Configure the augmented epsilon-constraint method (AUGMECON)

Explore trade-offs among registered objectives using AUGMECON. One objective is optimised directly, while the others are constrained by epsilon levels and contribute small, normalised improvement terms to discourage weakly efficient solutions.

Usage

set_method_augmecon(
  x,
  primary,
  aliases = NULL,
  runs = NULL,
  grid = NULL,
  n_points = NULL,
  include_extremes = NULL,
  lexicographic = TRUE,
  lexicographic_tol = 1e-09,
  augmentation = 0.001,
  slack_upper_bound = 1e+06,
  control = NULL
)

Arguments

x: A Problem with at least two registered objective aliases.
primary: Character scalar naming the objective optimised directly.
aliases: Optional character vector of registered objective aliases. By default, all registered objectives are included; primary must be one of them.
runs: A run design from set_runs_grid() or set_runs_manual(). Manual designs contain eps_<alias> columns for all secondary objectives.
grid, n_points, include_extremes: Deprecated run-design arguments; use runs instead. include_extremes is ignored because automatic grids always include their endpoints.
lexicographic: Logical. Whether to use lexicographic anchoring when computing automatic-grid extreme points. Defaults to TRUE.
lexicographic_tol: Non-negative tolerance for the lexicographic extreme-point calculation, in the objective's original units.
augmentation: Positive coefficient for the normalised slack reward. Defaults to 1e-3.
slack_upper_bound: Positive upper bound for AUGMECON slack variables. Defaults to 1e6; it must not exclude feasible values.
control: Optional run-failure settings from set_runs_control().

Value

An updated Problem with the method stored in x$data$method. Payoff-table calculations and optimisation runs take place in solve().

Details

Register at least two objectives with distinct alias values. The primary objective is optimised directly; the remaining objectives selected by aliases become epsilon-constrained objectives. For example, with woodland improvement as the primary objective and cost as secondary, each run seeks the greatest improvement while keeping cost below a ceiling: $$\max\ f_{\mathrm{woodland}}(x), \qquad f_{\mathrm{cost}}(x) \le \varepsilon_{\mathrm{cost}}.$$

What augmentation changes. Ordinary epsilon-constraint may return a weakly efficient plan when another objective could improve without changing the primary objective. AUGMECON introduces a non-negative slack for every secondary constraint and rewards normalised slack through a small positive coefficient augmentation (Mavrotas, 2009). For a minimised cost objective, the slack measures how far actual cost lies below its epsilon ceiling. Schematically, with a maximised primary objective, AUGMECON uses $$\max\ f_p(x) + \rho\sum_k \frac{s_k}{R_k},$$ where $\rho$ is augmentation, $s_k$ is the non-negative slack, and $R_k$ is the payoff-table range of the corresponding secondary objective. The implementation handles the signs of minimisation and maximisation objectives internally. Choose a small augmentation so that secondary improvements do not unintentionally outweigh primary performance.

Automatic epsilon grids. Use runs = set_runs_grid(n = 5) to obtain five epsilon levels, including the endpoints, for each secondary objective. Levels and payoff-table ranges are calculated during solve(). With three objectives, for example, two secondary grids of five levels yield up to 25 combinations. When lexicographic = TRUE, extreme-point calculations use lexicographic anchoring, with lexicographic_tol specifying the tolerance.

Manual thresholds. Use runs = set_runs_manual() to supply explicit epsilon values, one row per run and one column named eps_<alias> for each secondary objective. For a minimised cost objective, eps_cost = 20 means cost at most 20, not exactly 20. Unlike automatic grids, manual rows are not expanded into all possible combinations; use expand.grid() if needed.

Interpreting solutions. The augmentation reduces weak-efficiency problems, but a finite epsilon grid does not enumerate every Pareto-optimal plan, and different runs can produce identical spatial decisions. Numerical tolerances and finite solver time limits also matter. A numerical refinement pass may improve secondary slacks after an optimal augmented solve; solver settings and epsilon-bound checks are available in the solution diagnostics. slack_upper_bound must be large enough for valid slacks, without being unnecessarily loose. Use set_runs_control() to manage infeasible or unsuccessful runs.

Only one multi-objective method can be configured on a Problem; compare methods by starting from the same problem before calling set_method_*(). The older grid, n_points, and include_extremes arguments are deprecated in favour of runs.

References

Mavrotas, G. (2009). Effective implementation of the $\varepsilon$-constraint method in multi-objective mathematical programming problems. Applied Mathematics and Computation, 213(2), 455-465. doi:10.1016/j.amc.2009.03.037 .

Mavrotas, G., and Florios, K. (2013). An improved version of the augmented $\varepsilon$-constraint method (AUGMECON2) for finding the exact Pareto set in multi-objective integer programming problems. Applied Mathematics and Computation, 219(18), 9652-9669. doi:10.1016/j.amc.2013.03.002 . (Related extension; not the same method.)

Examples

# EXAMPLE: Maximise woodland improvement under alternative cost limits.
# Use the same 64-unit landscape and objectives as in the other MO methods.
sim <- load_sim_multiaction()

base <- create_problem(
  pu = sim$planning_units,
  features = sim$features,
  dist_features = sim$dist_features,
  cost = "cost"
) |>
  add_actions(sim$actions, cost = sim$action_costs) |>
  add_effects(sim$effect_assumptions) |>
  # Require some improvement to avoid an empty low-cost solution.
  add_constraint_targets_relative(0.05, features = "woodland") |>
  add_objective_max_effect(
    features = "woodland", alias = "woodland"
  ) |>
  add_objective_min_cost(
    include_pu_cost = FALSE, alias = "cost"
  )

# Explore five cost ceilings. Small slack rewards help distinguish
# plans with similar woodland improvements but different costs.
augmecon <- set_method_augmecon(
  base,
  primary = "woodland",
  aliases = c("woodland", "cost"),
  runs = set_runs_grid(n = 5),
  lexicographic = TRUE,
  augmentation = 1e-3
)

if (requireNamespace("rcbc", quietly = TRUE) &&
    requireNamespace("ggplot2", quietly = TRUE)) {
  solutions <- solve(
    set_solver_cbc(augmecon, time_limit = 30, verbose = FALSE)
  )

  # Actual cost may be lower than the epsilon ceiling.
  get_objectives(solutions, format = "wide")

  # Inspect the trade-off frontier generated by AUGMECON.
  plot_tradeoff(
    solutions, objectives = c("cost", "woodland"),
    label_runs = TRUE
  )

  # Compare selected management actions for three epsilon levels.
  # The three runs need not produce three different spatial plans.
  plot_spatial_actions(
    solutions, solutions = c(1, 3, 5), layout = "single", ncol = 3
  )
}



# ALTERNATIVE: use three manually specified cost ceilings.
# Start again from 'base', before configuring a multi-objective method.
manual <- set_method_augmecon(
  base,
  primary = "woodland",
  aliases = c("woodland", "cost"),
  runs = set_runs_manual(data.frame(eps_cost = c(10, 20, 40))),
  augmentation = 1e-3
)