## 4 Optimal Solution
### Nnsga-Ii  
The conventional NSGA-II includes two principal components: a non-dominated sorting solution and crowding distance (CD) sorting procedure for preserving the solution diversity [30]. The NSGA-II uses crossover and mutation operators to create the offspring population and adopts a rapid non-dominated sorting method to decide the non-dominated rank of individuals. Since all members of the previous solution injecting the new population may not be compiled, only several individuals corresponding to the number of available fronts can be chosen from the last solution based on the CD. Parents are also picked from the population using the crowded tournament selection method based on the rank and CD. The crowded tournament selection in the NSGA-II randomly identifies any two objects and selects the one in the less crowded region, when the two objects have the same non-domination level. The object with the lower rank or higher CD is decided. The adopted population makes offspring according to the crossover and mutation operators. Furthermore, this algorithm employs an elite preservation strategy to choose the new generation from the parent and offspring population. The CD sorting procedure calculates the dispersion of the solutions in each solution and maintains the Pareto solution diversity.  
\[CD_{i}=\frac{1}{N_{obj}}\sum_{g=1}^{N_{obj}}\left|f_{i+1}^{g}-f_{i-1}^{g}\right| \tag{34}\]  
where \(f_{i+1}^{g}\) and \(f_{i-1}^{g}\) are the \(g\)th objective of the \(i+1\)th and \(i-1\)th individuals, respectively.  
Although the NSGA-II contains improved concepts such as elitism, rapid non-dominated sorting, and diversity maintenance along the Pareto-optimal solution, it remains insufficient with respect to preserving the lateral diversity and a uniform distribution of the non-dominated solutions. An emphasis on lateral diversity is necessary to prevent excessive exploitation and thereby ensures that the search algorithm converges more rapidly. A stable distribution of non-dominated solutions is necessary to include the optimal Pareto solutions. To address the disadvantages of the NSGA-II, controlled elitism and the dynamic CD (DCD) are applied as the criteria for the optimal Pareto solution [31]. Therefore, in this study, it was ensured that the criterion for the multi-objective optimization process was satisfied by the convergence process, based on the criteria for assessing the Pareto optimal solution, which involves the controlled elitism and DCD of the proposed MNSGA-II.  
\[DCD_{i}=\frac{CD_{i}}{log(\frac{1}{Var_{i}})} \tag{35}\]  
where \(Var_{i}\) is the variance of CDs calculated by Equation (36). \(Var_{i}\) is based on  
\[Var_{i}=\frac{1}{N_{obj}}\sum_{g}^{N_{obj}}\left(\left|f_{i+1}^{g}-f_{i-1}^{g} \right|-CD_{i}\right)^{2} \tag{36}\]  
Regarding controlled elitism, the MNSGA-II regulates the number of objects in the optimal selection adaptively and preserves a pre-defined number of distributed objects in each solution. Firstly, the integrated parent and offspring population \(R_{h}=Pop_{h}\cup Off_{h}\) is divided for non-domination. Let \(Nf\) be the number of non-dominated solutions in the incorporated population (of size 2 \(M\)). According to the geometric distribution, the maximum available number of objects decided in the \(y\)th case (\(y=1\), 2,..., \(Nf\)) in the new population of size M\({}_{y}\) is expressed:  
\[M_{y}=M\cdot\frac{1-\gamma}{1-\gamma^{Nf}}\cdot\gamma^{y-1} \tag{37}\]  
where \(M_{y}\) is the new population size; \(M\) is the population size; \(\gamma\) is the reduction rate.  
Since \(\gamma<1\), the maximum available number of objects in the solution is the highest, and other solutions are permitted to contain an exponentially decreasing number of solutions.