Tae Use oF OPTIMISATION MeTHoDs
For Poticy Design IN A
System Dynamics MoDEL
497
R. G. Coyle - University of Bradford
R. K. Appiah - University of Zambia
August, 1982,
Introduction
In a previous paper (1), one of us: presented.a fairly traditional
System Dynamics analysis of a production and rav materials system.
Tt was shown that the system was remarkably unstable, in the sense
that the Rav Material Sector of the system responded explosively to an
‘exogenfous sine wave simulating a bi-annual seasonality in external
orders, It was also shown that bringing to bear the attitude of mind
of a control engineer, allied to soue simple rules of thunb steming
from that discipline,led to the identification of three alternative
control structures ,labelled Options I~ III, each of vhich was capable
of giving major improvements in the performance of that system. It was
suggested that the system dynamics modeller could achieve quite
significant improvements by that simple approach, without necessarily
knowing very much about control engineering. Much of this ability to
effect considerable improvement very easily way, hovever, derive
from the fact that many mamoged systens are very ill-behaved in the
fizst place, s0 that. alnost anything is bound to be an inprovenent.
The real problem for the enalyst is, hovever, to identify and evaluate
design alternatives and it is with that problem that the bulk of this
paper will deal.
It was readily conceded in that first paper that practitioners
of System Dynamics, who seek to analyse the controllability of managed
systems, could derive much benefit from a closer acquaintance with control
theory. However, one must accept that, as the analyst of managed systems
ete.
must also know a good deal about management, accounting, economics
there is, in the real world, inadequate time available for 2 really deep
study of control engineering. We therefore sought, in a second paper
(2) to bring to bear our respective backgrounds in System Dynamics and
3
Jane
Control Engineering to analyse the source system from the point of
view of the control engineer. We showed that the nodel could be
expressed ss analogue patch diagrans, and in the form of a control
matrix and a plant matrix.
Armed vith that formulation of the model we were to suggest @
fourth control strategy, Option IV. It was shown that the control
matrix formulation vas a fruitful means of generating alternative
configuration for the design of control policies.
= It was demonstrated that System Dynamics uodels are formally
reducible to the more conventional control engineering format, thereby
making then nore accessible for the control engineer to deploy his
considerable analytical methods for the design of control strategies.
Im particular we proposed that Adsptive Model Following Control was
Likely to be a fruitful Line of approach.
Im that paper we were, hovever, careful to point out the
important theoretical and practical differences between system dynanics
and control engineering as well as drawing on their simlarities,as it
hhad been suggested in reference 1 thet aysten dynamicists ought to do.
In particular, ve emphasised that the plant and the controller are not
independent, as they usually are for the analysis of engineering control
problems. The implication is that parts of the plant, and usually the
major part of it, are freely changeable by the system dynanicistf, a
freedom not enjoyed, apart from a major redesign and then only to a
Limited extent, by the control engineer. Indeed, this interdependence
of plant and controller is so persuasive that system dynaniciétey have
habitually made little or no distinction between the tvo. It is,
however, a renarkablé coincidence that the distinction has re~energed in
recent work by Coyle and Wolstenholme, (3) in which the ‘plant’ is
represented by flow and behavioural modules, and the ‘controller! is
explicitly separated intof information and control wodules.
Equally, recent vork by Coyle and King (4) has related the
control rules and the changeable parts of the plant to the concepts
of policy and strategy used in the Business Policy literature.
If these three lines of research from control engineering, the
reformulation of Systen Dynanics, and the business policy area,can
be successfully developed and integrated, then some very stimilating
developments tovards 2 more fully satisfactory theory of the behav! sur
” of managed systems become possible.
The present paper carries these converging strands of work
forward to address the problem of paraneter selection for a given
control structure and the compariaon of the resultant performance of
two competing policy designs.
