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add_boundary_penalties.html
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add_boundary_penalties.html
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<h1>Add boundary penalties</h1>
<small class="dont-index">Source: <a href='https://github.com/prioritizr/prioritizr/blob/master/R/add_boundary_penalties.R'><code>R/add_boundary_penalties.R</code></a></small>
<div class="hidden name"><code>add_boundary_penalties.Rd</code></div>
</div>
<div class="ref-description">
<p>Add penalties to a conservation planning <code><a href='problem.html'>problem()</a></code> to favor solutions
that spatially clump planning units together based on the overall
boundary length (perimeter).</p>
</div>
<div class="ref-usage sourceCode"><pre class='sourceCode r'><code><span class='fu'>add_boundary_penalties</span><span class='op'>(</span>
<span class='va'>x</span>,
<span class='va'>penalty</span>,
edge_factor <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/rep.html'>rep</a></span><span class='op'>(</span><span class='fl'>0.5</span>, <span class='fu'><a href='number_of_zones.html'>number_of_zones</a></span><span class='op'>(</span><span class='va'>x</span><span class='op'>)</span><span class='op'>)</span>,
zones <span class='op'>=</span> <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='fu'><a href='number_of_zones.html'>number_of_zones</a></span><span class='op'>(</span><span class='va'>x</span><span class='op'>)</span><span class='op'>)</span>,
data <span class='op'>=</span> <span class='cn'>NULL</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>penalty</th>
<td><p><code>numeric</code> penalty that is used to scale the importance
of selecting planning units that are spatially clumped together compared
to the main problem objective (e.g. solution cost when the argument to
<code>x</code> has a minimum set objective per <code><a href='add_min_set_objective.html'>add_min_set_objective()</a></code>).
Higher <code>penalty</code> values prefer solutions with a higher degree of spatial
clumping, and smaller <code>penalty</code> values prefer solutions with a smaller
degree of clumping. Note that negative <code>penalty</code> values prefer
solutions that are more spread out. This parameter is equivalent to
the boundary length modifier (BLM)
parameter in <a href='https://marxansolutions.org/'><em>Marxan</em></a>.</p></td>
</tr>
<tr>
<th>edge_factor</th>
<td><p><code>numeric</code> proportion to scale
planning unit edges (borders) that do not have any neighboring planning
units. For example, an edge factor of <code>0.5</code> is commonly used to
avoid overly penalizing planning units along a coastline.
Note that this argument must have an element for each zone in the argument
to <code>x</code>.</p></td>
</tr>
<tr>
<th>zones</th>
<td><p><code>matrix</code> or <code>Matrix</code> object describing the
clumping scheme for different zones. Each row and column corresponds to a
different zone in the argument to <code>x</code>, and cell values indicate the
relative importance of clumping planning units that are allocated to
a combination of zones. Cell values along the diagonal of the matrix
represent the relative importance of clumping planning units that are
allocated to the same zone. Cell values must range between 1 and -1, where
negative values favor solutions that spread out planning units. The default
argument to <code>zones</code> is an identity
matrix (i.e. a matrix with ones along the matrix diagonal and zeros
elsewhere), so that penalties are incurred when neighboring planning units
are not assigned to the same zone. If the cells along
the matrix diagonal contain markedly smaller values than those found
elsewhere in the matrix, then solutions are preferred that surround
planning units with those allocated to different zones
(i.e. greater spatial fragmentation).</p></td>
</tr>
<tr>
<th>data</th>
<td><p><code>NULL</code>, <code>data.frame</code>, <code>matrix</code>, or <code>Matrix</code>
object containing the boundary data. These data describe the total
amount of boundary (perimeter) length for each planning unit,
and the amount of boundary (perimeter) length shared between different
planning units (i.e. planning units that are adjacent to each other).
See the Data format section for more information.</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 penalties
added to it.</p>
<h2 class="hasAnchor" id="details"><a class="anchor" href="#details"></a>Details</h2>
<p>This function adds penalties to a conservation planning problem
to penalize fragmented solutions. It was is inspired by Ball <em>et al.</em>
(2009) and Beyer <em>et al.</em> (2016). The <code>penalty</code> argument is
equivalent to the boundary length modifier (<code>BLM</code>) used in
<a href='https://marxansolutions.org'><em>Marxan</em></a>.
Note that this function can only
be used to represent symmetric relationships between planning units. If
asymmetric relationships are required, use the
<code><a href='add_connectivity_penalties.html'>add_connectivity_penalties()</a></code> function.</p>
<h2 class="hasAnchor" id="data-format"><a class="anchor" href="#data-format"></a>Data format</h2>
<p>The argument to <code>data</code> can be specified using the following formats.
