p_median: P-Median Problem

In spopt: Spatial Optimization for Regionalization, Facility Location, and Market Analysis

P-Median Problem

Description

Solves the P-Median problem: minimize total weighted distance from demand points to their assigned facilities by locating exactly p facilities. This is an efficiency-focused objective that minimizes overall travel burden.

Usage

p_median(
  demand,
  facilities,
  n_facilities,
  weight_col,
  cost_matrix = NULL,
  distance_metric = "euclidean",
  fixed_col = NULL,
  verbose = FALSE
)

Arguments

demand	An sf object representing demand points.
facilities	An sf object representing candidate facility locations.
n_facilities	Integer. Number of facilities to locate (p).
weight_col	Character. Column name in demand containing demand weights.
cost_matrix	Optional. Pre-computed distance matrix.
distance_metric	Distance metric: "euclidean" (default) or "manhattan".
fixed_col	Optional column name in facilities indicating which facilities are pre-selected. The column should be logical (TRUE for fixed) or character ("required"/"candidate"). Fixed facilities are always selected; the solver optimizes the remaining slots.
verbose	Logical. Print solver progress.

Details

The p-median problem minimizes the total weighted distance (or travel cost) between demand points and their nearest assigned facility. It is the most widely used location model for efficiency-oriented facility siting.

The integer programming formulation is:

\min \sum_i \sum_j w_i d_{ij} x_{ij}

Subject to:

\sum_j y_j = p

\sum_j x_{ij} = 1 \quad \forall i

x_{ij} \leq y_j \quad \forall i,j

x_{ij}, y_j \in \{0,1\}

Where w_i is the demand weight at location i, d_{ij} is the distance from demand i to facility j, x_{ij} = 1 if demand i is assigned to facility j, and y_j = 1 if facility j is selected.

Value

A list with two sf objects:

• ⁠$demand⁠: Original demand sf with .facility column (assigned facility)

• ⁠$facilities⁠: Original facilities sf with .selected and .n_assigned columns

Metadata is stored in the "spopt" attribute.

Use Cases

P-median is appropriate when minimizing total travel cost or distance:

• Public facilities: Schools, libraries, or community centers where the goal is to minimize total student/patron travel

• Warehouses and distribution: Locating distribution centers to minimize total shipping costs to customers

• Healthcare: Positioning clinics to minimize aggregate patient travel time across a population

• Service depots: Locating maintenance facilities to minimize total technician travel to service calls

For equity-focused objectives where no demand point should be too far, consider p_center() instead.

References

Hakimi, S. L. (1964). Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph. Operations Research, 12(3), 450-459. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1287/opre.12.3.450")}

See Also

p_center() for minimizing maximum distance (equity objective)

Examples

library(sf)

demand <- st_as_sf(data.frame(
  x = runif(100), y = runif(100), population = rpois(100, 500)
), coords = c("x", "y"))
facilities <- st_as_sf(data.frame(x = runif(20), y = runif(20)), coords = c("x", "y"))

# Locate 5 facilities minimizing total weighted distance
result <- p_median(demand, facilities, n_facilities = 5, weight_col = "population")

# Mean distance to assigned facility
attr(result, "spopt")$mean_distance

spopt documentation built on April 22, 2026, 9:07 a.m.