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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en">
<head>
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<meta name="author" content="Cory Whitney" />
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<title>Decision Modeling for Agroecology</title>
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<body>
<a class='sr-only sr-only-focusable visually-hidden-focusable' href='#learnr-tutorial-content'>Skip to Tutorial Content</a>
<div class="pageContent band">
<main class="bandContent page">
<article class="topics" id="learnr-tutorial-content">
<div id="section-introduction" class="section level2">
<h2><strong>Introduction</strong></h2>
<p>Welcome to the interactive tutorial on <strong>Decision Modeling for
Agroecology & Conservation</strong>. This tutorial will guide you
through:</p>
<ul>
<li><strong>Theoretical foundations</strong> of decision modeling</li>
<li><strong>Identifying decision options and outcomes</strong></li>
<li><strong>Structuring causal models</strong> using Directed Acyclic
Graphs (DAGs)</li>
<li><strong>Monte Carlo simulation</strong> for decision-making</li>
<li><strong>Structured simulation</strong> with
<code>decisionSupport</code> and <code>mcSimulation</code></li>
<li><strong>Expert-elicited probabilities</strong> and model
validation</li>
<li><strong>Pareto-optimality</strong> (optional advanced topic)</li>
</ul>
<div class="form-group shiny-input-container">
<label class="control-label" id="intro_question-label" for="intro_question">What do you already know about decision modeling?</label>
<input id="intro_question" type="text" class="shiny-input-text form-control" value="" placeholder="Write a short response..."/>
</div>
<hr />
</div>
<div id="section-decision-modeling" class="section level2">
<h2><strong>1. Decision Modeling</strong></h2>
<p>Decision-making in agroecology often requires navigating trade-offs,
for example between productivity and biodiversity. Decisions are further
complicated by system complexity and uncertainty, i.e. the impacts of
climate variability or market fluctuations. In this section, we will
explore why decision modeling matters for agroecology, how it can guide
sustainable land management, and what tools can help us make informed
decisions.</p>
<div class="panel-heading tutorial-quiz-title"><span data-i18n="text.quiz">Quiz</span></div>
<div class="panel panel-default tutorial-question-container">
<div data-label="question1-1" class="tutorial-question panel-body">
<div id="question1-1-answer_container" class="shiny-html-output"></div>
<div id="question1-1-message_container" class="shiny-html-output"></div>
<div id="question1-1-action_button_container" class="shiny-html-output"></div>
<script>if (Tutorial.triggerMathJax) Tutorial.triggerMathJax()</script>
</div>
</div>
<hr />
</div>
<div id="section-identifying-decision-options-and-outcomes"
class="section level2">
<h2><strong>2. Identifying Decision Options and Outcomes</strong></h2>
<p><strong>Activity:</strong> Identify and list <strong>decision
options</strong> and <strong>outcomes of interest</strong> from the
perspective of decision-makers.</p>
<div class="form-group shiny-input-container">
<label class="control-label" id="decision_options-label" for="decision_options">List the decision options relevant to an agroecological problem:</label>
<input id="decision_options" type="text" class="shiny-input-text form-control" value="" placeholder="Example: Adopting agroforestry, expanding irrigation, switching to organic inputs..."/>
</div>
<div class="form-group shiny-input-container">
<label class="control-label" id="decision_outcomes-label" for="decision_outcomes">List the key outcomes of interest for decision-makers:</label>
<input id="decision_outcomes" type="text" class="shiny-input-text form-control" value="" placeholder="Example: Yield improvement, biodiversity conservation, financial returns..."/>
</div>
<hr />
</div>
<div id="section-structuring-a-causal-decision-model-dags"
class="section level2">
<h2><strong>3. Structuring a Causal Decision Model (DAGs)</strong></h2>
<p><strong>Activity:</strong> Sketch a decision model using the
interactive diagram tool or upload a hand-drawn image.</p>
<div id="section-option-1-interactive-dag-drawing"
class="section level3">
<h3><strong>Option 1: Interactive DAG Drawing</strong></h3>
<p>Share a link of a graph you build in <a
href="https://ncase.me/loopy/">Loopy</a> by <a
href="http://explorableexplanations.com/">Explorable
Explanations</a></p>
</div>
<div id="section-option-2-upload-a-hand-drawn-model"
class="section level3">
<h3><strong>Option 2: Upload a Hand-Drawn Model</strong></h3>
<p>Sketch out a decision model on a piece of paper and share with
us.</p>
<div class="form-group shiny-input-container">
<label class="control-label" id="model_upload-label" for="model_upload">Upload your decision model (optional)</label>
<div class="input-group">
<label class="input-group-btn input-group-prepend">
<span class="btn btn-default btn-file">
Browse...
