Add an objective to a project prioritization problem based on minimizing the cost of the solution, whilst ensuring that the target thresholds for each feature is met (Chadés et al. 2015). In some project prioritization exercises, decision makers may have a target threshold level of expected outcome for each feature (e.g., a 90% chance of persistence for each feature). This objective is especially useful for identifying solutions that ensure that each feature achieves, at least, some minimum level of expected outcome based on the target thresholds.
Arguments
- x
problem()object.
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 minimum set objective seeks to find the set of actions that minimizes the overall cost of the prioritization, while ensuring that the funded projects meet a set of targets for the expected outcomes (e.g., probabilities of persistence, population sizes, amount of habitat) associated with conservation features (e.g., populations, species, ecosystems). Let \(I\) represent the set of conservation actions (indexed by \(i\)). Let \(C_i\) denote the cost for funding action \(i\). Also, let \(F\) represent each feature (indexed by \(f\)), \(T_f\) represent the target for feature \(f\), and \(E_f\) denote the expected outcome for each feature 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 outcome for each feature
\(f \in F\) associated with the project \(j \in J\)
assuming that all of the actions comprising project \(j\) are funded
and that project is allocated to feature \(f\). For convenience,
let \(Q_{fj}\) denote the expected outcome for each
\(f \in F\) associated with the project \(j \in J\)
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 expected outcome
for a feature given that a project is funded and allocated to the feature
depends on (i) the probability of the project succeeding, (ii) the
outcome for the feature if the project succeeds,
and (iii) the outcome for the feature if the project fails
(per the baseline project).
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 expected outcome for feature \(f\).
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).
$$ \mathrm{Minimize} \space \sum_{i = 0}^{I} C_i X_i \space \mathrm{(eqn \space 1a)} \\ \mathrm{Subject \space to} \space \\ E_f \geq T_f \space \forall f \in F \space \mathrm{(eqn \space 1b)} \\ E_f = \sum_{j = 0}^{J} Z_{fj} Q_{fj} \space \forall \space f \in F \space \mathrm{(eqn \space 1c)} \\ Z_{fj} \leq Y_{j} \space \forall \space j \in J \space \mathrm{(eqn \space 1d)} \\ \sum_{j = 0}^{J} Z_{fj} \times \mathrm{ceil}(Q_{fj}) = 1 \space \forall \space f \in F \space \mathrm{(eqn \space 1e)} \\ A_{ij} Y_{j} \leq X_{i} \space \forall \space i \in I, j \in J \space \mathrm{(eqn \space 1f)} \\ E_{f} \geq 0 \space \forall \space f \in F \space \mathrm{(eqn \space 1g)} \\ X_{i}, Y_{j}, Z_{fj} \in \{0, 1\} \space \forall \space i \in I, j \in J, f \in F \space \mathrm{(eqn \space 1h)} $$
The objective (eqn 1a) is to minimize the cost of the funded actions. Constraints (eqn 1b) ensure that the targets are met. Constraints (eqn 1c) calculate the expected outcome for each feature 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 expected outcome variables (\(E_f\)) are greater than zero. Constraints (eqns 1h) ensure that the action funding (\(X_i\)), project funding (\(Y_j\)), and project allocation (\(Z_{fj}\)) variables are binary.
References
Chadés 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.
See also
See targets for an overview of functions for adding targets.
Other objectives:
add_max_phylo_div_objective(),
add_max_richness_objective(),
add_max_targets_met_objective(),
add_max_wtd_sum_objective()
Examples
# load the ggplot2 R package to customize plot
library(ggplot2)
# load data
data(sim_projects, sim_features, sim_actions)
# build problem with minimum set objective and targets that require each
# feature to have a 30% chance of persisting into the future
p <-
problem(
sim_projects, sim_actions, sim_features,
"name", "success", "name", "cost", "name"
) %>%
add_min_set_objective() %>%
add_absolute_targets(0.3) %>%
add_binary_decisions()
# solve problem
s <- solve(p)
#> 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 26 rows, 22 columns and 52 nonzeros (Min)
#> Model fingerprint: 0x21fb06c6
#> Model has 5 linear objective coefficients
#> Variable types: 0 continuous, 22 integer (22 binary)
#> Coefficient statistics:
#> Matrix range [9e-02, 1e+00]
#> Objective range [9e+01, 1e+02]
#> Bounds range [1e+00, 1e+00]
#> RHS range [3e-01, 1e+00]
#>
#> Found heuristic solution: objective 497.7671458
#> Presolve removed 25 rows and 20 columns
#> Presolve time: 0.00s
#> Presolved: 1 rows, 2 columns, 2 nonzeros
#> Variable types: 0 continuous, 2 integer (2 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(s)
#> # 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 498. 498. TRUE TRUE TRUE 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, and add a dashed line to indicate the feature targets
plot(p, s) + geom_hline(yintercept = 0.3, linetype = "dashed")
