Choosing a multi-objective optimisation method

The decision comes before the method

Multi-objective optimisation is useful when no feasible spatial plan is best in every respect. A plan with greater ecological benefit may also cost more; a compact plan may protect less of a particular feature; and a plan that performs well for one objective may perform poorly for another.

A multi-objective method does not remove these conflicts. Instead, it determines how trade-offs are represented and which solutions are generated for subsequent interpretation or decision making.

No method is universally preferable. Weighted sum, epsilon-constraint, and AUGMECON answer different decision questions and require different forms of preference information. Before choosing a method, ask:

• Are preferences sufficiently clear to combine the objectives?
• Are any objectives naturally expressed as limits, such as a budget or a minimum ecological requirement?
• Is the aim to select one plan now, or to learn about trade-offs before selecting?
• How many optimisation runs can reasonably be solved?

This article focuses on choosing among the multi-objective methods implemented in multiscape. Practical worked examples are provided in separate articles, while the corresponding set_method_*() functions document the complete argument structure and implementation details.

Choosing a method

Preferences before or after optimisation

Multi-objective workflows are often described as a priori or a posteriori.

• A priori: preferences enter before optimisation. This is appropriate when relative priorities, exchange rates, acceptable losses, or policy limits are already defensible.
• A posteriori: efficient alternatives are generated first, and stakeholders select among them after inspecting their ecological, spatial, and economic consequences. This is appropriate when preferences are uncertain, contested, or deliberately left open during model construction.

These terms describe the workflow rather than an immutable property of an algorithm. A single weighted-sum model with agreed weights is an a priori analysis, whereas a grid of weights may be used exploratorily. Similarly, epsilon-constraint is a priori when a fixed policy threshold is imposed, but becomes a posteriori when multiple epsilon levels are used to trace a frontier.

Quick choice

The following table provides a practical starting point. It is not a strict rule; the appropriate method depends on how preferences can be expressed and on the purpose of the analysis.

Decision situation	Suggested start	Why
Relative preferences are defensible	Weighted sum	Directly represents relative trade-offs through weights
One objective is primary and the others have meaningful limits	Epsilon-constraint	Keeps policy, ecological, or budget limits explicit
Preferences are unsettled and a frontier is needed	AUGMECON	Generates a systematic set of strongly efficient alternatives

A useful strategy is to begin with individual-objective solutions, use a coarse exploration to understand the scale and conflict among objectives, and refine only the region that is relevant for decision making.

The methods

Weighted sum: state relative preferences

Decision question

Can the relative importance or exchange rate among objectives be stated explicitly?

Main idea

Weighted sum combines the objectives into a single scalar objective. For objectives transformed conceptually to minimisation, the model is

\[ \min_{x \in X} \sum_{k=1}^{K} w_k\,\widetilde{f}_k(x), \]

where \(X\) is the feasible set, \(\widetilde{f}_k(x)\) is a suitably scaled version of objective \(k\), and \(w_k \geq 0\) is its weight.

The weights express relative importance after scaling. They are not automatically percentages of final objective performance. If objective scales differ substantially, a numerically large objective may dominate the weighted sum even when assigned a small weight.

Use weighted sum when

• stakeholders can defend relative trade-offs among objectives;
• only a small number of preference scenarios is required;
• computational simplicity is important;
• the analysis aims to generate one or a few plans from explicit priorities.

Use caution when

• objective scales have not been examined;
• an objective has a legal, ecological, or physical threshold that should remain explicit;
• the efficient frontier may be non-convex;
• weights would be chosen arbitrarily or interpreted as percentages without justification.

Weighted sum can recover supported efficient solutions, but it may miss efficient solutions in non-convex regions of the frontier. This limitation is especially relevant in discrete spatial optimisation problems.

In multiscape, weighted-sum models are configured with set_method_weighted_sum(). See the function reference and the dedicated worked examples for argument details and complete workflows.

Epsilon-constraint: state acceptable limits

Decision question

Is one objective naturally primary while the remaining objectives can be expressed as meaningful limits?

Main idea

Epsilon-constraint optimises one objective and converts the others into explicit constraints. For a primary minimisation objective \(f_p(x)\) and secondary minimisation objectives \(f_k(x)\), the model is

\[ \begin{aligned} \min_{x \in X} \quad & f_p(x) \\ \text{s.t.} \quad & f_k(x) \leq \varepsilon_k, && k \neq p. \end{aligned} \]

For a secondary objective expressed as maximisation, the inequality reverses:

\[ f_k(x) \geq \varepsilon_k. \]

The epsilon values retain concrete meanings. They may represent a maximum budget, a minimum ecological benefit, a maximum acceptable loss, a fragmentation limit, or another interpretable guarantee.

