4.2 ANN-Optimization model
MOPSO showed significant improvement in terms of convergence, diversity,
and uniformity of the solutions (Figure 7). MOEAD showed improved
diversity and uniformity whereas convergence was still similar to the
Pareto search (without ANN). Although GA showed no major improvement in
the convergence, the uniformity of Pareto solution distribution is
increased. The Pareto search solution without the use of ANN can be
taken as the reference for the comparison (Fig. 5 and Fig. 7). The
diversity and uniformity of the GA and MOPSO are close and comparable.
Spread and Generalized spread depict that the MOPSO and GA have a
uniform distribution of the solutions. Whereas the MOEAD shows a polar
and non-uniform distribution of solutions. The MOPSO is superior to
other algorithms, in terms of convergence, suggested by epsilon and
generational distance. The combined effect of convergence and diversity,
measured with hypervolume and inverted GD, is also best for MOPSO. GA,
PS, and MOEAD show a close value of hypervolume and inverted GD
indicating a nearly equivalent performance in terms of convergence and
diversity. The percentage increase in hypervolume and decrease in
inverted GD, both show that MOPSO in model 1and MOEAD in model 3, has
significant improvement (Fig. 9). Whereas GA and MOEAD in model 1 show
the least improvement. However, the GA and MOPSO (in Model 3) showed
conflict in the change in values. Inverted GD is less reliable because
it is strongly influenced by the distribution of the approximate Pareto
front. Figure 10(a) and Figure 10(b) shows the box-whisker plot of the
decision variables from the Pareto solutions obtained from each
optimization technique and two (1 and 3) model domain size. It can be
observed that the distribution of decision variables for the solutions
is dependent on the optimization algorithm rather than the model domain
size. Further, in GA the solution decision variables have a smaller
number of outliers compared based on well zone distribution. A similar
trend is shown in MOPSO and Pareto searches, with an increased number of
outliers. MOEAD shows a broader distribution with the length of the box
(25 and 75 percentiles bound) greater than other optimization
techniques. MOPSO solutions have a large number of outliers depicting a
large number of solutions focused in the quartile range. The mean
discharge for GA solutions is less than or close to 1000 for nearly all
of the zones. In Pareto Search, the mean values are close to the bounds
of the box (25 or 75 percentile) of the solutions and the mean discharge
largely varies with the decision variable number (i.e., location of well
zone).
Furthermore, figure 10 provides the spatial variation of the optimized
well discharge. In total, there are 30 well zones in both the domains
and half of them are classified as “far away” or “close” depending
upon their shortest distance from the river. The average optimal
discharge of faraway wells is observed to be slightly higher than those
close to the river. In Model 3, the value means optimal discharge of
faraway well is 860.8 m3/day and that of wells close
to the river is 832.9 m3/day. However, in Model 1 the
gap between the two values narrows down further, with far away well avg.
discharge as 905.4 m3/day and close well avg.
discharge as 901.2 m3/day.
5. Discussions :
The solutions of GA are highly diverse in model 1 and model 3, even at a
low number of simulations runs. Other than the optimization algorithm,
the Pareto fronts trends are determined by the model demarcation and its
boundary conditions. It shows that adding the Rhone River in the model
alters the groundwater flow correspondingly water budgeting of the area
and gives the least groundwater flow into the river due to excess
pumping. Results show that Model-2 which consists of the shortest area
along with the dominated constant boundary inflow is providing less
groundwater to the river in comparison to Domain-1 which consists of a
larger area in the simulation along with a one-sided watershed boundary.
It can be observed that even though all three domains extract the same
amount of groundwater, the modelled effect on the river varies for each
domain. In addition, the data from the Pareto front suggest that in
model 2, the extraction from the well and the river gain are heavily
related to each other. This relation will not only influence the
simulation results but will also affect the decision-making in
groundwater management. In ANN Optimization model, it is thus evident
that in terms of convergence, the solution points of MOPSO dominate
every other solution, and solution points of GA are dominated by most of
the other points. PS and MOEAD solutions lie in between in the GA and
MOPSO front for both models. MOPSO solutions largely consist of outliers
with either too small or too large box bounds. MOEAD solutions are
largely uniform and cover the lower and upper bound of decision
variables. Regardless of the small difference, the overall trend is:
wells close to rivers extract less groundwater than far away. Another
factor, that can contribute to the high discharge of a close well is
spatial heterogeneity in the aquifer properties.