# A Soft Computing Framework to Support Consumers in Obtaining Sustainable Appliances from the Market
## 4 Results and Discussion  
The proposed model was then applied to the case study considered here. After defining the calculations according to LCCA, and regarding each individual solution as well as the corresponding attributes according to MAVT (Appendices A and B), the optimization process took place using the NSGAII algorithm. The corresponding algorithm was then coded on MATLAB software, by accounting for the following parameters:  
* Selection method: tournament
* Crossover method: double point
* Mutation method: random mutation (one point)  
The remaining parameters, namely the initial population, crossover, and mutation rate, were established after several trials. The first parameter to be tested was the stopping criterion \"max number of generations\", where several runs were performed considering the corresponding values of 80 and 90 (Figure 6a,b respectively). A maximum number of iterations (generations) of 80 was also defined, which was achieved given the neglectable difference obtained between the corresponding Pareto fronts (Figure 6a,b) regarding both scenarios, i.e., Economics & Environment (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.00\), and \(\omega_{\rm C}=0.35\)) and Economics & Social (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.35\), and \(\omega_{\rm C}=0.00\)). Other parameters were also determined, such as the size of population (150 individuals), the size of tournament (10), the rate of crossover (0.75), and the rate of mutation (0.25). In order to better analyze the fitness behavior considering different values of mutation and the crossover rate, a robustness test was performed considering two scenarios and regarding the considered  
Figure 5: Non-dominated sorting genetic algorithm II (NSGAII) flowchart. case study, i.e., Economics & Environment (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.00\), and \(\omega_{\rm C}=0.35\)) and Economics & Social (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.35\), and \(\omega_{\rm C}=0.00\))). The fixed parameters were the size of population (100 individuals) and the size of the tournament (12 individuals). The rates regarding the mutation and crossover operators were then changed by performing several trials of crossover and mutation values (Table 4). All the trials shown in Table 4, were executed by setting a maximum number of iterations (90). The respective results are shown on Figure 7a,b, for each considered scenario. It is noted that a small change in the value of each parameter, has a negligible effect in the obtained results, considering both scenarios. Therefore, the parameters NSGAII that were used to show the sustainable results obtained in this case study, were tournament size (10), max iteration (90), population size (100), mutation rate (0.1) and crossover rate (0.9). After performing NSGAII calculations, a Pareto frontier is obtained by accounting for the scenarios described above for Economics vs. Social (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.35\), and \(\omega_{\rm C}=0.00\)) and Economics vs. Environment (\(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.00\), and \(\omega_{\rm C}=0.35\)) (Figure 8a,b respectively). Each point (or node) represents a global sustainable solution of the problem formed by a set of sustainable (and individual) solutions (household appliances) regarding each appliance type to be acquired by the consumer. Although the Economic wellbeing decreases, the Social one increases (Figure 8a), with the same trade-off, being observed in Figure 8b, by considering only Economic and Environmental wellbeing dimensions. Regarding the case study considered here and based on both trade-offs presented above, a scenario was considered with three dimensions and their corresponding consumer's relative importance, represented by the corresponding weights, i.e., \(\omega_{\rm A}=0.65\), \(\omega_{\rm B}=0.25\), and \(\omega_{\rm C}=0.10\).