4.1 Simulation-optimization model
Each combination of the simulation model and optimization technique was evaluated up to 5000 points depending upon the optimization algorithm and its parameters. In all the three models, the Pareto fronts by MOPSO were more converged, whereas the results obtained from GA were more diverse for model 1 and model 3 (Fig. 5). The Pareto search and MOEAD were more converged in comparison to GA, but they lack diversity. Only in model 2, the diversity of the PF of MOPSO outperforms that of GA. 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.
To understand the impact of different model domains on optimal outputs, the best-performed MOPSO front was studied. Pareto fronts show that Domain-1 is giving a high value of leakage out in comparison of the optimal value of groundwater discharge through wells whereas the output of Domain-2 is higher than Domain-3 ex. for discharge value of 2000000 m3/day leakage out in Domain-1, Domain-2 and Domain-3 are as follows 153560 m3/day,134940 m3/day and 100800 m3/day. The Pareto fronts (Fig. 5) also depict the effect of a model domain on the optimal discharge and river gain relation. For instance, in the case of MOPSO, an increase of discharge by 50,000 m3/day (from 200,000 m3/day to 250,000 m3/day) leads to a decrease in river gain by 15,820 m3/day, 37,657 m3/day, and 19,535 m3/day for the model domain 1, 2, and 3 respectively. Correspondingly, the Pareto search demonstrates a decrease of river gain by 22,278.9 m3/day, 40,116 m3/day, and 20,457.4 m3/day for the previously mentioned discharge increase.
All the six metrics, for the three model domains and four optimization techniques, are shown in Figure 6. The combined effect of diversity and convergence from inverted GD and hypervolume also suggests the superiority of MOPSO, in all three domains, over other optimization techniques. Further, the epsilon metric and generational distance suggest that convergence of MOPSO is best. Both the parameters, epsilon, and generational distance, show conflict in the comparison of other optimization techniques. In terms of uniformity of solutions, spread and generalized spread suggests the superiority of GA for model 1. In model 3, even though the Pareto front of GA is diverse, the solutions are not uniform due to which the spread of GA is high. For model 3, MOPSO is much more uniformly distributed and hence has a low value of the spread.