R/add_min_set_objective.R

In prioritizr/ppr: Optimal Project Prioritization

#' @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()))
    }))
}