---
title: "Calibrating trade-offs tutorial"
output:
  rmarkdown::html_vignette:
    toc: true
    fig_caption: true
fontsize: 11pt
documentclass: article
bibliography: references.bib
csl: reference-style.csl
vignette: >
  %\VignetteIndexEntry{Calibrating trade-offs tutorial}
  %\VignetteEngine{knitr::rmarkdown_notangle}
---

```{r, include = FALSE}
# define dummy variables so that vignette passes package checks
prelim_penalty <- rep(NA_real_, 100)
threshold <- rep(NA_real_, 100)
topsis_results <- data.frame(
  alt.row = seq_len(3), score = runif(3), rank = seq_len(3)
)
```

```{r, include = FALSE}
# define variables for vignette figures and code execution
h <- 3.5
w <- 3.5
is_check <-
  ("CheckExEnv" %in% search()) ||
  any(c("_R_CHECK_TIMINGS_", "_R_CHECK_LICENSE_") %in% names(Sys.getenv())) ||
  !identical(Sys.getenv("MY_UNIVERSE"), "")
knitr::opts_chunk$set(
  fig.align = "center",
  eval = !is_check, purl = !is_check
)
```

```{r, include = FALSE}
# set up print method
print <- function(x, ...) {
  if (inherits(x, "ConservationProblem")) {
    prioritizr::knit_print.ConservationProblem(x)
  } else if (inherits(x, "OptimizationProblem")) {
    prioritizr::knit_print.OptimizationProblem(x)
  } else {
    base::print(x)
  }
}
```

## Introduction

Systematic conservation planning requires making trade-offs [@r4; @r36]. Since different criteria may conflict with one another -- or not align perfectly -- prioritizations need to make trade-offs between different criteria [@r37]. Although some criteria can easily be accounted for by using locked constraints or representation targets [e.g., @r40; @r39], this is not always the case [e.g., @r38]. For example, prioritizations often need to balance overall cost with the overall level spatial fragmentation among reserves [@r41; @r42]. Additionally, prioritizations often need to balance the overall level of connectivity among reserves against other criteria [@r43]. Since the best trade-off depends on a range of factors -- such as available budgets, species' connectivity requirements, and management capacity -- finding the best balance can be challenging.

The _prioritizr R_ package provides multi-objective optimization methods to help identify the best trade-offs between different criteria. To achieve this, a conservation planning problem can be formulated with a primary objective (e.g., `add_min_set_objective()`) and penalties (e.g., `add_boundary_penalties()`) that relate to such criteria. When building the problem, the nature of the trade-offs can be specified using certain parameters (e.g., the `penalty` parameter of the `add_boundary_penalties()` function). To identify a prioritization that finds the best balance between different criteria, the trade-off parameters can be tuned using a calibration analysis. These analyses -- in the context of systematic conservation planning -- typically involve generating a set of candidate prioritizations based on different parameters, measuring their performance according to each of the criteria, and then selecting a prioritization (or set of prioritizations) based on how well they achieve the criteria [@r41; @r42; @r43]. For example, the _Marxan_ decision support tool has a range of parameters (e.g., species penalty factors, boundary length modifier) that are calibrated to balance cost, species' representation, and spatial fragmentation [@r45].

The aim of this tutorial is to provide guidance on calibrating trade-offs when using the _prioritizr R_ package. Here we will explore a couple of different approaches for generating candidate prioritizations, and methods for finding the best balance between different criteria. Specifically, we will try to generate prioritizations that strike the best balance between total cost and spatial fragmentation (measured as total boundary length). As such, the code used in this vignette will be directly applicable when performing a boundary length calibration analysis.