Note that boundary data must always describe symmetric relationships
between planning units.</p>
<dl>
<dt><code>data</code> as a <code>NULL</code> value</dt><dd><p>indicating that the data should be
automatically calculated using the <code><a href='boundary_matrix.html'>boundary_matrix()</a></code> function.
This argument is the default.
Note that the boundary data must be supplied
using one of the other formats below if the planning unit data
in the argument to <code>x</code> do not explicitly contain spatial information
(e.g. planning unit data are a <code>data.frame</code> or <code>numeric</code> class).</p></dd>
<dt><code>data</code> as a <code>matrix</code>/<code>Matrix</code> object</dt><dd><p>where rows and columns represent
different planning units and the value of each cell represents the
amount of shared boundary length between two different planning units.
Cells that occur along the matrix diagonal represent the amount of
exposed boundary associated with each planning unit that has
no neighbor (e.g. these value might pertain to boundaries along a
coastline).</p></dd>
<dt><code>data</code> as a <code>data.frame</code> object</dt><dd><p>with the columns <code>"id1"</code>,
<code>"id2"</code>, and <code>"boundary"</code>. The <code>"id1"</code> and <code>"id2"</code> columns contain
identifiers (indices) for a pair of planning units, and the <code>"boundary"</code>
column contains the amount of shared boundary length between these
two planning units.
This format follows the the standard <em>Marxan</em> format for boundary
data (i.e. per the "bound.dat" file).</p></dd>
</dl>
<h2 class="hasAnchor" id="mathematical-formulation"><a class="anchor" href="#mathematical-formulation"></a>Mathematical formulation</h2>
<p>The boundary penalties are implemented using the following equations. Let
\(I\) represent the set of planning units
(indexed by \(i\) or \(j\)), \(Z\) represent
the set of management zones (indexed by \(z\) or \(y\)), and
\(X_{iz}\) represent the decision
variable for planning unit \(i\) for in zone \(z\) (e.g. with binary
values one indicating if planning unit is allocated or not). Also, let
\(p\) represent the argument to <code>penalty</code>, \(E_z\) represent the
argument to <code>edge_factor</code>, \(B_{ij}\) represent the matrix argument
to <code>data</code> (e.g. generated using <code><a href='boundary_matrix.html'>boundary_matrix()</a></code>), and
\(W_{zz}\) represent the matrix argument to <code>zones</code>.</p>
<p>$$
\sum_{i}^{I} \sum_{j}^{I} \sum_{z}^{Z} (\mathit{ifelse}(i == j, E_z, 1)
\times p \times W_{zz} B_{ij}) + \sum_{i}^{I} \sum_{j}^{I} \sum_{z}^{Z}
\sum_{y}^{Z} (-2 \times p \times X_{iz} \times X_{jy} \times W_{zy} \times
B_{ij})$$</p>
<p>Note that when the problem objective is to maximize some measure of
benefit and not minimize some measure of cost, the term \(p\) is
replaced with \(-p\).</p>
<h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>
<p>Ball IR, Possingham HP, and Watts M (2009) <em>Marxan and relatives:
Software for spatial conservation prioritisation</em> in Spatial conservation
prioritisation: Quantitative methods and computational tools. Eds Moilanen
A, Wilson KA, and Possingham HP. Oxford University Press, Oxford, UK.</p>
<p>Beyer HL, Dujardin Y, Watts ME, and Possingham HP (2016) Solving
conservation planning problems with integer linear programming.
