Optimal facility location

Facility location problems, often called location analysis, are a vital branch of operations research and computational geometry. This field focuses on finding the most efficient placement for facilities on a plane or a network. The primary goal is often to minimize transportation costs. However, researchers also consider social and safety factors. For example, planners must avoid placing hazardous materials near residential housing. They must also consider the location of competitors. This mathematical study also applies to cluster analysis, where data points are grouped together.

To understand the mechanism, imagine a set of potential facility sites, called L, and a set of demand points, called D. Demand points are the locations that require service. The goal is to select a subset of sites, called F, to open as facilities. To find the best solution, mathematicians calculate the sum of the distances from every demand point to its nearest open facility. They then add this to the total cost of opening the facilities. The objective is to find the specific subset F that makes this entire sum as small as possible.

There are several distinct types of facility location problems. The minisum problem, such as the Weber problem, seeks to minimize the weighted sum of distances. This is useful when you want to lower total transportation costs. In contrast, the minimax problem focuses on the most affected community. Instead of looking at the total distance, it seeks to minimize the maximum distance any single site must travel. A third type is the maxmin problem, also known as the obnoxious facility location problem. This seeks to maximize the minimum distance to sites, which is helpful for placing unpleasant facilities like landfills.

History shows how these ideas evolved from mechanical tools to complex math. The study is often traced back to the Weber problem, formulated by Alfred Weber. He framed the problem as finding a location on a Euclidean plane to minimize ton-mileage costs. This involved a firm shipping goods to three fixed locations in straight lines. Before modern computers, people used mechanical methods like the Varignon frame to solve these problems. In 1937, E. Weiszfeld performed work similar to Weber's. Later, in 1957 and 1962, the first algorithms were created for versions with more than three points.

Complexity is a major factor in these calculations. Solving the facility location problem on general graphs is NP-hard, which means it is very difficult to find an exact solution quickly. This difficulty is linked to the set cover problem. When distances do not follow the triangle inequality, it is called a non-metric facility location problem. These can be approximated within a factor of O(log n). If distances do satisfy the triangle inequality, it is a metric facility location (MFL) problem. Even the MFL is NP-hard and hard to approximate better than a factor of 1.463.

Different algorithms provide different levels of accuracy. For the k-center problem, which is a type of minimax problem, an exact solution is often impractical. Instead, many use the farthest-point clustering (FPC) algorithm. This is a simple approximation with a factor of 2. It works by picking one point as a center, then repeatedly finding and picking the point farthest from the current set of centers. Another approach uses the coreset concept to solve problems with a complexity of n^{1+o(1)}. For the largest empty sphere problem, which relates to maxmin location, the planar case can be solved in Θ(n log n) time.

These mathematical models have significant real-world applications. In healthcare, facility location is critical because poor decisions can impact morbidity and mortality. If clinics are hard to access, community health suffers. In solid waste management, these models help optimize the location of landfills to manage waste costs. Finally, in computer science, these problems relate to centroid-based clustering. In this context, demand points are treated as data to be clustered. By solving the location problem, researchers can find the best centroids to partition data into equivalence classes, or colors.