Set the objective of a conservation planning problem to represent at least one instance of as many features as possible within a given budget. This type of objective does not use targets, and feature weights should be used instead to increase the representation of different features in solutions. Note that the mathematical formulation underpinning this function is different from versions prior to 3.0.0.0. See the Details section for more information on the changes since this version.

add_max_cover_objective(x, budget)

## Arguments

x ConservationProblem-class object. numeric value specifying the maximum expenditure of the prioritization. For problems with multiple zones, the argument to budget can be a single numeric value to specify a budget for the entire solution or a numeric vector to specify a budget for each each management zone.

## Value

ConservationProblem-class object with the objective added to it.

## 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 an error message when attempting to solve problem.

The maximum coverage objective 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 at least a single instance of a feature (i.e. an amount greater than 1). This formulation has often been used in conservation planning problems dealing with binary biodiversity data that indicate the presence/absence of suitable habitat (e.g. Church & Velle 1974). Additionally, weights can be used to favor the representation of certain features over other features (see add_feature_weights). Check out the add_max_features_objective for a more generalized formulation which can accommodate user-specified representation targets.

This formulation is based on the historical maximum coverage reserve selection formulation (Church & Velle 1974; Church et al. 1996). 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:

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

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. Please note that in versions prior to 3.0.0.0, this objective function implemented a different mathematical formulation. To the add_max_utility_objective function.

## 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.

add_feature_weights, objectives.

## Examples

# load data
data(sim_pu_raster, sim_pu_zones_stack, sim_features, sim_features_zones)

# threshold the feature data to generate binary biodiversity data
sim_binary_features <- sim_features
thresholds <- raster::quantile(sim_features, probs = 0.95, names = FALSE,
na.rm = TRUE)
for (i in seq_len(raster::nlayers(sim_features)))
sim_binary_features[[i]] <- as.numeric(raster::values(sim_features[[i]]) >
thresholds[[i]])

# create problem with maximum utility objective
p1 <- problem(sim_pu_raster, sim_binary_features) %>%
s1 <- solve(p1)#> Optimize a model with 6 rows, 95 columns and 120 nonzeros
#> Variable types: 0 continuous, 95 integer (95 binary)
#> Coefficient statistics:
#>   Matrix range     [1e+00, 2e+02]
#>   Objective range  [1e-04, 1e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [5e+02, 5e+02]
#> Found heuristic solution: objective -0.0000000
#> Presolve removed 5 rows and 90 columns
#> Presolve time: 0.00s
#> Presolved: 1 rows, 5 columns, 5 nonzeros
#> Variable types: 0 continuous, 5 integer (5 binary)
#> Presolved: 1 rows, 5 columns, 5 nonzeros
#>
#>
#> Root relaxation: objective 1.999790e+00, 1 iterations, 0.00 seconds
#>
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#>
#> *    0     0               0       1.9997896    1.99979  0.00%     -    0s
#>
#> Explored 0 nodes (1 simplex iterations) in 0.00 seconds
#> Thread count was 1 (of 4 available processors)
#>
#> Solution count 2: 1.99979 -0
#>
#> Optimal solution found (tolerance 1.00e-01)
#> Best objective 1.999789559459e+00, best bound 1.999789559459e+00, gap 0.0000%
# plot solution
plot(s1, main = "solution", axes = FALSE, box = FALSE)
# threshold the multi-zone feature data to generate binary biodiversity data
sim_binary_features_zones <- sim_features_zones
for (z in number_of_zones(sim_features_zones)) {
thresholds <- raster::quantile(sim_features_zones[[z]], probs = 0.95,
names = FALSE, na.rm = TRUE)
for (i in seq_len(number_of_features(sim_features_zones))) {
sim_binary_features_zones[[z]][[i]] <- as.numeric(
raster::values(sim_features_zones[[z]][[i]]) > thresholds[[i]])
}
}