<input id="model_upload" class="shiny-input-file" name="model_upload" type="file" style="position: absolute !important; top: -99999px !important; left: -99999px !important;" accept="image/png,image/jpeg"/>
</span>
</label>
<input type="text" class="form-control" placeholder="No file selected" readonly="readonly"/>
</div>
<div id="model_upload_progress" class="progress active shiny-file-input-progress">
<div class="progress-bar"></div>
</div>
</div>
<p>Or you can adjust this one if you are familiar with DOT language.</p>
<div class="tutorial-exercise" data-label="decision_model"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>DiagrammeR::grViz("
digraph decision_model {
# Define node styles
node [shape=ellipse, style=filled, fillcolor=lightyellow] Apple_Income Sheep_Income Total_incomes ;
node [shape=ellipse, style=filled, fillcolor=lightpink] Sheep_cost Apple_cost Total_costs;
node [shape=ellipse, style=filled, fillcolor=lightblue] Outcome_Profit;
# Add layout
graph [rankdir=LR]
# Add labels
Apple_Income [label = 'Apple Income (€)']
Sheep_Income [label = 'Sheep Income (€)']
Total_incomes [label = 'Total Income']
Apple_cost [label = 'Apple Cost (€)']
Sheep_cost [label = 'Sheep Cost (€)']
Total_costs [label = 'Total Cost']
Outcome_Profit [label = 'Net Profit (€)']
# Define relationships
Sheep_cost -> Total_costs;
Apple_cost -> Total_costs;
Apple_Income -> Total_incomes;
Sheep_Income -> Total_incomes;
Total_costs -> Outcome_Profit;
Total_incomes -> Outcome_Profit
}
")</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<hr />
</div>
</div>
<div id="section-implementing-a-monte-carlo-model"
class="section level2">
<h2><strong>4. Implementing a Monte Carlo Model</strong></h2>
<p>Now for a very simple <strong>Monte Carlo Simulation</strong> in R.
We will analyze a <strong>decision</strong>. The choice of a farmer to
start agroforestry (apple trees + sheep) or maintain a treeless sheep
system. Use <code>set.seed</code> and change the number of simulations
to 1000.</p>
<div class="tutorial-exercise" data-label="apples_model"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># set.seed(42)
num_simulations <- 100
# Define ranges per ha per year Euro
# lower and upper
apple_income <- runif(n = num_simulations,
min = 3000,
max = 60000)
apple_costs <- runif(n = num_simulations,
min = 15000,
max = 30000)
apple_profits <- apple_income - apple_costs</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support"
data-label="apples_model-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>set.seed(42)
num_simulations <- 1000
# Define ranges per ha per year Euro
# lower and upper
apple_income <- runif(n = num_simulations,
min = 3000,
max = 60000)
apple_costs <- runif(n = num_simulations,
min = 15000,
max = 30000)
apple_profits <- apple_income - apple_costs</code></pre>
</div>
<p>Add sheep to the horticulture system. So we can judge the relative
benefits of transition to a silvopastoral agroforestry system. Add the
sheep and apple profits as a last step in the function.</p>
<div class="tutorial-exercise" data-label="sheep_model"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># Euro per ha per year
sheep_income <- runif(n = num_simulations,
min = 2000,
max = 5000)
sheep_costs <- runif(n = num_simulations,
min = 1000,
max = 2500)
sheep_profits <- sheep_income - sheep_costs</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support" data-label="sheep_model-solution"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># Euro per ha per year
sheep_income <- runif(n = num_simulations,
min = 2000,
max = 5000)
sheep_costs <- runif(n = num_simulations,
min = 1000,
max = 2500)
sheep_profits <- sheep_income - sheep_costs
total_profits <- apple_profits + sheep_profits</code></pre>
</div>
<p>Plot a histogram of the profits. Change the color of the
apple_profits to grey.</p>
<div class="tutorial-exercise" data-label="plot_profits_overlay"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>hist(total_profits, col = "white")
hist(apple_profits, add = TRUE, col = "pink")
hist(sheep_profits, add = TRUE, col = "black")</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support"
data-label="plot_profits_overlay-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>hist(total_profits, col = "white")
hist(apple_profits, add = TRUE, col = "grey")
hist(sheep_profits, add = TRUE, col = "black")</code></pre>
</div>
<div class="panel-heading tutorial-quiz-title"><span data-i18n="text.quiz">Quiz</span></div>
<div class="panel panel-default tutorial-question-container">
<div data-label="question2-1" class="tutorial-question panel-body">
