## 4 Genetic Algorithm
### Genetic Operators
#### 4.5.4 Tournament Selection
The resulting convergence graphs are given in Figure 15. From these graphs, one can see that parallelization brings performance improvements in both RGA and 2 SGA. However, the crucial finding in this investigation is that the sequential 2 SGA using a single CPU outperforms the parallel RGA that uses 80 CPUs. This finding asserts the superiority of 2 SGA over RGA in solving the proposed multi-objective F

Figure 13: The convergence of 2 SGA and RGA in solving Problems 4, 5, and 6. (Each convergence graph is an average of 40 trials. The histograms are for the final values of the objective function in 40 trials). Figure 14: Randomly connected topologies for a given communication matrices (adopted from [47]). Note: The communication matrix and topology is generated every time before solution migration occurs. Figure 15: Improvements of convergence behaviors of RGA and 2 SGA using the parallelization technique. ### Empirical Analysis of the Algorithm Parameters

#### 5.3.1 Selection Operators

In this section, we present comparative empirical studies on the various selection operators presented in Section 4.5. The comparisons are presented in terms of the convergence behavior of 2 SGA in solving Problem 4, whereas similar results were obtained in solving several other problems. The first of these imperial studies is aimed at comparing the three fitness transformation functions in Equations (60)-(62) used in the proportional selection method. Figure 16 provides the average convergence from ten test runs using these three different fitness transformation equations. As can be seen from this figure, Equations (60) and (61) resulted in similar convergence behaviors of the algorithm. In contrast, the transformation function in Equation (62) resulted in a much better convergence of 2 SGA when using the proportional selection method. The second empirical study investigates the impact of tournament size in tournament selection. Figure 17 depicts the results of this study. This figure shows that tournament selection with a smaller tournament size was preferred in solving the proposed mathematical model using 2 SGA. Lastly, a comparison of proportional, linear ranking, and tournament selection was conducted, and the resulting convergence graphs are given in Figure 18. This figure shows that the tournament selection resulted in an improved convergence of the proposed 2 SGA. Hence, tournament selection with a small tournament size is the preferred selection operator in the proposed algorithm. Figure 16: The average convergence from ten test runs of 2 SGA using proportional selection under three different fitness transformation while solving Problem-4. Figure 17: The average convergence from ten test runs of 2 SGA using tournament selection under different tournament sizes while solving Problem-4. #### 5.3.2 Crossover and Mutation Probabilities

In the proposed algorithm, there are seven crossover and six mutation operators. Assigning probabilities individually for these thirteen operators and simultaneously tuning them can be a daunting task. Instead, in this paper, we suggested the crossover and mutation operators be assigned one crossover and one mutation probability, respectively. With this scheme, we performed an Analysis of Variance (ANOVA), where mutation and crossover probabilities are the only two factors, and the objective function is the response. We chose six levels for each of these factors. The levels for the mutation and crossover probabilities were {0.05, 0.15, 0.25, 0.35, 0.45, 0.55} and {0.75, 0.80, 0.85, 0.90, 0.95, 1.00}, respectively. For each factor level combination, we conducted five replications. Hence, the experiment required solving a problem 180 times. The genetic algorithm used a different seed for its random number generator in each replication of the experiment.