Add constraints to a conservation planning problem
to ensure
that all selected planning units in the solution have at least a certain
number of neighbors that are also selected in the solution.
# S4 method for ConservationProblem,ANY,ANY,ANY add_neighbor_constraints(x, k, zones, data) # S4 method for ConservationProblem,ANY,ANY,data.frame add_neighbor_constraints(x, k, zones, data) # S4 method for ConservationProblem,ANY,ANY,matrix add_neighbor_constraints(x, k, zones, data) # S4 method for ConservationProblem,ANY,ANY,array add_neighbor_constraints(x, k, zones, data)
x 


k 

zones 

data 

ConservationProblemclass
object with the constraint
added to it.
This function uses neighborhood data identify solutions that surround planning units with a minimum number of neighbors. It was inspired by the mathematical formulations detailed in Billionnet (2013) and Beyer et al. (2016).
The argument to data
can be specified in several ways:
NULL
neighborhood data should be calculated automatically
using the connected_matrix
function. This is the default
argument. Note that the neighborhood data must be manually defined
using one of the other formats below when the planning unit data
in the argument to x
is not spatially referenced (e.g.
in data.frame
or numeric
format).
matrix
, Matrix
where rows and columns represent
different planning units and the value of each cell indicates if the
two planning units are neighbors or not. Cell values should be binary
numeric
values (i.e. one or zero). Cells that occur along the
matrix diagonal have no effect on the solution at all because each
planning unit cannot be a neighbor with itself.
data.frame
containing the fields (columns)
"id1"
, "id2"
, and "boundary"
. Here, each row
denotes the connectivity between two planning units following the
Marxan format. The field boundary
should contain
binary numeric
values that indicate if the two planning units
specified in the fields "id1"
and "id2"
are neighbors
or not. This data can be used to describe symmetric or
asymmetric relationships between planning units. By default,
input data is assumed to be symmetric unless asymmetric data is
also included (e.g. if data is present for planning units 2 and 3, then
the same amount of connectivity is expected for planning units 3 and 2,
unless connectivity data is also provided for planning units 3 and 2).
If the argument to x
contains multiple zones, then the columns
"zone1"
and "zone2"
can optionally be provided to manually
specify if the neighborhood data pertain to specific zones. The fields
"zone1"
and "zone2"
should contain the character
names of the zones. If the columns "zone1"
and "zone2"
are present, then the argument to zones
must be NULL
.
array
containing fourdimensions where binary
numeric
values indicate if planning unit should be treated
as being neighbors with every other planning unit when they
are allocated to every combination of management zone. The first two
dimensions (i.e. rows and columns) correspond to the planning units,
and second two dimensions correspond to the management zones. For
example, if the argument to data
had a value of 1 at the index
data[1, 2, 3, 4]
this would indicate that planning unit 1 and
planning unit 2 should be treated as neighbors when they are
allocated to zones 3 and 4 respectively.
Beyer HL, Dujardin Y, Watts ME, and Possingham HP (2016) Solving conservation planning problems with integer linear programming. Ecological Modelling, 228: 1422.
Billionnet A (2013) Mathematical optimization ideas for biodiversity conservation. European Journal of Operational Research, 231: 514534.
