Add objective: minimize fragmentation

Define an objective that minimizes planning-unit fragmentation over a stored spatial relation.

This objective acts on the planning-unit selection pattern through the binary planning-unit variables \(w_i\). It is therefore appropriate when spatial cohesion is to be encouraged at the level of the selected planning-unit set as a whole.

Usage

add_objective_min_fragmentation_pu(
  x,
  relation_name = "boundary",
  weight_multiplier = 1,
  alias = NULL
)

Arguments

x

A Problem object.

relation_name

Character string giving the name of the spatial relation to use. The relation must already exist in x$data$spatial_relations.

weight_multiplier

Numeric scalar greater than or equal to zero. Global multiplier applied to the relation weights when the objective is built.

alias

Optional identifier used to register this objective for multi-objective workflows.

Value

An updated Problem object.

Details

Use this function when spatial cohesion should be encouraged at the level of the selected planning-unit set as a whole.

Let \(\mathcal{I}\) denote the set of planning units and let \(w_i \in \{0,1\}\) indicate whether planning unit \(i \in \mathcal{I}\) is selected.

Let the chosen spatial relation define a set of weighted pairs with weights \(\omega_{ij} \ge 0\). These relation weights are interpreted by the model builder after scaling by \(\lambda =\) weight_multiplier.

The internal preparation step constructs one auxiliary variable \(y_{ij} \in [0,1]\) for each unique non-diagonal undirected edge \((i,j)\) with \(i < j\). The intended semantics is: $$ y_{ij} = w_i \land w_j. $$

This is enforced by the standard linearization: $$ y_{ij} \le w_i, $$ $$ y_{ij} \le w_j, $$ $$ y_{ij} \ge w_i + w_j - 1. $$

Thus, \(y_{ij}=1\) if and only if both planning units \(i\) and \(j\) are selected, and \(y_{ij}=0\) otherwise.

The exact objective coefficients are assembled later by the model builder from:

• the planning-unit variables \(w_i\),

• the edge-conjunction variables \(y_{ij}\),

• the stored relation weights \(\omega_{ij}\),

• and the multiplier \(\lambda\).

Conceptually, the resulting objective is a boundary- or relation-based compactness functional that penalizes exposed or weakly connected selected patterns while rewarding adjacency among selected planning units.

In the common case where relation_name = "boundary" and the relation was built with add_spatial_boundary, the objective corresponds to a boundary-length-style fragmentation penalty.

Setting weight_multiplier = 0 removes the contribution of the spatial relation from the objective after scaling.

This objective does not distinguish between different actions within the same planning unit. If action-specific spatial cohesion is required, use add_objective_min_fragmentation_action instead.

See also

add_spatial_boundary, add_spatial_relations, add_objective_min_fragmentation_action

Examples

pu_tbl <- data.frame(
  id = 1:4,
  cost = c(1, 2, 3, 4)
)

feat_tbl <- data.frame(
  id = 1:2,
  name = c("feature_1", "feature_2")
)

dist_feat_tbl <- data.frame(
  pu = c(1, 1, 2, 3, 4),
  feature = c(1, 2, 2, 1, 2),
  amount = c(5, 2, 3, 4, 1)
)

actions_df <- data.frame(
  id = c("conservation", "restoration"),
  name = c("conservation", "restoration")
)

bound_df <- data.frame(
  id1 = c(1, 1, 2, 1, 2, 3, 4),
  id2 = c(1, 2, 2, 3, 4, 4, 4),
  boundary = c(4, 1, 4, 1, 1, 1, 4)
)

p <- create_problem(
  pu = pu_tbl,
  features = feat_tbl,
  dist_features = dist_feat_tbl,
  cost = "cost"
) |>
  add_actions(actions_df, cost = c(conservation = 1, restoration = 2))

p <- add_spatial_boundary(
  x = p,
  boundary = bound_df,
  name = "boundary",
  include_self = TRUE,
  edge_factor = 1
)

p <- add_objective_min_fragmentation_pu(
  p,
  relation_name = "boundary"
)

p$data$model_args
#> $model_type
#> [1] "minimizeFragmentation"
#> 
#> $objective_id
#> [1] "min_fragmentation"
#> 
#> $objective_args
#> $objective_args$relation_name
#> [1] "boundary"
#> 
#> $objective_args$weight_multiplier
#> [1] 1
#> 
#>