azp: Automatic Zoning Procedure (AZP)

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

Automatic Zoning Procedure (AZP)

Description

Performs regionalization using the Automatic Zoning Procedure algorithm. AZP uses local search to minimize within-region heterogeneity while maintaining spatial contiguity. Three variants are available: basic (greedy), tabu search, and simulated annealing.

Usage

azp(
  data,
  attrs = NULL,
  n_regions,
  weights = "queen",
  bridge_islands = FALSE,
  method = c("tabu", "basic", "sa"),
  max_iterations = 100L,
  tabu_length = 10L,
  cooling_rate = 0.99,
  initial_temperature = 0,
  scale = TRUE,
  seed = NULL,
  verbose = FALSE
)

Arguments

data	An sf object with polygon or point geometries.
attrs	Character vector of column names to use for clustering (e.g., c("var1", "var2")). If NULL, uses all numeric columns.
n_regions	Integer. Number of regions (clusters) to create.
weights	Spatial weights specification. Can be: "queen" (default): Polygons sharing any boundary point are neighbors "rook": Polygons sharing an edge are neighbors An nb object from spdep or created with sp_weights() A list for other weight types: list(type = "knn", k = 6) for k-nearest neighbors, or list(type = "distance", d = 5000) for distance-based weights
bridge_islands	Logical. If TRUE, automatically connect disconnected components (e.g., islands) using nearest-neighbor edges. If FALSE (default), the function will error when the spatial weights graph is disconnected.
method	Character. Optimization method: "basic" (greedy local search), "tabu" (tabu search), or "sa" (simulated annealing). Default is "tabu".
max_iterations	Integer. Maximum number of iterations (default 100).
tabu_length	Integer. Length of tabu list for tabu method (default 10).
cooling_rate	Numeric. Cooling rate for SA method, between 0 and 1 (default 0.99).
initial_temperature	Numeric. Initial temperature for SA method. If 0 (default), automatically set based on initial objective.
scale	Logical. If TRUE (default), standardize attributes before clustering.
seed	Optional integer for reproducibility.
verbose	Logical. Print progress messages.

Details

The Automatic Zoning Procedure (AZP) was introduced by Openshaw (1977) and refined by Openshaw & Rao (1995). It is a local search algorithm that:

1. Starts with an initial random partition into n_regions

2. Iteratively moves border areas between regions to reduce heterogeneity

3. Maintains spatial contiguity throughout

4. Terminates when no improving moves are found

Three variants are available:

• basic: Greedy local search that only accepts improving moves

• tabu: Tabu search that can accept non-improving moves to escape local optima, with a tabu list preventing cycling

• sa: Simulated annealing that accepts worse moves with decreasing probability as temperature cools

Value

An sf object with a .region column containing cluster assignments. Metadata is stored in the "spopt" attribute, including:

• algorithm: "azp"

• method: The optimization method used

• n_regions: Number of regions created

• objective: Total within-region sum of squared deviations

• solve_time: Time to solve in seconds

References

Openshaw, S. (1977). A geographical solution to scale and aggregation problems in region-building, partitioning and spatial modelling. Transactions of the Institute of British Geographers, 2(4), 459-472.

Openshaw, S., & Rao, L. (1995). Algorithms for reengineering 1991 Census geography. Environment and Planning A, 27(3), 425-446.

Examples

library(sf)
nc <- st_read(system.file("shape/nc.shp", package = "sf"))

# Basic AZP with 8 regions
result <- azp(nc, attrs = c("SID74", "SID79"), n_regions = 8)

# Tabu search (often finds better solutions)
result <- azp(nc, attrs = c("SID74", "SID79"), n_regions = 8,
              method = "tabu", tabu_length = 15)

# Simulated annealing
result <- azp(nc, attrs = c("SID74", "SID79"), n_regions = 8,
              method = "sa", cooling_rate = 0.95)

# View results
plot(result[".region"])

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