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<!DOCTYPE html>
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<h1>Add maximum feature representation objective</h1>
<small class="dont-index">Source: <a href="https://github.com/prioritizr/prioritizr/blob/HEAD/R/add_max_features_objective.R" class="external-link"><code>R/add_max_features_objective.R</code></a></small>
<div class="hidden name"><code>add_max_features_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
fulfill as many targets as possible while ensuring that the cost of the
solution does not exceed a budget.</p>
</div>
<div id="ref-usage">
<div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">add_max_features_objective</span><span class="op">(</span><span class="va">x</span>, <span class="va">budget</span><span class="op">)</span></span></code></pre></div>
</div>
<div id="arguments">
<h2>Arguments</h2>
<dl><dt>x</dt>
<dd><p><code><a href="problem.html">problem()</a></code> (i.e., <code><a href="ConservationProblem-class.html">ConservationProblem</a></code>) object.</p></dd>
<dt>budget</dt>
<dd><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></dd>
</dl></div>
<div id="value">
<h2>Value</h2>
<p>Object (i.e., <code><a href="ConservationProblem-class.html">ConservationProblem</a></code>) with the objective
added to it.</p>
</div>
<div id="details">
<h2>Details</h2>
<p>The maximum feature representation objective is an enhanced version of the
maximum coverage objective <code><a href="add_max_cover_objective.html">add_max_cover_objective()</a></code> because
targets can be used to ensure that a certain amount of each feature is
required in order for them to be adequately represented (similar to the
minimum set objective (see <code><a href="add_min_set_objective.html">add_min_set_objective()</a></code>). This
objective finds the set of planning units that meets representation targets
for as many features as possible while staying within a fixed budget
(inspired by Cabeza and Moilanen 2001). Additionally, weights can be used
<code><a href="add_feature_weights.html">add_feature_weights()</a></code>). If multiple solutions can meet the same
number of weighted targets while staying within budget, the cheapest
solution is returned.</p>
</div>
<div id="mathematical-formulation">
<h2>Mathematical formulation</h2>
<p>This objective can be expressed mathematically for a set of planning units
\(i\)) and a set of features (\(J\) indexed by \(j\)) as:</p>
<p>$$\mathit{Maximize} \space \sum_{i = 1}^{I} -s \space c_i \space x_i +
\sum_{j = 1}^{J} y_j w_j \\
\mathit{subject \space to} \\
\sum_{i = 1}^{I} x_i r_{ij} \geq y_j t_j \forall j \in J \\
\sum_{i = 1}^{I} x_i c_i \leq B$$</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 \(i\), \(t_j\) is the representation target for feature
\(j\), \(y_j\) indicates if the solution has meet
the target \(t_j\) for feature \(j\), and \(w_j\) is the
weight for feature \(j\) (defaults to 1 for all features; see
<code><a href="add_feature_weights.html">add_feature_weights()</a></code> to specify weights). Additionally,
\(B\) is the budget allocated for the solution, \(c_i\) is the
cost of planning unit \(i\), and \(s\) is a scaling factor used
to shrink the costs so that the problem will return a cheapest solution
when there are multiple solutions that represent the same amount of all
features within the budget.</p>
</div>
<div id="references">
<h2>References</h2>
<p>Cabeza M and Moilanen A (2001) Design of reserve networks and the
persistence of biodiversity. <em>Trends in Ecology & Evolution</em>,
16: 242--248.</p>
</div>
<div id="see-also">
<h2>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_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_largest_shortfall_objective.html">add_min_largest_shortfall_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>
</div>
<div id="ref-examples">
<h2>Examples</h2>
<div class="sourceCode"><pre class="sourceCode r"><code><span class="r-in"><span><span class="co"># \dontrun{</span></span></span>
<span class="r-in"><span><span class="co"># load data</span></span></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/utils/data.html" class="external-link">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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># create problem with maximum features objective</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span> <span class="fu">add_max_features_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>
<span class="r-in"><span> <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>
<span class="r-in"><span> <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>
<span class="r-in"><span> <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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># solve problem</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># plot solution</span></span></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">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>
<span class="r-plt img"><img src="add_max_features_objective-1.png" alt="" width="700" height="433"></span>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># create multi-zone problem with maximum features objective,</span></span></span>
<span class="r-in"><span><span class="co"># with 10% representation targets for each feature, and set</span></span></span>
<span class="r-in"><span><span class="co"># a budget such that the total maximum expenditure in all zones</span></span></span>
<span class="r-in"><span><span class="co"># cannot exceed 3000</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span> <span class="fu">add_max_features_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>
<span class="r-in"><span> <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" class="external-link">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>
<span class="r-in"><span> <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>
<span class="r-in"><span> <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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># solve problem</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># plot solution</span></span></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">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>
<span class="r-plt img"><img src="add_max_features_objective-2.png" alt="" width="700" height="433"></span>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># create multi-zone problem with maximum features objective,</span></span></span>
<span class="r-in"><span><span class="co"># with 10% representation targets for each feature, and set</span></span></span>
<span class="r-in"><span><span class="co"># separate budgets for each management zone</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span> <span class="fu">add_max_features_objective</span><span class="op">(</span><span class="fu"><a href="https://rdrr.io/r/base/c.html" class="external-link">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>
<span class="r-in"><span> <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" class="external-link">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>
<span class="r-in"><span> <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>
<span class="r-in"><span> <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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># solve problem</span></span></span>
<span class="r-in"><span><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>
<span class="r-in"><span></span></span>
<span class="r-in"><span><span class="co"># plot solution</span></span></span>
<span class="r-in"><span><span class="fu"><a href="https://rdrr.io/r/base/plot.html" class="external-link">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>
<span class="r-plt img"><img src="add_max_features_objective-3.png" alt="" width="700" height="433"></span>
<span class="r-in"><span><span class="co"># }</span></span></span>
</code></pre></div>
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