# load data data(sim_pu_raster, sim_features, sim_pu_zones_stack, sim_features_zones) # create minimal problem p1 < problem(sim_pu_raster, sim_features) %>% add_min_set_objective() %>% add_relative_targets(0.1) # create problem with constraints that require 1 neighbor # and neighbors are defined using a rookstyle neighborhood p2 < p1 %>% add_neighbor_constraints(1) # create problem with constraints that require 2 neighbor # and neighbors are defined using a rookstyle neighborhood p3 < p1 %>% add_neighbor_constraints(2) # create problem with constraints that require 3 neighbor # and neighbors are defined using a queenstyle neighborhood p4 < p1 %>% add_neighbor_constraints(3, data = connected_matrix(sim_pu_raster, directions = 8))#> Optimize a model with 5 rows, 90 columns and 450 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Coefficient statistics: #> Matrix range [2e01, 9e01] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [3e+00, 8e+00] #> Found heuristic solution: objective 2337.9617505 #> Presolve time: 0.00s #> Presolved: 5 rows, 90 columns, 450 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Presolved: 5 rows, 90 columns, 450 nonzeros #> #> #> Root relaxation: objective 1.931582e+03, 12 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 1931.58191 0 4 2337.96175 1931.58191 17.4%  0s #> H 0 0 1985.6818841 1931.58191 2.72%  0s #> #> Explored 1 nodes (12 simplex iterations) in 0.00 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 2: 1985.68 2337.96 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 1.985681884076e+03, best bound 1.931581908865e+03, gap 2.7245% #> Optimize a model with 95 rows, 90 columns and 828 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Coefficient statistics: #> Matrix range [2e01, 1e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [3e+00, 8e+00] #> Found heuristic solution: objective 2183.0924422 #> Presolve time: 0.00s #> Presolved: 95 rows, 90 columns, 828 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Presolved: 95 rows, 90 columns, 828 nonzeros #> #> #> Root relaxation: objective 1.932118e+03, 25 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 1932.11830 0 6 2183.09244 1932.11830 11.5%  0s #> H 0 0 1993.6247371 1932.11830 3.09%  0s #> #> Explored 1 nodes (25 simplex iterations) in 0.00 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 2: 1993.62 2183.09 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 1.993624737145e+03, best bound 1.932118304023e+03, gap 3.0852% #> Optimize a model with 95 rows, 90 columns and 828 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Coefficient statistics: #> Matrix range [2e01, 2e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [3e+00, 8e+00] #> Found heuristic solution: objective 3605.4363089 #> Presolve removed 2 rows and 4 columns #> Presolve time: 0.00s #> Presolved: 93 rows, 86 columns, 798 nonzeros #> Variable types: 0 continuous, 86 integer (86 binary) #> Presolved: 93 rows, 86 columns, 798 nonzeros #> #> #> Root relaxation: objective 1.939110e+03, 50 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 1939.10993 0 14 3605.43631 1939.10993 46.2%  0s #> H 0 0 2416.7011836 1939.10993 19.8%  0s #> 0 0 1941.28796 0 16 2416.70118 1941.28796 19.7%  0s #> H 0 0 2019.4822081 1941.28796 3.87%  0s #> #> Cutting planes: #> Cover: 7 #> MIR: 2 #> #> Explored 1 nodes (69 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 2019.48 2416.7 3605.44 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 2.019482208068e+03, best bound 1.941287964291e+03, gap 3.8720% #> Optimize a model with 95 rows, 90 columns and 1084 nonzeros #> Variable types: 0 continuous, 90 integer (90 binary) #> Coefficient statistics: #> Matrix range [2e01, 3e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [3e+00, 8e+00] #> Found heuristic solution: objective 2746.9251892 #> Presolve removed 1 rows and 1 columns #> Presolve time: 0.00s #> Presolved: 94 rows, 89 columns, 1074 nonzeros #> Variable types: 0 continuous, 89 integer (89 binary) #> Presolved: 94 rows, 89 columns, 1074 nonzeros #> #> #> Root relaxation: objective 1.933734e+03, 46 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 1933.73436 0 12 2746.92519 1933.73436 29.6%  0s #> H 0 0 2416.4661768 1933.73436 20.0%  0s #> 0 0 1938.57643 0 15 2416.46618 1938.57643 19.8%  0s #> H 0 0 2018.7760790 1938.57643 3.97%  0s #> #> Cutting planes: #> Cover: 5 #> MIR: 6 #> #> Explored 1 nodes (59 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 2018.78 2416.47 2746.93 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 2.018776078984e+03, best bound 1.938576429999e+03, gap 3.9727%# plot solutions plot(s1, box = FALSE, axes = FALSE, main = c("basic solution", "1 neighbor", "2 neighbors", "3 neighbors"))# create minimal problem with multiple zones p5 < problem(sim_pu_zones_stack, sim_features_zones) %>% add_min_set_objective() %>% add_relative_targets(matrix(0.1, ncol = 3, nrow = 5)) # create problem where