Multi-objective forest restoration planning in a highly productive landscape
Source:vignettes/Integrated_planning.Rmd
Integrated_planning.RmdOverview
Restoration planning is especially difficult in highly productive landscapes. Most land is already committed to agriculture, forestry, or other economically valuable uses, so restoration cannot be treated as though it were being placed on vacant land. Every new restoration site competes with an existing use, requires public or private expenditure, and may alter areas that already provide important ecosystem services.
At the same time, ecosystem-service provision is often uneven across the landscape. Some locations already store substantial carbon, regulate water flows, retain sediment, or support biological control, whereas other locations provide comparatively little of these functions. In this example, the planning goal is therefore not to restore the sites with the highest current service values. Instead, it is to identify eligible sites with relatively low baseline levels of ecosystem services. These low-service locations are interpreted as areas of current ecological deficit where restoration may be most justified, while avoiding the displacement of functions that are already being supplied elsewhere.
This creates a genuine many-objective problem. A site with low carbon stock may still have high seasonal water yield, and a site with low sediment retention may be important for rodent and lagomorph control. Moreover, the least expensive sites are not necessarily those with the largest ecosystem-service deficits. Collapsing the four services into a single index would hide these conflicts and would require fixed exchange weights before decision makers have seen their consequences.
This vignette uses multiscape to extend an existing
conservation network with a fixed amount of forest restoration. Existing
conservation areas are treated as prior commitments and remain
unchanged. The optimizer decides where to add restoration within the
surrounding productive matrix while balancing five separate objectives:
implementation cost and the baseline levels of four ecosystem services
inside the selected restoration portfolio.
The example addresses the following planning question:
Which additional sites should be restored to complete the landscape commitment at low cost, while directing restoration towards locations with comparatively low current provision of multiple ecosystem services?
The analysis produces a set of alternative spatial plans rather than one predefined answer. These alternatives show how the restoration pattern changes when greater emphasis is placed on different ecosystem-service deficits.
Planning formulation
Let \(i \in \mathcal{I}_R\) index planning units that are eligible for forest restoration. The binary decision variable \(r_i\) equals one when unit \(i\) is selected for restoration and zero otherwise. The existing conservation network is fixed in every solution. It provides the spatial foundation of the plan, but the decisions compared in this vignette concern only the additional restoration units.
The total restoration cost is
\[ C(r)=\sum_{i \in \mathcal{I}_R}c_i^R r_i, \]
where \(c_i^R\) is the local cost of restoring unit \(i\).
For each ecosystem service \(f\), the model also minimizes
\[ O_f(r)=\sum_{i \in \mathcal{I}_R}q_{if}r_i, \]
where \(q_{if}\) is the normalized baseline amount of service \(f\) currently provided by unit \(i\). This quantity can be interpreted as the total baseline service level contained in the selected restoration portfolio. Minimizing it directs restoration towards units with lower current provision of that service.
This interpretation is important. A low value does not mean that a site is ecologically unimportant, nor does it prove that restoration will generate a large future gain. It indicates only that the site is comparatively deficient in the service represented by that layer and that selecting it avoids placing restoration in areas where that service is already strongly expressed.
The complete objective vector is
\[ \min \left\{ C(r), O_{\mathrm{control}}(r), O_{\mathrm{carbon}}(r), O_{\mathrm{water}}(r), O_{\mathrm{sediment}}(r) \right\}. \]
The four ecosystem-service objectives are retained separately because their spatial patterns differ. The model therefore searches for low-cost restoration portfolios that target several types of ecological deficit without assuming that one unit of carbon, water yield, sediment retention, and biological control are directly interchangeable.
These objectives describe baseline conditions at candidate
restoration sites. They do not estimate the changes caused by
restoration. Consequently, add_effects() is not required
here. A model intended to predict post- restoration gains would need
action-specific effects and one or more benefit objectives.
Landscape and ecosystem-service data
The example represents a highly productive landscape divided into 30,496 planning units. Within this matrix, only a subset of units is available for new forest restoration, while other units are already conserved or excluded from intervention.
