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#' @include internal.R pproto.R Objective-proto.R star_phylogeny.R
NULL
#' Add minimum set objective
#'
#' Set the objective of a project prioritization [problem()] to
#' 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).
#'
#' @param x [ProjectProblem-class] object.
#'
#' @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 persistence targets for the conservation
#' features (e.g. populations, species, ecosystems). Let \eqn{I} represent
#' the set of conservation actions (indexed by \eqn{i}). Let \eqn{C_i} denote
#' the cost for funding action \eqn{i}. Also, let \eqn{F} represent each
#' feature (indexed by \eqn{f}), \eqn{T_f} represent the persistence target
#' for feature \eqn{f}, and \eqn{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 \eqn{J} denote the set of conservation projects
#' (indexed by \eqn{j}), and let \eqn{A_{ij}} denote which actions
#' \eqn{i \in I}{i in I} comprise each conservation project
#' \eqn{j \in J}{j in J} using zeros and ones. Next, let \eqn{P_j} represent
#' the probability of project \eqn{j} being successful if it is funded. Also,
#' let \eqn{B_{fj}} denote the enhanced probability that each feature
#' \eqn{f \in F}{f in F} associated with the project \eqn{j \in J}{j in J}
#' will persist if all of the actions that comprise project \eqn{j} are funded
#' and that project is allocated to feature \eqn{f}.
#' For convenience,
#' let \eqn{Q_{fj}} denote the actual probability that each
#' \eqn{f \in F}{f in F} associated with the project \eqn{j \in J}{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
#' \eqn{Q_{fj} = (P_{j} \times B_{fj}) + \bigg(\big(1 - (P_{j} B_{fj})\big)
#' \times (P_{n} \times B_{fn})\bigg)}{Q_{fj} = (P_j B_{fj}) + ((1 - (P_j
#' B_{fj})) * (P_n \times B_{fn}))}, 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
#' \eqn{Q_{fj} = P_{j} \times B_{fj}}{Q_{fj} = P_{j} * B_{fj}}.
#'
#' The binary control variables \eqn{X_i} in this problem indicate whether
#' each project \eqn{i \in I}{i in I} is funded or not. The decision
#' variables in this problem are the \eqn{Y_{j}}, \eqn{Z_{fj}}, and \eqn{E_f}
#' variables.
#' Specifically, the binary \eqn{Y_{j}} variables indicate if project \eqn{j}
#' is funded or not based on which actions are funded; the binary
#' \eqn{Z_{fj}} variables indicate if project \eqn{j} is used to manage
#' feature \eqn{f} or not; and the semi-continuous \eqn{E_f} variables
#' denote the probability that feature \eqn{f} will go extinct.
#'
#' 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 \eqn{F}
#' represent the set of ten features and also the number ten).
#'
#' \deqn{
#' \mathrm{Minimize} \space \sum_{i = 0}^{I} C_i X_i \space
#' \mathrm{(eqn \space 1a)} \\
#' \mathrm{Subject \space to} \space \\
#' (1 - E_f) \geq T_f \space \forall f \in F \space
#' \mathrm{(eqn \space 1b)} \\
#' E_f = 1 - \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 b \in B \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)}
#' }{
#' Maximize sum_i^I C_i X_i (eqn 1a);
#' Subject to:
#' E_f <= T_f for all f in F (eqn 1b),
#' E_f = 1 - sum_j^J Y_{fj} Q_{fj} for all f in F (eqn 1c),
#' Z_{fj} <= Y_j for all j in J (eqn 1d),
#' sum_j^J Z_{fj} * ceil(Q_{fj}) = 1 for all f in F (eqn 1e),
#' A_{ij} Y_{j} <= X_{i} for all i I, j in J (eqn 1f),
#' E_f >= 0, E_f <= 1 for all f in F (eqn 1g),
#' X_i, Y_j, Z_{fj} in [0, 1] for all i in I, j in J, f in F (eqn 1h)
#' }
#'
#' The objective (eqn 1a) is to minimize the cost of the funded actions.
#' Constraints (eqn 1b) ensure that the persistence targets are met.
#' 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
#' (\eqn{E_f}) are bounded between zero and one. Constraints (eqns 1h) ensure
#' that the action funding (\eqn{X_i}), project funding (\eqn{Y_j}), and
#' project allocation (\eqn{Z_{fj}}) variables are binary.
#'
#' @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.
#'
#' @seealso [objectives], [targets].
#'
#' @inherit add_max_richness_objective return
#'
#' @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()
#'
#' \dontrun{
#' # solve problem
#' s <- solve(p)
#'
#' # print solution
#' print(s)
#'
#' # plot solution, and add a dashed line to indicate the feature targets
#' plot(p, s) +
#' geom_hline(yintercept = 0.3, linetype = "dashed")
#' }
#' @name add_min_set_objective
NULL
#' @rdname add_min_set_objective
#' @export
add_min_set_objective <- function(x) {
# assert argument is valid
assertthat::assert_that(inherits(x, "ProjectProblem"))
# add objective to problem
x$add_objective(pproto(
"MinimumSetObjective",
Objective,
name = "Minimum set objective",
data = list(feature_names = feature_names(x)),
feature_phylogeny = function(self) {
star_phylogeny(self$data$feature_names)
},
default_feature_weights = function(self) {
stats::setNames(rep(NA_real_, length(self$data$feature_names)),
self$data$feature_names)
},
evaluate = function(self, y, solution) {
assertthat::assert_that(inherits(y, "ProjectProblem"),
inherits(solution, "tbl_df"))
fp <- y$feature_phylogeny()
bm <- branch_matrix(fp, FALSE)
bo <- rcpp_branch_order(bm)
rcpp_evaluate_min_set_objective(
y$action_costs(), y$pa_matrix(),
y$epf_matrix()[, y$feature_phylogeny()$tip.label, drop = FALSE],
bm[, bo, drop = FALSE], fp$edge.length[bo],
rep(0, y$number_of_features()), rep(0, y$number_of_features()),
as_Matrix(as.matrix(solution), "dgCMatrix"))
},
apply = function(self, x, y) {
assertthat::assert_that(inherits(x, "OptimizationProblem"),
inherits(y, "ProjectProblem"))
invisible(rcpp_apply_min_set_objective(x$ptr, y$feature_targets(),
y$action_costs()))
}))
}
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