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add_min_largest_shortfall_objective.html
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add_min_largest_shortfall_objective.html
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<meta property="og:title" content="Add minimum largest shortfall objective — add_min_largest_shortfall_objective" />
<meta property="og:description" content="Set the objective of a conservation planning problem() to
minimize the largest target shortfall while ensuring that
the cost of the solution does not exceed a budget. Note that if the
target shortfall for a single feature cannot be decreased beyond a certain
point (e.g. because all remaining planning units occupied by that feature
are too costly or are locked out), then solutions may only use a small
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<h1>Add minimum largest shortfall objective</h1>
<small class="dont-index">Source: <a href='https://github.com/prioritizr/prioritizr/blob/master/R/add_min_largest_shortfall_objective.R'><code>R/add_min_largest_shortfall_objective.R</code></a></small>
<div class="hidden name"><code>add_min_largest_shortfall_objective.Rd</code></div>
</div>
<div class="ref-description">
<p>Set the objective of a conservation planning <code><a href='problem.html'>problem()</a></code> to
minimize the largest target shortfall while ensuring that
the cost of the solution does not exceed a budget. Note that if the
target shortfall for a single feature cannot be decreased beyond a certain
point (e.g. because all remaining planning units occupied by that feature
are too costly or are locked out), then solutions may only use a small
proportion of the specified budget.</p>
</div>
<div class="ref-usage sourceCode"><pre class='sourceCode r'><code><span class='fu'>add_min_largest_shortfall_objective</span><span class='op'>(</span><span class='va'>x</span>, <span class='va'>budget</span><span class='op'>)</span></code></pre></div>
<h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
<table class="ref-arguments">
<colgroup><col class="name" /><col class="desc" /></colgroup>
<tr>
<th>x</th>
<td><p><code><a href='problem.html'>problem()</a></code> (i.e. <code><a href='ConservationProblem-class.html'>ConservationProblem</a></code>) object.</p></td>
</tr>
<tr>
<th>budget</th>
<td><p><code>numeric</code> value specifying the maximum expenditure of
the prioritization. For problems with multiple zones, the argument
to <code>budget</code> can be (i) a single <code>numeric</code> value to specify a single budget
for the entire solution or (ii) a <code>numeric</code> vector to specify
a separate budget for each management zone.</p></td>
</tr>
</table>
<h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>
<p>Object (i.e. <code><a href='ConservationProblem-class.html'>ConservationProblem</a></code>) with the objective
added to it.</p>
<h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
<p>The minimum largest shortfall objective aims to
find the set of planning units that minimize the largest
shortfall for any of 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. This objective is different from the
minimum shortfall objective (<code><a href='add_min_shortfall_objective.html'>add_min_shortfall_objective()</a></code>) because this
objective minimizes the largest (maximum) target shortfall,
whereas the minimum shortfall objective
minimizes the total (weighted sum) of the target shortfalls.
Note that this objective function is not compatible with feature weights
(<code><a href='add_feature_weights.html'>add_feature_weights()</a></code>).</p>
<h2 class="hasAnchor" id="mathematical-formulation"><a class="anchor" href="#mathematical-formulation"></a>Mathematical formulation</h2>
<p>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:</p>
<p>$$\mathit{Minimize} \space l \\
\mathit{subject \space to} \\
\sum_{i = 1}^{I} x_i r_{ij} + y_j \geq t_j \forall j \in J \\
l \geq \frac{y_j}{t_j} \forall j \in J \\
\sum_{i = 1}^{I} x_i c_i \leq B$$</p>
<p>Here, \(x_i\) is the <a href='decisions.html'>decisions</a> 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 \(i\), and \(t_j\) is the representation target for feature
\(j\).
Additionally, \(y_j\) denotes the target shortfall for
the target \(t_j\) for feature \(j\), and
\(l\) denotes the largest target shortfall.
Furthermore, \(B\) is the budget allocated for the solution,
\(c_i\) is the cost of planning unit \(i\). Note that
\(y_j\) and \(s\) are continuous variables bounded between zero
and infinity.</p>
<h2 class="hasAnchor" id="see-also"><a class="anchor" href="#see-also"></a>See also</h2>
<div class='dont-index'><p>See <a href='objectives.html'>objectives</a> for an overview of all functions for adding objectives.
