## 5 Metaheuristics
The distribution is performed according to \(CHIMinc_{m}\in[1,\infty)\) and \(crow_{m}\in(0,\infty)\) parameters, where \(CHIMinc_{m}\) defines how the extreme points of the CHIM are reassigned to increase the search area and \(crow_{m}\) defines the distance between two any reference points. Each reference point is assigned a set of reference points in its neighbouring according to the Euclidean distance, where the cardinal of this set is given by the \(neigh_{m}\) parameter. On this basis, MOEA/D considers a regular population \(P_{m}^{g}\), where each individual is associated with a different reference point in the CHIM, and an auxiliary population \(F_{m}^{g}\) of undefined size. Thus, \(P_{m}^{g}\) contains the individuals considered to generate solutions in the \(g\)-th iteration and \(F_{m}^{g}\) saves the non-dominated solutions found until \(g\). Over generations and for each solution in \(P_{m}^{g}\), two neighbouring solutions are selected (based on the neighbouring structure previously generated) to then produce a solution based on the crossover and mutation operators defined for NSGA-II, with \(cro_{m}\) and \(mut_{m}\), respectively. The solution generated replaces the previous one only if it is better in fitness value for the corresponding single-objective sub-problem. Some of the previously exposed metaheuristics were not considered in prior works for solving any of the two problems addressed in the present proposal. These algorithms are:  
* MO-FA and MOEA/D for the bi-objective approach. * MO-GSA and MO-VNS* for the three-objective approach. Without loss of generality, MO-FA, MO-GSA, and MO-VNS* can be implemented with minimal changes independently of the number of objectives. Most changes are related to the implementation of \(\prec_{n}\) as detailed in [45; 46; 47]. On the contrary, the implementation of MOEA/D for the bi-objective case requires further explanation. Suppose a bi-objective optimization problem maximizing \(f_{1}\) and \(f_{2}\). Let \(F_{1}=(\max\)\(F(f_{1}),\min\)\(F(f_{2}))\) and \(F_{2}=(\min\)\(F(f_{1}),\max\)\(F(f_{2}))\) be the two extreme points delimiting the objective space, where \(\max\)\(F(\cdot)\) and \(\min\)\(F(\cdot)\) denote the upper and lower bounds of a fitness function. Let \(\Upsilon=\{r^{1},\ldots,r^{ps_{m}}\}\) be a set of points evenly distributed on the plane I, for \(r^{i}=(r_{1}^{m},r_{2}^{m})\in\Upsilon\), \(m\in 1,\ldots,ps_{m}\), where \(ps_{m}\) is the cardinal of \(\Upsilon\) and \(F_{1},F_{2}\in\Upsilon\). Let \(\overline{v}_{n}^{\star}=(n_{1},n_{2})\) be a normal vector to the plane I. On this basis, the bi-objective optimization problem is decomposed into \(ps_{m}\) single-objective minimization subproblems, where the \(m\)-th subproblem optimizes the function \(g(x:r^{m},\overrightarrow{v_{n}})\) given by  
\[g(x:r^{m},\overrightarrow{v_{n}})=\max\{n_{1}\;(f_{1}(x)-r_{1}^{m}),n_{2}\;(f _{2}(x)-r_{2}^{m})\}, \tag{16}\]  
where \(x\) is a solution to the optimization problem. In addition of how to decompose the optimization problem, another important aspect in MOEA/D is the distribution of the reference points Y in the plane I. Algorithm 1 shows the procedure considered for distributing such points for the bi-objective approach by following a straight line defined by \(F_{1}\) and \(F_{2}\). This algorithm considers as input \(F_{1}\), \(F_{2}\), \(crow_{m}\), and \(CHIMinc_{m}\). Initially, at step 0 (lines 1-3), the new extreme points \(E_{1}\) and \(E_{2}\) depending on \(F_{1}\) and \(F_{2}\) are calculated solving the expression given by  
\[d(E_{1},E_{2})=d(F_{1},F_{2})\;CHIMinc_{m}, \tag{17}\]  
where \(d(\cdot)\) provides the Euclidean distance between two points. At step 1 (line 4), the number of divisions \(n_{E_{1},E_{2}}\) in the segment \(\overline{E_{1}E_{2}}\) is calculated. Finally, at step 2 (lines 5-8), reference points in the segment \(\overline{E_{1}E_{2}}\) are generated. From this point, MOEA/D can be implemented with minimal changes independently of the number of objectives as detailed in [47].