Improving Insight and Understanding by Optimising System Dynamics
Models
Eric F. Wolstenholme and Sattar A. Al-Alusi
University of Bradford Management Centre
Emm Lane, Bradford, West Yorkshire, BD9 4JL
United Kingdom
ABSTRACT
This paper will outline the concept of system dynamics optimisation using
the DYSMOD software and present a case study of its use to analyse a
defence problem. The insights into the problem, which were generated
from a conventional system dynamics model and its policy design
experiments, will be given. This will be followed by the presentation of
results from a set of optimisation experiments, utilising a range of
objective functions and structural design parameters. The paper will
focus on the value added to the understanding of the problem which
resulted from this process. The overall conclusion is that optimisation
subsumes conventional sensitivity analysis as well as providing an holistic
interpretation of the behaviour of a system dynamics model.
INTRODUCTION
Traditionally, system dynamics has relied on the use of intuition and
experience by system owners and analysts to help design policies for
improving system behaviour over time. This situation is now changing and
much effort is being expounded in the development of policy design
methods. Basically, two schools of thought are emerging. The first of
these concerns the application of control theoretic methods and the second
simulation by optimisation (Keloharju (1983). This approach relies
fundamentally on computer software and the software to be described and
applied for optimisation in this paper was originally developed as an
appendage to DYSMAP (Dynamic Simulation Model Application Programme)
(Cavana and Coyle (1982)) and is known as DYSMOD (Dynamic Simulation
Model Optimiser and Developer).
The DYSMOD software uses a hill climbing routine to heuristically
determine the optimum values for any number of model parameters
relative to predefined objective funetions or performance measures using
a system dynamics model as a starting point. Optimisation in parameter
space is achieved by interleaving simulation and optimisation. One
iteration of the procedure consists, firstly, of a DYSMAP simulation run, in
which the value of the objective function is calculated and secondly in a
tun of the optimiser to choose parameter values which might improve the
objective function. Subsequent iterations consist of rerunning DYSMAP to
test out the resultant improvement in the objective function under, the
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new parameters and further refinement of. them by optimisation. Any one
experiment with the software might take a 100 or more iterations.
A DEFENCE MODEL
The model to be developed here using optimisation is a defence model
(referred to as the armoured advance mode! (Wolstenholme 1987)). The
amoured advance model was constructed initially to examine the effects
of alternative formation change strategies by an attacking force (red),
under a variety of fire detivery strategies on the part of the defending
force (blue).
The red strategies considered were to change formation at a fixed
distance of advance or to change formation at a variable distance of
advance using a range of different variables on which to base this
decision; such as its speed and force size (number of units advancing).
Basically, the purpose of red delaying it's formation change point was to
facilitate recovery of speed lost (due to blue's fire) by staying in a fast
moving but dense battalion formation. Conversely, the purpose of red
advancing its formation change point was to conserve it's force size by
changing to a slow moving but less dense company column formation.
A pseudo parameter SZS (speed/size switch) was defined in the model to
allow any combination of weightings to be attached to the speed and size
variables in the formation change strategy equation. In the base model
this parameter was set to 0.5. When speed was used as a determinant of
the formation change point, a speed multiplier function was invoked,
which progressively increased the planned distance to the formation
change point. When force size was used, a size multiplier was invoked,
which progressively decreased the planned distance to the formation
change point. 1
The blue fire strategies were based on criteria involving red’s distance of
advance, red's speed of advance and red's momentum (speed * numbers).
Blue's fire was assumed to reduce both the number of red's units and their
speed of advance. However, speed was assumed to be recovered by red
when blue fire ceased. Both the speed and attrition effects were modelled
as the product of the rate of blue fire (shells/min) and the productivity of
fire (attrition rate/shell or speed reduction rate/shell). The latter was
assumed to depend on the density of the target [that is, the formation in
which red was advancing] and the accuracy of fire, which was made to
increase as the distance between the contestants reduced.
