Choice-based optimization problems are the family of optimization problems that incorporate the stochasticity of individual preferences according to discrete choice models to make planning decisions. This integration brings non-convexity and nonlinearity to the associated mathematical formulations. Previously, the authors have tackled these issues by introducing a simulation-based approximation of the choice model with the aim of linearizing it. Nevertheless, already existing exact methods and state-of-the-art commercial solvers fail to solve relevant instances. In this paper, we propose a novel Lagrangian decomposition method inspired by scenario decomposition and scenario grouping in the stochastic programming framework for the purpose of solving choice-based optimization problems. In addition, we develop a tailored algorithm to generate feasible solutions to the original problem from the solution of the Lagrangian subproblem. Hence, at each iteration of the subgradient method, which is used to solve the Lagrangian dual, we provide both an upper and a lower bound to the original problem. This enables the calculation of the duality gap to assess the quality of the generated solutions. Computational results show that the decomposition method provides solutions with optimality gaps below 0.5% and restricted duality gaps within low computational times. We also show that scenario grouping leads to high-quality feasible solutions and lower duality gaps.

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In many transportation systems, a mismatch between the associated design and planning decisions and the demand is typically encountered. A tailored system is not only appealing to operators, which could have a better knowledge of their operational costs, but also to users, since they would benefit from an increase in the level of service and satisfaction. Hence, it is important to explicitly allow for the interactions between the two in the model governing the decisions of the system. Discrete choice models (DCM) provide a disaggregate demand representation that is able to capture the impact on the behavior of these decisions by taking into account the heterogeneity of tastes and preferences of the users, as well as subjective aspects related to attitudes or perceptions. Despite their advantages, the demand expressions derived from DCM are non-linear and non-convex in the explanatory variables, which restricts their integration in optimization problems. In this paper, we overcome the probabilistic nature of DCM by relying on simulation in order to specify the demand directly in terms of the utility functions (instead of the choice probabilities). This allows us to define a mixed-integer linear formulation that characterizes the preference structure and the behavioral assumption of DCM, which can then be embedded in a mixed-integer linear programming (MILP) model. We provide an overview of the extent of the framework with an illustrative MILP model that is designed to solve a profit maximization problem of a parking services operator. The obtained results show the potential of the proposed methodology to adjust supply-related decisions to the users.@en