# Qom--A New Hydroologic Prediction Model Enhanced with Multi-Objective Optimization
## 5 Experimentation
### Solution Selection Strategy  
As commented before, the results of optimizing each problem with the seven multi-objective metaheuristics constitute a reference front composed of the non-dominated solutions produced by all the algorithms. Now, we proceed to describe and apply a solution strategy to select from the reference front that solution providing the best trade-off, according to the two objective functions, to properly set the Qom model with a high accurate prediction capability. Figure 4a,b show the reference fronts obtained for problem instances 1 and 2, respectively, including information about the number and percentage of solutions contributed by each algorithm to the reference front. We use a different combination of symbols and colors for each solution to associate to the algorithm that produced it. We start by analyzing the Asua river basin (a) with calibration series 1.1 CC. We observe that all the algorithms have contributed to the reference front, with SMS-EMOA being the one contributing with more solutions by far (70.81%), followed by MOEA/D (12.69%), NSGA-II (6.25%), AbYSS (5.26%), MOCell (2.76%), OMOPSO (1.64%), and SMPSO (0.58%). However, it is worth noting that the solutions in the extreme regions of the reference front have been obtained by algorithm AbYSS (green points with symbol \(\times\)). We observe also that the solutions of MOEA/D (blue points with symbol \(\triangledown\)) are concentrated on the bottom part of the front, which corresponds to higher values of the first objective (NSE) and lower values of the second objective function (PBIAS). The interesting fact is that all the algorithms complement each other, as the reference front has no gaps. We focus now on the numerical dispersion of results obtained for the calibration series with regards to validation ones on the Asua river data. Figure 5 includes the boxplots of distribution of results for each objective function and four data series of the problem. We observe that there are not high differences between calibration and validation. For objective function f1 (1-NSE measure), quarterlies of series 1.1 CC, 1.3 CV and 1.4 CV are in the range of 0.9 and 0.7 (NSE = 1 - f1) and only 1.2 CV shows higher dispersion with quarterlies 0.57 and 0.47, but without outliers. These results are classified in the reference literature [38] from very good to excellent. In the case of PBIAS (f2), series 1.1 CC, 1.2 CV and 1.3 CV show predicted results between 0% and 25% also without outliers, whereas for validation series 1.4 CV, results alternate from 45% to 60%, but with low dispersion between maximum and minimum limits. As AbYSS yielded the extreme solutions in the calibration, we have outlined the solutions having the lowest values for the two objectives in both, the calibration and validation results. We observe that AbYSS also gets the lowest values in all the cases but one, the 1.3 CV dataset with the second objective function. Keeping this in mind, we can now plot the different prediction error distributions of the two objective function values (1-NSE versus PBIAS), to show the influence of calibrated variables when applied to validation data series. This way, as shown in Figure 6 for problem instance 1 (Asua  
Figure 4: Reference Front approximations obtained by evaluated algorithms for Spanish river instances. The contribution of each algorithm (calibrator) to the reference front are computed in terms of percentage and number of non-dominated solutions. Figure 5: Boxplot diagrams of distributions of objective values obtained by AbYSS, for problem instance 1.1 CC (Asua river basin continuous series for calibration). The boxplots of objectives f1 = (1-NSE) and f2 = PBIAs are plotted in the left and right sub-figures, respectively. river basin), calibrated solutions remain accurate when applied to the corresponding validation data, although better results are obtained in general when selecting the best solution in 1.1 CC according to f2 (Figure 6 right).