Add neighbor penalties

Add penalties to a conservation planning problem to penalize solutions that have few neighboring planning units. These penalties can be used to promote spatial clustering in solutions. In particular, they are recommended for reducing spatial fragmentation in large-scale problems or when using open source solvers.

Usage

# S4 method for class 'ConservationProblem,ANY,ANY,matrix'
add_neighbor_penalties(x, penalty, zones, data)

# S4 method for class 'ConservationProblem,ANY,ANY,data.frame'
add_neighbor_penalties(x, penalty, zones, data)

# S4 method for class 'ConservationProblem,ANY,ANY,ANY'
add_neighbor_penalties(x, penalty, zones, data)

# S4 method for class 'ConservationProblem,ANY,ANY,array'
add_neighbor_penalties(x, penalty, zones, data)

Arguments

x: problem() object.
penalty: numeric value denoting the importance of selecting planning units with strong connectivity between them compared to the main problem objective (e.g., solution cost if x has a minimum set objective set using add_min_set_objective()). Higher penalty values can be used to obtain solutions with a high degree of connectivity, and smaller penalty values can be used to obtain solutions with a small degree of connectivity. Note that negative penalty values can be used to obtain solutions that avoid connectivity.
zones: matrix or Matrix object describing the neighborhood scheme for different zones. Each row and column corresponds to a different zone in x, and cell values must contain binary numeric values (i.e., one or zero) that indicate if neighboring planning units (per data) should be treated as neighbors if they are allocated to different zones. The cell values along the diagonal of the matrix indicate if planning units that are allocated to the same zone should be considered neighbors or not. Defaults to an identity matrix (i.e., a matrix with ones along the matrix diagonal and zeros elsewhere), so that planning units are only considered neighbors if they are both allocated to the same zone.
data: NULL, matrix, Matrix, data.frame, or array object showing which planning units are neighbors with each other. Defaults to NULL such that the neighborhood data are calculated automatically using the adjacency_matrix() function. See the Data format section for more information.

Value

An updated problem() object with the penalties added to it.

Details

This function adds penalties to conservation planning problem to penalize solutions that have low spatial clustering. Specifically, it favors pair-wise connections between planning units that have high connectivity values (based on Önal and Briers 2002).

Mathematical formulation

The neighbor penalties are implemented using the following equations. Let $I$ represent the set of planning units (indexed by $i$ or $j$), $Z$ represent the set of management zones (indexed by $z$ or $y$), and $X_{iz}$ represent the decision variable for planning unit $i$ for in zone $z$ (e.g., with binary values one indicating if planning unit is allocated or not). Also, let $p$ represent penalty, $D$ represent data, and $W$ represent zones.

If data is specified as a matrix or Matrix object, then the penalties are calculated as:

$$ \sum_{i}^{I} \sum_{j}^{I} \sum_{z}^{Z} \sum_{y}^{Z} (-p \times X_{iz} \times X_{jy} \times D_{ij} \times W_{zy})$$

Otherwise, if data is specified as a data.frame or array object, then the penalties are calculated as:

$$ \sum_{i}^{I} \sum_{j}^{I} \sum_{z}^{Z} \sum_{y}^{Z} (-p \times X_{iz} \times X_{jy} \times D_{ijzy})$$

Note that when the problem objective is to maximize some measure of benefit and not minimize some measure of cost, the term $-p$ is replaced with $p$.

Data format

The following formats can be used to specify data.