Four normalized raster layers describe current ecosystem-service provision: rodent and lagomorph control, carbon stock, seasonal water yield, and sediment retention. Values range from 0 to 1 within each layer. Higher values indicate stronger current provision of the corresponding service; lower values indicate areas where that service is comparatively weak and where restoration may help address an existing ecological deficit.
library(multiscape)
library(dplyr)
library(ggplot2)
library(terra)
data("sim_pu_sf", package = "multiscape")
ecosystem_services <- load_sim_features_raster()
service_labels <- c(
L1_control_inv = "Rodent and lagomorph control",
L1_stock_inv = "Carbon stock",
L1_rend_inv = "Seasonal water yield",
L1_reten_inv = "Sediment retention"
)
nrow(sim_pu_sf)
#> [1] 30496
names(ecosystem_services)
#> [1] "L1_control_inv" "L1_stock_inv" "L1_rend_inv" "L1_reten_inv"The maps show that the four deficits do not occur in the same places. An area with low carbon stock may not have low water yield, and an area with weak sediment retention may still contribute substantially to biological control. This lack of spatial congruence is the central reason for retaining four separate objectives. A restoration plan that performs well for one service can still redirect intervention towards high-value areas for another.
names(ecosystem_services) <- unname(service_labels[names(ecosystem_services)])
terra::plot(ecosystem_services)
Define the restoration decision context
The planning-unit attributes divide the landscape into three
management roles. Units marked locked_in constitute the
existing conservation network and are included in every plan. Units that
are neither locked in nor locked out form the candidate pool for
additional forest restoration. Units marked locked_out are
unavailable because of land-use, institutional, or other planning
restrictions represented in the example.
The action table includes both conservation and restoration so that the final maps display the complete intervention system. However, only restoration is a variable decision. The conservation network is inherited from previous policy commitments and remains spatially fixed across all solutions.
planning_attributes <- sim_pu_sf |>
sf::st_drop_geometry() |>
select(id, cost, area, locked_in, locked_out)
actions <- data.frame(
id = c("conservation", "restoration"),
name = c("Conservation", "Restoration")
)
conservation_units <- planning_attributes |>
filter(locked_in) |>
transmute(pu = id, action = "conservation")
restoration_units <- planning_attributes |>
filter(!locked_in, !locked_out) |>
transmute(pu = id, action = "restoration")
feasible_actions <- bind_rows(
conservation_units,
restoration_units
)
action_costs <- feasible_actions |>
left_join(
planning_attributes |> select(id, cost),
by = c("pu" = "id")
)
feasible_actions |> count(action)
#> action n
#> 1 conservation 3988
#> 2 restoration 25089The problem is therefore not a competition between conservation and restoration inside the same planning unit. It is a network-completion problem: given an existing conservation foundation, which additional units should be restored to meet the remaining area commitment?
This distinction also clarifies the role of the many-objective analysis. AUGMECON does not decide whether existing conservation should be replaced. It compares alternative ways of placing the same amount of new restoration across the eligible productive matrix.
Build the planning problem
create_problem() links the planning-unit geometries to
the four ecosystem- service layers and stores the baseline amount of
each service in every unit. add_actions() then registers
the feasible conservation and restoration assignments and their
associated local costs.
The planning-unit cost field remains available in the problem object for inspection. The optimization objective, however, uses the explicit action costs defined for restoration. This avoids counting the same cost component twice.
problem <- create_problem(
pu = sim_pu_sf,
features = ecosystem_services,
cost = "cost"
) |>
add_actions(
actions = actions,
include_pairs = feasible_actions,
cost = action_costs
)The landscape-level commitment is set at 20% of all 30,496 planning units. The existing conservation network already contributes 3,988 units towards this commitment. The remaining gap is therefore 2,112 units, which must be supplied through new forest restoration.
Every solution selects exactly these 2,112 restoration units. Holding restoration area constant is essential for interpretation: differences in cost or ecosystem-service objectives arise from where restoration is allocated, not from one solution restoring more land than another. Because all planning units have the same area, an equality constraint with a tolerance smaller than one unit enforces the same restoration effort in every plan.