## Data

Let's load the packages and dataset used in this tutorial. Since this tutorial uses the _prioritizrdata R_ package along with several other _R_ packages (see below), please ensure that they are all installed. This particular dataset comprises two objects: `tas_pu` and `tas_features`. Although we will briefly describe this dataset below, please refer `?prioritizrdata::tas_data` for further details.

```{r, message = FALSE}
# load packages
library(prioritizrdata)
library(prioritizr)
library(sf)
library(terra)
library(dplyr)
library(tibble)
library(ggplot2)
library(topsis)
library(withr)

# load planning unit data
tas_pu <- get_tas_pu()
print(tas_pu)

# load feature data
tas_features <- get_tas_features()
print(tas_features)
```

The `tas_pu` object contains planning units represented as spatial polygons (i.e., converted to a `sf::st_sf()` object). This object has three columns that denote the following information for each planning unit: a unique identifier (`id`), unimproved land value (`cost`), and current conservation status (`locked_in`). Specifically, the conservation status column indicates if at least half the area planning unit is covered by existing protected areas (denoted by a value of 1) or not (denoted by a value of zero).

```{r, fig.width = w, fig.height = h}
# plot map of planning unit costs
plot(tas_pu[, "cost"])

# plot map of planning unit statuses
plot(tas_pu[, "locked_in"])
```

The `tas_features` object describes the spatial distribution of different vegetation communities (using presence/absence data). We will use the vegetation communities as the biodiversity features for the prioritization.

```{r, fig.width = 4.5, fig.height = 4.5}
# plot map of the first four vegetation classes
plot(tas_features[[1:4]])
```

We can use this dataset to generate a prioritization. Specifically, we will use the minimum set objective so that the optimization process minimizes total cost. We will add representation targets to ensure that prioritizations cover 17% of each vegetation community. Additionally, we will add constraints to ensure that planning units covered by existing protected areas are selected (i.e., locked in). Finally, we will specify that the conservation planning exercise involves binary decisions (i.e., selecting or not selecting planning units for protected area establishment).

```{r}
# define a problem
p0 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions()

# print problem
print(p0)

# solve problem
s0 <- solve(p0)

# print result
print(s0)
```

```{r, fig.width = w, fig.height = h, results = "hide"}
# create column for making a map of the prioritization
s0$map_1 <- case_when(
  s0$locked_in > 0.5 ~ "locked in",
  s0$solution_1 > 0.5 ~ "priority",
  TRUE ~ "other"
)

# plot map of prioritization
plot(
  s0[, "map_1"], pal = c("purple", "grey90", "darkgreen"),
  main = NULL, key.pos = 1
)
```

We can see that the priority areas identified by the prioritization are scattered across the study area (shown in green). Indeed, none of the priority areas are connected to existing protected areas (shown in purple), and very of them are connected with other priority areas. As such, the prioritization has a high level of spatial fragmentation. If it is important to avoid such levels of spatial fragmentation, then we will need to explicitly account for spatial fragmentation in the optimization process.

## Preliminary processing

We need to conduct some preliminary processing procedures to prepare the data for subsequent analysis. This is important to help make it easier to find suitable trade-off parameters, and avoid numerical scaling issues that can result in overly long run times (see `presolve_check()` for further information). These processing steps are akin to data scaling (or normalization) procedures that are applied in statistical analysis to improve model convergence.

The first processing procedure involves setting the cost values for all locked in planning units to zero. This is so that the total cost estimates of the prioritization reflects the total cost of establishing new protected areas -- not just total land value. In other words, we want the total cost estimate for a prioritization to reflect the cost of implementing conservation actions. **This procedure is especially important when using the hierarchical approach described below, so that cost thresholds are based on percentage increases in the cost of establishing new protected areas.**

```{r, fig.width = w, fig.height = h}
# set costs for planning units covered by existing protected areas to zero
tas_pu$cost[tas_pu$locked_in > 0.5] <- 0

# plot map of planning unit costs
plot(tas_pu[, "cost"])
```

The second procedure involves pre-computing the boundary length data and manually re-scaling the boundary length values. This procedure is important because boundary length values are often very high, which can cause numerical issues that result in excessive run times (see `presolve_check()` for further details).

```{r}
# generate boundary length data for the planning units
tas_bd <- boundary_matrix(tas_pu)

# manually re-scale the boundary length values
tas_bd <- rescale_matrix(tas_bd)
```

After applying these procedures, our data is ready for subsequent analysis.

