Integrating advanced demand models within the framework of mixed integer linear problems: A Lagrangian relaxation method for the uncapacitated case

The integration of customer behavioral models in optimization provides a better understanding of the preferences of clients (the demand) to operators while planning for their systems (the supply). These preferences are formalized with discrete choice models, which are the state-of-the-art for the mathematical modeling of demand. However, their complexity leads to mathematical formulations that are highly nonlinear and nonconvex in the variables of interest, and are therefore difficult to be included in (mixed) integer linear problems (MILP). These problems correspond to the optimization models that are considered to design and configure a system. In this work, we present a general framework that integrates advanced discrete choice models in MILP. Nevertheless, a linear formulation comes with a high dimension of the problem. To address this issue, and given the underlying structure of the model, decomposition techniques such as Lagrangian decomposition can be applied. Two subproblems with common variables have been identified: one regarding the user and one regarding the operator. In the former, the user has to perform a decision based on what the operator is offering, whereas in the latter, the operator needs to decide about the features of the supply to attract the users.