Also, see <a href='targets.html'>targets</a> for an overview of all functions for adding targets, and
<code><a href='add_feature_weights.html'>add_feature_weights()</a></code> to specify weights for different features.</p>
<p>Other objectives:
<code><a href='add_max_cover_objective.html'>add_max_cover_objective</a>()</code>,
<code><a href='add_max_features_objective.html'>add_max_features_objective</a>()</code>,
<code><a href='add_max_phylo_div_objective.html'>add_max_phylo_div_objective</a>()</code>,
<code><a href='add_max_phylo_end_objective.html'>add_max_phylo_end_objective</a>()</code>,
<code><a href='add_max_utility_objective.html'>add_max_utility_objective</a>()</code>,
<code><a href='add_min_set_objective.html'>add_min_set_objective</a>()</code>,
<code><a href='add_min_shortfall_objective.html'>add_min_shortfall_objective</a>()</code></p></div>
<h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
<div class="ref-examples sourceCode"><pre class='sourceCode r'><code><span class='r-in'><span class='co'># load data</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/r/utils/data.html'>data</a></span><span class='op'>(</span><span class='va'>sim_pu_raster</span>, <span class='va'>sim_pu_zones_stack</span>, <span class='va'>sim_features</span>, <span class='va'>sim_features_zones</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with minimum largest shortfall objective</span></span>
<span class='r-in'><span class='va'>p1</span> <span class='op'><-</span> <span class='fu'><a href='problem.html'>problem</a></span><span class='op'>(</span><span class='va'>sim_pu_raster</span>, <span class='va'>sim_features</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'>add_min_largest_shortfall_objective</span><span class='op'>(</span><span class='fl'>1800</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_relative_targets.html'>add_relative_targets</a></span><span class='op'>(</span><span class='fl'>0.1</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_binary_decisions.html'>add_binary_decisions</a></span><span class='op'>(</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_default_solver.html'>add_default_solver</a></span><span class='op'>(</span>verbose <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-in'><span class='co'># \dontrun{</span></span>
<span class='r-in'><span class='co'># solve problem</span></span>
<span class='r-in'><span class='va'>s1</span> <span class='op'><-</span> <span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p1</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># plot solution</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/pkg/raster/man/plot.html'>plot</a></span><span class='op'>(</span><span class='va'>s1</span>, main <span class='op'>=</span> <span class='st'>"solution"</span>, axes <span class='op'>=</span> <span class='cn'>FALSE</span>, box <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-plt'><img src='add_min_largest_shortfall_objective-1.png' alt='' width='700' height='433' /></span>
<span class='r-in'><span class='co'># }</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create multi-zone problem with minimum largest shortfall objective,</span></span>
<span class='r-in'><span class='co'># with 10% representation targets for each feature, and set</span></span>
<span class='r-in'><span class='co'># a budget such that the total maximum expenditure in all zones</span></span>
<span class='r-in'><span class='co'># cannot exceed 3000</span></span>
<span class='r-in'><span class='va'>p2</span> <span class='op'><-</span> <span class='fu'><a href='problem.html'>problem</a></span><span class='op'>(</span><span class='va'>sim_pu_zones_stack</span>, <span class='va'>sim_features_zones</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'>add_min_largest_shortfall_objective</span><span class='op'>(</span><span class='fl'>3000</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_relative_targets.html'>add_relative_targets</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fl'>0.1</span>, ncol <span class='op'>=</span> <span class='fl'>3</span>, nrow <span class='op'>=</span> <span class='fl'>5</span><span class='op'>)</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_binary_decisions.html'>add_binary_decisions</a></span><span class='op'>(</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_default_solver.html'>add_default_solver</a></span><span class='op'>(</span>verbose <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-in'><span class='co'># \dontrun{</span></span>
<span class='r-in'><span class='co'># solve problem</span></span>
<span class='r-in'><span class='va'>s2</span> <span class='op'><-</span> <span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p2</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># plot solution</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/pkg/raster/man/plot.html'>plot</a></span><span class='op'>(</span><span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='va'>s2</span><span class='op'>)</span>, main <span class='op'>=</span> <span class='st'>"solution"</span>, axes <span class='op'>=</span> <span class='cn'>FALSE</span>, box <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-plt'><img src='add_min_largest_shortfall_objective-2.png' alt='' width='700' height='433' /></span>
<span class='r-in'><span class='co'># }</span></span>
<span class='r-in'><span class='co'># create multi-zone problem with minimum largest shortfall objective,</span></span>
<span class='r-in'><span class='co'># with 10% representation targets for each feature, and set</span></span>
<span class='r-in'><span class='co'># separate budgets for each management zone</span></span>
<span class='r-in'><span class='va'>p3</span> <span class='op'><-</span> <span class='fu'><a href='problem.html'>problem</a></span><span class='op'>(</span><span class='va'>sim_pu_zones_stack</span>, <span class='va'>sim_features_zones</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'>add_min_largest_shortfall_objective</span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/c.html'>c</a></span><span class='op'>(</span><span class='fl'>3000</span>, <span class='fl'>3000</span>, <span class='fl'>3000</span><span class='op'>)</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_relative_targets.html'>add_relative_targets</a></span><span class='op'>(</span><span class='fu'><a href='https://rdrr.io/r/base/matrix.html'>matrix</a></span><span class='op'>(</span><span class='fl'>0.1</span>, ncol <span class='op'>=</span> <span class='fl'>3</span>, nrow <span class='op'>=</span> <span class='fl'>5</span><span class='op'>)</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_binary_decisions.html'>add_binary_decisions</a></span><span class='op'>(</span><span class='op'>)</span> <span class='op'><a href='pipe.html'>%>%</a></span></span>
<span class='r-in'> <span class='fu'><a href='add_default_solver.html'>add_default_solver</a></span><span class='op'>(</span>verbose <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-in'><span class='co'># \dontrun{</span></span>
<span class='r-in'><span class='co'># solve problem</span></span>
<span class='r-in'><span class='va'>s3</span> <span class='op'><-</span> <span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p3</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># plot solution</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/pkg/raster/man/plot.html'>plot</a></span><span class='op'>(</span><span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='va'>s3</span><span class='op'>)</span>, main <span class='op'>=</span> <span class='st'>"solution"</span>, axes <span class='op'>=</span> <span class='cn'>FALSE</span>, box <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-plt'><img src='add_min_largest_shortfall_objective-3.png' alt='' width='700' height='433' /></span>
<span class='r-in'><span class='co'># }</span></span>
</code></pre></div>
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