Gontrol Matrixes for Policy Optic
In terms of the Fotarton introduced in the previous paper (2)
ones
the original model devcseeed a control matrix of the form
OO 1/zaHL o "10,0
’
2 0 0 ~1/tAmst o 1) 0
ee a
‘The sub-matrix in the small box is better written as
4
azo 1-%
x,
as) ‘>
6
499
5
“a 8B
with % “70 °2
Option III from reference 1 has a control matrix
oe *
. © 0 1ytam,~i/rams “@, 8) 0)
4 @
9 0 ' 0 ~1yrams G-5,) 2, 0
é 6 .
ap
vith % 3, <1
and Option IV in reference 2 has the matrix
%
- © 0 1/tm- yams 3, (18, 0
ro)
© 0 1/taBL - 1/raRNs (3) 2,0
eat
tt @
in which £,, 3,4 0
ALL three options (and we do not consider Options I and II from
reference 1, for the sake of brevity) also have associated with them
5 other parameters which we may group as follovs:~
@) K and WAP which are respectively the number of veeks of
Average Orders which are regarded as forming an acceptable
order backlog, and the number of weeks of average production
vhich are to form the desired raw material stock. To a
control engineer these are gain elements and reduction of
WAP vas the basix of Option TZ, vhich did indeed ‘ead to
increased damping in the system. ,
b) —TAOR and TAPR which are respectively the wean times for
Order Rate and Production Rate before the operation of K and
WAP, Control engineering practice suggested, in reference 1,
that increasing TAPR would stabilise the system
which proved to be the case.
) DDEL ~ the delay in receiving new supplies of raw
materials.
It ie natural to regard all these as paranetere of the plant
and that vas the viewpoint adopted in reference'2. However it is
to see that although they play no direct part in the control
equationSwhich determine the Production Rate and the Raw Material
>
Order Rate, all except DBEL are freely choosable. Even the latter
could be changed, by moving to another cupplier, and that would be ~
precisely analoggous to the control engineer's drastic option of a
major redesign of, say, the aircraft.
e In terms of the business policy concepts referred to earlier, K
and TAOR specify the company’s strategy for dealing with its
customers by defining the Order Backlofg which would be regarded as
acceptable in a given set of circumstances ie. for a given Order Rate.
Similarly WAP and TAPR reflect the strategy adopted for maintaining
internal stbcke of raw materials. Those strategies serve as the
governing factors on the streams of individual decisions on Production
Bate and Raw Material Order Rate, and those decisions ‘are provided by
the regulatory, or strategy achievement, policies specified in
whichever @My matrix the company finds it expedient to use.
The design problem is then to choose a matrix Go, G, oF G,,
and
jlues of its parameter, and perhaps also to choose values for
K, WAP, TAPR and TAOR,s0 as to maximise sone measure of perfornance.
Performance Assessment in $.D. Models
The tradition in System Dynamics for assessing the effects of
policy changes has been a visual comparison of plotted output, and,
generally, little attention has been paid to numerical couparisons
between simulations. Such a practiceis not necessarily a bad things
any managed systems are so badly controlled in any case that
it Se relatively easy to make such a large inprovenent that a
nunerical Figure of Merit, or Performance Index, would be
superfibus. In addition, experience{modellers are acutely avare of
the dangers of oversinpltying the couplexities of managenent by
forcing then into a nuserical index. Work hee, hovever, been done on ~
nethods for the formulation of Performance Indices for Systen
Pynanics models (5, 6, 7) and such indices are valuable in those cases,
Ga this is one such) vhere one is making comparisons of petformance
between relatively subtle alternatives.
Developing a Perfo: Index
3a this case there ar¢ four factors to be veighed in the index.
8) The need to prevent sharp variations in production from ite
previous average.
b) The corresponding need to avoid variation between the Rav
Material Order Rate and ite average.
©) The requirement that Order Backlong should not depart too
tar from its Desired Value. :
4) A corresponding wish that Raw Material Stocks should be
matched to their desired value.