# create multi-zone problem with maximum utility objective that
# has a single budget for all zones
p2 <- problem(sim_pu_zones_stack, sim_binary_features_zones) %>%
s2 <- solve(p2)#> Optimize a model with 106 rows, 285 columns and 1478 nonzeros
#> Variable types: 0 continuous, 285 integer (285 binary)
#> Coefficient statistics:
#>   Matrix range     [2e-01, 2e+02]
#>   Objective range  [3e-05, 1e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [1e+00, 8e+02]
#> Found heuristic solution: objective -0.0000000
#> Presolve removed 0 rows and 67 columns
#> Presolve time: 0.00s
#> Presolved: 106 rows, 218 columns, 1344 nonzeros
#> Variable types: 0 continuous, 218 integer (218 binary)
#> Presolved: 106 rows, 218 columns, 1344 nonzeros
#>
#>
#> Root relaxation: objective 9.471221e+00, 29 iterations, 0.00 seconds
#>
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#>
#>      0     0    9.47122    0   11   -0.00000    9.47122      -     -    0s
#> H    0     0                       5.9998641    9.47122  57.9%     -    0s
#> H    0     0                       6.9998616    9.47122  35.3%     -    0s
#>      0     0    9.46244    0   12    6.99986    9.46244  35.2%     -    0s
#>      0     0    9.44779    0   11    6.99986    9.44779  35.0%     -    0s
#>      0     0    9.22110    0   11    6.99986    9.22110  31.7%     -    0s
#> H    0     0                       7.9998572    9.22110  15.3%     -    0s
#>      0     0    9.21941    0   10    7.99986    9.21941  15.2%     -    0s
#>      0     0    9.21099    0   10    7.99986    9.21099  15.1%     -    0s
#>      0     0    9.20750    0   13    7.99986    9.20750  15.1%     -    0s
#>      0     0    9.20646    0   14    7.99986    9.20646  15.1%     -    0s
#>      0     0    9.19887    0   13    7.99986    9.19887  15.0%     -    0s
#>      0     0    9.19795    0   12    7.99986    9.19795  15.0%     -    0s
#>      0     0    9.19636    0   12    7.99986    9.19636  15.0%     -    0s
#>      0     0    9.19632    0   13    7.99986    9.19632  15.0%     -    0s
#>      0     0    9.19300    0   14    7.99986    9.19300  14.9%     -    0s
#>      0     0    9.19031    0   17    7.99986    9.19031  14.9%     -    0s
#>      0     0    9.18573    0   15    7.99986    9.18573  14.8%     -    0s
#>      0     0    9.18570    0   16    7.99986    9.18570  14.8%     -    0s
#>      0     0    9.16782    0   15    7.99986    9.16782  14.6%     -    0s
#>      0     0    9.16558    0   16    7.99986    9.16558  14.6%     -    0s
#>      0     0    9.15341    0   17    7.99986    9.15341  14.4%     -    0s
#>      0     0    9.15305    0   18    7.99986    9.15305  14.4%     -    0s
#>      0     0    9.15027    0   14    7.99986    9.15027  14.4%     -    0s
#>      0     0    9.14971    0   15    7.99986    9.14971  14.4%     -    0s
#>      0     0    9.14922    0   16    7.99986    9.14922  14.4%     -    0s
#>      0     0    9.14884    0   14    7.99986    9.14884  14.4%     -    0s
#>      0     0    9.14293    0   16    7.99986    9.14293  14.3%     -    0s
#>      0     0    9.14270    0   17    7.99986    9.14270  14.3%     -    0s
#>      0     0    9.12731    0   12    7.99986    9.12731  14.1%     -    0s
#>      0     0    9.12669    0   13    7.99986    9.12669  14.1%     -    0s
#>      0     0    9.12449    0   13    7.99986    9.12449  14.1%     -    0s
#>      0     0    9.12437    0   14    7.99986    9.12437  14.1%     -    0s
#>      0     0    9.11853    0   17    7.99986    9.11853  14.0%     -    0s
#>      0     0    9.09748    0   12    7.99986    9.09748  13.7%     -    0s
#>      0     0    9.09710    0   15    7.99986    9.09710  13.7%     -    0s
#>      0     0    8.94560    0   15    7.99986    8.94560  11.8%     -    0s
#>      0     0    8.93949    0   16    7.99986    8.93949  11.7%     -    0s
#>      0     0    8.92501    0   17    7.99986    8.92501  11.6%     -    0s
#>      0     0    8.92371    0   19    7.99986    8.92371  11.5%     -    0s
#>      0     0    8.92324    0   18    7.99986    8.92324  11.5%     -    0s
#>      0     0    8.91598    0   19    7.99986    8.91598  11.5%     -    0s
#>      0     0    8.91200    0   20    7.99986    8.91200  11.4%     -    0s
#>      0     0    8.90474    0   20    7.99986    8.90474  11.3%     -    0s
#>      0     0    8.89779    0   20    7.99986    8.89779  11.2%     -    0s
#>      0     0    8.35877    0   11    7.99986    8.35877  4.49%     -    0s
#>
#> Cutting planes:
#>   Gomory: 1
#>   Cover: 5
#>   MIR: 14
#>   StrongCG: 2
#>
#> Explored 1 nodes (325 simplex iterations) in 0.07 seconds
#> Thread count was 1 (of 4 available processors)
#>
#> Solution count 4: 7.99986 6.99986 5.99986 -0
#>
#> Optimal solution found (tolerance 1.00e-01)
#> Best objective 7.999857206152e+00, best bound 8.358770248057e+00, gap 4.4865%
# 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_binary_features_zones) %>%
s3 <- solve(p3)#> Optimize a model with 108 rows, 285 columns and 1478 nonzeros
#> Variable types: 0 continuous, 285 integer (285 binary)
#> Coefficient statistics:
#>   Matrix range     [2e-01, 2e+02]
#>   Objective range  [3e-05, 1e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [1e+00, 4e+02]
#> Found heuristic solution: objective -0.0000000
#> Presolve removed 3 rows and 74 columns
#> Presolve time: 0.01s
#> Presolved: 105 rows, 211 columns, 1026 nonzeros
#> Variable types: 0 continuous, 211 integer (211 binary)
#> Presolve removed 18 rows and 26 columns
#> Presolved: 87 rows, 185 columns, 956 nonzeros
#>
#>
#> Root relaxation: objective 8.997455e+00, 14 iterations, 0.00 seconds
#>
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#>
#>      0     0    8.99746    0    5   -0.00000    8.99746      -     -    0s
#> H    0     0                       6.9998278    8.99746  28.5%     -    0s
#> H    0     0                       7.9998255    8.99746  12.5%     -    0s
#>      0     0     cutoff    0         7.99983    7.99983  0.00%     -    0s
#>
#> Cutting planes:
#>   Gomory: 3
#>   Cover: 4
#>   Clique: 1
#>
#> Explored 1 nodes (35 simplex iterations) in 0.01 seconds
#> Thread count was 1 (of 4 available processors)
#>
#> Solution count 3: 7.99983 6.99983 -0
#>
#> Optimal solution found (tolerance 1.00e-01)
#> Best objective 7.999825481935e+00, best bound 7.999825481935e+00, gap 0.0000%
# plot solution
plot(category_layer(s3), main = "solution", axes = FALSE, box = FALSE)