## 5 Metaheuristics  
This section describes the MO metaheuristics considered for solving the two problems. As introduced before, some of the metaheuristics were exposed in prior works by ourselves and then, we chose not to include the whole implementing information in the present proposal to avoid duplicity. Instead, we next detail some key aspects adapting the meheuristics to solve the problem, for further information we recommend readers going to the specific works [44; 45; 46; 47]:* NSGA-II: It considers two populations \(P_{n}^{\mathcal{S}}\) and \(Q_{n}^{\mathcal{S}}\) with \(ps_{n}\) individuals each, where \(P_{n}^{\mathcal{S}}\) saves the parents of the current iteration \(g\) and \(Q_{n}^{\mathcal{S}}\) saves the offspring generated based on \(P_{n}^{\mathcal{S}}\). The individuals in the populations follow the same chromosome structure, where each individual has many genes as RNs should be deployed in the solution. Note that a gene includes the 2D-coordinates of an RN. This structure is considered for the remaining algorithms exposed in this section. Each individual in \(Q_{n}^{\mathcal{S}}\) is generated by applying crossover and mutation operators based on two previously selected solutions from \(P_{n}^{\mathcal{S}}\). The crossover operator is the usual one-point-crossover with a crossover probability \(cro_{n}\). The mutation operator applies random changes in the genes of the solution generated by the crossover operator according to a mutation probability \(mut_{n}\). The populations for the next iteration are generated as follows. \(P_{n}^{\mathcal{S}+1}\) is generated by selecting the best \(ps_{n}\) solutions combining \(P_{n}^{\mathcal{S}}\) and \(Q_{n}^{\mathcal{S}}\) according to the crowded-comparison operator \(\prec_{n}\)[14] and \(Q_{n}^{\mathcal{S}+1}\) is initialized to empty. * SPEA2: It considers a regular population \(P_{n}^{\mathcal{S}}\) of size \(ps_{s}\) and an auxiliary population \(\overline{P}_{n}^{\mathcal{S}}\) of size \(\overline{ps}_{s}\), saving the best individuals found so far. The methodology followed by SPEA2 is similar to NSGA-II, but considering a different selection strategy to \(\prec_{n}\) when generating \(P_{n}^{\mathcal{S}+1}\). We consider the same crossover and mutation operators as for NSGA-II with probabilities \(cro_{s}\) and \(mut_{s}\). * MO-ABC: It is an MO approach of ABC, which was adapted by ourselves based on the \(\prec_{n}\) concept. The algorithm considers a population \(P_{n}^{\mathcal{S}}\) with size \(ps_{a}\). The parameter \(se_{a}\) determines the percentage of solutions in \(P_{n}^{\mathcal{S}}\) managed by employed forager bees and the remaining ones are managed by onlooker bees. An employed forager bee tries to improve the solution by looking in its surrounding, i.e., RN coordinates are lightly modified. Note that the new solution is only accepted if it is better in fitness value; otherwise, it is discarded. If an employed forager bee tries to improve the solution \(limit_{a}\) times without any improvement in fitness value, then the solution is supposed exhausted, being mandatory be managed by an scout bee. An onlooker bee tries to improve the solution by looking in the surrounding of a randomly selected employed forager bee. As before, the solution is only accepted it it is better in fitness value. An scout bee generates a new solution based on a randomly selected onlooker bee solution from the two first Pareto fronts in \(P_{n}^{\mathcal{S}}\). Next, the Euclidean distance between the solution selected and all other solutions in \(P_{n}^{\mathcal{S}}\) is calculated. The new solution is obtained by combining the \(k_{a}\)-nearest solutions to the selected one, being \(k_{a}\) a random value in \(2,\ldots,11\). On the contrary that for employed forager and onlooker bees, the solution generated by the scout bee is directly accepted without analyzing the improvement in fitness value. \(P_{n}^{\mathcal{S}+1}\) is generated by including the solutions generated by the corresponding bees.