Mixed planar and network single-facility location problems

We consider the problem of optimally locating a single facility anywhere in a network to serve both on‐network and off‐network demands. Off‐network demands occur in a Euclidean plane, while on‐network demands are restricted to a network embedded in the plane. On‐network demand points are serviced using shortest‐path distances through links of the network (e.g., on‐road travel), whereas demand points located in the plane are serviced using more expensive Euclidean distances. Our base objective minimizes the total weighted distance to all demand points. We develop several extensions to our base model, including: (i) a threshold distance model where if network distance exceeds a given threshold, then service is always provided using Euclidean distance, and (ii) a minimax model that minimizes worst‐case distance. We solve our formulations using the ‘Big Segment Small Segment’ global optimization method, in conjunction with bounds tailored for each problem class. Computational experiments demonstrate the effectiveness of our solution procedures. Solution times are very fast (often under one second), making our approach a good candidate for embedding within existing heuristics that solve multi‐facility problems by solving a sequence of single‐facility problems.