# Add minimum shortfall objective

Source:`R/add_min_shortfall_objective.R`

`add_min_shortfall_objective.Rd`

Set the objective of a conservation planning problem to minimize the overall shortfall for as many targets as possible while ensuring that the cost of the solution does not exceed a budget.

## 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 minimum shortfall objective aims to
find the set of planning units that minimize the overall
(weighted sum) shortfall for the
representation targets---that is, the fraction of each target that
remains unmet---for as many features as possible while staying within a
fixed budget (inspired by Table 1, equation IV, Arponen *et al.*
2005). 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:

$$\mathit{Minimize} \space \sum_{j = 1}^{J} w_j \frac{y_j}{t_j} \\ \mathit{subject \space to} \\ \sum_{i = 1}^{I} x_i r_{ij} + y_j \geq t_j \forall j \in J \\ \sum_{i = 1}^{I} x_i c_i \leq B$$

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
\(j\), \(y_j\) denotes the representation shortfall for
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 \(i\). Note that \(y_j\) is a continuous
variable bounded between zero and infinity, and denotes the shortfall
for target \(j\).

## References

Arponen A, Heikkinen RK, Thomas CD, and Moilanen A (2005) The value of
biodiversity in reserve selection: representation, species weighting, and
benefit functions. *Conservation Biology*, 19: 2009--2014.

## See also

See objectives for an overview of all functions for adding objectives.
Also, see targets for an overview of all functions for adding targets, and
`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_max_utility_objective()`

,
`add_min_largest_shortfall_objective()`

,
`add_min_set_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 minimum shortfall objective
p1 <-
problem(sim_pu_raster, sim_features) %>%
add_min_shortfall_objective(1800) %>%
add_relative_targets(0.1) %>%
add_binary_decisions() %>%
add_default_solver(verbose = FALSE)
# solve problem
s1 <- solve(p1)
# plot solution
plot(s1, main = "solution", axes = FALSE)
# create multi-zone problem with minimum shortfall 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_zones_pu_raster, sim_zones_features) %>%
add_min_shortfall_objective(3000) %>%
add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) %>%
add_binary_decisions() %>%
add_default_solver(verbose = FALSE)
# solve problem
s2 <- solve(p2)
# plot solution
plot(category_layer(s2), main = "solution", axes = FALSE)
# create multi-zone problem with minimum shortfall objective,
# with 10% representation targets for each feature, and set
# separate budgets for each management zone
p3 <-
problem(sim_zones_pu_raster, sim_zones_features) %>%
add_min_shortfall_objective(c(3000, 3000, 3000)) %>%
add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) %>%
add_binary_decisions() %>%
add_default_solver(verbose = FALSE)
# solve problem
s3 <- solve(p3)
# plot solution
plot(category_layer(s3), main = "solution", axes = FALSE)
# }
```