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.