Returns the total benefit induced by the corresponding solution. The total benefit is computed as the sum of the benefits obtained, for all features, across all the units in the planning area.

getSolutionBenefit(x, type = "total")

## Arguments

x

Solution-class or Portfolio-class.

type

character. Output format of the benefits matrix; total shows the total benefit by feature, while local format shows the benefit achieved per feature and planning unit.

## Details

For a given feature $$s$$, let $$I_s$$ be the set of planning units associated with $$s$$, let $$r_{is}$$ is the amount of feature $$s$$ in planning unit $$i$$, let $$K_{s}$$ be the set of threats associated with $$s$$, and let $$K_{i}$$ be the set of threats associated with $$i$$. The local benefit associated with $$s$$ in a unit $$i$$ is given by:

$$b_{is} = p_{is} r_{is} \\ b_{is} = \frac{ \sum_{k \in K_i \cap K_s}{x_{ik}}}{|K_i \cap K_s|} r_{is}$$

Where $$x_{ik}$$ is a decision variable such that $$x_{ik} = 1$$ if an action againts threat $$k$$ is applied in unit $$i$$, and $$x_{ik} = 0$$, otherwise. This expression for the probability of persistence of the feature ($$p_{is}$$) is defined only for the cases where we work with values of binary intensities (presence or absence of threats). See the sensitivities vignette to know the work with continuous intensities.

While the total benefit is calculated as the sum of the local benefits per feature:

$$b_{s} = \sum_{i \in I_{s}}\frac{ \sum_{k \in K_i \cap K_s}{x_{ik}}}{|K_i \cap K_s|} r_{is}$$

## Examples

# \donttest{
# set seed for reproducibility
set.seed(14)

data(sim_pu_data, sim_features_data, sim_dist_features_data,
sim_threats_data, sim_dist_threats_data, sim_sensitivity_data,
sim_boundary_data)

## Create data instance
problem_data <- inputData(
pu = sim_pu_data, features = sim_features_data, dist_features = sim_dist_features_data,
threats = sim_threats_data, dist_threats = sim_dist_threats_data,
sensitivity = sim_sensitivity_data, boundary = sim_boundary_data
)

## Get maximum benefits to obtain
getPotentialBenefit(problem_data)
#>   feature dist dist_threatened maximum.conservation.benefit
#> 1       1   47              47                            0
#> 2       2   30              28                            2
#> 3       3   66              56                           10
#> 4       4   33              33                            0
#>   maximum.recovery.benefit maximum.benefit
#> 1                       47              47
#> 2                       28              30
#> 3                       56              66
#> 4                       33              33

## Create optimization model
problem_model <- problem(x = problem_data)
#> Warning: The blm argument was set to 0, so the boundary data has no effect
#> Warning: Some blm_actions argument were set to 0, so the boundary data has no effect for these cases

## Solve the optimization model
s <- solve(a = problem_model, time_limit = 2, output_file = FALSE, cores = 2)
#> Rcplex: num variables=396 num constraints=284

# get local benefits of solution
local_benefit <- getSolutionBenefit(s, type = "local")
#>     solution_name pu feature benefit.conservation benefit.recovery
#> 201           sol  1       3                    0                0
#> 202           sol  2       3                    0                0
#> 203           sol  3       3                    0                0
#> 204           sol  4       3                    0                0
#> 205           sol  5       3                    0                0
#> 206           sol  6       3                    0                0
#>     benefit.total
#> 201             0
#> 202             0
#> 203             0
#> 204             0
#> 205             0
#> 206             0

# get total benefits of solution
total_benefit <- getSolutionBenefit(s, type = "total")