Tn each case, the penalty function ean be expressed as the
integral overtine of the equared deviation. The factors are manifestly
not of equa importance, and they are assigned weights of 4, 1, 3, and
5 respectively, These reflect a judgement that it is far more
importance to keep activity in the factory stable than it is too
smooth the pattern of orders to outside suppliers, that it is quite
important to keep Raw Material Stock close to target because of the
500
.
effect on corporate liquidity, but that it is most important of ell -
fo wegulate the Order Backlog becewre of the effect on the eontomers.
These relative weights ere easily sfaled to make the Index, PI, equal
to 100 for the Base Caseg for the original model, sinusoidal of a
regular input. The behaviour in that case was rather awful but only
for the Raw Material Order Rate, which we have penalised very lightly.
g_the Index
‘The purpose of anlysis is to suggest alternatives rather than
to provide ansvers, Matrix analysis is a highly systematic way of
generating alternatives and, if it can be coupled with an equally
systematic way of numerically aad ‘that ‘Alternative Space’
(Policy Space’ is a better term), thjn the path to real system inprovenent
should be open. *
Such an approach may well be rather at variance with traditional
System Dynamics practice but recent work by Keloharju (8) has provided
very powerful tools for applying it. This has involved the complete
integration into the DYSMAP simulaton package (9) of a hill-climbing
optinisation facility thereby rendering automatic a process which is
otherwise exeedingly cumbersome, namely interfacing a simulation model
with sone very advanced optimisation facilities (10).
501
9
Armed with those tools, ve may nov return to the original system,
that is before applying any control redesifig/options at all. For that
version of the systen the matrix in ¢,, equation 1, snd the orfginel
Parameter values are TABLA, TARUSCh, do-Po=1.
We therefore attempt to minimise the performance index for
satrix Go alone, that ih leaving the "plant" peraueters at their infifal
values of Kn6, WAP=8, TAOR=4, TAPR=4,
Taitiatiy ve coven TABL end TARMS such that .
Sras.,ra008$10 vatues which were judged to represent the maxizua
extent to which management would depart fron established practice.
The result of the optimisation, which converges after about 70
iterations, iA that the PI ia reduced from 100 to 8.74 with the
optisel values of the parameters determined to be TARISWTANIf+10,
Yowd and for0.3f- This is Interesting, in that both TARIS end TAR,
have been forced to ther extren¢y as has Bo. The latter euggests
that the Raw Material Order policy has in the past been fundamentally
incorrect in using Zo=l. The rationale for that was that the Rav :
Material Manager took account of Average Production Rate on the
plausible grounds that it was his job to see that raw material stock”.
was avajlable to be used up in production, He ignored Average Order
Rate orf ageing reasonable, basis that orders were whit the Production
Manager dealt with. The optiniser, in driving 3o to 0, and therefore
setting (1-fo) to 1 ie indicated that those apparently sensible attitudes
are fundamentally misconcieved.
Similarly, the optiniser casts doubt on the Production Manager's
attitude that he ought to attempt to correct Backlog errors fairly
quickly and keep up with Average Order, which he has inplenented by
putting TABL§~4 and Dov. the optinisation instead indicates a much
nore relaxed attitude to order backlog, TABL=10 and far wore attention
paid to Average Production than to Average Order Rate, The last is not
surprising, given that an index attaches a heavy weight to achieving
10
stable production patterns but the former is surprising, given that
the matching of Gfier Backiog to ite desized value is the most heavily
weighted component of the Index.
The reason ie the relatively heavy veight attached to
maintaining rav material stocke at their desired level a requizenent
which seems as sensible, but may be as wrong, as the policies discussed
‘above. ‘The ability of the optimisation to cast doubt on its ovn
objective function is a fruitful source of improvement inthe
difficult area of creating andeoudening a good objective function.
For the purposes of this paper ve shall, hovever, stay with the one
we bave.
ta
The real significance of the seed a, and & of chat eke
optiniser has token us far deyond the depth of analysic achieved’ in
reference 1, by the application of simple heuristics. That approach
largely accepts the structure of the’ policy as fixed, as long as that
structure scons reasonably SOEtIe asa socusces tbe uain attention
on parameter values with relatively limited structural change. The more
sweeping search done by the optiniser has produced a more profound
result bearing on the nature of the policies to be applied.