<div id="question2-1-answer_container" class="shiny-html-output"></div>
<div id="question2-1-message_container" class="shiny-html-output"></div>
<div id="question2-1-action_button_container" class="shiny-html-output"></div>
<script>if (Tutorial.triggerMathJax) Tutorial.triggerMathJax()</script>
</div>
</div>
<hr />
</div>
<div id="section-monte-carlo-simulation-with-decisionsupport"
class="section level2">
<h2><strong>5. Monte Carlo Simulation with
<code>decisionSupport</code></strong></h2>
<p>Now we use the <code>decisionSupport</code> package for a structured
Monte Carlo simulation. This approach uses probability distributions for
inputs and a decision model function to compute Net Present Value (NPV)
outcomes, giving us a principled way to compare the agroforestry and
treeless systems.</p>
<div id="section-define-input-estimates" class="section level3">
<h3><strong>Define Input Estimates</strong></h3>
<p>Adjust the lower bounds to reflect realistic minimum values.</p>
<div class="tutorial-exercise" data-label="apple_sheep_estimates"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>input_estimates <- data.frame(
variable = c("sheep_income", "sheep_cost", "apple_income", "apple_cost", "discount_rate"),
lower = c(3000, 1000, 30000, 15000, 10),
median = NA,
upper = c(5000, 2500, 60000, 30000, 10),
distribution = c("posnorm", "posnorm", "posnorm", "posnorm", "const"),
label = c("Income from sheep (euro/year)", "Cost of sheep (euro/year)",
"Income from apple (euro/year)", "Cost of apple (euro/year)", "Discount Rate")
)</code></pre>
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</div>
<div class="tutorial-exercise-support"
data-label="apple_sheep_estimates-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>input_estimates <- data.frame(
variable = c("sheep_income", "sheep_cost", "apple_income", "apple_cost", "discount_rate"),
lower = c(300, 100, 3000, 1500, 1),
median = NA,
upper = c(5000, 2500, 60000, 30000, 10),
distribution = c("posnorm", "posnorm", "posnorm", "posnorm", "const"),
label = c("Income from sheep (euro/year)", "Cost of sheep (euro/year)",
"Income from apple (euro/year)", "Cost of apple (euro/year)", "Discount Rate")
)</code></pre>
</div>
</div>
<div id="section-build-the-decision-model-function"
class="section level3">
<h3><strong>Build the Decision Model Function</strong></h3>
<p>Adjust as you see fit. Add the income from agroforestry
<code>AF_income</code> to the return for example.</p>
<div class="tutorial-exercise" data-label="apple_sheep_model_function"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>model_function <- function() {
AF_income <- sheep_income + apple_income
AF_cost <- sheep_cost + apple_cost
AF_final_result <- AF_income - AF_cost
sheep_only <- sheep_income - sheep_cost
Decision_benefit <- AF_final_result - sheep_only
AF_NPV <- discount(AF_final_result, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_sheep_only <- discount(sheep_only, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_decision <- discount(Decision_benefit, discount_rate = discount_rate, calculate_NPV = TRUE)
return(list(
NPV_Agroforestry_System = AF_NPV,
NPV_Treeless_System = NPV_sheep_only,
NPV_decision = NPV_decision
))
}</code></pre>
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</div>
<div class="tutorial-exercise-support"
data-label="apple_sheep_model_function-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>model_function <- function() {
AF_income <- sheep_income + apple_income
AF_cost <- sheep_cost + apple_cost
AF_final_result <- AF_income - AF_cost
sheep_only <- sheep_income - sheep_cost
Decision_benefit <- AF_final_result - sheep_only
AF_NPV <- discount(AF_final_result, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_sheep_only <- discount(sheep_only, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_decision <- discount(Decision_benefit, discount_rate = discount_rate, calculate_NPV = TRUE)
return(list(
NPV_Agroforestry_System = AF_NPV,
NPV_Treeless_System = NPV_sheep_only,
NPV_decision = NPV_decision,
AF_income = AF_income
))
}</code></pre>
</div>
</div>
<div id="section-run-the-monte-carlo-simulation" class="section level3">
<h3><strong>Run the Monte Carlo Simulation</strong></h3>
<p>Adjust the number of simulations to <code>1000</code>.</p>
<div class="tutorial-exercise" data-label="apple_sheep_mc"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>apple_sheep_mc_simulation <- mcSimulation(
estimate = as.estimate(input_estimates),
model_function = model_function,
numberOfModelRuns = 100,
functionSyntax = "plainNames"
)</code></pre>
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</div>
