Models and Relaxations for Two-Level Uncapacitated Facility Location with Single-Assignment
Résumé
In this talk, we study and compare several formulations for the two-level uncapacitated
facility location problem with single assignment constraints (TUFLP-S), an extension of the
uncapacitated facility location problem (UFLP) [4]. The UFLP consists in selecting a set of
depots from potential locations in order to minimize an objective function that includes fixed
costs associated with each depot and transportation costs from any depot to each customer.
In the two-level uncapacitated facility location problem (TUFLP), the single set of locations is
substituted with two tiers of locations (depots and satellites), and the path to each customer
must begin at a depot and transit by a satellite. The objective function includes fixed costs
associated with the depots and the satellites, fixed costs for establishing connections between
depots and satellites, and transportation costs from any depot to each customer, i.e., each path
of the form depot-satellite-customer has a corresponding transportation cost. The TUFLP-S
imposes the additional restriction that each satellite can be connected to at most one depot.
These single assignment constraints appear in a number of applications, most notably in trans-
portation [5] and telecommunications [2]. Note also that, for a large class of TUFLP instances
for which the single assignment constraints are not explicitly enforced, there is an optimal
solution that satisfies these constraints, due to the structure of the objective function [2].
First, we present a general formulation for the TUFLP [1], and we adapt this model to
derive an initial MIP formulation for the TUFLP-S. We then propose five additional MIP
formulations based on reformulation techniques and on the relaxation of the integrality of some
of the variables associated with location decisions. One of these formulations was previously
considered in [3] to derive a Lagrangian relaxation for the TUFLP-S. We theoretically compare
the LP relaxations of these models. Then, we compare the models by solving a large number
of various instances with a state-of-the-art MIP solver. The results show that, whenever fixed
costs at the depots (at the satellites) are significant, it is beneficial to keep the integrality of the
corresponding binary variables, but to relax the integrality of the binary variables associated
with the satellites (with the depots). In our experiments, poor results are obtained by the
reformulation that minimizes the number of binary variables by relaxing the integrality of the
two types of location variables.