No feature-representation target is imposed. The policy requirement concerns the amount of land restored, while the four ecosystem-service layers are used to compare alternative locations for that fixed effort.
unit_area <- stats::median(sim_pu_sf$area)
total_commitment_units <- ceiling(0.20 * nrow(sim_pu_sf))
restoration_target_units <- total_commitment_units - nrow(conservation_units)
restoration_target_area <- restoration_target_units * unit_area
problem <- problem |>
add_constraint_area(
area = restoration_target_area,
sense = "equal",
tolerance = 0.25 * unit_area,
actions = "restoration"
) |>
add_constraint_locked_planning_units(
locked_in = "locked_in",
locked_out = "locked_out"
)Register five restoration objectives
The model registers five atomic objectives, each of which retains a distinct interpretation.
The first objective minimizes the explicit cost of the selected restoration actions. The other four objectives minimize the current amount of each ecosystem service contained within the restoration portfolio. In practical terms, they encourage the optimizer to direct restoration towards sites with low baseline carbon stock, low seasonal water yield, low sediment retention, or low rodent and lagomorph control.
Each ecosystem-service objective is restricted to the
restoration action. The fixed conservation network is
therefore visible in the maps but does not add a constant value to every
solution. This makes the reported objective values directly comparable
as properties of the additional restoration portfolio.
Keeping the four services separate is deliberate. A site selected because it has low carbon stock may still have high water yield, so no single restoration portfolio is expected to minimize all four service totals simultaneously.
problem <- problem |>
add_objective_min_cost(
alias = "cost",
include_pu_cost = FALSE,
include_action_cost = TRUE,
actions = "restoration"
) |>
add_objective_min_intervention_impact(
features = "L1_control_inv",
actions = "restoration",
alias = "rodent_control"
) |>
add_objective_min_intervention_impact(
features = "L1_stock_inv",
actions = "restoration",
alias = "carbon_stock"
) |>
add_objective_min_intervention_impact(
features = "L1_rend_inv",
actions = "restoration",
alias = "water_yield"
) |>
add_objective_min_intervention_impact(
features = "L1_reten_inv",
actions = "restoration",
alias = "sediment_retention"
)
print(problem)Generate alternatives with AUGMECON
The purpose of the analysis is to reveal the consequences of alternative restoration priorities before decision makers specify a preferred trade-off. A weighted-sum model would require weights for cost and the four services at the outset. Those weights would embed an exchange rule before stakeholders had seen how it changes the spatial plan.
AUGMECON is used instead as an a posteriori method. Cost is treated as the primary objective, while the four ecosystem-service objectives become explicit epsilon constraints. Each run asks a different version of the same management question: what is the least-cost restoration portfolio that also satisfies a particular combination of limits on the baseline services contained in the selected sites?
set_runs_grid(3) constructs three epsilon levels for
each of the four secondary objectives from their payoff ranges. Their
Cartesian product produces \(3^4=81\)
requested configurations. These configurations sample contrasting
combinations of ecosystem-service requirements, from relatively
permissive limits to combinations that force restoration strongly
towards low-service areas.
The grid is intentionally small for a vignette involving more than
30,000 planning units. It is sufficient to demonstrate the
many-objective workflow, but it should not be interpreted as an
exhaustive representation of the five-dimensional Pareto frontier. A
decision-support application could refine the grid or introduce
policy-defined thresholds with set_runs_manual().
Lexicographic anchoring is used to construct the payoff table, and the AUGMECON augmentation rewards unused slack after cost has been optimized. This helps avoid weakly efficient plans that improve one criterion only trivially while performing unnecessarily poorly on others.
Some combinations of stringent service limits may be infeasible. In this context, infeasibility has a substantive interpretation: the fixed restoration area cannot always be concentrated simultaneously in locations with very low levels of all four services. Such runs reveal conflicts among the spatial deficits rather than representing a failure of the analysis.
problem <- problem |>
set_method_augmecon(
primary = "cost",
aliases = objective_aliases,
runs = set_runs_grid(3),
lexicographic = TRUE,
augmentation = 1e-3,
control = set_runs_control(
stop_on_infeasible = FALSE,
stop_on_no_solution = FALSE
)
) |>
set_solver_gurobi(gap_limit = 0)The complete optimization requires Gurobi and a valid licence. It took about 18 minutes on the system used to prepare this vignette. Re-solving all 81 configurations whenever the vignette or package website is built would add substantial and unnecessary computational cost.
