Add maximum richness objective

Add an objective to a project prioritization problem based on maximizing the number of features likely to persist, whilst ensuring that the cost of the solution is within a pre-specified budget (Joseph, Maloney & Possingham 2009). Weights can be used to specify the relative importance of conserving specific features (see add_feature_weights()). Although this objective is conceptually similar to maximizing the weighted sum of expected outcomes for the features (add_max_richness_objective() it is designed to work for features that are associated with probabilistic outcomes.

Usage

add_max_richness_objective(x, budget)

Arguments

x: problem() object.
budget: numeric value representing the maximum amount of total expenditure for funding actions.

Value

A problem() with the objective added to it.

Details

A problem objective is used to specify the overall goal of the project prioritization problem. Here, the maximum richness objective seeks to find the set of actions that maximizes the total number of features (e.g., populations, species, ecosystems) that are expected to persist within a pre-specified budget. Let $I$ represent the set of conservation actions (indexed by $i$). Let $C_i$ denote the cost for funding action $i$, and let $m$ denote the maximum expenditure (i.e., the budget). Also, let $F$ represent each feature (indexed by $f$), $W_f$ represent the weight for each feature $f$ (defaults to one for each feature unless specified otherwise), and $E_f$ denote the probability that each feature will go extinct given the funded conservation projects.

To guide the prioritization, the conservation actions are organized into conservation projects. Let $J$ denote the set of conservation projects (indexed by $j$), and let $A_{ij}$ denote which actions $i \in I$ comprise each conservation project $j \in J$ using zeros and ones. Next, let $P_j$ represent the probability of project $j$ being successful if it is funded. Also, let $B_{fj}$ denote the probability that each feature $f \in F$ associated with the project $j \in J$ will persist if all of the actions that comprise project $j$ are funded and that project is allocated to feature $f$. For convenience, let $Q_{fj}$ denote the actual probability that each $f \in F$ associated with the project $j \in J$ is expected to persist if the project is funded. If the argument to adjust_for_baseline in the problem function was set to TRUE, and this is the default behavior, then $Q_{fj} = (P_{j} \times B_{fj}) + \bigg(\big(1 - (P_{j} B_{fj})\big) \times (P_{n} \times B_{fn})\bigg)$, where n corresponds to the baseline "do nothing" project. This means that the probability of a feature persisting if a project is allocated to a feature depends on (i) the probability of the project succeeding, (ii) the probability of the feature persisting if the project does not fail, and (iii) the probability of the feature persisting even if the project fails. Otherwise, if the argument is set to FALSE, then $Q_{fj} = P_{j} \times B_{fj}$.

The binary control variables $X_i$ in this problem indicate whether each project $i \in I$ is funded or not. The decision variables in this problem are the $Y_{j}$, $Z_{fj}$, and $E_f$ variables. Specifically, the binary $Y_{j}$ variables indicate if project $j$ is funded or not based on which actions are funded; the binary $Z_{fj}$ variables indicate if project $j$ is used to manage feature $f$ or not; and the continuous $E_f$ variables denote the probability that feature $f$ will persist.

Now that we have defined all the data and variables, we can formulate the problem. For convenience, let the symbol used to denote each set also represent its cardinality (e.g., if there are ten features, let $F$ represent the set of ten features and also the number ten).

$$ \mathrm{Maximize} \space \sum_{f = 0}^{F} E_f W_f \space \mathrm{(eqn \space 1a)} \\ \mathrm{Subject \space to} \sum_{i = 0}^{I} C_i \leq m \space \mathrm{(eqn \space 1b)} \\ E_f = \sum_{j = 0}^{J} Z_{fj} Q_{fj} \space \forall \space f \in F \space \mathrm{(eqn \space 1c)} \\ Z_{fj} \leq Y_{j} \space \forall \space j \in J \space \mathrm{(eqn \space 1d)} \\ \sum_{j = 0}^{J} Z_{fj} \times \mathrm{ceil}(Q_{fj}) = 1 \space \forall \space f \in F \space \mathrm{(eqn \space 1e)} \\ A_{ij} Y_{j} \leq X_{i} \space \forall \space i \in I, j \in J \space \mathrm{(eqn \space 1f)} \\ E_{f} \geq 0, E_{f} \leq 1 \space \forall \space f \in F \space \mathrm{(eqn \space 1g)} \\ X_{i}, Y_{j}, Z_{fj} \in \{0, 1\} \space \forall \space i \in I, j \in J, f \in F \space \mathrm{(eqn \space 1h)} $$

The objective (eqn 1a) is to maximize the weighted number of features that are expected to persist. Constraint (eqn 1b) limits the maximum expenditure (i.e., ensures that the cost of the funded actions do not exceed the budget). Constraints (eqn 1c) calculate the probability that each feature will go extinct according to their allocated project. Constraints (eqn 1d) ensure that feature can only be allocated to projects that have all of their actions funded. Constraints (eqn 1e) state that each feature can only be allocated to a single project. Constraints (eqn 1f) ensure that a project cannot be funded unless all of its actions are funded. Constraints (eqns 1g) ensure that the probability variables ($E_f$) are bounded between zero and one. Constraints (eqns 1h) ensure that the action funding ($X_i$), project funding ($Y_j$), and project allocation ($Z_{fj}$) variables are binary.