When we allow the 'plant’, or strategy parameters to enter the
optimisation, containing K, WAP, TAOR and TAPE to lie between 2 and
10, a further substantifjnal inprovenent in the index is achieved. The
final value of the index decreases yet again to 1.02 with the optimal
Paraneter values
iL
TARMS=10 Boraray bps 0-38
elo @ Bo=o.084+
‘TABL=9.1
TAPR=10
whore Wd
i warez
M11 of the parameters in the fixet colum have been driven to
Mduek ace these a. 2
their extrene, except TABL/which io near Hts extrenge In particular,
the high value of TAPR and the low value of WAP confirm the control
engineering rule of thunb, used in Option II in reference, (1) that
Smereasing gain and reducing delay are good ways to increase stability,
This is, hovever, slightly contradicted by the fact that K, which is
also a gain elenent, has risen fron its initial value of 6 to its
Limit of 10, This is the number of weeks that custoners vill have to
wait for their orders, snd it has been driven to the value which manage
ment feel is the safe linit, bearing in mind that reliability of
promised delivery is often more important that the magn{tude of the
wait. In fact, the high value of K, and the low value of WAP,
correspond to the umegerial heuristic of making the custoners wait as
Jong as one dares, and holding as little stock-as possible.
The question to which ve now tum is vhether G, and G, can be
made to perform equally well, or better.
The first optimisation in this section shoved that vhen the
Constant
policy natrix Go was optimised for a comtvest strategy, the Performance
Index, improved from 100 to 8.73, and the second shoved that when policy
and strategy vere optinised together there was a further decrease to
14,02. Finally, therefore, ve optimise the strategies alone and find
the minizum attainable index to be 12.0 when WAP=2, Ke10, TAOR-8.44
and TAPR10.0. This ic very close to the high-backlog-low stock strategy
but the main point of these results is that neither strategy nor structure
is of dominant importance, and each has to be tuned to the requirements
of the other. c
502
12
Optimisation of Option 111
The performance of Option III is already known to be better than
the base run on the original model, so that the parameter values in G5,
equation 2, are set by rule of thuab (or guesswork) to
tanu-tamsed and YyeP got
the value of the PI is 8.01 dow
which is-better than the minimum for G,, when these form parameters
are optimised, with the sane constraints, the PI reduces to 4.84 with
TARNS"10, TABL~9.6, 1¥470.337 and B,-O, These are practically the sane as
the values for Go confirming the earlier indication that the Production
and Rav Materials policies are fundamentally wrong.
For matrix G5, joint optimisation of strategy and policy reduces the
185 with the paraneter values TARHS=10, TAHL=3.99, °
PI still further to
Ke10, WAP=2, TAORS.77, TAPR*10.0, 490.456 and 34°.
‘This is further confirmation of the kind of strategy fpolicy balance
appropriate for this system, but the change in TABL, which has moved
very nearly back to its original value of 4 shows how very carefully
one has to watch the strategy/policy balance.
Ia the previous section we also tested strategies alone but that
is clearly rather realistic, in general, and ve make no further
attempt to do so.
Optimisation of Option IV
Finally, we examine matrix ¢,, equation 3, With initial values
4
of TARYSATABL4 and j5B,c1 its PI is 15.5, certainly better than the
original system, but by no means as good as Option III. If
15.0.5, the PI changes negligibly to 15.3 suggesting that TARMS and
TABL are more significant parameters.
13
This
‘porne out by the optimisation, which leads, for 6,
\ “
optimised alone, to a PI of 6.29 when TABL=TARMS=10, HG50032 and
B 4,0.
When strategy and policy are jointly optimised, the PI decreases
: @
te 1.57 with all parameters driven to,-ct-etesmto, their extrem#,
Bye Out
a8 . 5
except 10224 and@,-0.5f. These last tvo values are interesting
as they inply the Production and Raw Materials Managers really
co-ordinating very closely and each giving nearly equal weight to
Average Orders and Average Production in their respective decisions.
