Calibrating trade-offs tutorial


Systematic conservation planning requires making trade-offs (Margules & Pressey 2000; Vane-Wright et al. 1991). Since different criteria may conflict with one another – or not align perfectly – prioritizations need to make trade-offs between different criteria (Klein et al. 2013). Although some criteria can easily be accounted for by using locked constraints or representation targets (e.g., Dorji et al. 2020; Hermoso et al. 2018), this is not always the case (e.g., Beger et al. 2010). For example, prioritizations often need to balance overall cost with the overall level spatial fragmentation among reserves (Hermoso et al. 2011; Stewart & Possingham 2005). Additionally, prioritizations often need to balance the overall level of connectivity among reserves against other criteria (Hermoso et al. 2012). 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 (Hermoso et al. 2011, 2012; Stewart & Possingham 2005). 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 (Ardron et al. 2010).

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.


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.

# load packages

# load planning unit data
tas_pu <- get_tas_pu()
## Simple feature collection with 1130 features and 4 fields
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: 298809.6 ymin: 5167775 xmax: 613818.8 ymax: 5502544
## Projected CRS: WGS 84 / UTM zone 55S
## # A tibble: 1,130 × 5
##       id  cost locked_in locked_out                                         geom
##    <int> <dbl> <lgl>     <lgl>                                <MULTIPOLYGON [m]>
##  1     1 60.2  FALSE     TRUE       (((328497 5497704, 326783.8 5500050, 326775…
##  2     2 19.9  FALSE     FALSE      (((307121.6 5490487, 305344.4 5492917, 3053…
##  3     3 59.7  FALSE     TRUE       (((321726.1 5492382, 320111 5494593, 320127…
##  4     4 32.4  FALSE     FALSE      (((304314.5 5494324, 304342.2 5494287, 3043…
##  5     5 26.2  FALSE     FALSE      (((314958.5 5487057, 312336 5490646, 312339…
##  6     6 51.3  FALSE     FALSE      (((327904.3 5491218, 326594.6 5493012, 3284…
##  7     7 32.3  FALSE     FALSE      (((308194.1 5481729, 306601.2 5483908, 3066…
##  8     8 38.4  FALSE     FALSE      (((322792.7 5483624, 319965.3 5487497, 3199…
##  9     9  3.55 FALSE     FALSE      (((334896.6 5490731, 335610.4 5492490, 3357…
## 10    10  1.83 FALSE     FALSE      (((356377.1 5487952, 353903.1 5487635, 3538…
## # ℹ 1,120 more rows
# load feature data
tas_features <- get_tas_features()
## class       : SpatRaster 
## dimensions  : 398, 359, 33  (nrow, ncol, nlyr)
## resolution  : 1000, 1000  (x, y)
## extent      : 288801.7, 647801.7, 5142976, 5540976  (xmin, xmax, ymin, ymax)
## coord. ref. : WGS 84 / UTM zone 55S (EPSG:32755) 
## source      : tas_features.tif 
## names       : Banks~lands, Bould~marks, Calli~lands, Cool ~orest, Eucal~hyll), Eucal~torey, ... 
## min values  :           0,           0,           0,           0,           0,           0, ... 
## max values  :           1,           1,           1,           1,           1,           1, ...

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).

# 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.

# plot map of the first four vegetation classes

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).

