Facility Location

321. Facility Location

A family of combinatorial optimization problems: where should warehouses / stores / hospitals / cell towers be located to best serve known demand?

321.1. Variants

The variants differ by:

Objective: minimize total cost, minimize maximum distance, maximize coverage
Constraints: facility capacity, customer assignment rules
Discrete vs continuous: candidate locations are finite ( $𝑝$ -median) or anywhere in space (center of gravity)

Variant	Objective	Constraint
UFLP	min $\sum 𝑓_{𝑗} 𝑦_{𝑗} + \sum 𝑐_{𝑖 𝑗} 𝑥_{𝑖 𝑗}$	uncapacitated
CFLP	same	capacity per facility
$𝑝$ -median	min total assigned distance	exactly $𝑝$ facilities
$𝑝$ -center	min max distance	exactly $𝑝$ facilities
Set covering	min number of facilities	every demand covered
Max covering	max demand covered	exactly $𝑝$ facilities
Center of gravity	min weighted sum of distances	single facility, continuous

321.2. Common decision variables

For discrete location problems:

𝑦_{𝑗} = {\begin{cases} 1 if facility 𝑗 is opened \\ 0 otherwise \end{cases}

𝑥_{𝑖 𝑗} = fraction of demand at 𝑖 served by facility 𝑗 (or 0/1 for single-source)

321.3. Trade-offs

Many facilities: low transport cost, high fixed cost, less risk pooling
Few facilities: high transport cost, low fixed cost, more pooling benefit
Capacity-constrained: forces some demand to nearer facilities even when distant ones are cheaper

321.4. Where it shows up

Supply chain network design — DC placement
Public service — fire stations, ambulances, schools ( $𝑝$ -center for response time)
Telecom — cell tower placement (set covering)
Retail — store location (max covering under budget)
Hub-and-spoke logistics — depots, airports

321.5. See also

UFLP / CFLP — fixed-charge formulations
$𝑝$ -median / $𝑝$ -center
Set Covering / Max Covering
Center of Gravity — continuous case
Daganzo Continuous — strategic scale