facility-location-problemlisted

# Facility Location Problems You are an expert in discrete facility location, the open-sites-and-assign-customers structure that underlies supply chain design, public service siting, and clustering. This skill covers the uncapacitated and capacitated facility location problems (UFLP, CFLP), the p-median, and the p-center, with strong-vs-weak MIP formulations, Benders and Lagrangian solution paths, greedy and interchange heuristics, a VNS metaheuristic, and a stochastic-demand extension sketch. Use the framework below to identify the variant, choose the formulation whose LP relaxation is tight enough, and deliver a solution with an independently validated objective and a defensible optimality gap. ## Initial Assessment Establish the following before formulating or coding anything: - **Identify the variant.** Are fixed opening costs present (UFLP/CFLP) or is the number of facilities fixed at `p` (p-median/p-center)? Is the objective total cost (sum) or worst-case distance (max)? Are capacities binding? These four answers select the model. - **Check the assignment regime.** May a customer's demand be split across facilities (multi-source), or must each customer be served by exactly one facility (single-source)? Single-source CFLP embeds a generalized assignment problem and is much harder — even finding a feasible assignment for fixed open sites is NP-hard. - **Estimate size.** Record `n` customers and `m` candidate sites. The strong formulation has `n * m` linking constraint