In this note we will briefly present one problem common in operation research. The practical needs that formulated this problem are quite obvious: choosing the best location to build some important services for communities.

The optimal minimax facility location refers to the placement of a facility (such as a warehouse, hospital, or service center) in such a way that the maximum distance or cost between the facility and any of the demand points (such as customers, patients, or users) is minimized. This approach is particularly useful when the goal is to ensure that no demand point is too far from the facility, thus providing a form of equity in service delivery.

Key Concepts:

  1. Minimax Objective: The primary goal is to minimize the maximum distance (or cost) from the facility to any demand point. This is in contrast to other objectives, such as minimizing the total or average distance.

  2. Facility Location Problem: This is a classic problem in operations research where the aim is to determine the best location for one or more facilities to serve a set of demand points efficiently.

  3. Demand Points: These are the locations that the facility needs to serve. They could be customers, population centers, or other entities requiring service.

  4. Distance Metric: The distance can be measured in various ways, such as Euclidean distance, Manhattan distance, or travel time, depending on the context.

Mathematical Formulation:

Given a set of demand points $D = \{d_1, d_2, \ldots, d_n\}$ and a potential facility location $F$, the minimax facility location problem can be formulated as:

$$ \text{Minimize } \max_{i} \text{distance}(F, d_i) $$

Where:

  • $\text{distance}(F, d_i)$ is the distance between the facility $F$ and the demand point $d_i$.

Solution Approaches:

  1. Geometric Median: In some cases, especially with Euclidean distances, the optimal minimax facility location can be found using the geometric median, which minimizes the maximum distance to any point.

  2. Voronoi Diagrams: These can be used to partition the space into regions based on distance to the facility, helping to identify the optimal location.

  3. Linear Programming: For more complex scenarios, linear programming techniques can be employed to find the optimal location.

  4. Heuristics and Metaheuristics: For large-scale problems, heuristic methods like genetic algorithms, simulated annealing, or tabu search might be used to find a near-optimal solution.

Applications:

  • Emergency Services: Placing hospitals or fire stations so that the maximum response time is minimized.
  • Logistics: Locating warehouses to ensure that no customer is too far from a distribution center.
  • Telecommunications: Positioning cell towers to minimize the maximum distance to any user.

One view that we need is in ProbCover, explained in Active Learning.