R/add_max_richness_objective.R

In oppr: Optimal Project Prioritization

#' @include internal.R pproto.R Objective-proto.R star_phylogeny.R
NULL

#' Add maximum richness objective
#'
#' Set the objective of a project prioritization [problem()] to
#' 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). This objective is conceptually similar
#' to maximizing species richness in a study area. Furthermore, weights can
#' also be used to specify the relative importance of conserving specific
#' features (see [add_feature_weights()]).
#'
#' @param x [ProjectProblem-class] object.
#'
#' @param budget `numeric` budget for funding actions.
#'
#' @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 is expected to persist within a pre-specified budget.
#'   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}, and
#'   let \eqn{m} denote the maximum expenditure (i.e. the budget). Also,
#'   let \eqn{F} represent each feature (indexed by \eqn{f}), \eqn{W_f}
#'   represent the weight for each feature \eqn{f} (defaults to one for
#'   each feature unless specified otherwise), 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 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{Maximize} \space \sum_{f = 0}^{F} (1 - 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 = 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_f^F (1 - E_f) W_f (eqn 1a);
#'   Subject to:
#'   sum_i^I C_i X_i <= m 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 maximize the weighted persistence of all the
#' species. 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
#' (\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
#' Joseph LN, Maloney RF & Possingham HP (2009) Optimal allocation of
#' resources among threatened species: A project prioritization protocol.
#' *Conservation Biology*, **23**, 328--338.
#'
#' @return [ProjectProblem-class] object with the objective
#'   added to it.
#'
#' @seealso [objectives].
#'
#' @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()
#'
#' \dontrun{
#' # solve problem
#' s1 <- solve(p1)
#'
#' # print solution
#' print(s1)
#'
#' # plot solution
#' plot(p1, s1)
#' }
#'
#' # build another problem that includes feature weights
#' p2 <- p1 %>%
#'      add_feature_weights("weight")
#'
#' \dontrun{
#' # solve problem with feature weights
#' s2 <- solve(p2)
#'
#' # print solution based on feature weights
#' print(s2)
#'
#' # plot solution based on feature weights
#' plot(p2, s2)
#' }
#' @name add_max_richness_objective
NULL

#' @rdname add_max_richness_objective
#' @export
add_max_richness_objective <- function(x, budget) {
  # assert argument is valid
  assertthat::assert_that(inherits(x, "ProjectProblem"),
                          assertthat::is.number(budget),
                          assertthat::noNA(budget),
                          isTRUE(budget >= 0))
  # add objective to problem
  x$add_objective(pproto(
    "MaximumRichnessObjective",
    Objective,
    name = "Maximum richness objective",
    data = list(feature_names = feature_names(x)),
    parameters = parameters(numeric_parameter("budget", budget,
                                              lower_limit = 0)),
    feature_phylogeny = function(self) {
      star_phylogeny(self$data$feature_names,
                     rep(0, length(self$data$feature_names)))
    },
    default_feature_weights = function(self) {
      stats::setNames(rep(1, length(self$data$feature_names)),
                      self$data$feature_names)
    },
    replace_feature_weights = function(self) {
      TRUE
    },
    evaluate = function(self, y, solution) {
      assertthat::assert_that(inherits(y, "ProjectProblem"),
                              inherits(solution, "tbl_df"))
      fp <- y$feature_phylogeny()
      bm <- branch_matrix(fp, FALSE)
      bo <- rcpp_branch_order(bm)
      w <- y$feature_weights()[y$feature_phylogeny()$tip.label]
      rcpp_evaluate_max_phylo_div_objective(
        y$action_costs(), y$pa_matrix(),
        y$epf_matrix()[, y$feature_phylogeny()$tip.label, drop = FALSE],
        bm[, bo, drop = FALSE], rep(0, ncol(bm)),
        rep(0, y$number_of_features()), w,
        as_Matrix(as.matrix(solution), "dgCMatrix"))
    },
    apply = function(self, x, y) {
      assertthat::assert_that(inherits(x, "OptimizationProblem"),
                              inherits(y, "ProjectProblem"))
      fp <- y$feature_phylogeny()
      bo <- rcpp_branch_order(branch_matrix(fp, FALSE))
      invisible(rcpp_apply_max_phylo_div_objective(
        x$ptr, y$action_costs(), self$parameters$get("budget"),
        fp$edge.length[bo]))
    }))
}