Use epsilon-constraint when

• one objective is clearly primary;
• secondary objectives have meaningful policy, ecological, or management limits;
• the analysis must retain thresholds in their original units;
• non-convex regions of the frontier may be relevant;
• a set of scenarios is to be generated by varying one or more limits.

Use caution when

• epsilon values are arbitrary or poorly justified;
• combinations of epsilon levels may be infeasible;
• a dense grid would require too many optimisation runs;
• the selected primary objective imposes an interpretation that is not appropriate for the decision context.

An infeasible epsilon combination is not necessarily an error. It may reveal that the requested guarantees cannot be achieved simultaneously.

In multiscape, epsilon-constraint models are configured with set_method_epsilon_constraint(). Automatic grids can be used for exploratory analyses, while manually supplied epsilon values are appropriate when policy or ecological limits are already known.

AUGMECON: construct a cleaner efficient frontier

Decision question

Is the aim to generate a systematic set of strongly efficient alternatives for later inspection and selection?

Main idea

Basic epsilon-constraint can return weakly efficient solutions. A solution is weakly efficient when a secondary objective could improve without worsening the primary objective. AUGMECON reduces this problem by augmenting the primary objective with normalised slack terms that reward improvement beyond the epsilon limits.

For a primary minimisation objective and secondary minimisation objectives, a simplified form is

\[ \begin{aligned} \min_{x,s} \quad & f_p(x) - \rho \sum_{k \neq p}\frac{s_k}{R_k} \\ \text{s.t.} \quad & f_k(x) + s_k = \varepsilon_k, && k \neq p, \\ & s_k \geq 0, && k \neq p, \\ & x \in X, \end{aligned} \]

where \(s_k\) measures improvement beyond the epsilon limit, \(R_k\) is the payoff-table range of objective \(k\), and \(\rho > 0\) is a small augmentation coefficient. The exact signs depend on the objective senses; multiscape constructs the corresponding formulation internally.

Normalisation by \(R_k\) prevents measurement units from dominating the augmentation term. The augmentation coefficient should remain small enough that the primary objective retains its intended role while still discouraging weakly efficient solutions.

Use AUGMECON when

• preferences are not yet settled;
• a frontier is required for later discussion or stakeholder selection;
• strongly efficient alternatives are preferred;
• the analysis seeks systematic coverage of the trade-off space;
• the additional computational cost is acceptable.

Use caution when

• the number of secondary objectives is large;
• each objective is assigned many epsilon levels;
• the resulting Cartesian grid becomes computationally prohibitive;
• objective ranges are poorly estimated;
• the augmentation coefficient is chosen without considering numerical scale.

With \(m\) secondary objectives and \(q\) levels for each, a full Cartesian design may require up to

\[ q^m \]

optimisation runs, in addition to the solves required to establish objective ranges. A coarse grid should normally be used first, followed by targeted refinement in regions of interest.

In multiscape, AUGMECON models are configured with set_method_augmecon().

A practical workflow

A robust multi-objective analysis usually follows five stages.

1. Establish objective ranges

Optimise each objective separately. These anchor solutions reveal the attainable range of each objective and help identify:

• redundant objectives;
• objectives with very different numerical scales;
• strongly conflicting objectives;
• ties or alternative optima at the extremes;
• modelling errors or unexpectedly inactive constraints.

Automatic epsilon grids use extreme-point and payoff-table information during solve().

2. Decide how preferences can be expressed

Use:

• weights for defensible relative compensation;
• epsilon bounds for meaningful guarantees;
• a frontier when neither weights nor limits can yet be agreed.

3. Start coarse

Avoid dense grids before the broad structure of the trade-off space is understood. A coarse analysis often identifies the relevant region at a fraction of the computational cost.

4. Inspect both objective values and spatial plans

Two solutions may be close in objective space yet spatially very different. Conversely, visibly different plans may have nearly identical objective values. Multi-objective interpretation should therefore consider both numerical trade-offs and the spatial configuration of the selected planning units or actions.

5. Document the decision process

Report:

• objective definitions and senses;
• any scaling or normalisation;
• the source of weights or epsilon bounds;
• the primary objective, when applicable;
• the run design;
• lexicographic settings;
• solver tolerances and stopping criteria;
• infeasible or failed runs;
• the rule used to select any final representative solution.