data as a NULL value: Here the neighborhood data are calculated automatically using the adjacency_matrix() function. This is the default for data. Note that the neighborhood data must be manually defined using one of the other formats below if the planning unit data in x is not spatially referenced (e.g., data.frame or numeric format).
data as a matrix/Matrix object: Here rows and columns correspond to different planning units and cell values indicate if two planning units are neighbors or not. Cells must have binary numeric values (i.e., one or zero). Note that cells along the matrix diagonal have no effect on the solution because each planning unit cannot be a neighbor with itself.
data as a data.frame object: Here rows correspond to a pair of planning units and columns provide information about each pair of planning units. In particular, data must have the columns: "id1", "id2", and "boundary". The "id1" and "id2" columns contain identifiers (indices) for a pair of planning units, and the "boundary" column contains binary numeric values that indicate if the two planning units specified in the "id1" and "id2" columns should be treated as neighbors or not. These data can be used to describe symmetric or asymmetric relationships between planning units. By default, input data is assumed to be symmetric unless asymmetric data is specified (e.g., if data is present for planning units 2 and 3, then the same amount of connectivity is expected for planning units 3 and 2, unless connectivity data is also provided for planning units 3 and 2). If x has multiple zones, then the "zone1"and"zone2"columns can optionally be provided to manually specify that the neighborhood data pertain to specific zones. The"zone1"and"zone2"columns should contain thecharacternames of the zones. Note that if the columns"zone1"and"zone2"are present, thenzonesmust beNULL`.
data as an array object: Here a four-dimension array containing binary numeric values is used to specify if planning unit should be treated as neighbors with every other planning unit when they are allocated to every combination of management zone. The first two dimensions (i.e., rows and columns) correspond to the planning units, and second two dimensions correspond to the management zones. For example, if data had a value of 1 at the index data[1, 2, 3, 4], this would indicate that planning unit 1 and planning unit 2 should be treated as neighbors when they are allocated to zones 3 and 4 (respectively).

References

Williams JC, ReVelle CS, and Levin SA (2005) Spatial attributes and reserve design models: A review. Environmental Modeling and Assessment, 10: 163–181.

Examples

# load data
sim_pu_raster <- get_sim_pu_raster()
sim_features <- get_sim_features()
sim_zones_pu_raster <- get_sim_zones_pu_raster()
sim_zones_features <- get_sim_zones_features()

# create minimal problem
p1 <-
  problem(sim_pu_raster, sim_features) %>%
  add_min_set_objective() %>%
  add_relative_targets(0.1) %>%
  add_default_solver(verbose = FALSE)

# create problem with low neighbor penalties and
# using a rook-style neighborhood (the default neighborhood style)
p2 <- p1 %>% add_neighbor_penalties(0.001)

# create problem with high penalties
# using a rook-style neighborhood (the default neighborhood style)
p3 <- p1 %>% add_neighbor_penalties(0.01)

# create problem with high penalties and using a queen-style neighborhood
p4 <-
  p1 %>%
  add_neighbor_penalties(
    0.01, data = adjacency_matrix(sim_pu_raster, directions = 8)
  )

# solve problems
s1 <- c(solve(p1), solve(p2), solve(p3), solve(p4))
names(s1) <- c("basic solution", "low (rook)", "high (rook)", "high (queen")

# plot solutions
plot(s1, axes = FALSE)


# create minimal problem with multiple zones
p5 <-
  problem(sim_zones_pu_raster, sim_zones_features) %>%
  add_min_set_objective() %>%
  add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) %>%
  add_default_solver(verbose = FALSE)

# create problem with low neighbor penalties, a rook style neighborhood,
# and planning units are only considered neighbors if they are allocated to
# the same zone
z6 <- diag(3)
print(z6)
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
p6 <- p5 %>% add_neighbor_penalties(0.001, zones = z6)

# create problem with high penalties and the same neighborhood as above
p7 <- p5 %>% add_neighbor_penalties(0.01, zones = z6)

# create problem with high neighborhood penalties, a queen-style
# neighborhood, neighboring planning units that are allocated to zones 1
# or 2 are treated as neighbors
z8 <- diag(3)
z8[1, 2] <- 1
z8[2, 1] <- 1
print(z8)
#>      [,1] [,2] [,3]
#> [1,]    1    1    0
#> [2,]    1    1    0
#> [3,]    0    0    1
p8 <- p5 %>% add_neighbor_penalties(0.01, zones = z8)

# create problem with high neighborhood penalties, a queen-style
# neighborhood, and here we want to promote spatial fragmentation
# within each zone, so we use negative zone values.
z9 <- diag(3) * -1
print(z9)
#>      [,1] [,2] [,3]
#> [1,]   -1    0    0
#> [2,]    0   -1    0
#> [3,]    0    0   -1
p9 <- p5 %>% add_neighbor_penalties(0.01, zones = z9)

# solve problems
s2 <- list(p5, p6, p7, p8, p9)
s2 <- lapply(s2, solve)
s2 <- lapply(s2, category_layer)
s2 <- terra::rast(s2)
names(s2) <- c("basic problem", "p6", "p7", "p8", "p9")

# plot solutions
plot(s2, main = names(s2), axes = FALSE)