Problem objective

An objective is used to specify the overall goal of a project prioritization problem(). All project prioritization problems involve minimizing or maximizing some kind of objective. For instance, the decision maker may require a funding scheme that maximizes the total number of species that are expected to persist into the future whilst ensuring that the total cost of the funded actions does not exceed a budget. Alternatively, the planner may require a solution that ensures that each species meets a target level of persistence whilst minimizing the cost of the funded actions. A project prioritization problem() must have a specified objective before it can be solved, and attempting to solve a problem which does not have a specified objective will throw an error.

Details

The following objectives can be added to a conservation planning problem():

add_max_richness_objective()

Maximize the total number of features that are expected to persist, whilst ensuring that the cost of the solution is within a pre-specified budget (Joseph, Maloney & Possingham 2009).

add_max_targets_met_objective()

Maximize the total number of persistence targets met for the features, whilst ensuring that the cost of the solution is within a pre-specified budget (Chades et al. 2015).

add_max_phylo_div_objective()

Maximize the phylogenetic diversity that is expected to persist into the future, whilst ensuring that the cost of the solution is within a pre-specified budget (Bennett et al. 2014, Faith 2008).

add_min_set_objective()

Minimize the cost of the solution whilst ensuring that all targets are met. This objective is conceptually similar to that used in Marxan (Ball, Possingham & Watts 2009).

References

Ball IR, Possingham HP & Watts M (2009) Marxan and relatives: software for spatial conservation prioritisation. Spatial conservation prioritisation: Quantitative methods and computational tools, 185-195.

Bennett JR, Elliott G, Mellish B, Joseph LN, Tulloch AI, Probert WJ, Di Fonzo MMI, Monks JM, Possingham HP & Maloney R (2014) Balancing phylogenetic diversity and species numbers in conservation prioritization, using a case study of threatened species in New Zealand. Biological Conservation, 174: 47--54.

Chades I, Nicol S, van Leeuwen S, Walters B, Firn J, Reeson A, Martin TG & Carwardine J (2015) Benefits of integrating complementarity into priority threat management. Conservation Biology, 29, 525--536.

Faith DP (2008) Threatened species and the potential loss of phylogenetic diversity: conservation scenarios based on estimated extinction probabilities and phylogenetic risk analysis. Conservation Biology, 22: 1461--1470.

Joseph LN, Maloney RF & Possingham HP (2009) Optimal allocation of resources among threatened species: A project prioritization protocol. Conservation Biology, 23, 328--338.

See also

constraints, decisions, problem(), solvers, targets, weights.

Examples

# load data
data(sim_projects, sim_features, sim_actions, sim_tree)

# build problem with maximum richness objective and $200 budget
p1 <- problem(sim_projects, sim_actions, sim_features,
             "name", "success", "name", "cost", "name") %>%
     add_max_richness_objective(budget = 200) %>%
     add_binary_decisions()

# \dontrun{
# solve problem
s1 <- solve(p1)
#> Set parameter Username
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter NumericFocus to value 3
#> Set parameter Presolve to value 2
#> Set parameter Threads to value 1
#> Set parameter PoolSolutions to value 1
#> Set parameter PoolSearchMode to value 2
#> Academic license - for non-commercial use only - expires 2025-04-21
#> Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "Ubuntu 22.04.4 LTS")
#> 
#> CPU model: 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz, instruction set [SSE2|AVX|AVX2|AVX512]
#> Thread count: 4 physical cores, 8 logical processors, using up to 1 threads
#> 
#> Optimize a model with 47 rows, 47 columns and 102 nonzeros
#> Model fingerprint: 0x193cb636
#> Variable types: 0 continuous, 42 integer (42 binary)
#> Semi-Variable types: 5 continuous, 0 integer
#> Coefficient statistics:
#>   Matrix range     [9e-02, 1e+02]
#>   Objective range  [1e+00, 1e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [1e+00, 2e+02]
#> Found heuristic solution: objective 1.4456093
#> Presolve removed 16 rows and 12 columns
#> Presolve time: 0.00s
#> Presolved: 31 rows, 35 columns, 65 nonzeros
#> Variable types: 0 continuous, 35 integer (35 binary)
#> Root relaxation presolved: 31 rows, 35 columns, 65 nonzeros
#> 
#> 
#> Root relaxation: objective 2.190381e+00, 11 iterations, 0.00 seconds (0.00 work units)
#> 
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#> 
#> *    0     0               0       2.1903807    2.19038  0.00%     -    0s
#> 
#> Explored 1 nodes (11 simplex iterations) in 0.00 seconds (0.00 work units)
#> Thread count was 1 (of 8 available processors)
#> 
#> Solution count 1: 2.19038 
#> No other solutions better than 2.19038
#> 
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 2.190380737245e+00, best bound 2.190380737245e+00, gap 0.0000%