## Generating candidate prioritizations

Here we will start the calibration analysis by generating a set of candidate prioritizations. Specifically, these prioritizations will be generated using different parameters to specify different trade-offs between the different criteria. Since this tutorial involves navigating trade-offs between the overall cost of a prioritization and the level of spatial fragmentation associated with a prioritization (as measured by total boundary length), we will generate prioritizations using different parameters related to these criteria. We will examine two approaches for generating candidate prioritizations based on multi-objective optimization procedures.
**Although we'll be examining both approaches in this tutorial, you would normally only use one of these approaches when conducting your own analysis**

### Blended approach

The blended approach for multi-objective optimization involves combining separate criteria (e.g., total cost and total boundary length) into a single joint criterion. To achieve this, a trade-off (or scaling) parameter is used to specify the relative importance of each criterion. This approach is the default approach provided by the _prioritizr R_ package. Specifically, each of the functions for adding a penalty to a problem formulation (e.g., `add_boundary_penalties()`) contains a parameter to control the relative importance of the penalties (i.e., the `penalty` parameter). For example, when using the `add_boundary_penalties()` function, setting a high `penalty` value will indicate that it is important to reduce the overall exposed boundary (perimeter) of the prioritization.

The main challenge with the blended approach is identifying a range of suitable `penalty` values to generate candidate prioritizations. If we set a `penalty` value that is too low, then the penalties will have no effect (e.g., boundary length penalties would have no effect on the prioritization). If we set a `penalty` value too high, then the prioritization will effectively ignore the primary objective. In such cases, the prioritization will be overly spatially clustered -- because the planning unit cost values have no effect --- and contain a single reserve. Thus we need to find a suitable range of `penalty` values before we can generate a set of candidate prioritizations.

We can find a suitable range of `penalty` values by generating a set of preliminary prioritizations. These preliminary prioritizations will be based on different `penalty` values -- similar to the process for generating the candidate prioritizations -- but solved using customized settings that sacrifice optimality for fast run times (see below for details). This is especially important because specifying a `penalty` value that is too high will cause the optimization process to take a very long time to generate a solutions (due to the numerical scaling issues mentioned previously). To find a suitable range of `penalty` values, we need to identify an upper limit for the `penalty` value (i.e., the highest `penalty` value that result in a prioritization containing a single reserve). Let's create some preliminary `penalty` to identify this upper limit. **Please note that you might need to adjust the `prelim_upper` value to find the upper limit when analyzing different datasets.**

```{r}
# define a range of different penalty values
## note that we use a power scale to avoid focusing on very high penalty values
prelim_lower <- -5  # change this for your own data
prelim_upper <- 1.75 # change this for your own data
prelim_penalty <- round(10^seq(prelim_lower, prelim_upper, length.out = 9), 5)

# print penalty values
print(prelim_penalty)
```

Next, let's use the preliminary `penalty` values to generate preliminary prioritizations. As mentioned earlier, we will generate these preliminary prioritizations using customized settings to reduce runtime. Specifically, we will set a time limit of 10 minutes per run, and relax the optimality gap to 20%. Although we would not normally use such settings -- because the resulting prioritizations are not guaranteed to be near-optimal (the default gap is 10%) -- this is fine because our goal here is to tune the preliminary `penalty` values. Indeed, none of these preliminary prioritizations will be considered as candidate prioritizations. **Please note that you might need to set a higher time limit, or relax the optimality  gap even further (e.g., 40%) when analyzing larger datasets.**

```{r, results = "hide"}
# define a problem without boundary penalties
p0 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions()

# generate preliminary prioritizations based on each penalty
## note that we specify a relaxed gap and time limit for the solver
prelim_blended_results <- lapply(prelim_penalty, function(x) {
  s <-
    p0 %>%
    add_boundary_penalties(penalty = x, data = tas_bd) %>%
    add_default_solver(gap = 0.2, time_limit = 10 * 60) %>%
    solve()
  s <- data.frame(s = s$solution_1)
  names(s) <- with_options(list(scipen = 30), paste0("penalty_", x))
  s
})

# format results as a single spatial object
prelim_blended_results <- cbind(
  tas_pu, do.call(bind_cols, prelim_blended_results)
)

# preview results
print(prelim_blended_results)
```

