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.

## Arguments

- x
`problem()`

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

An updated `problem()`

object 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()`

). It is essentially calculated as a weighted
sum of the feature data inside the selected planning units.

## 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 also

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{
# 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 problem with maximum utility objective
p1 <-
problem(sim_pu_raster, sim_features) %>%
add_max_utility_objective(5000) %>%
add_binary_decisions() %>%
add_default_solver(gap = 0, verbose = FALSE)
# solve problem
s1 <- solve(p1)
# plot solution
plot(s1, main = "solution", axes = FALSE)
# create multi-zone problem with maximum utility objective that
# has a single budget for all zones
p2 <-
problem(sim_zones_pu_raster, sim_zones_features) %>%
add_max_utility_objective(5000) %>%
add_binary_decisions() %>%
add_default_solver(gap = 0, verbose = FALSE)
# solve problem
s2 <- solve(p2)
# plot solution
plot(category_layer(s2), main = "solution", axes = FALSE)
# create multi-zone problem with maximum utility objective that
# has separate budgets for each zone
p3 <-
problem(sim_zones_pu_raster, sim_zones_features) %>%
add_max_utility_objective(c(1000, 2000, 3000)) %>%
add_binary_decisions() %>%
add_default_solver(gap = 0, verbose = FALSE)
# solve problem
s3 <- solve(p3)
# plot solution
plot(category_layer(s3), main = "solution", axes = FALSE)
# }
```