# print solution
print(s1)
#> # A tibble: 1 × 21
#>   solution status    obj  cost F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl>     <dbl>     <dbl>     <dbl>     <dbl>     <dbl>
#> 1        1 OPTIMAL  2.19  195.         1         1         0         0         0
#> # ℹ 12 more variables: baseline_action <dbl>, F1_project <dbl>,
#> #   F2_project <dbl>, F3_project <dbl>, F4_project <dbl>, F5_project <dbl>,
#> #   baseline_project <dbl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution
plot(p1, s1)

# }

# build problem with maximum phylogenetic diversity objective and $200 budget
p2 <- problem(sim_projects, sim_actions, sim_features,
             "name", "success", "name", "cost", "name") %>%
     add_max_phylo_div_objective(budget = 200, tree = sim_tree) %>%
     add_binary_decisions()

# \dontrun{
# solve problem
s2 <- solve(p2)
#> Set parameter Username
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter NumericFocus to value 3
#> Set parameter Presolve to value 2
#> Set parameter Threads to value 1
#> Set parameter PoolSolutions to value 1
#> Set parameter PoolSearchMode to value 2
#> Academic license - for non-commercial use only - expires 2025-04-21
#> Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "Ubuntu 22.04.4 LTS")
#> 
#> CPU model: 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz, instruction set [SSE2|AVX|AVX2|AVX512]
#> Thread count: 4 physical cores, 8 logical processors, using up to 1 threads
#> 
#> Optimize a model with 50 rows, 50 columns and 123 nonzeros
#> Model fingerprint: 0x8b6e070d
#> Model has 3 piecewise-linear objective terms
#> Variable types: 3 continuous, 42 integer (42 binary)
#> Semi-Variable types: 5 continuous, 0 integer
#> Coefficient statistics:
#>   Matrix range     [9e-02, 1e+02]
#>   Objective range  [2e-01, 2e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [1e+00, 2e+02]
#>   PWLObj x range   [6e-01, 5e+00]
#>   PWLObj obj range [5e-03, 1e+00]
#> Found heuristic solution: objective 1.7230501
#> Presolve removed 16 rows and 12 columns
#> Presolve time: 0.00s
#> Presolved: 37 rows, 2530 columns, 2572 nonzeros
#> Variable types: 2495 continuous, 35 integer (35 binary)
#> Root relaxation presolved: 34 rows, 2527 columns, 2566 nonzeros
#> 
#> 
#> Root relaxation: objective 2.638346e+00, 23 iterations, 0.00 seconds (0.01 work units)
#> 
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#> 
#>      0     0    2.63835    0    6    1.72305    2.63835  53.1%     -    0s
#> H    0     0                       1.9924876    2.62002  31.5%     -    0s
#> H    0     0                       2.0656903    2.62002  26.8%     -    0s
#> H    0     0                       2.1724148    2.62002  20.6%     -    0s
#> H    0     0                       2.5726722    2.62002  1.84%     -    0s
#>      0     0 infeasible    0         2.57267    2.57267  0.00%     -    0s
#> 
#> Explored 1 nodes (28 simplex iterations) in 0.02 seconds (0.04 work units)
#> Thread count was 1 (of 8 available processors)
#> 
#> Solution count 1: 2.57267 
#> No other solutions better than 2.57267
#> 
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 2.572672188340e+00, best bound 2.572672188340e+00, gap 0.0000%

# print solution
print(s2)
#> # A tibble: 1 × 21
#>   solution status    obj  cost F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl>     <dbl>     <dbl>     <dbl>     <dbl>     <dbl>
#> 1        1 OPTIMAL  2.57  194.         1         0         0         0         1
#> # ℹ 12 more variables: baseline_action <dbl>, F1_project <dbl>,
#> #   F2_project <dbl>, F3_project <dbl>, F4_project <dbl>, F5_project <dbl>,
#> #   baseline_project <dbl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution
plot(p2, s2)

# }
# build problem with maximum targets met objective, $200 budget, and
# 40% persistence targets
p3 <- problem(sim_projects, sim_actions, sim_features,
             "name", "success", "name", "cost", "name") %>%
     add_max_targets_met_objective(budget = 200) %>%
     add_absolute_targets(0.4) %>%
     add_binary_decisions()