After generating the preliminary prioritizations, let's create some maps to visualize them. In particular, we want to understand how different penalty values influence the spatial fragmentation of the prioritizations.

```{r, fig.width = 7, fig.height = 5.0}
# plot maps of prioritizations
plot(
  x =
    prelim_blended_results %>%
    dplyr::select(starts_with("penalty_")) %>%
    mutate_if(is.numeric, function(x) {
      case_when(
        prelim_blended_results$locked_in > 0.5 ~ "locked in",
        x > 0.5 ~ "priority",
        TRUE ~ "other"
      )
    }),
  pal = c("purple", "grey90", "darkgreen")
)
```

```{r "verify-correct-best-guess", echo = FALSE}
# this code is used to ensure consistency between the results and the
# text in the vignette. Specifically, the text is written assuming that
# the 7th solution in the preliminary set of prioritizations is the
# prioritization that (1) was generated with the lowest penalty value and
# (2) has all planning units selected

# preliminary calculations
result_nms <- names(prelim_blended_results)
result_nms <- result_nms[startsWith(result_nms, "penalty_")]
result_mtx <- sf::st_drop_geometry(prelim_blended_results)[, result_nms]
result_n_selected <- apply(result_mtx, 2, sum, na.rm = TRUE)

# compute best guess
best_guess_idx <- min(which(result_n_selected == max(result_n_selected)))

# run checks
invisible(
    assertthat::assert_that(
    best_guess_idx == 8,
    msg = paste(
      "inconsistency between results and text,",
      "index of best guess index should be",
      best_guess_idx
    )
  )
)
```

We can see that as the `penalty` value used to generate the prioritizations increases, the spatial fragmentation of the prioritizations decreases. In particular, we can see that a `penalty` value of `r prelim_penalty[8]` results in a single reserve -- meaning this is our best guess of the upper limit. Using this `penalty` value as an upper limit, we will now generate a second series of prioritizations that will be the candidate prioritizations. Critically, these candidate prioritizations will not be generated using with time limit and be generated using a more suitable gap (i.e., default gap of 10%).

```{r, fig.width = 7, fig.height = 7.5, results = "hide"}
# define a new set of penalty values
penalty <- round(10^seq(-5, log10(prelim_penalty[8]), length.out = 9), 5)

# generate prioritizations based on each penalty
blended_results <- lapply(penalty, function(x) {
  ## generate solution
  s <-
    p0 %>%
    add_boundary_penalties(penalty = x, data = tas_bd) %>%
    solve()
  ## return data frame with solution
  s <- data.frame(s = s$solution_1)
  names(s) <- with_options(list(scipen = 30), paste0("penalty_", x))
  s
})

# format results as a single spatial object
blended_results <- cbind(tas_pu, do.call(bind_cols, blended_results))

# plot maps of prioritizations
plot(
  x =
    blended_results %>%
    dplyr::select(starts_with("penalty_")) %>%
    mutate_if(is.numeric, function(x) {
      case_when(
        blended_results$locked_in > 0.5 ~ "locked in",
        x > 0.5 ~ "priority",
        TRUE ~ "other"
      )
    }),
  pal = c("purple", "grey90", "darkgreen")
)
```

We now have a set of candidate prioritizations generated using the blended approach. The main advantages of this approach is that it is similar calibration analyses used by other decision support tools for conservation (i.e., _Marxan_) and it is relatively straightforward to implement. However, this approach also has a key disadvantage. Because the `penalty` parameter is a unitless trade-off parameter -- meaning that we can't leverage existing knowledge to specify a suitable range of `penalty` values -- we first have to conduct a preliminary analysis to identify an upper limit. Although finding an upper limit was fairly simple for the example dataset, it can be difficult to find for more realistic data. In the next section, we will show how to generate a set of candidate prioritizations using the hierarchical approach -- which does not have this disadvantage.

### Hierarchical approach

The hierarchical approach for multi-objective optimization involves generating a series of incremental prioritizations -- using a different objective at each increment to refine the previous solution -- until the final solution achieves all of the objectives. The advantage with this approach is that we can specify trade-off parameters for each objective based on a percentage from optimality. This means that we can leverage our own knowledge -- or that of decision maker -- to generate a range of suitable trade-off parameters. As such, this approach does not require us to generate a series of preliminary prioritizations.

