## 5 Metaheuristics
* MO-FA: It is an MO approach of the Firefly Algorithm (FA), which was adapted by ourselves based on the \(\prec_{n}\) concept. In this algorithm, a firefly is a possible solution to the problem and its brightness is defined by its solution quality. The attractiveness that a brighter firefly causes in a less bright one implies a movement in its RNs controlled by \(r_{f}\in[0,1]\), \(\beta_{0f}\in[0,1]\), and \(\lambda_{f}\in(0,\infty)\) parameters. Thus, MO-FA considers two populations \(P_{n}^{\mathcal{S}}\) and \(Q_{f}^{\mathcal{S}}\) with size \(ps_{f}\), where \(P_{f}^{\mathcal{S}}\) saves the fireflies at the beginning of the iteration \(g\) and \(Q_{f}^{\mathcal{S}}\) contains the resulting fireflies after applying the attractiveness mechanism in \(P_{f}^{\mathcal{S}}\). The populations for the next iterations are generated as follows. \(Q_{f}^{\mathcal{S}+1}\) is initialized to empty and \(P_{f}^{\mathcal{S}+1}\) is generated by selecting the \(ps_{f}\) best solutions combining \(P_{f}^{\mathcal{S}}\) and \(Q_{f}^{\mathcal{S}}\). Finally, in case that the percentage of non-dominated solutions in \(Q_{f}^{\mathcal{S}}\), regarding \(P_{f}^{\mathcal{S}}\) is lower than \(when\_sc_{f}\in[0,1]\), then the mutation operator discussed for NSGA-II is applied with mutation probability \(mut_{f}\) to each solution in \(P_{f}^{\mathcal{S}+1}\). * MO-VNS: It considers two populations \(P_{n}^{\mathcal{S}}\) and \(S_{n}^{\mathcal{S}}\) with unlimited size, where \(P_{n}^{\mathcal{S}}\) keeps only non-dominated solutions and \(S_{n}^{\mathcal{S}}\) saves the solutions from \(P_{n}^{\mathcal{S}}\) considered to explore the search space during the current iteration \(g\), i.e., \(S_{n}^{\mathcal{S}}\) is put to empty at the beginning of the iteration. A non-considered solution is selected from \(P_{n}^{\mathcal{S}}\) until all solutions were selected. The solution is used to generate new individuals in its surrounding based on \(neigh_{v}\) neighbourhood structures. Each structure determines how different could be the new solution compared to the initial one in terms of maximum displacement of the RNs, which is limited by the \(ns_{v}\in[1,\infty)\) parameter. Thus, the neighborhood structures are iteratively applied from higher to lower displacement, generating new solutions to be included in \(P_{v}^{g+1}\) if they fulfill the non-dominated requirement. * MO-VNS*: It considers the same focus as for MO-VNS but including a perturbation mechanism at the end of each iteration. This mechanism is performed for each solution in \(P_{v}^{g+1}\) by applying the mutation operator discussed for NSGA-II and SPEA2 with mutation probability \(mut_{v}\). * MO-GSA: It is an MO approach of the Gravitational Search Algorithm (GSA), which was adapted by ourselves based on the \(\prec_{n}\) concept. In this algorithm, an object is a possible solution to the problem and its mass is defined by its solution quality. All objects are mutually attracted by the Newtonian gravity force, causing a global movement of all objects towards heavier masses, corresponding the better solutions. The algorithm considers two populations \(P_{ga}^{g}\) and \(Q_{ga}^{g}\) with \(ps_{ga}\) individuals, where \(P_{ga}^{g}\) saves the objects at the beginning of \(g\), before acting gravitational forces, and \(Q_{ga}^{g}\) contains the resulting objects after applying the forces in \(P_{ga}^{g}\). \(P_{ga}^{g+1}\) is generated by selecting the best \(ps_{ga}\) solutions combining \(P_{ga}^{g}\) and \(Q_{ga}^{g}\). Finally, in case that the percentage of non-dominated solutions in \(Q_{ga}^{g}\), regarding \(P_{ga}^{g}\) is lower than \(when\_sc_{ga}\), then the mutation operator for NSGA-II is applied with mutation probability \(mut_{ga}\) to each solution in \(P_{ga}^{g+1}\). * MOEA/D: It decomposes an MO optimization problem into several single-objective sub-problems by distributing reference points on the Convex Hull of Individual Minima (CHIM), based on the NBI-Tchebycheff approach [59].