The strategy of blue delivering fire on the basis of red's distance of
advance, was achieved by defining the percentage of the distance of
advance over which fire would take place. This was initially chosen as the
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first and last 10% of the distance advanced by red in each formation. The
strategy of blue delivering fire on the basis of red's speed and momentum,
was achieved by defining upper and lower limits of those variables; fire
being delivered whenever the upper limit (or trigger value) was achieved
by red and switched fire off when the value of the variable had been driven
down to the lower limit. The blue strategies were replicated for two
different firing rates (light and heavy).
The results from this model suggested that from red's point of view it was
better to deploy a variable formation change strategy and to delay the
formation change point. This gave a shorter arrival time at the blue
position whilst not sacrificing numbers too much and hence a generally
improved arrival momentum (numbers arriving/arrival time). This
suggests the emphasis by red should be on staying as long as possible in
battalion formation. The rationale for this was thought to be the higher
speed achievable in this formation and because speed unlike size is
recoverable if blue firing stops.
From blue's point of view the results suggested that it was best to first
deliver fire on a criterion of red momentum, secondly to use a criterion of
red speed and thirdly to use a criterion of red distance. Here, the aim was
to prolong the red, advance and reduce as much as possible the numbers
arriving and their arrival momentum.
It should be appreciated that the initial results from the mode! depended
on the assumptions made about the strategies used. For example, red's
formation change point depended on the value of SZS chosen or on the
shape of the multiplier functions. Likewise, blue's fire delivery strategies
depended on the percentage of the distance of red's advance over which
firing took place [and, indeed, the location of that percentage] and the
parameters chosen for the upper and lower limits of speed and momentum
at which fire was switched off and on.
The conventional approach to this problem in system dynamics is to
perform sen: ity analysis. However, this is only possible to achieve in
a limited way for extreme parameter values due to the enormous number
of permutations of parameters involved and hence the enormous number of
computer runs of the model necessary. Optimisation can help here by
seeking out that permutation of parameters which gives the best outcome
according to given objective functions.
Optimisation of the Defence Model
The approach in applying optimisation to the. armoured advance model was
to firstly define relevant objective functions and then to define for each
objective function the parameters whose values would be chosen by the
optimisation procedure; and the upper and lower limits of the feasible
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ranges for these values. In general the guideline followed in defining
parameters was that only red's strategy parameters could be involved in
experiments using red's objective functions and only blue's strategy
parameters could be involved in experiments using blue's objective
functions.
The Optimisation of Red Strategies
From red's point of view the main objectives are to maximise it's size
and momentum on arrival at the blue position or to minimise it's total
advance time. For all three of these objective functions, the relevant
parameters to be chosen are those involving the red formation change
decisions. These are whether red's speed or size is used as a basis for
modifying the distance to company column deployment, and within this,
the shape of the speed or size multiplier table used. Freedom was given to
the optimiser to choose the value of the speed/size parameter and each of
the y-coordinates for the speed and size multiplier functions.
Results from Experiments with Red's Objective Functions
When red's arrival size was maximised it was found that in every case
that the optimised arrival size significantly exceeded the arrival size
attained in the base model. This was achieved by the optimiser employing
size as the sole determinant of the formation change point and adjusting
the shape of the size multiplier to produce a very aggressive response in
the planned distance to the formation change point, as soon as the actua!
red size fell below the planned red size. The consequence was that the
whole of the red advance was carried out in company columns. In all cases
the focus on a long advance in company columns meant that red’s total
advance time was much slower. Additionally, blue used much ammunition
and the efficiency of use was correspondingly low.
This result, of carrying out the whole advance in company columns,
represents the maximum extent to which red can protect itself against
losses incurred in force size. It might be expected that savings in size
resulting from this extreme strategy would at least compensate for the
longer advance time and hence not seriously affect the momentum of red's
arrival at blue's position. However, this was not the case and the
deterioration in momentum can best be explained in terms of the
additional exposure time of red to blue's fire, which results from the
strategy.
When red's arrival time was minimised it was found in every case that the
optimised arrival time was significantly less than that recorded from the
base model. This improvement was achieved in the optimisation process
by employing speed as the sole determinant of the formation change point
and by again adjusting the shape of the speed multipler to give a much
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more aggressive response in the planned distance to the formation change
point, as soon as red's actual speed fell below the planned speed. The
consequence was that the whole of the red advance was now carried out in
battalion formation.
