# A Microbial Screening in Silico Method for the Fitness Step Evaluation in Evolutionary Algorithms
## 3 Results  
The results obtained in the experiments with the genetic algorithms (Figures 10 and 11) opens the possibility of evaluating fitness in evolutionary algorithms by applying a methodology inspired by synthetic biology techniques. In consequence, the proposed screening procedures are helpful for the fitness evaluation solving optimization problems with genetic algorithms. For example, in the 0/1 knapsack problem, with the microbial screening procedure, we obtained the following fitness function:  
\[\textbf{Fitness}\ (y_{i})\ =\ \left\{ \begin{array}{ll}w\leq W,&352.37\,x_{i}-14 2.71\\ w>W,&29.96\,x_{i}+30.49\end{array} \right.\]  
where \(x_{i}\) is the value of a set of items selected for the knapsack, \(w\) is the weight of the knapsack and \(W\) is the maximum weight capacity. Finally, the function was included in the genetic algorithm (in our case, written in Python language). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Gro Version** & **Real Time** & **Simulated Time** \\ \hline
[14] & 35 min & 248 min \\ \hline
[16] & 4 min & 278 min \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation times in the Gro simulator. Figure 10: Performance graph (average fitness per generation) for the 0/1 knapsack problem obtained under (**a**) SGA with antibiotic screening procedure, (**b**) SMGA with antibiotic screening procedure, (**c**) SGA with the Lac operon screening procedure, (**d**) SMGA with the Lac operon screening procedure, (**e**) SGA with chemotaxis screening procedure and (**f**) SMGA with chemotaxis screening procedure. In the SAT optimization problem, the evaluation function is a table (Table 3) that includes the number of satisfied clauses and a fitness average score. In general, screening methods in which the selection is based on an antibiotic or the Lac operon are more effective in evaluating fitness than the screening inspired by bacterial chemotaxis. Regarding the two evolutionary algorithms, we can conclude that there are no differences in the evolutionary convergence between the SGA algorithm and the SMGA algorithm. According to Figures 10 and 11, we can conclude that the kind of crossover operator does not affect the evolution of the population, and in consequence, the algorithm performance. Figure 11: Performance graph (average fitness per generation) for the 2-SAT problem obtained under (**a**) SGA with antibiotic screening procedure, (**b**) SMGA with antibiotic screening procedure, (**c**) SGA with Lac operon screening procedure, (**d**) SMGA with Lac operon screening procedure, (**e**) SGA with chemotaxis screening procedure and (**f**) SMGA with chemotaxis screening procedure. \begin{table}
\begin{tabular}{c c c} \hline
**Satisfied Clauses** & **Gene** & **Fitness (y\({}_{1}\))** \\ \hline
3 & \(\{0,0,0,0,0\}\) & 1669.2 \\ \hline
3 & \(\{0,0,0,0,0\}\) & 1793.75 \\ \hline
3 & \(\{0,0,0,0,0\}\) & 1848.92 \\ \hline
2 & \(\{0,0,0,0,1\}\) & 290.837 \\ \hline
2 & \(\{0,0,0,0,1\}\) & 282.283 \\ \hline
2 & \(\{0,0,0,0,1\}\) & 319.313 \\ \hline
1 & \(\{0,0,0,1,0\}\) & 36.958 \\ \hline
1 & \(\{0,0,0,1,0\}\) & 31.8889 \\ \hline \end{tabular}
\end{table} 
Table 3: In silico screening output for the 2-SAT optimization problem. In the 0/1 knapsack problem, the evolutionary surfaces (Figure 10) obtained are very similar to each other, not depending on the screening procedure. However, the most efficient screening method is the one that simulates the Lac operon. In general, we observed a surface that rises from the solution with weight 0 and profit 0 to the optimal solution for the instance with weight 100 and profit 55. Afterwards, the surface underwent a sharp decrease, once the capacity of the knapsack was exceeded (maximum tolerated weight of 100). Therefore, it can be concluded that the proposed screening procedures are capable of generating a representative evolutionary surface (Figure 12) for the addressed problems.