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We study the problem of the optimal design of routes and frequencies in urban public transit systems, the Transit Network Design Problem (TNDP), which is modeled as a multi-objective combinatorial optimization problem. A new heuristic based on the GRASP metaheuristic is proposed to solve the TNDP. As a multiobjective metaheuristic, it produces in a single run a set of non-dominated solutions representing different trade-off levels between the conflicting objectives of users and operators. Previous approaches have dealt with the multi-objective nature of the problem by weighting the different objectives into a single objective function. The case proposed by Mandl is used to show that the multi-objective metaheuristic is capable of producing a diverse set of solutions, which are compared with solutions obtained by other authors. We show that the proposed algorithm produces more non-dominated solutions than the Weighted Sum Method with the same computational effort, using the case of Mandl and another real test case.

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Frequency setting takes place at the strategic and tactical planning stages of public transportation systems. The problem consists in determining the time interval between subsequent vehicles for a given set of lines, taking into account interests of users and operators. The result of this stage is considered as input at the operational level. In general, the problem faced by planners is how to distribute a given fleet of buses among a set of given lines. The corresponding decisions determine the frequency of each line, which impacts directly on the waiting time of the users and operator costs. In this work, we consider frequency setting as the problem of minimizing simultaneously users' total travel time and fleet size, which represents the interest of operators. There is a trade-off between these two measures; therefore, we face a multiobjective problem. We extend an existing single-objective formulation to account explicitly for this tradeoff, and propose a Tabu Search solving method to handle efficiently this multi-objective variant of the problem. The proposed methodology is then applied to a real medium-sized problem instance, using data of Puerto Montt, Chile. We consider two data sets corresponding to morning-peak and off-peak periods. The results obtained show that the proposed methodology is able to improve the current solution in terms of total travel time and fleet size. In addition, the proposed method is able to efficiently suggest (in computational terms) different trade-off solutions regarding the conflicting objectives of users and operators.to the users and minimize the level of required resources delivered by the operators. Problem data is given by the itinerary of each line and origin-destination (OD) demand within a specific time horizon. An important component of the model is the assignment sub-model, which represents the behavior of the users with respect to a set of lines and frequencies. This sub-model is needed to measure the performance of the system with respect to the users, that is, the level of service.The literature concerning transit frequency optimization can be classified into (i) analytical models that admit closed-form solutions and (ii) mathematical programming formulations either explicit or not, with associated solution algorithms. In the first group, there are formulations that characterize the system in terms of few variables and allow getting a full description of the optimal solution. Although these models make considerable simplifications of the real system, they allow obtaining practical guidelines that are theoretically well founded [3,4], for example, the well-known rule of the square root [5,6]. The other important stream of work is based on a detailed characterization of the transit system, in terms of the route network and the demand that should be transported over it. These studies formulate the optimization problem in terms of a graph model, where decision variables are the capacities of the arcs (represented by the frequencies) and the flows that represent the...

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