Facility location design under continuous traffic equilibrium

• We optimize facility location under customer traffic equilibrium in a continuous space.
• The problem is bi-level, mixed-integer, and involves nonlinear partial differential equations.
• Continuum approximation is used to effectively optimize facility density and traffic flux.
• A discrete model (solved by Lagrangian relaxation and finite element method) is also proposed.
• Numerical experiments illustrate applicability of both models, and also cast insights.

This paper presents two modeling approaches for median-type facility location design under elastic customer demand and traffic equilibrium in a continuous space. The first approach, following the continuum approximation scheme, builds upon the special case of an infinite homogeneous plane where traffic equilibrium can be described by an ordinary differential equation. The solution to this homogeneous case, sometimes in a closed form, is then used to develop approximate solutions to more general cases (e.g., those in a heterogeneous space). This model provides a computationally efficient way to obtain managerial insights and near-optimal solutions, especially for large problem instances. We also develop a more traditional discrete location model in the form of a mixed-integer program, which builds directly upon a nonlinear partial differential equation description of customer traffic equilibrium. We develop a Lagrangian relaxation based solution approach with an embedded finite-element method subroutine, to separate and solve the location decisions as well as the traffic equilibrium. Numerical experiments are conducted to illustrate applicability of the proposed models and to compare performance of the two complementing modeling approaches.