Set the objective of a conservation planning problem() to secure as much of the features as possible without exceeding a budget. This objective does not use targets, and feature weights should be used instead to increase the representation of certain features by a solution. Note that this objective does not aim to maximize as much of each feature as possible, and so often results in solutions that are heavily biased towards just a few features.

add_max_utility_objective(x, budget)

## Arguments

x

problem() (i.e., ConservationProblem) object.

budget

numeric value specifying the maximum expenditure of the prioritization. For problems with multiple zones, the argument to budget can be (i) a single numeric value to specify a single budget for the entire solution or (ii) a numeric vector to specify a separate budget for each management zone.

## Value

Object (i.e., ConservationProblem) with the objective added to it.

## Details

The maximum utility objective seeks to maximize the overall level of representation across a suite of conservation features, while keeping cost within a fixed budget. Additionally, weights can be used to favor the representation of certain features over other features (see add_feature_weights()).

## Mathematical formulation

This objective can be expressed mathematically for a set of planning units ($$I$$ indexed by $$i$$) and a set of features ($$J$$ indexed by $$j$$) as:

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

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 $$i$$, $$A_j$$ is the amount of feature $$j$$ represented in in the solution, 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.

## Notes

In early versions (< 3.0.0.0), this function was named as the add_max_cover_objective function. It was renamed to avoid confusion with existing terminology.

See objectives for an overview of all functions for adding objectives. Also, see add_feature_weights() to specify weights for different features.

Other objectives: add_max_cover_objective(), add_max_features_objective(), add_max_phylo_div_objective(), add_max_phylo_end_objective(), add_min_largest_shortfall_objective(), add_min_set_objective(), add_min_shortfall_objective()

## Examples

# \dontrun{
data(sim_pu_raster, sim_pu_zones_stack, sim_features, sim_features_zones)

# create problem with maximum utility objective
p1 <- problem(sim_pu_raster, sim_features) %>%
add_default_solver(gap = 0, verbose = FALSE)

# solve problem
s1 <- solve(p1)

# plot solution
plot(s1, main = "solution", axes = FALSE, box = FALSE)

# create multi-zone problem with maximum utility objective that
# has a single budget for all zones
p2 <- problem(sim_pu_zones_stack, sim_features_zones) %>%
add_default_solver(gap = 0, verbose = FALSE)

# solve problem
s2 <- solve(p2)

# plot solution
plot(category_layer(s2), main = "solution", axes = FALSE, box = FALSE)

# create multi-zone problem with maximum utility objective that
# has separate budgets for each zone
p3 <- problem(sim_pu_zones_stack, sim_features_zones) %>%
add_max_utility_objective(c(1000, 2000, 3000)) %>%