<em>Ecological Modelling</em>, 228: 14--22.</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='penalties.html'>penalties</a> for an overview of all functions for adding penalties.</p>
<p>Other penalties:
<code><a href='add_connectivity_penalties.html'>add_connectivity_penalties</a>()</code>,
<code><a href='add_feature_weights.html'>add_feature_weights</a>()</code>,
<code><a href='add_linear_penalties.html'>add_linear_penalties</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'># set seed for reproducibility</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/r/base/Random.html'>set.seed</a></span><span class='op'>(</span><span class='fl'>500</span><span class='op'>)</span></span>
<span class='r-in'></span>
<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_features</span>, <span class='va'>sim_pu_zones_stack</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 minimal problem</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'><a href='add_min_set_objective.html'>add_min_set_objective</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_relative_targets.html'>add_relative_targets</a></span><span class='op'>(</span><span class='fl'>0.2</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>
<span class='r-in'><span class='co'># create problem with low boundary penalties</span></span>
<span class='r-in'><span class='va'>p2</span> <span class='op'><-</span> <span class='va'>p1</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>50</span>, <span class='fl'>1</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with high boundary penalties but outer edges receive</span></span>
<span class='r-in'><span class='co'># half the penalty as inner edges</span></span>
<span class='r-in'><span class='va'>p3</span> <span class='op'><-</span> <span class='va'>p1</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>500</span>, <span class='fl'>0.5</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create a problem using precomputed boundary data</span></span>
<span class='r-in'><span class='va'>bmat</span> <span class='op'><-</span> <span class='fu'><a href='boundary_matrix.html'>boundary_matrix</a></span><span class='op'>(</span><span class='va'>sim_pu_raster</span><span class='op'>)</span></span>
<span class='r-in'><span class='va'>p4</span> <span class='op'><-</span> <span class='va'>p1</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>50</span>, <span class='fl'>1</span>, data <span class='op'>=</span> <span class='va'>bmat</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># \dontrun{</span></span>
<span class='r-in'><span class='co'># solve problems</span></span>
<span class='r-in'><span class='va'>s</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/pkg/raster/man/stack.html'>stack</a></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 class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p2</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 class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p4</span><span class='op'>)</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># plot solutions</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'>s</span>, main <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='st'>"basic solution"</span>, <span class='st'>"small penalties"</span>, <span class='st'>"high penalties"</span>,</span>
<span class='r-in'> <span class='st'>"precomputed data"</span><span class='op'>)</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_boundary_penalties-1.png' alt='' width='700' height='433' /></span>
<span class='r-in'><span class='co'># }</span></span>
<span class='r-in'><span class='co'># create minimal problem with multiple zones and limit the run-time for</span></span>
<span class='r-in'><span class='co'># solver to 10 seconds so this example doesn't take too long</span></span>
<span class='r-in'><span class='va'>p5</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'><a href='add_min_set_objective.html'>add_min_set_objective</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_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'>0.2</span>, nrow <span class='op'>=</span> <span class='fl'>5</span>, ncol <span class='op'>=</span> <span class='fl'>3</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>time_limit <span class='op'>=</span> <span class='fl'>10</span>, verbose <span class='op'>=</span> <span class='cn'>FALSE</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create zone matrix which favors clumping planning units that are</span></span>
<span class='r-in'><span class='co'># allocated to the same zone together - note that this is the default</span></span>
<span class='r-in'><span class='va'>zm6</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='fl'>3</span><span class='op'>)</span></span>
<span class='r-in'><span class='fu'><a href='print.html'>print</a></span><span class='op'>(</span><span class='va'>zm6</span><span class='op'>)</span></span>
<span class='r-out co'><span class='r-pr'>#></span> [,1] [,2] [,3]</span>
<span class='r-out co'><span class='r-pr'>#></span> [1,] 1 0 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [2,] 0 1 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [3,] 0 0 1</span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with the zone matrix and low penalties</span></span>
<span class='r-in'><span class='va'>p6</span> <span class='op'><-</span> <span class='va'>p5</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>50</span>, zone <span class='op'>=</span> <span class='va'>zm6</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create another problem with the same zone matrix and higher penalties</span></span>
<span class='r-in'><span class='va'>p7</span> <span class='op'><-</span> <span class='va'>p5</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>500</span>, zone <span class='op'>=</span> <span class='va'>zm6</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create zone matrix which favors clumping units that are allocated to</span></span>
<span class='r-in'><span class='co'># different zones together</span></span>
<span class='r-in'><span class='va'>zm8</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'>1</span>, ncol <span class='op'>=</span> <span class='fl'>3</span>, nrow <span class='op'>=</span> <span class='fl'>3</span><span class='op'>)</span></span>
<span class='r-in'><span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='va'>zm8</span><span class='op'>)</span> <span class='op'><-</span> <span class='fl'>0</span></span>
<span class='r-in'><span class='fu'><a href='print.html'>print</a></span><span class='op'>(</span><span class='va'>zm8</span><span class='op'>)</span></span>