This approach is slightly more complicated to implement within the _prioritizr R_ package then the blended approach. To start off, we generate an initial prioritization based on a problem formulation that does not consider any penalties. Critically, we will generate this prioritization by solving the problem to optimality (using the `gap` parameter of the `add_default_solver()` function).

```{r, fig.width = w, fig.height = h, results = "hide"}
# define a problem without boundary penalties
p1 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions() %>%
  add_default_solver(gap = 0)

# solve problem
s1 <- solve(p1)

# add column for making a map of the prioritization
s1$map_1 <- case_when(
  s1$locked_in > 0.5 ~ "locked in",
  s1$solution_1 > 0.5 ~ "priority",
  TRUE ~ "other"
)

# plot map of prioritization
plot(
  s1[, "map_1"], pal = c("purple", "grey90", "darkgreen"),
  main = NULL, key.pos = 1
)
```

Next, we will calculate the total cost of the initial prioritization.

```{r, fig.width = w, fig.height = h}
# calculate cost
s1_cost <- eval_cost_summary(p1, s1[, "solution_1"])$cost

# print cost
print(s1_cost)
```

Now we will calculate a series of cost thresholds. These cost thresholds will be calculated by inflating the cost of the initial prioritization by a range of percentage values. Since these values are percentages -- and not unitless values unlike those used in the blended approach -- we can use domain knowledge to specify a suitable range of cost thresholds. For this tutorial, let's assume that it would be impractical -- per our domain knowledge -- to expend more than four times the total cost of the initial prioritization to reduce spatial fragmentation.

```{r}
# calculate cost threshold values
threshold <- s1_cost + (s1_cost * seq(1e-5, 4, length.out = 9))
threshold <- ceiling(threshold)

# print cost thresholds
print(threshold)
```

After generating the cost thresholds, we can use them to generate prioritizations. Specifically, we will generate prioritizations that aim to minimize total boundary length as much as possible -- ignoring the total cost of the prioritizations -- whilst ensuring that the total cost of the prioritization does not exceed a given cost threshold and the other considerations (e.g., locked in constraints). To achieve this, we create a new column in the `tas_pu` object that contains only zero values (called `zeros`) and use this new column to specify the cost data for the prioritizations.
**Although we normally recommend against cost data that contain zero values -- because planning units with zero costs are often selected in prioritizations even if they are not needed -- here we use zero cost values so that the prioritization will focus exclusively on spatial fragmentation.** Additionally, when it comes to generating the prioritization, we will add linear constraints to ensure that the total cost of the prioritization does not exceed a given cost threshold (using the `add_linear_constraints()` function).

```{r, fig.width = 7, fig.height = 5.0, results = "hide"}
# add a column with zeros
tas_pu$zeros <- 0

# define a problem with zero cost values and boundary penalties
## note that because all the costs are all zero, it doesn't actually
## matter what penalty value is used (as long as the value is > 0)
## and so we just use a value of 1
p2 <-
  problem(tas_pu, tas_features, cost_column = "zeros") %>%
  add_min_set_objective() %>%
  add_boundary_penalties(penalty = 1, data = tas_bd) %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions()

# generate prioritizations based on each cost threshold
## note that the prioritizations are solved to within 10% of optimality
## (the default gap) because the gap is not specified
hierarchical_results <- lapply(threshold, function(x) {
  ## generate solution by adding a constraint based on the threshold and
  ## using the "real" cost values (i.e., not zeros)
  s <-
    p2 %>%
    add_linear_constraints(threshold = x, sense = "<=", data = "cost") %>%
    solve()
  ## return data frame with solution
  s <- data.frame(s = s$solution_1)
  names(s) <- paste0("threshold_", x)
  s
})

# format results as a single spatial object
hierarchical_results <- cbind(tas_pu, do.call(bind_cols, hierarchical_results))

# plot maps of prioritizations
plot(
  x =
    hierarchical_results %>%
    dplyr::select(starts_with("threshold_")) %>%
    mutate_if(is.numeric, function(x) {
      case_when(
        hierarchical_results$locked_in > 0.5 ~ "locked in",
        x > 0.5 ~ "priority",
        TRUE ~ "other"
      )
    }),
  pal = c("purple", "grey90", "darkgreen")
)
```

We now have a set of candidate prioritizations generated using the hierarchical approach. This approach can be much faster than the blended approach because it does not require generating a set of prioritizations to identify an upper limit for the `penalty` trade-off parameter. After generating a set of candidate prioritizations, we can then calculate performance metrics to compare the prioritizations.