# \dontrun{
# solve problem
s3 <- solve(p3)
#> Set parameter Username
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter NumericFocus to value 3
#> Set parameter Presolve to value 2
#> Set parameter Threads to value 1
#> Set parameter PoolSolutions to value 1
#> Set parameter PoolSearchMode to value 2
#> Academic license - for non-commercial use only - expires 2025-04-21
#> Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "Ubuntu 22.04.4 LTS")
#> 
#> CPU model: 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz, instruction set [SSE2|AVX|AVX2|AVX512]
#> Thread count: 4 physical cores, 8 logical processors, using up to 1 threads
#> 
#> Optimize a model with 47 rows, 47 columns and 102 nonzeros
#> Model fingerprint: 0xec4d8189
#> Variable types: 0 continuous, 47 integer (47 binary)
#> Coefficient statistics:
#>   Matrix range     [9e-02, 1e+02]
#>   Objective range  [1e+00, 1e+00]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [1e+00, 2e+02]
#> Found heuristic solution: objective 2.0000000
#> Presolve removed 11 rows and 7 columns
#> Presolve time: 0.00s
#> Presolved: 36 rows, 40 columns, 75 nonzeros
#> Variable types: 0 continuous, 40 integer (40 binary)
#> Root relaxation presolved: 36 rows, 40 columns, 75 nonzeros
#> 
#> 
#> Root relaxation: cutoff, 17 iterations, 0.00 seconds (0.00 work units)
#> 
#>     Nodes    |    Current Node    |     Objective Bounds      |     Work
#>  Expl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time
#> 
#>      0     0     cutoff    0         2.00000    2.00000  0.00%     -    0s
#> 
#> Explored 1 nodes (17 simplex iterations) in 0.00 seconds (0.00 work units)
#> Thread count was 1 (of 8 available processors)
#> 
#> Solution count 1: 2 
#> No other solutions better than 2
#> 
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 2.000000000000e+00, best bound 2.000000000000e+00, gap 0.0000%

# print solution
print(s3)
#> # A tibble: 1 × 21
#>   solution status    obj  cost F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl>     <dbl>     <dbl>     <dbl>     <dbl>     <dbl>
#> 1        1 OPTIMAL     2  195.         1         1         0         0         0
#> # ℹ 12 more variables: baseline_action <dbl>, F1_project <dbl>,
#> #   F2_project <dbl>, F3_project <dbl>, F4_project <dbl>, F5_project <dbl>,
#> #   baseline_project <dbl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution
plot(p3, s3)

# }

# build problem with minimum set objective, $200 budget, and 40%
# persistence targets
p4 <- problem(sim_projects, sim_actions, sim_features,
             "name", "success", "name", "cost", "name") %>%
     add_min_set_objective() %>%
     add_absolute_targets(0.4) %>%
     add_binary_decisions()

# \dontrun{
# solve problem
s4 <- solve(p4)
#> Set parameter Username
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter NumericFocus to value 3
#> Set parameter Presolve to value 2
#> Set parameter Threads to value 1
#> Set parameter PoolSolutions to value 1
#> Set parameter PoolSearchMode to value 2
#> Academic license - for non-commercial use only - expires 2025-04-21
#> Gurobi Optimizer version 11.0.2 build v11.0.2rc0 (linux64 - "Ubuntu 22.04.4 LTS")
#> 
#> CPU model: 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz, instruction set [SSE2|AVX|AVX2|AVX512]
#> Thread count: 4 physical cores, 8 logical processors, using up to 1 threads
#> 
#> Optimize a model with 46 rows, 42 columns and 92 nonzeros
#> Model fingerprint: 0x9cba05f4
#> Variable types: 0 continuous, 42 integer (42 binary)
#> Coefficient statistics:
#>   Matrix range     [9e-02, 1e+00]
#>   Objective range  [9e+01, 1e+02]
#>   Bounds range     [1e+00, 1e+00]
#>   RHS range        [4e-01, 1e+00]
#> Found heuristic solution: objective 497.7671458
#> Presolve removed 45 rows and 20 columns
#> Presolve time: 0.00s
#> Presolved: 1 rows, 22 columns, 2 nonzeros
#> Variable types: 0 continuous, 22 integer (22 binary)
#> 
#> Explored 0 nodes (0 simplex iterations) in 0.00 seconds (0.00 work units)
#> Thread count was 1 (of 8 available processors)
#> 
#> Solution count 1: 497.767 
#> No other solutions better than 497.767
#> 
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 4.977671458279e+02, best bound 4.977671458279e+02, gap 0.0000%

# print solution
print(s4)
#> # A tibble: 1 × 21
#>   solution status    obj  cost F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl>     <dbl>     <dbl>     <dbl>     <dbl>     <dbl>
#> 1        1 OPTIMAL  498.  498.         1         1         1         1         1
#> # ℹ 12 more variables: baseline_action <dbl>, F1_project <dbl>,
#> #   F2_project <dbl>, F3_project <dbl>, F4_project <dbl>, F5_project <dbl>,
#> #   baseline_project <dbl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution
plot(p4, s4)

# }