## 5 Issues for Numerical Solutions
In particular, a crossover value close to 1 indicates a higher probability of considering the value from a mutation process, and a value close to zero indicates a higher probability of using values from the previous iteration. A crossover parameter is compared against a value from a continuous random distribution between one and zero. Additionally, a discrete uniform distribution can be used to guarantee that at least one component is different from the previous vector. The mutation operator in Figure 6 is used after selecting three different candidates from a population in line 16 of Algorithm 1, which defines a search direction and a step size to move in that direction. DE operators are controlled by the crossover parameter as (\(DE_{CR}\in[0,1]\)) and the mutation parameter as (\(DE_{F}\in[0,1]\)). A crossover parameter can be selected at the beginning of execution and remains constant while the mutation factor is selected randomly for every design variable. Figure 6: A flowchart of a DE implemented algorithm. ```
1\(\mathit{It}=0\)
2 Initialize the external memory \({}^{\mathit{It}}EMdb=\varnothing\)
3 create a random initial population \({}^{\mathit{It}}\mathbf{x}_{i}\;\forall i=1,...NI\)
4 evaluate objective functions \({}^{\mathit{It}}\mathbf{\Phi}=\mathbf{\Phi}({}^{\mathit{It}}\mathbf{x}_{i})\)
5 evaluate constraints \({}^{\mathit{It}}\mathbf{G}=G({}^{\mathit{It}}\mathbf{x}_{i})\)
6 evaluate non-dominated rules in search of feasible-non-dominated solutions to add to the \(EMdb\)
7 for\(\mathit{It}:=1\) to \(\mathit{It}_{\mathit{max}}\)do
8 for\(\mathit{it}:=1\) to \(\mathit{NI}\)do
9 if\(\mathit{DE}_{\mathit{CRW}}{}^{\mathit{It}}_{\mathit{max}}>\mathit{It}\) & \({}^{\mathit{It}}EMdb\geq 3\)then
10 select from \({}^{\mathit{It}}EMdb\) the three individuals with higher crowding value \(\{r_{1}\
eq r_{2}\
eq r_{3}\}\)
11 else
12 select randomly \(\{r_{1}\
eq r_{2}\
eq r_{3}\}\) from current population
13
14 end if
15 for\(\mathit{f}:=1\) to \(\mathit{NV}\)do
16 if\(\mathit{IU}(0,1)<\mathit{DE}_{\mathit{CR}}\;\|\;\mathit{j}=\mathit{U}\{0,NI\}\)then
17\({}^{\mathit{It}}\mathbf{G}=G({}^{\mathit{It}}\mathbf{v}_{\mathit{ij}})\)
18 evaluate non-dominated rules in search of feasible-non-dominated solutions to add to the \(EMdb\)
19
20 end if
21
22 verify non-dominance among existing feasible-non-dominated solutions and recent added candidates in the \(EMdb\)
23
24 end for
```  
**Algorithm 1**Pseudo code of a multi-objective DE implementation. Once the candidates of a population are generated according to the DE algorithm approach, all the candidates are evaluated using objective functions, inequality, and equality constraints of Equations (1)-(7). The last computations in Figure 6 are executed to verify the Pareto dominance of the candidates versus the current population in order to update the EMdb and to verify the same rules for all that stored in the EMdb. The stop criterion is a maximum number of iterations that can be fixed as an additional algorithm parameter, and this procedure is executed until that value is reached. Algorithm 1 list the computations that are programmed in a routine to solve the formulated multi-objective optimization problem of Equations (1)-(9) using the above mentioned steps as summarized in the flowchart of Figure 6.