<div class="tutorial-exercise-support"
data-label="apple_sheep_mc-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>apple_sheep_mc_simulation <- mcSimulation(
estimate = as.estimate(input_estimates),
model_function = model_function,
numberOfModelRuns = 1000,
functionSyntax = "plainNames"
)</code></pre>
</div>
</div>
<div id="section-plot-the-results" class="section level3">
<h3><strong>Plot the Results</strong></h3>
<div class="tutorial-exercise" data-label="plot_mc_results"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>plot_distributions(mcSimulation_object = apple_sheep_mc_simulation,
vars = c("NPV_Agroforestry_System", "NPV_Treeless_System", "NPV_decision"))</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
</div>
<div id="section-expected-value-of-perfect-information-evpi"
class="section level3">
<h3><strong>Expected Value of Perfect Information (EVPI)</strong></h3>
<p>The EVPI tells us how much we would benefit from having perfect
information about each uncertain variable.</p>
<div class="tutorial-exercise" data-label="evpi_calc"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>mcSimulation_table <- data.frame(apple_sheep_mc_simulation$x, apple_sheep_mc_simulation$y[3])
evpi <- multi_EVPI(mc = mcSimulation_table, first_out_var = "NPV_decision")
evpi</code></pre>
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</div>
<hr />
</div>
</div>
<div id="section-the-reference-model-monte_carlo.r"
class="section level2">
<h2><strong>6. The Reference Model:
<code>monte_carlo.R</code></strong></h2>
<p><code>monte_carlo.R</code> is the <strong>foundational
model</strong>. It defines five uncertain variables, a simple decision
model function, and a single call to <code>mcSimulation()</code>. All
inputs are treated as <em>annual</em> values — there is no explicit time
horizon. The <code>discount()</code> function is applied to a single
number, so it converts an annual net benefit into a present value using
the discount rate.</p>
<div id="section-workflow" class="section level3">
<h3><strong>Workflow</strong></h3>
<ol style="list-style-type: decimal">
<li>Build an <code>input_estimates</code> data frame with probability
distributions for each variable.</li>
<li>Write a <code>model_function()</code> that computes outcomes from
those variables.</li>
<li>Call <code>mcSimulation()</code> to sample from the distributions
and run the model thousands of times.</li>
<li>Plot and interpret the resulting distributions.</li>
</ol>
</div>
<div id="section-step-1-define-the-input-estimates"
class="section level3">
<h3><strong>Step 1: Define the input estimates</strong></h3>
<p>The table below covers exactly five variables. <code>posnorm</code>
draws from a positive normal distribution bounded by <code>lower</code>
and <code>upper</code>. <code>const</code> fixes the variable at its
<code>lower</code> value with no uncertainty.</p>
<div class="tutorial-exercise" data-label="mc_basic_estimates"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>input_estimates <- data.frame(
variable = c("sheep_income", "sheep_cost", "apple_income", "apple_cost", "discount_rate"),
lower = c(3000, 1000, 30000, 15000, 10),
median = NA,
upper = c(5000, 2500, 60000, 30000, 10),
distribution = c("posnorm", "posnorm", "posnorm", "posnorm", "const"),
label = c("Income from sheep (euro/year)", "Cost of sheep (euro/year)",
"Income from apple (euro/year)", "Cost of apple (euro/year)", "Discount Rate")
)</code></pre>
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</div>
</div>
<div id="section-step-2-write-the-model-function"
class="section level3">
<h3><strong>Step 2: Write the model function</strong></h3>
<p>Each call to <code>model_function()</code> receives one draw of each
variable from the distributions above. The function computes the net
outcome for the agroforestry system, the treeless sheep system, and the
difference (the decision benefit). <code>discount()</code> converts each
annual result into a Net Present Value (NPV).</p>
<div class="tutorial-exercise" data-label="mc_basic_model"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>model_function <- function() {
AF_income <- sheep_income + apple_income
AF_cost <- sheep_cost + apple_cost
AF_final_result <- AF_income - AF_cost
sheep_only <- sheep_income - sheep_cost
Decision_benefit <- AF_final_result - sheep_only
AF_NPV <- discount(AF_final_result, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_sheep_only <- discount(sheep_only, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_decision <- discount(Decision_benefit, discount_rate = discount_rate, calculate_NPV = TRUE)