selected planning units require at least 2 neighbors # for each zone and planning units are only considered neighbors if they # are allocated to the same zone z6 < diag(3) print(z6)#> [,1] [,2] [,3] #> [1,] 1 0 0 #> [2,] 0 1 0 #> [3,] 0 0 1p6 < p5 %>% add_neighbor_constraints(rep(2, 3), z6) # create problem where the planning units in zone 1 don't explicitly require # any neighbors, planning units in zone 2 require at least 1 neighbors, and # planning units in zone 3 require at least 2 neighbors. As before, planning # units are still only considered neighbors if they are allocated to the # same zone p7 < p5 %>% add_neighbor_constraints(c(0, 1, 2), z6) # create problem given the same constraints as outlined above, except # that when determining which selected planning units are neighbors, # planning units that are allocated to zone 1 and zone 2 can also treated # as being neighbors with each other z8 < diag(3) z8[1, 2] < 1 z8[2, 1] < 1 print(z8)#> [,1] [,2] [,3] #> [1,] 1 1 0 #> [2,] 1 1 0 #> [3,] 0 0 1#> Optimize a model with 105 rows, 270 columns and 1620 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e01, 1e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [1e+00, 8e+00] #> Found heuristic solution: objective 7019.1222763 #> Presolve time: 0.01s #> Presolved: 105 rows, 270 columns, 1620 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Presolved: 105 rows, 270 columns, 1620 nonzeros #> #> #> Root relaxation: objective 5.935429e+03, 100 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 5935.42867 0 13 7019.12228 5935.42867 15.4%  0s #> H 0 0 6082.2792264 5935.42867 2.41%  0s #> #> Explored 1 nodes (100 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 2: 6082.28 7019.12 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 6.082279226435e+03, best bound 5.935428674960e+03, gap 2.4144% #> Optimize a model with 375 rows, 270 columns and 2760 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e01, 2e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [1e+00, 8e+00] #> Found heuristic solution: objective 10188.214728 #> Presolve removed 6 rows and 9 columns #> Presolve time: 0.01s #> Presolved: 369 rows, 261 columns, 2685 nonzeros #> Variable types: 0 continuous, 261 integer (261 binary) #> Presolved: 369 rows, 261 columns, 2685 nonzeros #> #> #> Root relaxation: objective 5.950461e+03, 264 iterations, 0.01 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 5950.46090 0 51 10188.2147 5950.46090 41.6%  0s #> H 0 0 7453.2524990 5950.46090 20.2%  0s #> H 0 0 6495.8274057 5950.46090 8.40%  0s #> #> Explored 1 nodes (264 simplex iterations) in 0.02 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 6495.83 7453.25 10188.2 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 6.495827405667e+03, best bound 5.950460895938e+03, gap 8.3956% #> Optimize a model with 375 rows, 270 columns and 2670 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e01, 2e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [1e+00, 8e+00] #> Found heuristic solution: objective 7339.4976993 #> Presolve removed 91 rows and 3 columns #> Presolve time: 0.01s #> Presolved: 284 rows, 267 columns, 2357 nonzeros #> Variable types: 0 continuous, 267 integer (267 binary) #> Presolved: 284 rows, 267 columns, 2357 nonzeros #> #> #> Root relaxation: objective 5.941652e+03, 197 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 5941.65239 0 25 7339.49770 5941.65239 19.0%  0s #> H 0 0 6738.7445245 5941.65239 11.8%  0s #> H 0 0 6326.6355905 5941.65239 6.09%  0s #> #> Explored 1 nodes (197 simplex iterations) in 0.01 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 6326.64 6738.74 7339.5 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 6.326635590527e+03, best bound 5.941652389062e+03, gap 6.0851% #> Optimize a model with 375 rows, 270 columns and 3250 nonzeros #> Variable types: 0 continuous, 270 integer (270 binary) #> Coefficient statistics: #> Matrix range [2e01, 2e+00] #> Objective range [2e+02, 2e+02] #> Bounds range [1e+00, 1e+00] #> RHS range [1e+00, 8e+00] #> Found heuristic solution: objective 7926.6786941 #> Presolve removed 91 rows and 3 columns #> Presolve time: 0.01s #> Presolved: 284 rows, 267 columns, 2647 nonzeros #> Variable types: 0 continuous, 267 integer (267 binary) #> Presolved: 284 rows, 267 columns, 2647 nonzeros #> #> #> Root relaxation: objective 5.941652e+03, 178 iterations, 0.00 seconds #> #> Nodes  Current Node  Objective Bounds  Work #> Expl Unexpl  Obj Depth IntInf  Incumbent BestBd Gap  It/Node Time #> #> 0 0 5941.65239 0 25 7926.67869 5941.65239 25.0%  0s #> H 0 0 6740.4618068 5941.65239 11.9%  0s #> H 0 0 6327.3313218 5941.65239 6.10%  0s #> #> Explored 1 nodes (178 simplex iterations) in 0.02 seconds #> Thread count was 1 (of 4 available processors) #> #> Solution count 3: 6327.33 6740.46 7926.68 #> #> Optimal solution found (tolerance 1.00e01) #> Best objective 6.327331321838e+03, best bound 5.941652389062e+03, gap 6.0954%