Suprisingly, the arrival time did not suffer too greatly and hence in most
cases the arrival momentum achieved by red was as good as, or better,
than that from the base model. Blue's use of ammunition was also low and
its efficiency of use high. It would, therefore, appear that the gain in
speed achieved by red by staying in battalion formation more than
compensates for the higher attrition. This result is again perhaps best
explained in terms of the low exposure time to blue's fire arising from red
maintaining a higher speed formation.
When red's arrival momentum was maximised it was found that in every
case the optimised value of red's arrival momentum was significantly
better than that recorded in the results from the base model. This result
was achieved by red predominently chosing speed as the criterion for
formation change with the shape of both the speed and size multiplier
functions chosen for aggressive responses. The results tended much more
towards those obtained in minimising red's arrival time rather than those
obtained from maximising red's arrival size. Whilst the whole advance
was not now in battalion formation the time to company column
deployment was extended with little effect on the arrival size.
These results are described in more detail elsewhere (Wolstenholme and
Al-Alusi, 1988) and provide substantial confirmation of the previous
conclusions from the non-optimised model, that red's best strategy is to
stay in battalion formation for as long as possible. The investigation of
what might be intuitively considered as trivial runs from two extreme
situations (maximising red's arrival size and minimising red's arrival
time), was shown to facilitate the formation of a perspective concerning
the trade off between the two, which is clearly confirmed in the run
involving maximisation of red's arrival momentum. The optimisation
approach added to the analysis by clearly highlighting a further effect by
which blue’s strategies interact with reds. Originally the red desire to
stay in battalion formation was seen primarily as being based on
increasing it's ability to recover speed, and hence momentum, when blue's
firing ceased. Additionally, however, as shown here, it is clear that this
strategy further minimises the time over which red is exposed to blue's
fire.
The Optimisation of Blue Strategies
From blue’s point of view it's main objectives are the opposite to those
defined for red. These are, to minimise red's arrival size, maximise red's
arrival time and to minimise red's arrival momentum. Additionally, it has
the objective of trying to achieve these objectives, particularly that of
minimising red's momentum with the minimum use of ammunition. A
further objective function is therefore relevant in blue's case. This
involves maximising the average reduction in red momentum per shell
fired. For all these objective functions the relevant parameters to be
chosen are those involving blue fire delivery. That is (i) the proportion of
the red advance over which blue delivers fire (where the latter is carried
out on a distance criteria); (ii) the upper and lower limits of red speed at
which blue fire is switched on and off (where blue fire is delivered on a
speed criterion}. and (iii) the upper and lower limits of red momentum at
which blue fire is switched on and off (where blue fire is delivered on a
momentum criterion). Optimisation experiments were carried out for each
of these objective functions and at two levels of fire delivery (light and
heavy).
Results from Experiments with Blue's Objective Functions
lt was found that the results for the first three of blue's objective
functions were identical. It would appear that if red’s flexibility
concerning its choice of formation change point is removed, then the same
effect is achievable by blue via any of these objectives.
Under the blue strategy of delivering fire on a distance criterion,
optimisation of red’s arrival size, time and momentum was achieved by
firing during the whole of the red advance; that is by deploying continuous
fire. Under the blue strategy of delivering fire on a speed or momentum
criteria, optimisation of red's arrival size, time and arrival momentum
was achieved by switching fire on and off at the lowest points within the
range defined for red's speed or momentum.
Overall blue achieved the longest time for red's advance and the lowest
values of red's arrival speed and momentum when delivering fire on a
distance criteria. This was because such a strategy not only allowed
continuous fire but also allowed continuity of fire delivery. When fire was
delivered on a speed or momentum criteria there were periods when fire
was switched off, which facilitated red speed recovery. This result
exposes an inherent weakness of these strategies.