return(list(
NPV_Agroforestry_System = AF_NPV,
NPV_Treeless_System = NPV_sheep_only,
NPV_decision = NPV_decision
))
}</code></pre>
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</div>
</div>
<div id="section-step-3-run-mcsimulation" class="section level3">
<h3><strong>Step 3: Run <code>mcSimulation()</code></strong></h3>
<p>Increase <code>numberOfModelRuns</code> to 1000 and observe how the
output distributions change.</p>
<div class="tutorial-exercise" data-label="mc_basic_sim"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>apple_sheep_mc <- mcSimulation(
estimate = as.estimate(input_estimates),
model_function = model_function,
numberOfModelRuns = 100,
functionSyntax = "plainNames"
)
plot_distributions(mcSimulation_object = apple_sheep_mc,
vars = c("NPV_Agroforestry_System", "NPV_Treeless_System", "NPV_decision"))</code></pre>
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</div>
<div class="tutorial-exercise-support"
data-label="mc_basic_sim-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>apple_sheep_mc <- mcSimulation(
estimate = as.estimate(input_estimates),
model_function = model_function,
numberOfModelRuns = 1000,
functionSyntax = "plainNames"
)
plot_distributions(mcSimulation_object = apple_sheep_mc,
vars = c("NPV_Agroforestry_System", "NPV_Treeless_System", "NPV_decision"))</code></pre>
</div>
<div class="panel-heading tutorial-quiz-title"><span data-i18n="text.quiz">Quiz</span></div>
<div class="panel panel-default tutorial-question-container">
<div data-label="question_mc_basic-1" class="tutorial-question panel-body">
<div id="question_mc_basic-1-answer_container" class="shiny-html-output"></div>
<div id="question_mc_basic-1-message_container" class="shiny-html-output"></div>
<div id="question_mc_basic-1-action_button_container" class="shiny-html-output"></div>
<script>if (Tutorial.triggerMathJax) Tutorial.triggerMathJax()</script>
</div>
</div>
<hr />
</div>
</div>
<div id="section-extending-the-model-with-time-monte_carlo_add.r"
class="section level2">
<h2><strong>7. Extending the Model with Time:
<code>monte_carlo_add.R</code></strong></h2>
<p>The model in Section 6 treats all inputs as single annual values —
useful for a first pass, but it cannot capture how outcomes unfold over
time. What if apple trees take 4–6 years to mature before producing any
income? What if there are large upfront costs in year one that dwarf
later annual costs? A single-year average glosses over all of that.</p>
<p>Here we extend the model to handle exactly that. Income and costs are
tracked <em>year by year</em> across a full time horizon, and
<code>discount()</code> processes a <strong>vector</strong> of yearly
values to return a true multi-year NPV.</p>
<div id="section-key-differences-from-monte_carlo.r"
class="section level3">
<h3><strong>Key differences from
<code>monte_carlo.R</code></strong></h3>
<table>
<thead>
<tr class="header">
<th>Feature</th>
<th><code>monte_carlo.R</code></th>
<th><code>monte_carlo_add.R</code></th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Number of variables</td>
<td>5</td>
<td>8</td>
</tr>
<tr class="even">
<td>Time horizon</td>
<td>None (single-year)</td>
<td><code>n_years</code> (e.g. 20 years)</td>
</tr>
<tr class="odd">
<td>Apple income delay</td>
<td>None</td>
<td>Zero until <code>time_to_harvest</code></td>
</tr>
<tr class="even">
<td>Establishment cost</td>
<td>None</td>
<td><code>cost_orchard_establishment</code> in year 1</td>
</tr>
<tr class="odd">
<td><code>make_variables()</code> helper</td>
<td>No</td>
<td>Yes — for testing a single draw</td>
</tr>
<tr class="even">
<td>Input to <code>discount()</code></td>
<td>Scalar</td>
<td>Year-vector (length <code>n_years</code>)</td>
</tr>
</tbody>
</table>
</div>
<div id="section-workflow-1" class="section level3">
<h3><strong>Workflow</strong></h3>
<ol style="list-style-type: decimal">
<li>Add the <code>make_variables()</code> helper to sample one draw and
inspect variable values.</li>
<li>Extend the <code>input_estimates</code> table with the three new
variables.</li>
<li>Modify the model function to build year-vectors and apply the
time-based rules.</li>
<li>Test the model function with a single draw before running the full
simulation.</li>
<li>Run <code>mcSimulation()</code> as before.</li>
</ol>
</div>
<div id="section-step-1-the-make_variables-helper"
class="section level3">
<h3><strong>Step 1: The <code>make_variables()</code>
helper</strong></h3>
<p><code>make_variables()</code> draws one sample from the estimate
distributions and assigns each variable into the global environment.
