add_max_cover_objective: Add Maximum Coverage Objective

Description

Set an objective to find the solution aims to represent one instance of as many features as possible within a given budget. This type of objective does not require the addition of targets. The mathematical formulation underpinning this function is different from versions prior to 3.0.0.0; see Details section for more information.

Usage

add_max_cover_objective(x, budget)

Arguments

ConservationProblem-class object.

budget

numeric value specifying the maximum expenditure of the prioritization.

Details

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 objectives. Failing to do so will return a default error message when solving.

The maximum coverage problem seeks to find the set of planning units that maximizes the number of represented features, while keeping cost within a fixed budget. Here, features are treated as being represented if the reserve system contains any non-zero amount of the feature (specifically, an amount greater than 1). One situation where these problem formulation could be useful is when dealing with binary biodiversity data which indicate the presence of suitable habitat. Check out the add_max_features_objective for a more generalized formulation which can accommodate user-specified representation targets.

In versions prior to 3.0.0.0, this objective function was implemented a different mathematical formulation. This formulation is based on the historical maximum coverage reserve selection formulation (Church & Velle 1974; Church et al. 1996). To access the formulation used in versions prior to 3.0.0.0, please see the add_max_utility_objective function.

The maximum coverage 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:

$$\mathit{Maximize} \space \sum_{i = 1}^{I} -s \space c_i + \sum_{j = 1}^{J} y_j w_j \\ \mathit{subject \space to} \\ \sum_{i = 1}^{I} x_i r_{ij} >= y_j \times 1 \forall j \in J \\ \sum_{i = 1}^{I} x_i c_i \leq B$$

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 $i$, $y_j$ 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.

Note that this formulation is functionally equivalent to the add_max_features_objective function with absolute targets set to 1.

References

Church RL and Velle CR (1974) The maximum covering location problem. Regional Science, 32: 101--118.

Church RL, Stoms DM, and Davis FW (1996) Reserve selection as a maximum covering location problem. Biological Conservation, 76: 105--112.

Examples

Run this code

# NOT RUN {
# load data
data(sim_pu_raster, sim_features)

# set a 90th percentile threshold to the feature data
sim_features_binary <- sim_features
thresholds <- raster::quantile(sim_features, probs = 0.9, names = FALSE,
                               na.rm = TRUE)
for (i in seq_len(raster::nlayers(sim_features)))
  sim_features_binary[[i]] <- as.numeric(raster::values(sim_features[[i]]) >
                                         thresholds[[i]])

# create problem
p <- problem(sim_pu_raster, sim_features_binary) %>%
     add_max_cover_objective(5000) %>%
     add_binary_decisions()
# }
# NOT RUN {
# solve problem
s <- solve(p)

# plot solution
plot(s, main = "solution", axes = FALSE, box = FALSE)
# }
# NOT RUN {

# }