Set the objective of a conservation planning problem() to
secure as much of the features as possible without exceeding a 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 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
specific features.
add_max_utility_objective(x, budget)
Arguments
| x |
|
|---|---|
| budget |
|
Value
Object (i.e. ConservationProblem) 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 utility objective seeks to find the set of planning units that
maximizes 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()). 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
Examples
# load data 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_max_utility_objective(5000) %>% add_binary_decisions() %>% add_default_solver(gap = 0) # \dontrun{ # solve problem s1 <- solve(p1)#> Gurobi Optimizer version 9.0.2 build v9.0.2rc0 (linux64) #> Optimize a model with 6 rows, 95 columns and 545 nonzeros #> Model fingerprint: 0x75ade7c1 #> Variable types: 5 continuous, 90 integer (90 binary) #> Coefficient statistics: #> Matrix range [2e-01, 2e+02] #> Objective range [1e-04, 1e+00] #> Bounds range [1e+00, 7e+01] #> RHS range [5e+03, 5e+03] #> Found heuristic solution: objective -0.0000000 #> Presolve removed 5 rows and 5 columns #> Presolve time: 0.00s #> Presolved: 1 rows, 90 columns, 90 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Presolved: 1 rows, 90 columns, 90 nonzeros #> #> #> Root relaxation: objective 7.435117e+01, 1 iterations, 0.00 seconds #> #> Nodes | Current Node | Objective Bounds | Work #> Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time #> #> 0 0 74.35117 0 1 -0.00000 74.35117 - - 0s #> H 0 0 74.2352258 74.35117 0.16% - 0s #> H 0 0 74.2714723 74.35117 0.11% - 0s #> 0 0 74.28961 0 2 74.27147 74.28961 0.02% - 0s #> 0 0 74.28961 0 1 74.27147 74.28961 0.02% - 0s #> 0 0 74.28961 0 2 74.27147 74.28961 0.02% - 0s #> 0 0 74.28596 0 2 74.27147 74.28596 0.02% - 0s #> 0 0 74.28498 0 4 74.27147 74.28498 0.02% - 0s #> 0 0 74.28417 0 2 74.27147 74.28417 0.02% - 0s #> 0 0 74.28410 0 6 74.27147 74.28410 0.02% - 0s #> 0 0 74.28247 0 6 74.27147 74.28247 0.01% - 0s #> 0 0 74.28247 0 6 74.27147 74.28247 0.01% - 0s #> 0 2 74.28226 0 6 74.27147 74.28226 0.01% - 0s #> #> Cutting planes: #> Cover: 3 #> MIR: 2 #> StrongCG: 1 #> #> Explored 21 nodes (55 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 74.2715 74.2352 -0 #> #> Optimal solution found (tolerance 0.00e+00) #> Best objective 7.427147227067e+01, best bound 7.427147227067e+01, gap 0.0000%# } # 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_max_utility_objective(5000) %>% add_binary_decisions() %>% add_default_solver(gap = 0) # \dontrun{ # solve problem s2 <- solve(p2)#> Gurobi Optimizer version 9.0.2 build v9.0.2rc0 (linux64) #> Optimize a model with 106 rows, 285 columns and 1905 nonzeros #> Model fingerprint: 0x1b273c19 #> Variable types: 15 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e-01, 2e+02] #> Objective range [3e-05, 1e+00] #> Bounds range [1e+00, 8e+01] #> RHS range [1e+00, 5e+03] #> Found heuristic solution: objective -0.0000000 #> Presolve removed 105 rows and 195 columns #> Presolve time: 0.00s #> Presolved: 1 rows, 90 columns, 90 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Presolved: 1 rows, 90 columns, 90 nonzeros #> #> #> Root relaxation: objective 7.691841e+01, 1 iterations, 0.00 seconds #> #> Nodes | Current Node | Objective Bounds | Work #> Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time #> #> 0 0 76.91841 0 1 -0.00000 76.91841 - - 0s #> H 0 0 74.5659436 76.91841 3.15% - 0s #> 0 0 76.90382 0 2 74.56594 76.90382 3.14% - 0s #> H 0 0 76.8663921 76.90382 0.05% - 0s #> 0 0 76.89002 0 3 76.86639 76.89002 0.03% - 0s #> 0 0 76.89002 0 1 76.86639 76.89002 0.03% - 0s #> 0 0 76.89002 0 1 76.86639 76.89002 0.03% - 0s #> 0 0 76.89002 0 3 76.86639 76.89002 0.03% - 0s #> 0 0 76.89002 0 4 76.86639 76.89002 0.03% - 0s #> 0 0 76.88913 0 5 76.86639 76.88913 0.03% - 0s #> 0 0 76.88879 0 6 76.86639 76.88879 0.03% - 0s #> 0 0 76.88233 0 6 76.86639 76.88233 0.02% - 0s #> 0 0 76.88112 0 6 76.86639 76.88112 0.02% - 0s #> 0 0 76.87939 0 7 76.86639 76.87939 0.02% - 0s #> 0 0 76.87914 0 8 76.86639 76.87914 0.02% - 0s #> 0 0 76.87832 0 8 76.86639 76.87832 0.02% - 0s #> 0 0 76.87640 0 5 76.86639 76.87640 0.01% - 0s #> 0 0 76.87625 0 6 76.86639 76.87625 0.01% - 0s #> 0 2 76.87622 0 6 76.86639 76.87622 0.01% - 0s #> #> Cutting planes: #> Cover: 1 #> MIR: 2 #> StrongCG: 2 #> #> Explored 4 nodes (37 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 76.8664 74.5659 -0 #> #> Optimal solution found (tolerance 0.00e+00) #> Best objective 7.686639207002e+01, best bound 7.686639207002e+01, gap 0.0000%# } # 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)) %>% add_binary_decisions() %>% add_default_solver(gap = 0) # \dontrun{ # solve problem s3 <- solve(p3)#> Gurobi Optimizer version 9.0.2 build v9.0.2rc0 (linux64) #> Optimize a model with 108 rows, 285 columns and 1905 nonzeros #> Model fingerprint: 0x0d15a0b2 #> Variable types: 15 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e-01, 2e+02] #> Objective range [3e-05, 1e+00] #> Bounds range [1e+00, 8e+01] #> RHS range [1e+00, 3e+03] #> Found heuristic solution: objective -0.0000000 #> Presolve removed 15 rows and 15 columns #> Presolve time: 0.00s #> Presolved: 93 rows, 270 columns, 540 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Presolved: 93 rows, 270 columns, 540 nonzeros #> #> #> Root relaxation: objective 8.792265e+01, 9 iterations, 0.00 seconds #> #> Nodes | Current Node | Objective Bounds | Work #> Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time #> #> 0 0 87.92265 0 3 -0.00000 87.92265 - - 0s #> H 0 0 86.3925730 87.92265 1.77% - 0s #> H 0 0 86.4414954 87.92265 1.71% - 0s #> 0 0 86.52619 0 1 86.44150 86.52619 0.10% - 0s #> H 0 0 86.4414968 86.52619 0.10% - 0s #> 0 0 86.51632 0 2 86.44150 86.51632 0.09% - 0s #> 0 0 86.48573 0 4 86.44150 86.48573 0.05% - 0s #> 0 0 86.48573 0 3 86.44150 86.48573 0.05% - 0s #> 0 0 86.48573 0 6 86.44150 86.48573 0.05% - 0s #> H 0 0 86.4668758 86.48573 0.02% - 0s #> H 0 0 86.4735445 86.48573 0.01% - 0s #> 0 0 86.48289 0 2 86.47354 86.48289 0.01% - 0s #> 0 0 86.48289 0 3 86.47354 86.48289 0.01% - 0s #> H 0 0 86.4735471 86.48289 0.01% - 0s #> 0 0 86.48289 0 4 86.47355 86.48289 0.01% - 0s #> 0 0 86.48289 0 2 86.47355 86.48289 0.01% - 0s #> 0 0 86.48263 0 6 86.47355 86.48263 0.01% - 0s #> 0 0 86.47613 0 4 86.47355 86.47613 0.00% - 0s #> 0 0 86.47595 0 6 86.47355 86.47595 0.00% - 0s #> 0 0 86.47567 0 6 86.47355 86.47567 0.00% - 0s #> 0 0 86.47552 0 2 86.47355 86.47552 0.00% - 0s #> 0 0 86.47551 0 6 86.47355 86.47551 0.00% - 0s #> #> Cutting planes: #> Gomory: 1 #> Cover: 1 #> MIR: 2 #> StrongCG: 2 #> RLT: 1 #> #> Explored 1 nodes (150 simplex iterations) in 0.02 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 7: 86.4735 86.4735 86.4669 ... -0 #> #> Optimal solution found (tolerance 0.00e+00) #> Best objective 8.647354705008e+01, best bound 8.647354705008e+01, gap 0.0000%# }
