IEEE Computational Intelligence Magazine - August 2022 - 62

single parametric experiment runs (see
Table IV). Figure 5 shows the Pareto
front for ENPIPIECON++
, which is the
experiment setting with the highest
overall HV. It can be observed, that
manipulating pqr (e.g., a lockdown) positively
correlates with f2, damaging the
economy. On the other hand, adjustments
to cr is beneficial to both main
objectives. Exploring correlations with f3
can be used as a proxy for the strength
of a parameter's influence. For instance,
pqr
has high costs on economy and
simultaneously low correlation with the
f3 values, which suggests that the optimized
solutions avoid heavy use of this
parameter. In general, solutions which
favor f2 tend to be good trade-off solutions.
The best solutions regarding f1 sacrifice
f2 to a considerable degree, but
accepting a slightly worse f1 makes it
possible to optimize f2 to a high degree
alongside it. This stands out when comparing
the positions of the extreme
solutions for f1 and f2, where those for f2
tend to be close to the prominent knee
point of the Pareto fronts, while those
for f1 tend to be on the far end of the
spectrum. The cost of the strategies (f3)
seems symmetrically distributed, which
means that the mentioned dynamic also
translates to cheaper solutions. It is evident
that getting close to optimal f1 or f2
values
is possible with little cost, as
indicated by the light colored solutions
that extend to the borders of the estimated
Pareto fronts (see Figure 5).
However, the less costly a containment
strategy is, the stronger is its inherent
trade-off. If both objectives are to be
optimized simultaneously, increased costs
are inevitable. This result suggests that
solving the HED dilemma successfully
for health and economy goals requires
high intervention efforts.
D. Containment Strategies
The extreme points in ENPI can be used
to illustrate possible decision-making
strategies (see Figure 6). More complex
experiments can achieve better results,
but a simpler setting aids the identification
of possible strategies. It is problematic
to nominate a single best strategy,
since the presented use case does not
refer to a real-world scenario and the
decision-makers' preferences are
unknown. The solution causing the lowest
infection peaks (
f .0 0497) modi1
=
fies mainly three of four available
parameters. Adjustments to pqer (e.g., the
length of preventive quarantine) are
largely neglected in this strategy. This
strategy has very pronounced oscillations,
with parameters non-overlapping
during their dominant phases. The average
cost of the strategy (f3) is below 0.45.
This is comparable to the second strategy
(see Figure 6(b)), which is optimized
for economic well-being. In this case,
the optimizer only utilized two parameters:
cr and pqer. Surprisingly, the latter
only builds up to full strength after the
first infection wave has already faded.
The third strategy in this figure represents
the one which utilizes the available
policies to the greatest extent f3
^ =
6
4
2
-2
-4
-6
0.1
0.2
f1
FIGURE 5 Comparison results. Combined Pareto front for ENPIPIECON++
, including the correlation
coefficients' matrix. Dark (bright) gray points indicate high (low) f3 values. The red,
orange, and green points show the minimal f1, f2 or maximal f3 values. × shows the results
obtained by [7].
0.3
0.4
0.5
f1 = 0.05
f2 = -5.51
f3 = 0.62
f1
f2
f3
cr
pqr
cd
dr
iqrr
vr
pqer
(GDP)pi
(GDP)pqi
(GDP)eqi
-1
1
0.68). From one perspective, it seems to
be an inefficient version of the healthoptimal
strategy, which achieves the
same minimal infection numbers at
lower cost. Upon closer inspection, it is
evident that the increased cost has been
utilized to improve upon the economy
objective without
sacr ificing the
obtained health objective. Without
tracking costs, this strategy would appear
to be strictly superior to the first, when
it is, in fact, a trade-off.
E. Statistical Evaluation
All trials are grouped by each experiment
setting E (see Figure 4) and by
the algorithms that were used on them
in the optimization process. This allows
us to assess
the performances of the
algorithms in comparison with each
other, while being able to identify effective
experiment settings. Both group
configurations are evaluated using a
Kruskal-Wallis test, followed by a MannWhitney-U
test. Containment strategies
whose optimized policies do not influence
the simulation fail the KruskalWallis
test and are not considered for
62 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | AUGUST 2022
f2
f1
f2
f3
cr
pqr
cd
dr
iqrr
vr
pqer
(GDP)pi
(GDP)pqi
(GDP)eqi
See Salgotra et al. 2021

IEEE Computational Intelligence Magazine - August 2022

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