R/add_max_features_objective.R
add_max_features_objective.Rd
Set the objective of a conservation planning problem()
to
fulfill as many targets as possible while ensuring that the cost of the
solution does not exceed a budget.
add_max_features_objective(x, budget)
x 


budget 

Object (i.e. ConservationProblem
) with the objective
added to it.
A problem objective is used to specify the overall goal of the conservation planning problem. Please note that all conservation planning problems formulated in the prioritizr package require the addition of objectivesfailing to do so will return an error message when attempting to solve problem.
The maximum feature representation objective is an enhanced version of the
maximum coverage objective add_max_cover_objective()
because
targets can be used to ensure that a certain amount of each feature is
required in order for them to be adequately represented (similar to the
minimum set objective (see add_min_set_objective()
). This
objective finds the set of planning units that meets representation targets
for as many features as possible while staying within a fixed budget
(inspired by Cabeza and Moilanen 2001). Additionally, weights can be used
add_feature_weights()
). If multiple solutions can meet the same
number of weighted targets while staying within budget, the cheapest
solution is returned.
The maximum feature objective for the reserve design problem can be expressed mathematically for a set of planning units (\(I\) indexed by \(i\)) and a set of features (\(J\) indexed by \(j\)) as:
$$\mathit{Maximize} \space \sum_{i = 1}^{I} s \space c_i \space x_i + \sum_{j = 1}^{J} y_j w_j \\ \mathit{subject \space to} \\ \sum_{i = 1}^{I} x_i r_{ij} \geq y_j t_j \forall j \in J \\ \sum_{i = 1}^{I} x_i c_i \leq B$$
Here, \(x_i\) is the decisions variable (e.g.
specifying whether planning unit \(i\) has been selected (1) or not
(0)), \(r_{ij}\) is the amount of feature \(j\) in planning
unit \(i\), \(t_j\) is the representation target for feature
\(j\), \(y_j\) indicates if the solution has meet
the target \(t_j\) for feature \(j\), and \(w_j\) is the
weight for feature \(j\) (defaults to 1 for all features; see
add_feature_weights()
to specify weights). Additionally,
\(B\) is the budget allocated for the solution, \(c_i\) is the
cost of planning unit \(i\), and \(s\) is a scaling factor used
to shrink the costs so that the problem will return a cheapest solution
when there are multiple solutions that represent the same amount of all
features within the budget.
Cabeza M and Moilanen A (2001) Design of reserve networks and the persistence of biodiversity. Trends in Ecology & Evolution, 16: 242248.
# load data data(sim_pu_raster, sim_pu_zones_stack, sim_features, sim_features_zones) # create problem with maximum features objective p1 < problem(sim_pu_raster, sim_features) %>% add_max_features_objective(1800) %>% add_relative_targets(0.1) %>% add_binary_decisions() %>% add_default_solver(verbose = FALSE) # \dontrun{ # solve problem s1 < solve(p1) # plot solution plot(s1, main = "solution", axes = FALSE, box = FALSE)# } # create multizone problem with maximum features objective, # with 10% representation targets for each feature, and set # a budget such that the total maximum expenditure in all zones # cannot exceed 3000 p2 < problem(sim_pu_zones_stack, sim_features_zones) %>% add_max_features_objective(3000) %>% add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) %>% add_binary_decisions() %>% add_default_solver(verbose = FALSE) # \dontrun{ # solve problem s2 < solve(p2) # plot solution plot(category_layer(s2), main = "solution", axes = FALSE, box = FALSE)# } # create multizone problem with maximum features objective, # with 10% representation targets for each feature, and set # separate budgets for each management zone p3 < problem(sim_pu_zones_stack, sim_features_zones) %>% add_max_features_objective(c(3000, 3000, 3000)) %>% add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) %>% add_binary_decisions() %>% add_default_solver(verbose = FALSE) # \dontrun{ # solve problem s3 < solve(p3) # plot solution plot(category_layer(s3), main = "solution", axes = FALSE, box = FALSE)# }