This lets you call <code>model_function()</code> interactively and
inspect intermediate values — useful for debugging before running
thousands of simulations.</p>
<div class="tutorial-exercise" data-label="mc_add_make_vars"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>make_variables <- function(est, n = 1) {
x <- random(rho = est, n = n)
for (i in colnames(x)) assign(i, as.numeric(x[1, i]), envir = .GlobalEnv)
}</code></pre>
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</div>
</div>
<div id="section-step-2-extend-the-input-estimates"
class="section level3">
<h3><strong>Step 2: Extend the input estimates</strong></h3>
<p>Three variables are added: <code>n_years</code> and
<code>discount_rate</code> are fixed constants;
<code>cost_orchard_establishment</code> and <code>time_to_harvest</code>
are uncertain. Notice <code>time_to_harvest</code> uses a
<code>unif</code> (uniform) distribution because the uncertainty is
bounded without a clear central tendency.</p>
<div class="tutorial-exercise" data-label="mc_add_estimates"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>input_estimates_add <- data.frame(
variable = c("sheep_income", "sheep_cost", "apple_income", "apple_cost",
"discount_rate", "n_years", "cost_orchard_establishment", "time_to_harvest"),
lower = c(3000, 1000, 30000, 15000, 10, 20, 50000, 4),
median = NA,
upper = c(5000, 2500, 60000, 30000, 10, 20, 70000, 6),
distribution = c("posnorm", "posnorm", "posnorm", "posnorm",
"const", "const", "posnorm", "unif"),
label = c("Income from sheep (euro/year)", "Cost of sheep (euro/year)",
"Income from apple (euro/year)", "Cost of apple (euro/year)",
"Discount Rate", "Simulated Years",
"Construction cost of orchard", "Time until fruit harvest")
)</code></pre>
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</div>
</div>
<div id="section-step-3-write-the-time-aware-model-function"
class="section level3">
<h3><strong>Step 3: Write the time-aware model function</strong></h3>
<p>The model now builds a vector of length <code>n_years</code> for each
income and cost stream. Apple income is set to zero for the first
<code>time_to_harvest</code> years (the establishment phase), and the
full orchard construction cost replaces the normal apple cost in year
1.</p>
<div class="tutorial-exercise" data-label="mc_add_model"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>model_function_add <- function() {
# time_to_harvest must be an integer for indexing
time_to_harvest <- round(time_to_harvest, digits = 0)
# Expand scalar inputs into year-vectors
apple_income <- rep(apple_income, n_years)
apple_cost <- rep(apple_cost, n_years)
sheep_cost <- rep(sheep_cost, n_years)
sheep_income <- rep(sheep_income, n_years)
# No apple income during the establishment phase
apple_income[1:time_to_harvest] <- 0
# Upfront orchard establishment cost replaces year-1 apple cost
apple_cost[1] <- cost_orchard_establishment
AF_income <- sheep_income + apple_income
AF_cost <- sheep_cost + apple_cost
AF_final_result <- AF_income - AF_cost
sheep_only <- sheep_income - sheep_cost
Decision_benefit <- AF_final_result - sheep_only
AF_NPV <- discount(AF_final_result, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_sheep_only <- discount(sheep_only, discount_rate = discount_rate, calculate_NPV = TRUE)
NPV_decision <- discount(Decision_benefit, discount_rate = discount_rate, calculate_NPV = TRUE)
return(list(
NPV_Agroforestry_System = AF_NPV,
NPV_Treeless_System = NPV_sheep_only,
NPV_decision = NPV_decision
))
}</code></pre>
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</div>
</div>
<div id="section-step-4-test-with-a-single-draw" class="section level3">
<h3><strong>Step 4: Test with a single draw</strong></h3>
<p>Use <code>make_variables()</code> to populate the global environment
and call the model function once before running the full simulation.