The vignette therefore loads a precomputed SolutionSet.
This object preserves the action assignments, objective values, solver
status, and run metadata needed for the remaining analysis.
The code below shows how to regenerate and save the complete solution
set after changing the formulation, run design, or solver configuration.
It is displayed for reproducibility but is not evaluated during ordinary
vignette or pkgdown builds.
Examine the many-objective solution set
Run design and objective values
The run table records the 81 requested combinations of ecosystem-service thresholds. A run describes an attempted model configuration; it does not necessarily have an associated solution. Some configurations may be infeasible, while several feasible configurations may lead to the same spatial allocation because the decision variables are discrete.
For interpretation, the complete result is filtered to feasible,
non-dominated solutions and repeated decision vectors are removed. The
resulting SolutionSet contains the distinct efficient
restoration portfolios generated by this experiment.
efficient_solutions <- solutions |>
solution_filter(feasible_only = TRUE, nondominated = TRUE) |>
solution_unique(by = "decisions")
run_summary <- data.frame(
requested_configurations = nrow(get_runs(solutions)),
retained_efficient_plans = nrow(get_runs(efficient_solutions))
)
run_summary
#> requested_configurations retained_efficient_plans
#> 1 81 16
get_runs(efficient_solutions)
#> run_id solution_id status runtime gap
#> 1 23 23 optimal 1.2710001 0.000000e+00
#> 2 24 24 optimal 2.1499999 0.000000e+00
#> 3 26 26 optimal 1.0330000 1.060691e-14
#> 4 27 27 optimal 2.2810001 9.624842e-15
#> 5 41 41 optimal 261.4949999 0.000000e+00
#> 6 42 42 optimal 8.2400000 0.000000e+00
#> 7 44 44 optimal 100.7430000 4.346440e-15
#> 8 45 45 optimal 19.3859999 4.729136e-15
#> 9 50 50 optimal 4.0480001 0.000000e+00
#> 10 51 51 optimal 0.9520001 1.184638e-15
#> 11 53 53 optimal 5.0090001 3.818109e-15
#> 12 54 54 optimal 1.1499999 0.000000e+00
#> 13 71 71 optimal 38.7410002 3.362206e-15
#> 14 72 72 optimal 15.8169999 6.628831e-15
#> 15 80 80 optimal 1.5079999 5.496960e-15
#> 16 81 81 optimal 0.3240001 0.000000e+00
get_objectives(efficient_solutions, format = "wide")
#> solution_id cost rodent_control carbon_stock water_yield
#> 1 23 258716.2 1495.492 8711.409 10027.200
#> 2 24 258701.0 1493.568 8711.404 10027.187
#> 3 26 148168.2 1511.004 10014.317 10027.200
#> 4 27 148167.4 1539.062 10014.318 10027.200
#> 5 41 215254.0 1511.468 8711.411 9148.542
#> 6 42 208536.5 2098.464 8711.411 9148.542
#> 7 44 147312.3 1511.469 10014.300 9148.542
#> 8 45 141545.5 2520.960 10014.303 9148.543
#> 9 50 150860.0 1511.470 8711.403 10027.165
#> 10 51 147406.2 2058.816 8711.412 10027.080
#> 11 53 121961.2 1511.469 10014.318 10027.198
#> 12 54 119665.0 2362.490 10009.051 10027.176
#> 13 71 147154.9 1511.467 10014.300 9148.543
#> 14 72 140495.7 2521.595 10014.310 9148.543
#> 15 80 121774.2 1511.470 10014.311 10027.172
#> 16 81 118753.5 2521.662 10014.318 10027.201
#> sediment_retention
#> 1 5.511997e-07
#> 2 7.745541e-07
#> 3 6.396928e-07
#> 4 6.396928e-07
#> 5 2.858626e+01
#> 6 3.174687e+01
#> 7 3.613481e+01
#> 8 3.613296e+01
#> 9 2.334568e+01
#> 10 3.310843e+01
#> 11 3.613522e+01
#> 12 3.613518e+01
#> 13 5.123982e+01
#> 14 7.186292e+01
#> 15 5.388135e+01
#> 16 7.227048e+01The reduction from 81 requested configurations to a smaller set of unique efficient plans is expected. Nearby epsilon combinations can select the same 2,112 restoration units, dominated solutions add no new trade-off information, and some combinations of very restrictive service limits cannot be achieved simultaneously.
