problem: Create mathematical model

Description

Create an optimization model for the multi-action conservation planning problem, following the mathematical formulations used in Salgado-Rojas et al. (2020).

Usage

problem(
  x,
  model_type = "minimizeCosts",
  budget = 0,
  blm = 0,
  curve = 1,
  segments = 3
)

Value

An object of class optimizationProblem.

Arguments

x: data object. Data used in a problem of prioritization of multiple conservation actions. This object must be created using the problem() function.
model_type: character. Name of the type of model to create. With two possible values: minimizeCosts and maximizeBenefits.
budget: numeric. Maximum budget allowed. This field is used only if a model of the type maximizeBenefits is applied.
blm: numeric. Weight factor applied to the sum of connectivity penalties for missed connections in a solution, similar to Boundary Length Modifier (BLM) in Marxan. This argument only has an effect when the boundary is available.
curve: integer. Type of continuous curve used to represent benefit expression. It can be a linear (1), quadratic (2) or cubic (3) function. See Details for more information.
segments: integer. Number of segments (1, 2, 3, 4 or 5) used to approximate the non-linear expression (curve) in the calculate benefits. See Details for more information.

Details

Currently the problem function allows you to create two types of mathematical programming models:

minimize cost (minimizeCosts):

This model seeks to find the set of actions that minimizes the overall planning costs, while meeting a set of representation targets for the conservation features.

This model can be expressed mathematically for a set of planning units $I$ indexed by $i$ a set of features $S$ indexed by $s$, and a set of threats $K$ indexed by $k$ as: $$ \min \space \sum_{i \in I}\sum_{k \in K_i} x_{ik} c_{ik} + \sum_{i \in I} x_{i \cdot} c'_{i} + blm \cdot connectivity\\ \mathit{s.t.} \\ \sum_{i \in I_s} p_{is} r_{is} \geq t_s \space \forall \space s \in S $$ Where, $x_{ik}$ is a decisions variable that specifies whether an action to abate threat $k$ in planning unit $i$ has been selected (1) or not (0), $c_{ik}$ is the cost of the action to abate the threat $k$ in the planning unit $i$, $c'_{i}$ is the monitoring cost of planning unit $i$, $p_{is}$ is the probability of persistence of the feature $s$ in the planning unit $i$ (ranging between 0 and 1), $r_{is}$ is the amount of feature $s$ in planning unit $i$. $t_s$ is the recovery target for feature $s$. In the case of working with conservation target, the following constraint is necessary:

$$ \sum_{i \in I_s: |K_{s} \cap K_{i}| \neq 0} z_{is} r_{is} \geq t'_s \space \forall \space s \in S $$ With, $z_{is}$ as the probability of persistence by conservation of the feature s in the planning unit i (ranging between 0 and 1). It is only present when there is no spatial co-occurrence between a feature and its threats (i.e. $|K_{s} \cap K_{i}| \neq 0$). In the case of binary threat intensities it is assumed as 1. $t'_s$ is the conservation target for feature $s$.

maximize benefits (maximizeBenefits):

The maximize benefits model seeks to find the set of actions that maximizes the sum of benefits of all features, while the cost of performing actions and monitoring does not exceed a certain budget. Using the terminology presented above, this model can be expressed mathematically as:

$$ \max \space \sum_{i \in I}\sum_{s \in S_i} b_{is} - blm \cdot connectivity\\ \mathit{s.t.} \\ \sum_{i \in I} \sum_{k \in K_i} x_{ik} c_{ik} + \sum_{i \in I} x_{i \cdot} c'_{i} \leq budget $$

Where $b_{is}$ is the benefit of the feature $s$ in a planning unit $i$ and it is calculated by multiplying the probability of persistence of the feature in the unit by its corresponding amount, i.e., $b_{is} = p_{is} r_{is}$. When we talk about recovering, the probability of persistence is a measure of the number of actions taken against the threats that affect said feature. For more information on its calculation, see the getSolutionBenefit() or getPotentialBenefit() functions references.

As a way of including the risk associated with calculating our probability of persistence of the features and in turn, avoiding that many low probabilities of persistence end up reaching the proposed targets, is that we add the curve parameter. That incorporates an exponent (values of 1: linear, 2: quadratic or 3: cubic) to the calculation of the probability of persistence. Thus penalizing the low probabilities in the sum of the benefits achieved. Since prioriactions works with linear models, we use a piecewise linearization strategy to work with non-linear curves in $b_{is}$. The segments parameter indicates how well the expression approximates the curved used in $b_{is}$. A higher number implies a better approximation but increases the resolution complexity. Note that for a linear curve (curve = 1) it is not necessary to set a segment parameter.

Parameters blm and blm_actions allow controlling the spatial connectivity of the selected units and of the deployed actions, respectively (similar to BLM in Marxan).

Examples

Run this code

## This example uses input files included into package.

## set seed for reproducibility
set.seed(14)

## Load data
data(sim_pu_data, sim_features_data, sim_dist_features_data,
sim_threats_data, sim_dist_threats_data, sim_sensitivity_data,
sim_boundary_data)

## Create data instance
problem_data <- inputData(
  pu = sim_pu_data, features = sim_features_data, dist_features = sim_dist_features_data,
  threats = sim_threats_data, dist_threats = sim_dist_threats_data,
  sensitivity = sim_sensitivity_data, boundary = sim_boundary_data
)

## Create minimizeCosts model
model_min <- problem(x = problem_data, blm = 1, model_type = "minimizeCosts")

#' ## Create maximazeBenefits model
model_max <- problem(x = problem_data, model_type = "maximizeBenefits", budget = 100)