‘This is rather different from the corresponding results for Option IIT
and the original system in which the Rav Material Manager was told to
ignore Average Production and pay attention instead to Average Orders,
which was exactly opposite to what he had.been doing in the original
model, and the Production Manager was recomended to do almost the
sane thing, and change his behaviour. :
The reasons is the much Closer co-ordination of their activities
implied dy the presence of TABL and TARMS in both colun@e/3 and 4 of
matrix G,. This mgfigh that the Production Manager takes into account
the Rav Material Stock position as well as the more obvious Order Backlog
which was all he considered in the original system. Similarly, the Rav
Materials Manager includes Order Backlog in his decision making, as well
as the obvious factor of Rav Material Stock, Paradoxically this seems to
be too tight a degree of co-ordination, as shown by the fact that the
optimal performance of Option IV is worse than that of Option IIT, and
is even somewhat worse than the optimal performance of the original
model, as shown in Table 1.
Soinn fi He Mambar a sate
Ae enna afpans tw ot, Filo ing diagrams
503
‘The simple conclusion to be drawn from Table 1 is that it
‘is feasible to over-control a system, which is what is happening in
Option IV, The additonal feedback mechanisms create such a tight net
of control that the system has too little freedom of manoeuvre,
The secoid, and perhaps nore subtle, conclusion is that very
major improvements can beg brought about by very small changes, though
those changes may run counter to established pratice and ‘eomon sens
The change from 100 to 8.73 in the original systen is achieved largely
iy acne a BAS Hagan KS atte uate piphaaes, THe mae
Material Manager should heve his ordering not on production, but on
customer orders, and the Production Manager should link his
production decisions far loss heavily to customer orders and fer more
0 to previous production. Naturally, once these results have been’
obtained, it is easy to advance common-sense arguments for then,
but it is far harder to reach them by such arguments in the first place.
The third conclusion is that, if policy and strategy are allowed
to enter the analysis, even further improvements are possible, and
‘that these improvements con converge so that éifferent structures are
able to give practically identical performances. That does not mean
that structure ic unimportant as, in this case, one gives slightly
worse results than the others.
; a 8 s
a
tis oa Fy
& 3 4 c
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g g a
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£ z |
2 2 #
3 2 2b
2 § 33
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a oe
a 92 #2
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504
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él
tion has worked vell in this
The matrix approach to policy gen
case in that, having generated and analysed a collection of options,
we can now be quite confident that there is very little further
improvement available for this system. It has been optimised in a
far more fund mental sense than simply minimising the Performance Index
for any one configuration. The process applied is more avopety called
‘simulation through repeated optimisation’ than ‘optimisation through
‘repeated simulation’. .
‘The matrix approach does, however, have two disadvanteges. The
fixer is that it is quite simply not accessible to many othervise well
qualified System Dyhamicists. For example, an accountant is well
qualified to analyse financial problems, and he can learn the required
simplation techniques quite’ raPidly, but it is rather unlikely that he
would have the tine or the inclination to learn fuch matrix algebra
Secondly, matrives have a quality of rigidity, which is probably
psychological rather than mathematical, which makes it clumsy to write
down all the possibilities in a compact form. More seriously, the
matrix formulation separates too rigidly the attributes of policy and
strategy which have been discussed in this paper.
An alternative is the Policy Option Diagram, the form of which
was prepated by Keloharju, through the English nane is due to
Wolstenholme. The link to policy and strategy explained below is
prepared for the first tine in thie paper.
The Policy Option Diagram, and one cannot resist using the acronym
POD, is a tree, the apex being a flow rate in the model, and the branches
being all factors which do, or could, form part of the determination of
that flow rate. ‘The POD for Production Rate appears in Fig. 1.
505
The colid lines at the left of the diagram show the factors taken
Desired
into account in determining Production Rate in the original system,
Raw Material
Stock DRS
'
y
L
1
'
Production
Rate
APR
Average
show
& That is they sttOw the policy option which management has hitherto
|
proincrcrereg
chosen to exercise, with what notably had results vas demonstrated in
reference 1, and indicated by the Performance Index of 100 used in thie
Stock
RHS
Paper.
