by a delay.
As we have defined coherence, it is a structural concept which derives
from the pattern of connections in the diagram and, as such, either existe
or does not. There is, however, a DEFINITIONAL COHERENCE which arises from
the nature of the variables whose types have been inferred from the starting
assumption. Thus, in our firet example, we assumed that our knowledge of
the character of X indicated that it could be a level, and from that we
inferred that variable Z had to be a rate.
Now, the statement 'X is a level' derived from our knowledge of X as a
aystem component, and the statement 'Z is a rate', was inferred from the
structural relationships in the influence diagram. Clearly, this second
statement must also marry with our knowledge of the nature of Z as a system
parts: 1; forinatanceyizohas ali chevanpeatance (od dn tacesratibnvehende
is very unlikely that it can be a rate, and the in€luence diagram, although
structurally coherent would be DEFINITIONALLY INCOHERENT,
The test for definitional coherence must be applied to those variables
which have been inferred to be levels or rates from the starting assumption,
once the test of structural coherence has been passed. If any of the inferred
levels or rates fails the definitional coherence test then a mistake has been
made in drawing up the influence diagram from the verbal description of the
system or a definitional mistake was made in the starting assumption, In
practice it is almost certain that something has been missed out, probably in
the list extension process, and the only solution is to check the diagram
Ww.
= 528 -
against the system in the hope of discovering the error.
In the remainder of this paper we shall asaune definitional coherence.
We shall conclude by working a fairly complex example. The working
and the results are given in Fig. 3. with a few explanatory remarke,
Despite the complexity of the influence diagram this is a simple
assignment problem, because 6 variables - C, E, J, K, V and T, are fixed
by the three delays, There are 6 successor levels to these rates, G, Ly
N, P, Q and W so that 12 variables are typed automatically. The large
number of ticks on the diagram shows the extent to which the system solves
itself. The 6 levels immediately give the other rates which precede them,
for example, atep 17 gives Was a level, which automatically makes Z
a rat
step 18. The high degree of self-determination of this system makes
it almost too easy to assign the types, and thie is not unconmon for apparent~
ly large and difficult influence diagrama. Thie is true for most large
models and the value of Type Assignment lies in unravelling the fine structure
of a model.
It is-important to make sure that'all Links are checked to ensure
system coherence ~ even if this is done at the end of the assignment
calculation as in steps 31 and 32 in this case.
Structural Incoherence in Influence Diagrams
The presence of incoherence means that a mistake has been made, either
in the type-assignment process or in the aystem description as it appeara in
the influence diagram, We shall devote the remainder of thie paper to a
treatment of the causes of, and remedies for, incoherence arising from the
second of these causes.
Consider the system of Fig. 4a)
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~ 529 ~ - 530 -
A Complex Type-Asaij
nt_Example
Fig. 3.
An Incoherent Influence Diagram
Fig. 4
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, ae y ne
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re .
~ 3 Zz
¢ 3 *
Ried AML
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- 531 -
With respect to Fig. 4a) this is closed and appears to be an acceptable ‘
influence diagram. If V is known to be a level we get 4b).
Step 3 in Fig. 4b) indicates that ¥ be a level, and step 4 shows
that one of the possibilities for ¥ ia that it may be a level, Ae far as
However Link V - ¥
links ¥ -2 and X - ¥ are concerned, ¥ is a level.
shows that ¥ cannot be a level, and the diagram is incoherent,
The real importance of the concept of incoherence lies in what it tells
us about the model as it has been developed. DYSMAP” sometimes allows one
to bend the strict rules of system structure far enough to permit the writing ©
of a computable program from an incoherent diagram, s0 that incoherence does
not always prevent apparent progress. It does, however, prevent real progress
because the model contains errors which should have been cleared up.
There are two possible causes of Incoherence~ either the starting
assumption was invalid, or there is some fundamental fault in the modelling. ve
This may be that a link has been put in which does not exist in the real
system, or that the influence diagram contains impermissible components.
Taking the first of these possibilities, we examine the other option
for V, namely that it could be a rate. Fig. 4c) shows that this assumption
alco leads to difficulties, Clearly X and V cannot both be rates, as there
is no delay recorded for the link V ~ X and the diagram is still structurally
incoherent, The inccherence is more .than'a matter of the starting assumption
and whether there is reason to regard V as a level or, for a different system
but the same diagram, to treat it as a rate, the diagram is coherent. In
neither case is the reason hard to find, and in both examples, it stems.
from a misunderstanding of the system structure,
For the first case, where V was assumed to be a level, the solution
lies in a further examination of the system, Certainly, the link from V to ¥
is suspect.
‘*DYSMAP - Dynamic System Modelling and Analysis Package ~ is a DYNAMO-type
language developed at Bradford.
~ 532 -
The sacond possibility, when V was a rate, is more easily disposed of,
Again there has been an error in drawing up the diagram from the investigation
of the system. The detected incoherence actually helps by suggesting that
there may be a delay in the link from X to V which has been overlooked. If
there is, and this can only be determined by further investigation of the
system, the diagram immediately becomes coherent.
We have used the same influence diagram to represent two different
an error has been made in drawing
systems each of which is incoherent becau
up the diagram from the system description, In both cases the solution of
the incoherence is a matter of ‘back to the drawing board', that is, further
investigation of the system itself is called for. We refer to such cases as
SYSTEN-RESOLVABLE.
Structurally Resolvable Incoherence
It is sometimes possible to dispose of incoherence by arguments deriving
from the fundamental concepts of system structure. This is called
STRUGTURALLY-RESOLVABLE INCOHERENCE, and we now consider an example of such a
@ituation shown in Fig. 5.