## Calculating performance metrics

Here we will calculate performance metrics to compare the prioritizations. Since we aim to navigate trade-offs between the total cost of a prioritization and the overall level of spatial fragmentation associated with a prioritization (as measured by total boundary length), we will calculate metrics to assess these criteria. Although we generated two sets of candidate prioritizations in the previous section; for brevity, here we will consider the candidate prioritizations generated using the hierarchical approach. **Please note that you could also apply the following procedures to candidate prioritizations generated using the blended approach.**

```{r}
# calculate metrics for prioritizations
## note that we use p0 and not p1 so that cost calculations are based
## on the cost values and not zeros
hierarchical_metrics <- lapply(
  grep("threshold_", names(hierarchical_results)), function(x) {
    x <- hierarchical_results[, x]
    data.frame(
      total_cost = eval_cost_summary(p0, x)$cost,
      total_boundary_length = eval_boundary_summary(p0, x)$boundary
    )
  }
)
hierarchical_metrics <- do.call(bind_rows, hierarchical_metrics)
hierarchical_metrics$threshold <- threshold
hierarchical_metrics <- as_tibble(hierarchical_metrics)

# preview metrics
print(hierarchical_metrics)
```

After calculating the metrics, let's we can use them to help select a prioritization.

## Selecting a prioritization

Now we need to decide on which candidate prioritization achieves the best trade-off. There are a range of qualitative and quantitative methods that are available to select a candidate prioritization [@r45]. Here we will consider three different methods. Since some of these methods use a set of candidate prioritizations, we will use the candidate prioritizations using the hierarchical approach for these methods. To keep track of the prioritizations selected by different methods, let's create a `results_data` table.

```{r}
# create data for plotting
result_data <-
  hierarchical_metrics %>%
  ## rename threshold column to value column
  rename(value = "threshold") %>%
  ## add column with column names that contain candidate prioritizations
  mutate(name = grep(
    "threshold_", names(hierarchical_results), value = TRUE, fixed = TRUE
  )) %>%
  ## add column with labels for plotting
  mutate(label = paste("Threshold =", value)) %>%
  ## add column to keep track prioritizations selected by different methods
  mutate(method = "none")

# print table
print(result_data)
```

Next, let's examine some different methods for selecting prioritizations.

### Visual method

One qualitative method involves plotting the relationship between the different criteria, and using the plot to visually select a candidate prioritization. This visual method is often used to help calibrate trade-offs among prioritizations generated using the _Marxan_ decision support tool [e.g., @r41; @r42]. So, let's create a plot to select a prioritization.

```{r, fig.width = 7, fig.height = 5.0}
# create plot to visualize trade-offs and show selected candidate prioritization
result_plot <-
  ggplot(
    data = result_data,
    aes(x = total_boundary_length, y = total_cost, label = label)
  ) +
  geom_line() +
  geom_point(size = 3) +
  geom_text(hjust = -0.15) +
  scale_color_manual(
    values = c("visual" = "blue", "not selected" ="black")
  ) +
  xlab("Total boundary length of prioritization") +
  ylab("Total cost of prioritization") +
  scale_x_continuous(expand = expansion(mult = c(0.05, 0.4))) +
  theme(legend.title = element_blank())

# render plot
print(result_plot)
```

We can see that there is a clear relationship between total cost and total boundary length. It would seem that in order to achieve a lower total boundary length -- and thus lower spatial fragmentation -- the prioritization must have a greater cost. Although we might expect the results to show a smoother curve -- in other words, only Pareto dominant solutions -- this result is expected because we generated candidate prioritizations using the default optimality gap of 10%. Typically, the visual method involves selecting a prioritization near the elbow of the plot. So, let's select the prioritization generated using a `threshold` value of `r threshold[3]`. To keep track of the prioritizations selected based on different methods, let's create a `method` column in the `result_data` table.

```{r}
# specify prioritization selected by visual method
result_data$method[3] <- "visual"
```

Next, let's consider a quantitative approach.

