Facility Location Problem

The foundational variant where any facility can serve an unlimited amount of demand. The objective is purely to balance the fixed cost of opening a facility against the variable transportation costs to serve customers.

• Each customer is assigned to exactly one open facility
• No limit on throughput per facility
• Often solved using Lagrangian relaxation or dual-based heuristics
• Forms the basis for more complex, constrained models

Introduces a hard capacity constraint on each facility, limiting the total demand it can serve. This transforms the problem from a simple assignment to a bin-packing hybrid, where both location and allocation must respect throughput limits.

• Facilities have a maximum service volume
• Often requires splitting customer demand across multiple facilities
• Solved via branch-and-cut or Benders decomposition
• Critical for modeling real-world warehouse and server farm constraints

Fixes the number of facilities to exactly P and seeks to minimize the total weighted distance between demand nodes and their assigned facilities. There are no fixed opening costs; the budget is expressed as a cardinality constraint.

• Pure focus on accessibility and coverage
• No facility cost trade-off; P is an input parameter
• Common in public sector: locating fire stations, schools, or clinics
• Solved using vertex substitution heuristics or exact LP methods

A minimax objective: fix the number of facilities to P and minimize the maximum distance any demand point must travel to its nearest facility. This optimizes for worst-case service equity rather than average efficiency.

• Guarantees a service radius for all customers
• Used for emergency response: ambulance depots, police stations
• Inherently a covering problem with a distance threshold
• More computationally challenging than the P-Median due to the bottleneck objective

Flips the constraint structure: given a fixed number of facilities and a service radius, maximize the total demand that falls within the coverage zone. Not all demand must be served; the goal is maximal population reach.

• Binary coverage: a customer is either covered or not
• Used for retail site selection and sensor placement
• Often formulated as an integer programming problem
• Can be extended to gradual coverage decay functions

Models multi-tier systems where facilities at different levels (e.g., regional distribution centers vs. local warehouses) interact. Flow moves sequentially through the hierarchy, and each tier may have distinct cost structures and capacities.

• Captures transshipment and consolidation points
• Facilities at one level serve as demand points for the level above
• Essential for modeling global supply chains with echelons
• Solved using multi-commodity flow formulations

Determining the optimal geographic placement for a new warehouse in an existing network. The model balances inbound freight costs from suppliers, outbound delivery costs to customers, and fixed facility operating expenses. Modern solvers incorporate geospatial demand density, labor availability, and tax incentive zones as weighted constraints. A typical MILP formulation minimizes the sum of transportation costs and facility amortization over a 10-year horizon, often revealing that a location 50 miles from the intuitive choice saves millions annually.

Urban retailers use FLP solvers to identify micro-fulfillment center locations that enable sub-2-hour delivery. The objective function minimizes the maximum delivery radius while respecting real estate cost ceilings. Constraints include zoning restrictions, dock availability, and the requirement that 95% of a city's population falls within a 15-minute drive time. This is a capacitated p-median problem variant, where the number of facilities p is traded off against service level targets.

Procurement teams apply FLP to the inbound supply network by modeling which suppliers to retain and where to locate consolidation centers. The model ingests supplier risk scores, volume commitments, and multi-modal freight rates. A fixed-charge facility location formulation determines whether keeping a regional consolidation hub reduces total landed cost, even after accounting for additional handling. This often reveals that reducing the supplier base by 30% and routing through a single cross-dock yields lower total cost of ownership.

Humanitarian agencies solve a stochastic facility location problem to pre-position emergency supplies before a hurricane season. The model uses scenario-based demand forecasts derived from storm surge models and population vulnerability indices. The objective minimizes the expected deprivation cost—the economic value of human suffering due to delayed aid. Facilities are placed to ensure that, under the worst-case scenario ensemble, critical supplies reach affected populations within 72 hours.

Global manufacturers re-optimize their plant footprint every 3-5 years using multi-echelon FLP models. The decision variables include which plants to open, close, or retool, and which products to assign to each site. Constraints capture tariff differentials, local content requirements, and carbon border taxes. The objective function minimizes total landed cost to customer, including manufacturing, inter-facility transfers, and duties. A recent re-optimization for a consumer electronics firm identified $120M in annual savings by shifting production from 12 plants to 8 strategically located mega-factories.

Electric vehicle fleet operators solve a flow-capturing facility location problem to place charging depots. Unlike traditional FLP, demand is not point-based but occurs along traffic flow corridors. The model maximizes the volume of vehicle-miles that can be electrified given a fixed budget for k stations. Constraints include grid capacity, charging dwell times, and vehicle range anxiety buffers. This ensures that a delivery fleet transitioning to EVs can complete all scheduled routes without stranding.

Extends the facility location problem by adding inventory stocking decisions across multiple tiers of a distribution network. Rather than simply deciding where to place facilities, this approach determines:

• Which echelon (factory, regional DC, local depot) should hold stock
• Optimal safety stock levels at each location
• How inventory policies affect total network cost

The integration of location and inventory theory is formalized in the Joint Location-Inventory Problem.

Two classical variants of facility location with distinct objective functions. The P-Median Problem minimizes the average distance between demand points and their assigned facilities, optimizing for overall efficiency. The P-Center Problem minimizes the maximum distance any demand point must travel, optimizing for equity and worst-case service guarantees. Both are NP-hard for general networks, requiring heuristic solutions at scale.

A foundational geographic concept used to estimate demand flows between facilities and customers. The model posits that interaction between two locations is proportional to their masses (e.g., facility capacity, population) and inversely proportional to the distance decay function. Modern facility location models incorporate gravity-based demand attraction to account for the reality that customers may not always choose the nearest facility.

The continuous-space precursor to discrete facility location, first formalized by Alfred Weber in 1909. It seeks the optimal coordinates for a single facility that minimizes the sum of weighted Euclidean distances to a set of demand points. The iterative Weiszfeld algorithm provides a numerical solution. While modern problems use discrete candidate locations, the Weber problem established the foundational geometric intuition for all subsequent location theory.