This is a good debugging practice.</p>
<div class="tutorial-exercise" data-label="mc_add_test"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>make_variables(as.estimate(input_estimates_add))
model_function_add()</code></pre>
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</div>
</div>
<div id="section-step-5-run-the-extended-simulation"
class="section level3">
<h3><strong>Step 5: Run the extended simulation</strong></h3>
<div class="tutorial-exercise" data-label="mc_add_sim"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>apple_sheep_mc_add <- mcSimulation(
estimate = as.estimate(input_estimates_add),
model_function = model_function_add,
numberOfModelRuns = 100,
functionSyntax = "plainNames"
)
plot_distributions(mcSimulation_object = apple_sheep_mc_add,
vars = c("NPV_Agroforestry_System", "NPV_Treeless_System", "NPV_decision"))</code></pre>
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</div>
<div class="panel-heading tutorial-quiz-title"><span data-i18n="text.quiz">Quiz</span></div>
<div class="panel panel-default tutorial-question-container">
<div data-label="question_mc_add-1" class="tutorial-question panel-body">
<div id="question_mc_add-1-answer_container" class="shiny-html-output"></div>
<div id="question_mc_add-1-message_container" class="shiny-html-output"></div>
<div id="question_mc_add-1-action_button_container" class="shiny-html-output"></div>
<script>if (Tutorial.triggerMathJax) Tutorial.triggerMathJax()</script>
</div>
</div>
<hr />
</div>
</div>
<div id="section-optional-pareto-optimality-advanced"
class="section level2">
<h2><strong>Optional: Pareto-Optimality (Advanced)</strong></h2>
<p>Now we will calcualte Pareto-Optimality. We will use the concept of
the Pareto front to find the best trade-offs between two competing
objectives. We can learn which solutions are <em>optimal</em> (better in
at least one way, without being worse in another) and which solutions
are <em>dominated</em> (we could improve at least one objective without
making the other worse).</p>
<p>First we need to increase the model complexity a little by adding
management options. We add stocking density (sheep per hectare), tree
density (trees per hectare), proportion of tree area, and proportion of
sheep area (1 - proportion_tree_area). <em>Adjust these numbers to
reflect your best estimate.</em></p>
<div class="tutorial-exercise" data-label="management_variables"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># Define management variables per hectare
max_stocking_density <- 20 # Max sheep per ha before overgrazing effects occur
stocking_density <- runif(num_simulations, min = 5, max = max_stocking_density)
max_tree_density <- 500 # Maximum trees per hectare
tree_density <- runif(num_simulations, min = 50, max = max_tree_density)
# Define proportion of tree area vs. sheep area
proportion_tree_area <- runif(num_simulations, min = 0.2, max = 0.8)
proportion_sheep_area <- 1 - proportion_tree_area # Remaining area for sheep</code></pre>
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</div>
<div class="tutorial-exercise-support"
data-label="management_variables-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code># Define management variables per hectare
max_stocking_density <- 40 # Max sheep per ha before overgrazing effects occur
stocking_density <- runif(num_simulations, min = 5, max = max_stocking_density)
max_tree_density <- 200 # Maximum trees per hectare
tree_density <- runif(num_simulations, min = 50, max = max_tree_density)
# Define proportion of tree area vs. sheep area
proportion_tree_area <- runif(num_simulations, min = 0.2, max = 0.8)
proportion_sheep_area <- 1 - proportion_tree_area # Remaining area for sheep
total_profits_agroforestry <- apple_profits + (sheep_profits * proportion_sheep_area)
total_profits_treeless <- sheep_profits # No tree component in treeless system</code></pre>
</div>
<p>Adjust these biodiversity numbers to reflect your best estimate. For
example, depending on the ecosystem, each tree likely adds far fewer
species and each sheep likely has far less of a negative impact.</p>
<div class="tutorial-exercise" data-label="biodiversity_variables"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># Define species richness and grazing impact factors per hectare
species_richness_per_tree <- 0.05 # Each tree adds 0.05 species per ha
species_loss_per_sheep <- 0.02 # Each sheep reduces 0.02 species per ha
# Calculate biodiversity per ha
tree_species_richness <- tree_density * species_richness_per_tree
grazing_species_loss <- stocking_density * species_loss_per_sheep
# Compute biodiversity outcomes per ha
biodiversity_agroforestry <- tree_species_richness - grazing_species_loss
biodiversity_treeless <- -grazing_species_loss # No trees, only grazing effect
# Calculate difference (decision impact of trees)
# measures the financial difference when switching to trees
total_profits <- total_profits_agroforestry - total_profits_treeless