### TOPSIS method

Multiple-criteria decision analysis is a discipline that uses analytical methods to evaluate trade-offs between multiple criteria [MCDA; reviewed in @r44]. Although this discipline contains many different methods, here we will use the the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method [@r46]. This method requires (i) data describing the performance of each prioritization according to the different criteria, (ii) weights to encode the relative importance of each criteria, and (iii) details on whether each criteria should ideally be minimized or maximized. Let's run the analysis, assuming that we want equal weighting for total cost and total boundary length.

```{r}
# calculate TOPSIS scores
topsis_results <- topsis(
  decision =
    hierarchical_metrics %>%
    dplyr::select(total_cost, total_boundary_length) %>%
    as.matrix(),
  weights = c(1, 1),
  impacts = c("-", "-")
)

# print results
print(topsis_results)
```

The candidate prioritization with the greatest TOPSIS score is considered to represent the best trade-off between total cost and total boundary length. So, based on this method, we would select the prioritization generated using a `threshold` value of `r threshold[which.max(topsis_results$score)]`. Let's update the `result_data` with this information.

```{r}
# add column indicating prioritization selected by TOPSIS method
result_data$method[which.max(topsis_results$score)] <- "TOPSIS"
```

Next, let's consider another quantitative method.

### Cohon *et al.* (1979) method

This method is based on an algorithm that was originally developed by Cohon *et al.* [@r49], and was later adapted for use in systematic conservation planning [@r50]. Specifically, it involves generating two ideal prioritizations -- with each prioritization representing the ideal prioritization for each criteria -- and then using the performance metrics calculated for these prioritizations to automatically derive a trade-off `penalty` value [@r45; @r49]. Thus, unlike the two other methods, this method does not require a set of candidate prioritizations. **As such, this method can be used to find a prioritization that meets multiple criteria in a much shorter period of time than the other methods.** To implement this method, we first need to generate the ideal prioritizations (note that we specify an gap of 0% to ensure optimality).

```{r, results = "hide"}
# generate ideal prioritization based on cost criteria
## note that this is simply the same as the s1 prioritization we generated
## for the hierarchical approach
p3 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions() %>%
  add_default_solver(gap = 0)

# solve problem
s3 <- solve(p3)

# generate ideal prioritization based on spatial fragmentation criteria
## note that any non-zero penalty value would here,
## so we just use a penalty  of 1
p4 <-
  problem(tas_pu, tas_features, cost_column = "zeros") %>%
  add_min_set_objective() %>%
  add_boundary_penalties(penalty = 1, data = tas_bd) %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions() %>%
  add_default_solver(gap = 0)

# solve problem
s4 <- solve(p4)
```

Next, let's calculate the performance metrics for these prioritizations.

```{r}
# generate problem formulation with costs and boundary penalties for
# calculating performance metrics
p5 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_boundary_penalties(penalty = 1, data = tas_bd) %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions()

# calculate performance metrics for ideal cost prioritization
s3_metrics <- tibble(
  total_cost = eval_cost_summary(p5, s3[, "solution_1"])$cost,
  total_boundary_length =
    eval_boundary_summary(p5, s3[, "solution_1"])$boundary
)

# calculate performance metrics for ideal boundary length prioritization
s4_metrics <- tibble(
  total_cost = eval_cost_summary(p5, s4[, "solution_1"])$cost,
  total_boundary_length =
    eval_boundary_summary(p5, s4[, "solution_1"])$boundary
)
```

After calculating these performance metrics, we can use them to automatically calculate a `penalty` value.

```{r}
# calculate penalty value based on Cohon et al. 1979
cohon_penalty <- abs(
  (s3_metrics$total_cost - s4_metrics$total_cost) /
  (s3_metrics$total_boundary_length - s4_metrics$total_boundary_length)
)

# round to 5 decimal places to avoid numerical issues during optimization
cohon_penalty <- round(cohon_penalty, 5)

# print penalty value
print(cohon_penalty)
```

Now that we have calculated a `penalty` value using this method, we can use it to generate a prioritization.

```{r, results = "hide"}
# generate prioritization using penalty value calculated using Cohon et al. 1979
p6 <-
  problem(tas_pu, tas_features, cost_column = "cost") %>%
  add_min_set_objective() %>%
  add_boundary_penalties(penalty = cohon_penalty, data = tas_bd) %>%
  add_relative_targets(0.17) %>%
  add_locked_in_constraints("locked_in") %>%
  add_binary_decisions()

# solve problem
s6 <- solve(p6)
```

Let's update the `results_data` table with results about the prioritization.

```{r}
# add new row with data for prioritization generated following Cohon et al. 1979
result_data <- bind_rows(
  result_data,
  tibble(
    total_cost = eval_cost_summary(p6, s6[, "solution_1"])$cost,
    total_boundary_length =
      eval_boundary_summary(p6, s6[, "solution_1"])$boundary,
    value = cohon_penalty,
    name = paste0("penalty_", cohon_penalty),
    label = paste0("Penalty = ",  cohon_penalty),
    method = "Cohon"
  )
)
```

Next, let's compare the prioritizations selected by different methods.