# 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") %>%

# print problem
## A conservation problem (<ConservationProblem>)
## ├•data
## │├•features:    "Banksia woodlands" , … (33 total)
## │└•planning units:
## │ ├•data:       <sftbl_dftbldata.frame> (1130 total)
## │ ├•costs:      continuous values (between 0.1925 and 61.9273)
## │ ├•extent:     298809.5764, 5167774.5993, 613818.7743, 5502543.7119 (xmin, ymin, xmax, ymax)
## │ └•CRS:        WGS 84 / UTM zone 55S (projected)
## ├•formulation
## │├•objective:   minimum set objective
## │├•penalties:   none specified
## │├•targets:     relative targets (between 0.17 and 0.17)
## │├•constraints:
## ││└•1:          locked in constraints (257 planning units)
## │└•decisions:   binary decision
## └•optimization
##  ├•portfolio:   shuffle portfolio (`number_solutions` = 1, …)
##  └•solver:      gurobi solver (`gap` = 0.1, `time_limit` = 2147483647, `first_feasible` = FALSE, …)
## # ℹ Use `summary(...)` to see complete formulation.
# solve problem
s0 <- solve(p0)
## Gurobi Optimizer version 10.0.1 build v10.0.1rc0 (linux64)
## CPU model: AMD Ryzen Threadripper 3960X 24-Core Processor, instruction set [SSE2|AVX|AVX2]
## Thread count: 24 physical cores, 48 logical processors, using up to 1 threads
## Optimize a model with 33 rows, 1130 columns and 8157 nonzeros
## Model fingerprint: 0xd205b612
## Variable types: 0 continuous, 1130 integer (1130 binary)
## Coefficient statistics:
##   Matrix range     [2e-06, 6e+01]
##   Objective range  [2e-01, 6e+01]
##   Bounds range     [1e+00, 1e+00]
##   RHS range        [2e-01, 2e+03]
## Found heuristic solution: objective 10337.581214
## Presolve removed 19 rows and 354 columns
## Presolve time: 0.01s
## Presolved: 14 rows, 776 columns, 2296 nonzeros
## Variable types: 0 continuous, 776 integer (776 binary)
## Found heuristic solution: objective 9018.7830067
## Explored 0 nodes (0 simplex iterations) in 0.01 seconds (0.01 work units)
## Thread count was 1 (of 48 available processors)
## Solution count 2: 9018.78 10337.6 
## Optimal solution found (tolerance 1.00e-01)
## Best objective 9.018783006750e+03, best bound 8.496084047117e+03, gap 5.7957%
# print result
## Simple feature collection with 1130 features and 5 fields
## Geometry type: MULTIPOLYGON
## Dimension:     XY
## Bounding box:  xmin: 298809.6 ymin: 5167775 xmax: 613818.8 ymax: 5502544
## Projected CRS: WGS 84 / UTM zone 55S
## # A tibble: 1,130 × 6
##       id  cost locked_in locked_out solution_1                              geom
##    <int> <dbl> <lgl>     <lgl>           <dbl>                <MULTIPOLYGON [m]>
##  1     1 60.2  FALSE     TRUE                0 (((328497 5497704, 326783.8 5500…
##  2     2 19.9  FALSE     FALSE               0 (((307121.6 5490487, 305344.4 54…
##  3     3 59.7  FALSE     TRUE                0 (((321726.1 5492382, 320111 5494…
##  4     4 32.4  FALSE     FALSE               0 (((304314.5 5494324, 304342.2 54…
##  5     5 26.2  FALSE     FALSE               0 (((314958.5 5487057, 312336 5490…
##  6     6 51.3  FALSE     FALSE               0 (((327904.3 5491218, 326594.6 54…
##  7     7 32.3  FALSE     FALSE               0 (((308194.1 5481729, 306601.2 54…
##  8     8 38.4  FALSE     FALSE               0 (((322792.7 5483624, 319965.3 54…
##  9     9  3.55 FALSE     FALSE               0 (((334896.6 5490731, 335610.4 54…
## 10    10  1.83 FALSE     FALSE               0 (((356377.1 5487952, 353903.1 54…
## # ℹ 1,120 more rows
# 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
  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.

# 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).

# 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.

## [1]  0.00001  0.00007  0.00049  0.00340  0.02371  0.16548  1.15478  8.05842
## [9] 56.23413

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.

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.

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 8.05842 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%).

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).

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

## [1] 354.3258

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.

## [1]  355  532  709  886 1063 1241 1418 1595 1772

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).

## Warning: `x[, cost_column]` has only zero values.