# Calculate difference (decision impact of trees on biodiversity)
# measures the net gain/loss in biodiversity from switching to trees
biodiversity_impact <- biodiversity_agroforestry - biodiversity_treeless
# Define a weighting factor for biodiversity impact (convert biodiversity into € equivalent)
biodiversity_value_per_species <- 500 # Assume each additional species is worth €500</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support"
data-label="biodiversity_variables-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code># Define species richness and grazing impact factors per hectare
species_richness_per_tree <- 0.01 # Each tree adds 0.001 species per ha
species_loss_per_sheep <- 0.001 # Each sheep reduces 0.001 species per ha
# Calculate biodiversity per ha
tree_species_richness <- tree_density * species_richness_per_tree
grazing_species_loss <- stocking_density * species_loss_per_sheep
# Compute biodiversity outcomes per ha
biodiversity_agroforestry <- tree_species_richness - grazing_species_loss
biodiversity_treeless <- -grazing_species_loss # No trees, only grazing effect
# Calculate difference (decision impact of trees)
# measures the financial difference when switching to trees
total_profits <- total_profits_agroforestry - total_profits_treeless
# Calculate difference (decision impact of trees on biodiversity)
# measures the net gain/loss in biodiversity from switching to trees
biodiversity_impact <- biodiversity_agroforestry - biodiversity_treeless
# Define a weighting factor for biodiversity impact (convert biodiversity into € equivalent)
biodiversity_value_per_species <- 500 # Assume each additional species is worth €500</code></pre>
</div>
<p>For the Pareto front plot we will first add the data together, store
results in dataframe and plot a scatter plot. Change the color of the
points in the scatter plot to green.</p>
<div class="tutorial-exercise" data-label="results_data"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code>results <- data.frame(
total_profits,
biodiversity_impact
)
# Scatter plot
plot(results$biodiversity_impact, results$total_profits,
col = "blue", pch = 19, xlab = "Biodiversity Impact (Species Change per ha)",
ylab = "Total Profits (€ per ha)", main = "Pareto Front: Agroforestry vs. Treeless System")</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support"
data-label="results_data-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code>results <- data.frame(
total_profits,
biodiversity_impact
)
# Scatter plot
plot(results$biodiversity_impact, results$total_profits,
col = "green", pch = 19, xlab = "Biodiversity Impact (Species Change per ha)",
ylab = "Total Profits (€ per ha)", main = "Pareto Front: Agroforestry vs. Treeless System")</code></pre>
</div>
<p>Now that we see what the scatter plot looks like what is your
impression?</p>
<div class="form-group shiny-input-container">
<label class="control-label" id="pareto_question-label" for="pareto_question">What do you expect the trade-offs to be between profit and biodiversity?</label>
<input id="pareto_question" type="text" class="shiny-input-text form-control" value="" placeholder="Write your prediction before running the analysis..."/>
</div>
<p>Now we can look for the non-dominated points where no other solution
is better in both biodiversity and total profits. A point (biodiversity,
profit) is Pareto-optimal if no other point has higher biodiversity and
higher profits. Change the color of the points to blue.</p>
<div class="tutorial-exercise" data-label="pareto_points"
data-completion="1" data-diagnostics="1" data-startover="1"
data-lines="0" data-pipe="|>">
<pre class="text"><code># Sort results by biodiversity impact
results <- results[order(results$biodiversity_impact, decreasing = TRUE),]
# Initialize Pareto front
pareto_front <- results[1, ] # Start with the first point
# Loop through sorted results and keep only non-dominated points
for (i in 2:nrow(results)) {
if (results$total_profits[i] > tail(pareto_front$total_profits, 1)) {
pareto_front <- rbind(pareto_front, results[i, ])
}
}
# Plot all points
plot(results$biodiversity_impact, results$total_profits,
col = "green", pch = 19, xlab = "Biodiversity Impact (Species Change per ha)",
ylab = "Total Profits (€ per ha)", main = "Pareto Front: Agroforestry vs. Treeless System")
# Highlight Pareto front points
points(pareto_front$biodiversity_impact, pareto_front$total_profits,
col = "red", pch = 19, type = "o", lwd = 2)</code></pre>
<script type="application/json" data-ui-opts="1">{"engine":"r","has_checker":false,"caption":"<span data-i18n=\"text.enginecap\" data-i18n-opts=\"{"engine":"R"}\">R Code<\/span>"}</script>
</div>
<div class="tutorial-exercise-support"
data-label="pareto_points-solution" data-completion="1"
data-diagnostics="1" data-startover="1" data-lines="0"
data-pipe="|>">
<pre class="text"><code># Sort results by biodiversity impact
results <- results[order(results$biodiversity_impact, decreasing = TRUE),]
# Initialize Pareto front
pareto_front <- results[1, ] # Start with the first point