References

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

Examples

# load data
data(sim_projects, sim_features, sim_actions)

# build problem with maximum richness objective and $300 budget
p1 <-
  problem(
    sim_projects, sim_actions, sim_features,
    "name", "success", "name", "cost", "name"
  ) %>%
  add_max_richness_objective(budget = 200) %>%
  add_binary_decisions()

# solve problem
s1 <- solve(p1)
#> Set parameter Username
#> Set parameter LicenseID to value 2806834
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter ScaleFlag to value 2
#> Set parameter NumericFocus to value 1
#> 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 2027-04-14
#> Gurobi Optimizer version 13.0.1 build v13.0.1rc0 (linux64 - "Ubuntu 24.04.2 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
#> 
#> Non-default parameters:
#> TimeLimit  2147483647
#> MIPGap  0
#> ScaleFlag  2
#> NumericFocus  1
#> Presolve  2
#> Threads  1
#> PoolSolutions  1
#> PoolSearchMode  2
#> 
#> Optimize a model with 27 rows, 27 columns and 62 nonzeros (Max)
#> Model fingerprint: 0x1ee444fb
#> Model has 5 linear objective coefficients
#> Variable types: 5 continuous, 22 integer (22 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 1.4456093
#> Presolve removed 16 rows and 12 columns
#> Presolve time: 0.00s
#> Presolved: 11 rows, 15 columns, 25 nonzeros
#> Variable types: 0 continuous, 15 integer (15 binary)
#> Root relaxation presolved: 11 rows, 15 columns, 25 nonzeros
#> 
#> 
#> Root relaxation: objective 2.190381e+00, 12 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 (12 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   cost   obj F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl> <lgl>     <lgl>     <lgl>     <lgl>     <lgl>    
#> 1        1 OPTIMAL  195.  2.19 TRUE      TRUE      FALSE     FALSE     FALSE    
#> # ℹ 12 more variables: baseline_action <lgl>, F1_project <lgl>,
#> #   F2_project <lgl>, F3_project <lgl>, F4_project <lgl>, F5_project <lgl>,
#> #   baseline_project <lgl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution
plot(p1, s1)


# build another problem that includes feature weights
p2 <- p1 %>% add_feature_weights("weight")

# solve problem with feature weights
s2 <- solve(p2)
#> Set parameter Username
#> Set parameter LicenseID to value 2806834
#> Set parameter TimeLimit to value 2147483647
#> Set parameter MIPGap to value 0
#> Set parameter ScaleFlag to value 2
#> Set parameter NumericFocus to value 1
#> 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 2027-04-14
#> Gurobi Optimizer version 13.0.1 build v13.0.1rc0 (linux64 - "Ubuntu 24.04.2 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
#> 
#> Non-default parameters:
#> TimeLimit  2147483647
#> MIPGap  0
#> ScaleFlag  2
#> NumericFocus  1
#> Presolve  2
#> Threads  1
#> PoolSolutions  1
#> PoolSearchMode  2
#> 
#> Optimize a model with 27 rows, 27 columns and 62 nonzeros (Max)
#> Model fingerprint: 0x307c39d4
#> Model has 5 linear objective coefficients
#> Variable types: 5 continuous, 22 integer (22 binary)
#> 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]
#> 
#> Found heuristic solution: objective 0.6654645
#> Presolve removed 16 rows and 12 columns
#> Presolve time: 0.00s
#> Presolved: 11 rows, 15 columns, 25 nonzeros
#> Variable types: 0 continuous, 15 integer (15 binary)
#> Root relaxation presolved: 11 rows, 15 columns, 25 nonzeros
#> 
#> 
#> Root relaxation: objective 1.511230e+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       1.5112297    1.51123  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: 1.51123 
#> No other solutions better than 1.51123
#> 
#> Optimal solution found (tolerance 0.00e+00)
#> Best objective 1.511229665304e+00, best bound 1.511229665304e+00, gap 0.0000%

# print solution based on feature weights
print(s2)
#> # A tibble: 1 × 21
#>   solution status   cost   obj F1_action F2_action F3_action F4_action F5_action
#>      <int> <chr>   <dbl> <dbl> <lgl>     <lgl>     <lgl>     <lgl>     <lgl>    
#> 1        1 OPTIMAL  199.  1.51 FALSE     FALSE     FALSE     TRUE      TRUE     
#> # ℹ 12 more variables: baseline_action <lgl>, F1_project <lgl>,
#> #   F2_project <lgl>, F3_project <lgl>, F4_project <lgl>, F5_project <lgl>,
#> #   baseline_project <lgl>, F1 <dbl>, F2 <dbl>, F3 <dbl>, F4 <dbl>, F5 <dbl>

# plot solution based on feature weights
plot(p2, s2)