## 3 Resolution Methodology  
In multi-objective optimization, it is not possible to find a unique optimal solution that simultaneously optimizes all the objective functions [51]. Thus, the goal is to identify a set of efficient solutions that cannot be compared mathematically and demonstrate the tradeoff between two objectives. Specifically, in bi-objective minimization problems, a solution vector \(s^{*}\) within the feasible space \(S\) is considered a non-dominated or efficient solution if there is no other solution \(s\in S\) that is strictly better, i.e., simultaneously holds \(f_{1}(s)\leq f_{1}(s^{*}),f_{2}(s)\leq f_{2}(s^{*})\) and \(f_{i}(s)<f_{i}(s^{*})\) for some objective \(i\) (where \(f_{i}(s)\) is the objective function of objective \(i\) evaluated on solution \(s\)). For example, Figure 3 illustrates a bi-objective minimization problem. In this depiction, solution \(c\) is evidently dominated by solutions \(a\), \(b\), and \(d\), as these three solutions exhibit superior values in both objectives. Subsequently, solution \(d\) is dominated by solution \(a\) since, despite having the same value for \(f_{2}\), solution \(a\) excels in terms of objective \(f_{1}\). Finally, solutions \(a\) and \(b\) emerge as non-dominated solutions, with neither being preferable over the other based on the provided information. The set of non-dominated or efficient solutions is the Pareto frontier. Figure 3: Example of non-dominated solutions in a bi-objective minimization problem. The most commonly used methods for handling multi-objective problems are the weighting or weighted-sum method and the \(\varepsilon\)-constraint method. These methods need to be executed multiple times to find in each run a non-dominated solution and iteratively construct the Pareto front. However, the weighting method usually requires a larger number of runs as many of them can result in repeated solutions. Additionally, this method cannot reach solutions located in non-convex regions of the Pareto front [51; 52]. In this work, the \(\varepsilon\)-constraint method [53] is chosen to address the target problem. This selection is based on the evidence in the related literature that has been successfully applied in other similar bi-objective problems related to transport planning. For example, [PERSON] et al. [54] solved a bi-objective optimization problem for the transit network design of public transport while minimizing the travel times and the monetary costs of users and while also considering the cost of the system as a constraint. They implemented the \(\varepsilon\)-constraint method, which was able to obtain approximations of the Pareto front for benchmark instances based on benchmark instances developed by [PERSON] [55] in reasonable computing times. Moreover, the \(\varepsilon\)-constraint method allowed them to study the tradeoff among both objectives, leading to relevant information for the decision-making process. [PERSON] et al. [56] used the \(\varepsilon\)-constraint to addressed a Dial-a-Ride public transport problem in which the customer requires a delivery to be made specifying the place of pickup (origin) and delivery (destination), time windows for the pickup and the delivery, and the quantity of a certain product that has to be transported. The model aims at assigning vehicles for the required trips while minimizing the maximum total ride time and the total waiting time for customers. The model is applied over benchmark instances [57]. Finally, in another transport planning problem, [PERSON] and [PERSON] [58] applied the \(\varepsilon\)-constraint method to design the route of a CityTour Bus service, minimizing the total investment cost and CO\({}_{2}\) emissions. Their study, focused on the city of Seoul, effectively explored the tradeoff among objectives, showcasing the versatility and applicability of the \(\varepsilon\)-constraint method in diverse transport planning scenarios.