## 6 Numerical Experiments
### Comprehensive Evaluation of the Algorithms
#### 6.4.5 Changes in Population Diversity  
The population diversity is considered to be one of the critical factors in the design of EAs. A diverse population is represented not only with the individuals that have high fitness values (a.k.a., \"super-individuals\"), but also with the individuals that have medium and low fitness values. The EA with a diverse population can explore various domains of the search space more efficiently [20; 21]. Lack of the population diversity, especially at the beginning of the algorithmic run, may negatively affect the EA performance and cause the \"premature convergence\". The \"premature convergence\" generally occurs due to lack of the explorative and exploitative capabilities of the algorithm when it is not able to identify any other promising domains of the search space and convergences at one of the local optima. The scope of numerical experiments includes evaluation of the population diversity for all the developed solution algorithms. This study adopted the standard deviation (STD) of the fitness function values of the population chromosomes as an indicator of the population diversity for each algorithm. The population fitness STD values were recorded at each generation of the developed solution algorithms for each replication and large size problem instance. Figure 14 illustrates changes in the population fitness STD values throughout evolution of the SAEA, AEA, DPCEA, and EA algorithms for the first replication of problem instances 55-60 (6 large size problem instances with the largest number of vessels, calling for service at the MCT). Note that similar tendencies in the algorithmic population diversity changes were observed for the rest of replications and problem instances.  
The conducted numerical experiments indicate that all the developed solution algorithms were able to maintain a diverse population, especially at the beginning of the search process (i.e., within the first \(\approx\) 50 generations). For certain solution algorithms, the population fitness STD values even exceeded 1000 h (e.g., the maximum population fitness STD values of the SAEA and EA algorithms comprised 1138.08 h and 1096.09 h, respectively, for problem instance 56). Such a high population diversity at the beginning of the search process can be justified by the adopted population initialization mechanism, where half of the population was initialized using the FCFS-SR heuristic, while the other half was generated randomly. As discussed in Section 4.3 of the manuscript, a random population initialization may negatively affect fitness of the generated chromosomes but allows maintaining a diverse population.  
Moreover, it can be noticed that the population fitness STD significantly decreased after \(\approx\)50 generations, as the developed solution algorithms identified a set of promising domains of the search space and primarily focused on the exploitation process. Although all the developed solution algorithms were able to maintain a diverse population, application of the self-adaptive parameter control strategy (within the SAEA algorithm) was found to be more efficient for the search process as compared to the other parameter control strategies and the parameter tuning strategy, as the SAEA  
Figure 13: The objective function coefficient of variation values for the developed solution algorithms and large size problem instances.  
algorithm outperformed the other considered solution algorithms in terms of the objective function values at termination (see Section 6.4.1 of the manuscript for more details).