### Method comparison

Let's create a plot to visualize the results from the different methods.

```{r, fig.width = 7, fig.height = 5.0}
# create plot to visualize trade-offs and show selected candidate prioritization
result_plot <-
  ggplot(
    data =
      result_data %>%
      mutate(vjust = if_else(method == "Cohon", -1, 0.5)),
    aes(x = total_boundary_length, y = total_cost, label = label)
  ) +
  geom_line() +
  geom_point(aes(color = method), size = 3) +
  geom_text(aes(vjust = vjust, color = method), hjust = -0.1) +
  scale_color_manual(
    name = "Method",
    values = c(
      "visual" = "#984ea3",
      "none" = "#000000",
      "TOPSIS" = "#e41a1c",
      "Cohon" = "#377eb8"
    )
  ) +
  xlab("Total boundary length of prioritization") +
  ylab("Total cost of prioritization") +
  scale_x_continuous(expand = expansion(mult = c(0.05, 0.4)))

# render plot
print(result_plot)
```

We can see that the different methods selected different prioritizations. To further compare the results from the different methods, let's create some maps showing the selected prioritizations.

```{r, fig.width = 7.0, fig.height = h}
# extract column names for creating the prioritizations
visual_name <- result_data$name[[which(result_data$method == "visual")]]
topsis_name <- result_data$name[[which(result_data$method == "TOPSIS")]]

# create object with selected prioritizations
solutions  <- bind_cols(
  tas_pu,
  hierarchical_results %>%
    st_drop_geometry() %>%
    dplyr::select(all_of(c(visual_name, topsis_name))) %>%
    setNames(c("Visual", "TOPSIS")),
  s6 %>%
    st_drop_geometry() %>%
    dplyr::select(solution_1) %>%
    rename(Cohon = "solution_1")
)

# plot maps of selected prioritizations
plot(
  x =
    solutions %>%
    dplyr::select(Visual, TOPSIS, Cohon) %>%
    mutate_if(is.numeric, function(x) {
      case_when(
        hierarchical_results$locked_in > 0.5 ~ "locked in",
        x > 0.5 ~ "priority",
        TRUE ~ "other"
      )
    }),
  pal = c("purple", "grey90", "darkgreen")
)
```

How do we determine which one is best? This is difficult to say. Ideally, additional information could be used to help select a prioritization, such as knowledge on available resources, species' connectivity requirements, and impacts of neighboring land use. However, from a practical perspective, prioritizations generated for academic contexts might find the quantitative approaches more useful because they have greater transparency and reproducibility. Ultimately, all of these methods are designed to support decision making. This means that they are intended to assist the decision making process, not serve as a replacement.

## Conclusion

Hopefully, this vignette has provided a useful introduction for resolving trade-offs in prioritizations. Although we only explored trade-offs between total cost and spatial fragmentation in this tutorial, this analysis could be adapted to explore trade-offs between a wide range of different criteria. For instance, instead of considering total cost as the primary objective, future analyses could explore trade-offs with feature representation (using the `add_min_shortfall_objective()` function). Additionally, instead of spatial fragmentation, future analyses could explore trade-offs that directly relate to connectivity (using the `add_connectivity_penalties()` function) or specific variables of interest -- such as ecosystem intactness or inverse human footprint index [@r47; @r48] -- to inform decision making (using the `add_linear_penalties()` function). Furthermore, after identifying the best `penalty` or `threshold` values to strike a balance between multiple criteria, you could generate a portfolio of prioritizations (e.g., using the `add_gap_portfolio_function()`) to find multiple options for achieving a similar balance. This might be helpful when you need to generate a set of prioritizations that have comparable performance -- in terms of how well they achieve different criteria -- but select different planning units.

## References