<span class='r-out co'><span class='r-pr'>#></span> [,1] [,2] [,3]</span>
<span class='r-out co'><span class='r-pr'>#></span> [1,] 0 1 1</span>
<span class='r-out co'><span class='r-pr'>#></span> [2,] 1 0 1</span>
<span class='r-out co'><span class='r-pr'>#></span> [3,] 1 1 0</span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with the zone matrix</span></span>
<span class='r-in'><span class='va'>p8</span> <span class='op'><-</span> <span class='va'>p5</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>500</span>, zone <span class='op'>=</span> <span class='va'>zm8</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create zone matrix which strongly favors clumping units</span></span>
<span class='r-in'><span class='co'># that are allocated to the same zone together. It will also prefer</span></span>
<span class='r-in'><span class='co'># clumping planning units in zones 1 and 2 together over having</span></span>
<span class='r-in'><span class='co'># these planning units with no neighbors in the solution</span></span>
<span class='r-in'><span class='va'>zm9</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='fl'>3</span><span class='op'>)</span></span>
<span class='r-in'><span class='va'>zm9</span><span class='op'>[</span><span class='fu'><a href='https://rdrr.io/r/base/lower.tri.html'>upper.tri</a></span><span class='op'>(</span><span class='va'>zm9</span><span class='op'>)</span><span class='op'>]</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'>0.3</span>, <span class='fl'>0</span>, <span class='fl'>0</span><span class='op'>)</span></span>
<span class='r-in'><span class='va'>zm9</span><span class='op'>[</span><span class='fu'><a href='https://rdrr.io/r/base/lower.tri.html'>lower.tri</a></span><span class='op'>(</span><span class='va'>zm9</span><span class='op'>)</span><span class='op'>]</span> <span class='op'><-</span> <span class='va'>zm9</span><span class='op'>[</span><span class='fu'><a href='https://rdrr.io/r/base/lower.tri.html'>upper.tri</a></span><span class='op'>(</span><span class='va'>zm9</span><span class='op'>)</span><span class='op'>]</span></span>
<span class='r-in'><span class='fu'><a href='print.html'>print</a></span><span class='op'>(</span><span class='va'>zm9</span><span class='op'>)</span></span>
<span class='r-out co'><span class='r-pr'>#></span> [,1] [,2] [,3]</span>
<span class='r-out co'><span class='r-pr'>#></span> [1,] 1.0 0.3 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [2,] 0.3 1.0 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [3,] 0.0 0.0 1</span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with the zone matrix</span></span>
<span class='r-in'><span class='va'>p9</span> <span class='op'><-</span> <span class='va'>p5</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>500</span>, zone <span class='op'>=</span> <span class='va'>zm9</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create zone matrix which favors clumping planning units in zones 1 and 2</span></span>
<span class='r-in'><span class='co'># together, and favors planning units in zone 3 being spread out</span></span>
<span class='r-in'><span class='co'># (i.e. negative clumping)</span></span>
<span class='r-in'><span class='va'>zm10</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/r/base/diag.html'>diag</a></span><span class='op'>(</span><span class='fl'>3</span><span class='op'>)</span></span>
<span class='r-in'><span class='va'>zm10</span><span class='op'>[</span><span class='fl'>3</span>, <span class='fl'>3</span><span class='op'>]</span> <span class='op'><-</span> <span class='op'>-</span><span class='fl'>1</span></span>
<span class='r-in'><span class='fu'><a href='print.html'>print</a></span><span class='op'>(</span><span class='va'>zm10</span><span class='op'>)</span></span>
<span class='r-out co'><span class='r-pr'>#></span> [,1] [,2] [,3]</span>
<span class='r-out co'><span class='r-pr'>#></span> [1,] 1 0 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [2,] 0 1 0</span>
<span class='r-out co'><span class='r-pr'>#></span> [3,] 0 0 -1</span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># create problem with the zone matrix</span></span>
<span class='r-in'><span class='va'>p10</span> <span class='op'><-</span> <span class='va'>p5</span> <span class='op'><a href='pipe.html'>%>%</a></span> <span class='fu'>add_boundary_penalties</span><span class='op'>(</span><span class='fl'>500</span>, zone <span class='op'>=</span> <span class='va'>zm10</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># \dontrun{</span></span>
<span class='r-in'><span class='co'># solve problems</span></span>
<span class='r-in'><span class='va'>s2</span> <span class='op'><-</span> <span class='fu'><a href='https://rdrr.io/pkg/raster/man/stack.html'>stack</a></span><span class='op'>(</span><span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p5</span><span class='op'>)</span><span class='op'>)</span>, <span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p6</span><span class='op'>)</span><span class='op'>)</span>,</span>
<span class='r-in'> <span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p7</span><span class='op'>)</span><span class='op'>)</span>, <span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p8</span><span class='op'>)</span><span class='op'>)</span>,</span>
<span class='r-in'> <span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p9</span><span class='op'>)</span><span class='op'>)</span>, <span class='fu'><a href='category_layer.html'>category_layer</a></span><span class='op'>(</span><span class='fu'><a href='solve.html'>solve</a></span><span class='op'>(</span><span class='va'>p10</span><span class='op'>)</span><span class='op'>)</span><span class='op'>)</span></span>
<span class='r-in'></span>
<span class='r-in'><span class='co'># plot solutions</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'>s2</span>, main <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='st'>"basic solution"</span>, <span class='st'>"within zone clumping (low)"</span>,</span>
<span class='r-in'> <span class='st'>"within zone clumping (high)"</span>, <span class='st'>"between zone clumping"</span>,</span>
<span class='r-in'> <span class='st'>"within + between clumping"</span>, <span class='st'>"negative clumping"</span><span class='op'>)</span>,</span>
<span class='r-in'> 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_boundary_penalties-2.png' alt='' width='700' height='433' /></span>
<span class='r-in'><span class='co'># }</span></span>
</code></pre></div>
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