The retained plans should therefore be read as a set of substantively different restoration strategies, not as 81 equally distinct alternatives.
Read the solution set as a sequence of decisions
Five-dimensional trade-offs are difficult to understand from a complete matrix of pairwise plots alone. The following diagnostics organize the results around a simpler management question:
How much additional expenditure is required to move restoration away from sites with high current ecosystem-service provision and towards sites with lower baseline service levels?
All five objectives are minimized. Accordingly, an improvement in an ecosystem- service objective means that the selected restoration portfolio contains less of that service at baseline than the least-cost portfolio. In the logic of this example, the plan is targeting a stronger current deficit for that service. It does not mean that the service has already increased after restoration.
How much deficit targeting accompanies additional cost?
Each panel compares an efficient plan with the least-cost restoration plan, which is located at zero on both axes. Moving to the right means accepting a higher implementation cost. Moving upwards means reducing the baseline amount of the corresponding ecosystem service inside the selected restoration sites.
A higher point therefore represents a portfolio that directs restoration more strongly towards locations where that service is currently low. Separate vertical scales are retained because the observed percentage range differs among services and because equal percentage changes need not have equal ecological or policy significance.
A point below zero would indicate that a more expensive plan selects sites with a higher baseline level of that service than the least-cost plan.
The points are not connected because they represent alternative spatial portfolios, not successive stages of one restoration trajectory. Labels identify the least-cost plan, the five-objective compromise, and the best observed plan for the service shown in each panel. The same solution identifiers are used in the later heatmap and maps.
The figure should first be read within each panel. A steep increase indicates that a relatively small additional expenditure can redirect restoration towards a substantially stronger deficit for that service. A long movement to the right with little vertical improvement indicates declining returns. Comparing panels then reveals whether the same costly plan addresses several deficits or whether each service requires a different spatial allocation.
Because the services are not aggregated, the graph preserves the central planning conflict: restoration sites that are attractive for one ecological reason may be unattractive for another.
Compare the complete performance profile of each plan
The heatmap summarizes the full five-objective profile of every retained plan. Solutions are ordered from lowest to highest cost. Within each objective column, values are normalized across the observed efficient set. Blue indicates the best observed value for a minimized objective, red the worst, and intermediate colours show relative performance between those extremes.
This normalization supports visual comparison but does not make the objectives commensurable. It does not imply that cost and ecosystem services share units, or that the four services should receive equal policy weight.
White circles identify the best observed plan for each objective. Rows with blue cells across several service columns represent restoration portfolios that simultaneously target multiple ecosystem-service deficits. Rows that combine blue and red cells expose specialization: the plan performs strongly for some services by accepting weaker performance for others.
The cost column completes the story. It shows whether a portfolio that targets several ecological deficits can be obtained with a moderate budget increase or whether it requires a substantially more expensive spatial reallocation.
Identify extremes and representative compromises
frontier_extremes() identifies the best and worst
observed solution for each objective. These extremes are useful
reference points because they show the maximum contrast generated by the
experiment. However, the plan that best targets one ecosystem-service
deficit may be costly or may select sites with high baseline values of
another service.
extremes <- frontier_extremes(
efficient_solutions,
objectives = objective_aliases,
ties = "first"
)
extremes
#> solution_id objective sense bound role value
#> 1 81 cost min min best 1.187535e+05
#> 2 23 cost min max worst 2.587162e+05
#> 3 24 rodent_control min min best 1.493568e+03
#> 4 81 rodent_control min max worst 2.521662e+03
#> 5 50 carbon_stock min min best 8.711403e+03
#> 6 81 carbon_stock min max worst 1.001432e+04
#> 7 41 water_yield min min best 9.148542e+03
#> 8 81 water_yield min max worst 1.002720e+04
#> 9 23 sediment_retention min min best 5.511997e-07
#> 10 81 sediment_retention min max worst 7.227048e+01With five objectives, a two-dimensional geometric knee is not
appropriate. frontier_knee(method = "ideal") instead
identifies the observed plan closest to the normalized five-objective
ideal. The ideal combines the best observed value of every criterion,
even though no single feasible plan may attain all of them
simultaneously.
