Algorithmic contributions to bilevel location problems with queueing and user equilibrium : exact and semi-exact approaches

While the location literature is vast, most papers consider simpler models, in which  a central authority assigns users to the closest facilities. More realistic traits, such as user behaviour, competition, and congestion are often overlooked, perhaps due to their `complicating' highly non-linear nature. A few papers did incorporate them, but separately, and only heuristic approaches have been proposed as solution methods.  The facility location problem consists in optimally locating a set of facilities in order to satisfy a given demand. In a congested user-choice environment, facilities are typically modeled as queues, and users select the facilities to patronize based on their perceived utility, which is, in general, written as linear combination of travel distance, waiting time at facilities, etc. The resulting bilevel model belongs to the class of mathematical programs with equilibrium constraints (MPECs), where the equilibrium can be expressed as a variational inequality.   Our work is focused on the \emph{competitive congested user-choice facility location problem} (CC-FLP), and we provide a number of strong contributions. From the modeling point of view, we propose various models that capture the key features of CC-FLP. For these NP-hard discrete nonlinear programs we designed exact and approximated algorithms, as well as tailored heuristics. Our work spans three papers. In the first article we consider different models that incorporate balking at facilities, due to limited places in the queue, while user behaviour can be either deterministic or stochastic. In the latter case, user behaviour fits Wardrop's equilibrium principle, while in the former case, customers distribute among facilities according to a Logit random utility choice model. Beyond the analysis of the model's theoretical properties,  we design a user-driven heuristic and a linear approximation algorithm, for which we prove a bound on the approximation error, for the M/M/1 queue.   The second paper is dedicated to the design of a novel exact branch-and-bound (B&B) algorithm for solving a more general subclass of MPECs, which is implemented and evaluated on a CC-FLP. The idea is to virtually treat each node of the B&B tree as a separate optimization problem, in oder to leverage the strength of the MILP solvers and their strong preprocessing at the root node. Our algorithmic approach is based on a combination of Mixed-Integer Linear Programming (MILP), linearization techniques and the iterative solution of convex subproblems, and requires a sophisticated tree management.   In the third paper we incorporate mill pricing into the CC-FLP. Price is a continuous decision variable, along with the location and service levels, and user incorporate it into their utility. Since concepts from network pricing and CC-FLP are merged into a single model, the problem becomes extremely challenging, also due to the presence of facility location and service level decision variables, as well as bivariate queueing delays. For this non-convex bilevel program we devise an algorithm based on linear approximations, that borrows from both location and network pricing literature.

Keyphrases: location, bilevel programming, mixed integer programming, equilibrium, queueing, global optimization, pricing