Singlelocation

Singlelocation, in the context of operations research and logistics, refers to the single-facility location problem. It asks where to place one facility to serve a set of demand points in a region while minimizing total transportation cost or distance. The problem can be posed in two main forms: a discrete version, where the facility must be located at one of a finite set of candidate sites, and a continuous version, where the facility can be placed anywhere within a region.

In the discrete formulation, the objective is to choose a candidate site that minimizes the sum of weighted distances to all demand points, possibly plus a fixed opening cost for that site. If weights represent demand and distances are measured, for each candidate site j the cost is the sum over i of w_i times the distance from j to i; the best site is the one with the smallest total. The continuous version is the Fermat-Weber problem, seeking a location x that minimizes the sum of weighted distances to the demand points. Solutions depend on the distance metric: with L1 distance, the optimal coordinates are the medians of the demand coordinates; with L2 distance, the solution is the geometric median, which generally lacks a closed form and is found by iterative methods such as Weiszfeld’s algorithm.

Applications include retail site selection, emergency and public-service facility placement, and distribution or manufacturing network design. The single-location problem provides a baseline for understanding how a single facility should be positioned relative to demand, though real-world considerations often require extensions to account for multiple facilities, capacities, costs, and dynamic demand.