The resulting plan is a descriptive compromise within the generated solution set. It is not a universally optimal restoration strategy and does not replace stakeholder preferences.
For ecological interpretation, the analysis also calculates distance to an ideal defined only by the four ecosystem-service objectives. We refer to this as joint ecosystem-service regret. A value of zero would mean that one plan simultaneously attains the best observed baseline-service total for all four services. Larger values indicate increasing distance from that joint deficit- targeting profile.
Plotting this distance against additional cost keeps the financial and ecological dimensions visible. Because each service is normalized to its observed range, the diagnostic gives the four services equal influence. This equal weighting is used only to describe the generated alternatives; it does not alter the AUGMECON formulation.
service_distances <- frontier_distances(
efficient_solutions,
objectives = service_aliases,
reference = "ideal"
)
service_distances$cost <- objective_values$cost[
match(service_distances$solution_id, objective_values$solution_id)
]
service_distances$additional_cost <- 100 * (
service_distances$cost / minimum_cost - 1
)
ecological_id <- service_distances$solution_id[
which.min(service_distances$distance_to_ideal)
]
specialist_id <- unname(service_best_ids[["sediment_retention"]])
compromise[, c(
"solution_id",
objective_aliases,
"knee_score",
"knee_rank"
)]
#> solution_id cost rodent_control carbon_stock water_yield sediment_retention
#> 1 41 215254 1511.468 8711.411 9148.542 28.58626
#> knee_score knee_rank
#> 1 0.644435 1
data.frame(
five_objective_compromise = compromise_id,
closest_to_ecological_ideal = ecological_id,
same_solution = compromise_id == ecological_id
)
#> five_objective_compromise closest_to_ecological_ideal same_solution
#> 1 41 41 TRUEIn this experiment, the five-objective compromise and the plan closest to the four-service ideal are both solution 41. This agreement is an empirical feature of the generated set, not a property guaranteed by AUGMECON.
The figure also highlights the best observed plan for each individual service. Because solution 41 is simultaneously the water-yield extreme, the highlighted roles correspond to the same five unique plans mapped below. The colour scheme is retained across figures: orange represents the least-cost plan, magenta the joint compromise, and green a service-specific extreme.
Connect objective performance to spatial decisions
Objective values describe how a plan performs, but they do not show which parts of the productive landscape would actually change. The maps therefore compare the least-cost plan, the best plan for each ecosystem-service objective, and the five-objective compromise.
Because the compromise is also the best observed plan for water yield, the set contains five unique spatial allocations rather than six. Together they represent three distinct planning logics: minimizing expenditure, targeting a particular ecosystem-service deficit, and balancing cost with all four deficits.
#> role solution_id
#> 1 Least-cost plan 81
#> 2 Five-objective compromise and water-yield best 41
#> 3 Rodent-control best 24
#> 4 Carbon-stock best 50
#> 5 Sediment-retention best 23
#> additional_cost_percent rodent_control_reduction_percent
#> 1 0.0 0.0
#> 2 81.3 40.1
#> 3 117.8 40.8
#> 4 27.0 40.1
#> 5 117.9 40.7
#> carbon_stock_reduction_percent water_yield_reduction_percent
#> 1 0 0.0
#> 2 13 8.8
#> 3 13 0.0
#> 4 13 0.0
#> 5 13 0.0
#> sediment_retention_reduction_percent
#> 1 0.0
#> 2 60.4
#> 3 100.0
#> 4 67.7
#> 5 100.0Positive percentages in the service columns represent reductions in baseline service levels inside the restoration portfolio relative to the least-cost plan. For example, a positive carbon-stock value means that the plan restores sites with lower current carbon stock than those selected by the least-cost portfolio. It should not be interpreted as a measured increase in carbon after restoration.
The table provides the numerical context for the maps by showing the additional cost required to achieve each spatial redirection.