The foregoing results generated a very high useage of ammunition. When
red's arrival momentum per-shell was maximised the ammunition useage
was much less. This improvement was achieved by blue only chosing to
fire whilst red was in battalion formation. This is an interesting insight
which results because of the density of this formation. The productivity
of the blue fire directed at red in battalion formation is higher than that
directed at red in company column formation. However, because of the
selectivity of blue fire, red's arrival size, time and momentum were all
improved which is, of course, detrimental to blue. Nevertheless, the
savings in ammunition were phenomenal, which might more than
compensate for this deterioration if ammunition was limited.
In addition, when red chose to delay its formation change point the
optimisation procedure led to a further intriguing-insight. Here, blue
chose only to fire on red at the most productive point during the red
advance. This was chosen by the optimiser to be at the very end of red's
battalion formation advance and this choice is explainable since it is when
red is not only in its most dense formation, but also at its closest point to
blue (hence blue's accuracy of fire is higher). This result provides a good
example of how optimisation facilitates an holistic appreciation and
intrepretation of results.
Yet another interesting insight, arising when blue delivered fire on a speed
or momentum criterion, was that the greatest reduction in momentum per
shell was achieved by blue deploying a light rate of fire rather than a
heavy rate. This is because not only does light fire save on ammunition,
but also it again results in more consistent fire. Heavy fire, by definition
of these fire delivery strategies, is more intermittant than light. It
quickly drives both red's speed and momentum down to the point at which
fire is switched off (thus allowing these attributes to recover) rather
than reducing them more gently but for longer periods of time.
CONCLUSIONS
It is concluded that a number of significant additional insights into both
red and blue strategies were achieved by optimisation of the armoured
advance model. These were:
- red should maintain a high density, high speed formation for as long as
possible since this retains the flexibility of speed recovery and minimises
the total advance time and hence the period of exposure to blue fire.
- it is important for blue to maintain continuity of fire. This is
facilitated by blue firing on a distance rather than a speed or momentum
criteria, since the latter results in periods of zero fire when the trigger
points, defined in these fire delivery strategies, come into play. This in
turn allows red to recover speed which shortens the advance time.
- when ammunition is limited it is more productive for blue to restrict
fire to periods when red is advancing in battalion formation and to apply
light but consistent fire. For optimum productivity from limited
ammunition blue should focus fire over the later stages of the red advance
in battalion formation.
the results from the situation where blue is allowed to manipulate red's
formation change point, confirm the previous results from the experiments
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on red's objective functions. That is, red’s speed is a more important
variable than red's size to both sides. It is best for red always to extend
its battalion formation advance as long as possible. This increases speed
and, by reducing the exposure time to blue fire, reduces attrition.
- when ammunition is limited it is to blue's advantage if red does prolong
it's advance in battalion formation, since biue can fire for longer in its
more productive mode.
The original insights into the armoured advance model were obtained from
12 DYSMAP computer runs (3 blue strategies at two levels of firing
against 2 red strategies). The optimisation experiments involved a total
of 8400 DYSMAP computer runs (the red experiments repeated the 12 runs
for 3 objective functions of 100 iterations each and the blue experiments
covered 4 objective functions of 100 iterations each). To achieve the
same level of analysis using conventional sensitivity analysis would have
required a repeat of the 12 original runs for all permutations of the values
of the parameters used in the optimisation runs. This is conservatively
estimated at over 10 times the number of optimiser runs. Further, given
the degree of automation of the optimisation process, the time saving over
coventional sensitivity analysis would be enormous.
REFERENCES
Cavana, R.Y. and R. G. Coyle (1982) DYSMAP User Manual, University of
Bradford.
Keloharju, R. (1983). Relativity Dynamics. Helsinki School of Economics,
Helsinki , Finland.
Wolstenholme, E. F. (1988). Defence Operational Analysis using System
Dynamics, European Journal of Operational Research, vol. 34 no. 1,
Feb. 1988, pp. 16-18.
Wolstenholme, E.F. and A. S. Al-Alusi (1988) System Dynamics and
Heuristic Optimsiation in Defence Analysis, System Dynamics
Review, vol. 3, no. 2, Winter 1987, pp. 102-116.
Acknowledgements
The authors wish to acknowledge that some of the work described has been
supported by the Procurement Executive, Ministry of Defence. However,
any views expressed are those of the authors and do not necessarily
represent those of either this Department or H.M. Government.