Raw Material
The top tier in the POD is labelled the Policy Level because these
47 are the factors used in the daily stream of decision}by which the firms
Avetage
Production
attempts to meet its targets for, in thie case, Order Backlog. The
ee
Rate
APR
lover tier is called the Strategy Level because it contains the governing
factors which produce that decision streamg. The parameters appear at
the sides of the branches to which they relate,
The lowest level is named the Exogenous Level because it is here
L
that driving forces enter the system.
TABI
The point about the POD is that it should make one think of what
might be made to enter into a decision rather than merely concentrating
Backlog Error
on what enter it. This is shown by the dotted: lines which imply
that Rav Material Stock Error might be used to control production (which
is
should not influence PJ€ as well as, or instead of, AOR. To do so would
jentially Option 111). For example, there is no reason why APR
g,
3g
i
4
“necessitate a new paraneter which creates a further dimension in the
wok
Parameter space.
Any state variable in the model could be used to govern PR and
that is implied by the ? at the extreme right hand edge. The only
Rate
‘AOR
Average Order
limit is the imagination of the analyst.’ There is, for example, no law
that says that Desired Raw Material Stock ‘should depend only, or at
PEE Se eee tee
all, on Average Production Rate. The POD is thus at least as effective
B a
i Pict i as control matrixes in genrating alternatives, has the advantage of
Pa g
as os as Linking policy and strategy,and is a good deal moré transparent to many
people.
¥ Policy
19
‘It has the major disadvantage that it does not connect readily to
the analytical methods of control engineering,which perhaps reinforces
the earlier coument about the pressing need for System Dynamicists to
Decome more knowledgeable about control theory and some of its
mathematics.
Conclusion
We make no attempt to generalise the results of thie analysis to
all preduction systems, though they do apply to a significant class of
auch systems, We argue, instead, that the diversity of production
environments is so great and the needs of different firms at differing
stages of their evolution is so varied, that any generalisation could be
highly misleading. .
‘Instead, it is argued that the simulation and optimisation
Procedures used in this analysis are so easy to learn and to use thet
it is more cost-effective to deal with each case as one encounters it,
A critic could say that that was solving problems ad hoc, though it
could equally well be called treating each case on its merits
506
ad
I krew
bern el
20
References
Coyle, R.
‘The Dynauics of a Production and
Raw Materials Systen’
IEEE Trans. on Cybernetics Man and Society.
Septenber, 1982,
Appiah, R-K, ‘ ‘Policy Design in System Dynamice Modele ~
and R,G, Coyle @ modern control engineering viewpoint!
Submitted to IEEE Trans. on Cybernetic Man.
and Society.
Coyle, RG. and
E.P, Wolstenholme Siptamn descrip
T Op Cer See Te appewr
Coyle, R.G. Equations for Systems
University of Bradiora Printer, 1979.
Coyle, B.C. + Modelling the Future of Mining Groupe
Trans. Inst. Min. Metal
Geet.A), 90, 1961, pp.A8i-88.
Montaldo, D.H.
AMPH.D Thesis, 1981. unpublished. Unvouily
Relativity Dynamics
Helsinki School “of Economica,
Working Paper F33, 1982.
Reloharju, Re
* Cavan, RY. DYSMAP Users Manual
and R.G. Coyle University of Bradford Printer, 1982,
Luostarinen, A,
and RG, Coyle
DYSMOD Users Manual
University of Bradford Printer, 1982.
Coyle, R.G. and
Towards a Structuralist Theory of
P.D.A. King Business Policy Issues a
Unp-udjlritved alu ing paper.
Fas ene Ge Te Grong pteae bk tener Ue eo
Nurae A by Adar. Ser y JT lave We we Olewe Ur
He be ety
Syl Dynami 40, Ldyoreug Pocadure
tPolicy Design for an Operating Mine’
ily 2% Bradford
CC BY-NC-SA 4.0