The first stages in the calculation are shown in Fig. 5a), from the
starting assumption that O is a level,
In the usual manner we make a provisional assignment of M as an auxiliary
and proceed with steps 4 and 5 on links M - P and 0 ~ P., as in Fig. 5b). °
‘There are tvo possible provisional assignments for P; as a rate or as an
Definitional conside:
auxiliary, tions will sometimes, but by no means alwa;
distinguish between a level and rate, but not between a rate and an auxiliary,
nor between a level and an auxiliary. In order, therefore, to reeolve this
apparent case of what may best be called semi-coherence we must attempt to
type Q and, since there are apparently two options for P, we have two steps for
the ‘link P - Q., as in Fig. Se),
- 533 - . > 53h -
Unfortunately, both steps 7 and 8 lead to a result which is coherent
Fig. 5." Structuralty-Resovable Incoherence 2 with step 6 and we can, it ceems,make two coherent type-assignments.
Variable Type
H A
N R
o L
P R A
q 3 t A
’ This outcome appears to defy the rule that type-assignment should lead
to a unique model order. The key lies in the feedback loops as the basic
components of system structure.
This system contains three loops, the possible structures of which
i are:~
Loop Variable Names Variable Types
. uM A
69) N R
° _ ob
P M A A
N R R
@) 0 Lor L
P R A
Q L A
4 A A
Q) P Ror A
Q L A
Cotter
vr
OO MERIT Uv tan 5
= 535 =
Now a feedback loop must contain at least one rate and one level.
Loop 3'will violate this requiremant if P and Q are typed as auxiliaries
and they must, therefore, be a rate and a level respectively, This reduces
the semi-coherence to unique coherence.
It will be seen from this example that coherence is a property which
id the
relates to and derives from the structure of the influence diagram
fundamental concepts of feedback-loop structure.
Double Coherence
It sometimes happens that a system has two or more coherent solutions
to definitional
and there is no way of distinguishing between them by appea!
points or by the use of the structure of feedback loops. For example,
consider the system of Figure 6 dropping some of the detailed assignment
steps. This is a case in which, between variables Z and W, it is possible
to formulate two equally coherent sets of types for the same structure.
The solution to this kind of dilemma lies in noticing that the parameters
of the system have not been included in the diagram. Normally they are not
needed because, as we have seen, quite complicated systems can be assigned
without reference to the parameters.
~ 536 -
Fig. 6. Double Coherence
7 Re gh --- On
\
Z ———) ¥ 9k — 5
‘
Poe eee
arr abies
ect ety tena
- 537 - = 538 =
ibitities
For the disputed top lino in this case, consider two possi " wi. apa(ilien nie 2 abe “eeeeanitiens
a oy Y dX we * ype A ignment nd
7 7 : Coherence Testing in Practical Modelling
D +
In many cases of practical modelling the influence diagram becomes
very large. There are two principal reasons for this
2 ) : :
z >Y¥ x .
7 7 a) Many practical problems really are very complicated with
/ 4 ¢ : large numbers of interacting variables, and it is not casy
» : to see which variables are essential and which. are, relatively
unimportant.
where D, E, F and G may be parameters or input variables (though not the
b) In order to have a chance of recommendations being implemented,
and that is always the real object of the exercise in actual
business situations, it is essential for the managers concerned
_solution interval, DT, because that is a parameter of the simulation, not
the system).
In the case 1 the implied equations are:- to have a high degree of confidence that the relevant factors
Y= £@, 2)
have been modelled.
The advantage of the influence diagram is that it makes the
= ov
‘ - te, x) model structure very clear and this is an aid to elucidating
: the system structure from management, ho are in the best
and thie means that X aust be a level as only a level is parameter-free. position tockade'what charcacruecirestes, “Wacdver} ‘shi/aTagven
A first-order information delay has a parameter, af course, but its correct also shows what has not been included, and this gives managers
influence diagram structure is the opportunity of being the arbiters of relevance.
Rate o\
Simulation modelling makes it easier to include a factor to
‘ — >,
Change of A . x
satisfy a manager who feels that it ia relevant to system
performance, than to convince him that it is not and may be
omitted from the model, In any case, the analyst will
ron . usually find it hard to adduce convincing arguments for the
thing
orhing irrelevance of a factor until it has been included and tested.
time ST
which is not the same as the above case of double coherence.
A little thought will show that in Case 2 the assignment must be
auxiliaries, and that this approach does not depend on the particular
number of parameters used in this example, Thus we may restate the position
that the type assignment must be unique providing enough detail is included
in the influence diagram,
wR TTY tan.
aa 2 = 530- i = Sho -
In practice then, we are likely to have lerge influence diagrams, and
Refere
this has three dravbacks:~
a) -The type-assignment and coherence-testing procedures become
5 Coyle, RG. 1976 Management System Dynamics
rather tedious yet if not applied there are more chances for
University of Bradford and John Wiley
errors to creep in.
b) The larger the model is, the more difficult, time-conauming
Coyle, R.G. &
and understand, 1976 System Dynamics Problems & Cases
Sharp, J.A.
“ . University of Bradford
and expensive i¢ will be to analy:
c) As the model size incre: the problem of conveying it and
ita results to management in a concise, comprehensible, fashion
in the short time which is usually available for presentation of
results becomes almost involvable. The virtue of simulation ~
getting management confidence in the model - almost proves ‘to
be ite downfall when to the influence diagram is added a welter
of computer printout and aystem analysis.
These methods have been applied in a number of very different practical Prepared: 15th April, 1976
projects in tanker chartering, metals manufacture, chenicala, and consumer
goods industries and have been found to be well worth the trouble involved.
Further details appear in Coyle (1976) and more examples in Coyle & Sharp,
(1976).
eat etn +
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