Set the objective of a conservation planning problem to minimize the overall shortfall for as many targets as possible while ensuring that the cost of the solution does not exceed a budget.

## Usage

add_min_shortfall_objective(x, budget)

## Arguments

x

problem() object.

budget

numeric value specifying the maximum expenditure of the prioritization. For problems with multiple zones, the argument to budget can be (i) a single numeric value to specify a single budget for the entire solution or (ii) a numeric vector to specify a separate budget for each management zone.

## Value

An updated problem() object with the objective added to it.

## Details

The minimum shortfall objective aims to find the set of planning units that minimize the overall (weighted sum) shortfall for the representation targets---that is, the fraction of each target that remains unmet---for as many features as possible while staying within a fixed budget (inspired by Table 1, equation IV, Arponen et al. 2005). Additionally, weights can be used to favor the representation of certain features over other features (see add_feature_weights().

## Mathematical formulation

This objective can be expressed mathematically for a set of planning units ($$I$$ indexed by $$i$$) and a set of features ($$J$$ indexed by $$j$$) as:

$$\mathit{Minimize} \space \sum_{j = 1}^{J} w_j \frac{y_j}{t_j} \\ \mathit{subject \space to} \\ \sum_{i = 1}^{I} x_i r_{ij} + y_j \geq t_j \forall j \in J \\ \sum_{i = 1}^{I} x_i c_i \leq B$$

Here, $$x_i$$ is the decisions variable (e.g., specifying whether planning unit $$i$$ has been selected (1) or not (0)), $$r_{ij}$$ is the amount of feature $$j$$ in planning unit $$i$$, $$t_j$$ is the representation target for feature $$j$$, $$y_j$$ denotes the representation shortfall for the target $$t_j$$ for feature $$j$$, and $$w_j$$ is the weight for feature $$j$$ (defaults to 1 for all features; see add_feature_weights() to specify weights). Additionally, $$B$$ is the budget allocated for the solution, $$c_i$$ is the cost of planning unit $$i$$. Note that $$y_j$$ is a continuous variable bounded between zero and infinity, and denotes the shortfall for target $$j$$.

Arponen A, Heikkinen RK, Thomas CD, and Moilanen A (2005) The value of biodiversity in reserve selection: representation, species weighting, and benefit functions. Conservation Biology, 19: 2009--2014.