## 6 Experimental Results
### Solving the FLP Using Genetic Algorithms  
The genetic algorithms used for solving the problems presented in Section 2 are parameterized as follows: a random seed is initially generated, and half of the problem instances are assigned to the MOPSDA algorithm, while the other half is assigned to the NSGA-II algorithm. The population size is set to 100, and the number of generations is fixed at 100. The number of runs is also set to 5. These multi-objective evolutionary algorithms (MOEAs) are used because they can handle non-continuous, non-convex, and/or non-linear objectives/constraints, as well as problems where the objective function is not explicitly known. The NSGA-II and MOPSDA algorithms are utilized to obtain near-optimal solutions by efficiently exploring a fraction of the entire solution space. The MOEAs are based on the process of evolution, where the best traits of a population are identified and used to generate the next generation or replace the population. To solve the first subproblem of Section 2, the optimal Pareto set is approximated using both NSGA-II and MOPSDA, and the resulting Pareto sets are merged into one. The decision variable \(y\) is represented using a chromosome, where position \(i\) has the bit \(y_{i}\). The chromosome length is equal to the number of distribution centers (\(|F|\)), and element \(y_{i}\) is equal to 1 if the facility is open, and 0 otherwise. Each chromosome is evaluated using two figures of merit: one to obtain the number of facilities and the other to obtain the total distance (Figures 5 and 6). Figure 5: Computed Pareto set for the Swain dataset using genetic algorithms. We chose the optimal solutions obtained from 20 independent runs of the genetic algorithms illustrated in Figure 5. This figure depicts the Pareto set obtained for the Swain dataset and presents a reasonable representation of the Pareto set. However, the genetic algorithms fail to find solutions for all possible instances. This results in three potential approaches to addressing the missing solutions. The first option is to re-run the genetic algorithms (as we carried out 20 times). Nevertheless, this is computationally expensive and consumes substantial resources (approximately 30 min for the Swain dataset and 3.5 h for the London dataset). The second option involves utilizing a crowdsourcing approach, as proposed, to fill the gaps left by the genetic algorithms. Finally, as demonstrated in the following sections, the third alternative uses machine learning for predictive purposes as a more efficient and timely solution to fill the gaps. We chose the optimal solutions obtained from 20 independent runs of the genetic algorithms illustrated in Figure 6. This figure depicts the Pareto set obtained for the London dataset and highlights the incompleteness of the solutions found by the Pareto set. Furthermore, NP-hard problems can become computationally intractable for larger instances of the FLP or other complex problems. As a result, it is imperative to have alternative approaches, such as the one proposed in this study, to estimate the solutions that would be obtained after multiple iterations of the genetic algorithms. This is particularly relevant for larger instances of NP-hard problems, as they require increased computational resources. The Pareto set is defined as the set of solutions that cannot be improved in at least one dimension without worsening in another dimension. However, in some cases, the relationships between different solution dimensions may not be monotonic; that is, there may be solutions that are not strictly dominated but do not belong to the Pareto set. In these non-monotonic situations, solutions may not have a clear order among themselves, making the identification of the Pareto set more challenging. Additionally, a simple post-processing of the output of the genetic algorithm may not be sufficient to resolve these non-monotonic situations.