A visually attractive knee point is not automatically the preferred plan. Its relevance depends on the decision context and on whether the implied compromise is acceptable to stakeholders.

Mathematical background

A common mathematical view

Let \(x\) denote a feasible spatial plan and let \(X\) contain all plans satisfying the model constraints. With \(K\) objectives,

\[ f(x) = \bigl(f_1(x), f_2(x), \ldots, f_K(x)\bigr). \]

Some objectives are minimised and others maximised. To express them under a common minimisation convention, define

\[ \widehat{f}_k(x) = \begin{cases} f_k(x), & \text{if objective } k \text{ is minimised},\\ -f_k(x), & \text{if objective } k \text{ is maximised}. \end{cases} \]

The multi-objective problem can then be written as

\[ \min_{x \in X} \left( \widehat{f}_1(x), \widehat{f}_2(x), \ldots, \widehat{f}_K(x) \right). \]

This notation is conceptual. multiscape handles objective senses internally, so users define each objective in its natural direction.

Pareto efficiency

A feasible plan \(x^\star \in X\) is Pareto efficient if there is no other feasible plan \(x \in X\) such that

\[ \widehat{f}_k(x) \leq \widehat{f}_k(x^\star) \qquad \text{for all } k, \]

with strict inequality for at least one objective.

Equivalently, no objective can be improved without worsening at least one other objective. The objective vectors associated with Pareto-efficient plans form the Pareto frontier.

A plan \(x^\star\) is weakly Pareto efficient if there is no feasible plan \(x\) satisfying

\[ \widehat{f}_k(x) < \widehat{f}_k(x^\star) \qquad \text{for all } k. \]

Weak efficiency is less restrictive: it permits a solution for which one or more objectives could improve without worsening the others. This distinction motivates the slack-based augmentation used by AUGMECON.

What does lexicographic = TRUE mean?

Before an automatic epsilon grid can be constructed, the range of each objective must be estimated from anchor solutions. A simple anchor is obtained by optimising one objective alone. The difficulty is that several plans may share the same optimal value for that objective while differing substantially in the others. A solver may return any tied optimum, producing unstable or unnecessarily poor endpoints for the payoff table.

Lexicographic anchoring resolves this ambiguity in two stages. For two minimisation objectives \(f_1\) and \(f_2\), the anchor associated with \(f_1\) is computed by first solving

\[ z_1^\star = \min_{x \in X} f_1(x), \]

and then solving

\[ \begin{aligned} \min_{x \in X} \quad & f_2(x) \\ \text{s.t.} \quad & f_1(x) \leq z_1^\star + \tau, \end{aligned} \]

where \(\tau\) is lexicographic_tol. The roles of the objectives are then reversed to obtain the other anchor. For maximisation objectives, multiscape applies the equivalent bound in the appropriate direction.

The first objective retains strict priority; the second objective only breaks ties among solutions that are optimal, or within \(\tau\) of optimal, for the first. This generally produces better-defined payoff-table ranges and avoids selecting an arbitrarily poor secondary value at an extreme.

In multiscape, lexicographic = TRUE is relevant to the automatic grid construction used by epsilon-constraint and AUGMECON. It does not:

• change a manually supplied epsilon design;
• turn weighted sum into a lexicographic method;
• define a permanent stakeholder ranking for every frontier solution;
• replace AUGMECON’s slack-based augmentation.

Set lexicographic = TRUE in most automatic-grid analyses, especially when multiple solutions tie on an objective. Set it to FALSE mainly to reduce the additional anchor solves or to reproduce non-lexicographic endpoints.

With lexicographic_tol = 0, the first optimum is preserved exactly, subject to solver numerics. A small positive tolerance may improve numerical robustness, but it also permits the first objective to deteriorate by that amount while the second improves. The tolerance is expressed in the original units of the first objective and should therefore be chosen relative to its scale and to the solver’s feasibility tolerance.

Summary

• Choose weighted sum when relative preferences are defensible and a compact, computationally simple analysis is sufficient.
• Choose epsilon-constraint when one objective is primary and the remaining objectives can be stated as meaningful limits.
• Choose AUGMECON when the aim is to generate a systematic set of strongly efficient alternatives for later discussion and selection.
• When uncertain, solve the individual objectives first, explore a coarse frontier, and refine only the region that matters for the decision.

The purpose of multi-objective optimisation is not to identify a universally best plan. It is to make trade-offs explicit, generate defensible alternatives, and support transparent decisions about the consequences of different priorities.