The maps reveal the spatial mechanism behind the objective trade-offs. They show whether improved targeting of a service deficit is achieved through a small number of local substitutions or through a broader relocation of the restoration portfolio across the landscape.
Because the conservation network is fixed, all differences among panels arise from the 2,112 restoration decisions. Comparing the service-specific extremes shows whether different deficits point to common restoration areas or to contrasting parts of the productive matrix. The compromise map identifies the allocation that balances these competing spatial signals rather than optimizing one of them in isolation.
Compare decision-space similarity
Two plans can have similar objective values while selecting different places. Jaccard similarity complements the performance diagnostics by measuring overlap in selected planning-unit–action pairs.
The heatmap follows the same cost ordering used above. Higher similarity means that two plans prescribe more of the same spatial actions. Exact duplicate decision vectors have already been removed, so values close to one indicate very similar but not necessarily identical portfolios.
The fixed conservation network contributes the same assignments to every solution and can therefore raise overall similarity. Differences among plans are generated only by the variable restoration component. Consequently, the heatmap should be interpreted as similarity of the complete intervention system in a setting where one large component is intentionally held constant.
Selection frequency provides a more local view of stability. It reports how often each planning-unit–action assignment appears across the retained non-dominated solutions.
For restoration units, a high frequency indicates that a location is selected under many different combinations of cost and ecosystem-service limits. Such a site is comparatively robust to the trade-off configuration explored here. For conservation units, frequency one is expected because those assignments are locked into every solution.
Frequency is therefore a measure of stability within this generated solution set. It should not be interpreted as ecological irreplaceability, restoration success, or the probability that a site is objectively optimal.
assignment_frequency <- selection_frequency(efficient_solutions)
head(
assignment_frequency[
order(-assignment_frequency$frequency),
],
15
)
#> pu action n_selected n_solutions frequency
#> 83 10075 restoration 16 16 1
#> 109 10099 restoration 16 16 1
#> 196 10180 restoration 16 16 1
#> 198 10182 restoration 16 16 1
#> 226 10208 restoration 16 16 1
#> 227 10209 restoration 16 16 1
#> 283 1026 conservation 16 16 1
#> 338 10314 restoration 16 16 1
#> 387 10360 restoration 16 16 1
#> 388 10361 restoration 16 16 1
#> 646 10605 restoration 16 16 1
#> 651 10610 restoration 16 16 1
#> 786 10736 restoration 16 16 1
#> 840 10786 restoration 16 16 1
#> 978 10919 restoration 16 16 1Interpretation and limitations
This vignette demonstrates how multiscape can prioritize
additional forest restoration in a highly productive landscape where
land is scarce, existing uses matter, and ecosystem-service deficits are
spatially heterogeneous. The existing conservation network is treated as
fixed, and every alternative adds the same 2,112 restoration units. The
decision problem is therefore entirely about spatial allocation: which
parts of the productive matrix should be restored?
The formulation gives a specific answer to that question. It favours locations with low implementation cost and low current levels of four ecosystem services. These low-service locations are treated as areas of ecological deficit, making them plausible candidates for restoration while avoiding sites where important functions are already strongly provided. By retaining the services as separate objectives, the model shows that there is no single universal map of ecological need: carbon, water, sediment retention, and biological control can point to different restoration priorities.
This interpretation must remain within the limits of the data and formulation. Low baseline service provision is not equivalent to high restoration potential. A degraded site may respond strongly to intervention, weakly, or only after a long time. Conversely, a site with high current service provision could still benefit from restoration. The model identifies where current service deficits and implementation costs favour intervention; it does not predict the ecological response produced by that intervention.
A more complete application could combine the present deficit-based objectives with action-specific estimates of expected restoration gain, feasibility, time to recovery, land-use opportunity cost, connectivity, or future climate risk. Those additions would distinguish more explicitly between where services are currently low and where restoration is expected to improve them most.
Finally, the generated alternatives depend on the epsilon design. The 81 requested boundary configurations provide a transparent first exploration of the five-dimensional trade-off space, not an exhaustive approximation of the Pareto frontier. A real decision process should refine the thresholds around policy-relevant regions, test the stability of the spatial recommendations, and use stakeholder input to identify acceptable compromises after the consequences of the alternatives have been made visible.