Relatively Robust Multicriteria Decisions

1. Introduction

In real-world decision making, evaluating alternatives often involves multiple, sometimes conflicting criteria. Whether considering investment portfolios under Environmental, Social, and Governance (ESG) parameters, selecting products based on bundles of attributes, or valuing companies for both profitability and sustainability, decision makers must navigate tradeoffs between competing objectives. Similarly, lifecycle environmental impact assessments—such as comparing vehicles with electric, combustion, or hybrid engines—require reconciling diverse metrics like emissions, cost, and resource consumption. These complex but often inevitable comparisons highlight the critical need for robust multicriteria optimization frameworks.

Multicriteria optimization involves identifying solutions that balance conflicting objectives in a manner consistent with the decision maker’s priorities. Traditional approaches often rely on scalarization techniques, where multiple criteria are combined into a single weighted objective function using weighted sums. However, these methods depend heavily on the precise specification of weights, which are rarely known a priori and can be difficult to justify. This uncertainty complicates the search for decisions that are both Pareto-efficient and robust to variations in tradeoff preferences. To address these challenges, we specialize the concept of relatively robust decisions by Weber (2023) so as to refer to decisions that achieve Pareto-efficiency among all relevant criteria while also maximizing a performance index designed to account for the ambiguity in weights. Specifically, the performance index measures the worst-case (WC) ratio attained by the weighted objective relative to its maximum value over all possible weight configurations. By focusing on worst-case performance, this framework provides guarantees of robustness and tradeoff transparency, which are particularly valuable in high-stakes or uncertain decision contexts.

1.1. Practical Examples

The approach developed here can be used for virtually all multicriteria decision problems, such as the following three example applications.¹

• ESG Investing: Investors face the challenge of balancing financial returns with social and environmental impact. For example, a fund manager might evaluate portfolios based on criteria such as profitability, carbon footprint, and diversity inclusion. Weighting these criteria is inherently subjective, and the optimal portfolio might vary widely depending on the chosen weights. The relatively robust optimization framework proposed here enables the identification of investment strategies that remain robust across different weight configurations, offering a performance guarantee regardless of the specific preferences.²

• Lifecycle Assessment of Vehicles: Consider evaluating the environmental impact of electric, combustion, and hybrid cars. Criteria might include greenhouse gas emissions, energy efficiency, and resource use (see, e.g., Hawkins et al. 2012). A relatively robust decision could pinpoint vehicle types or designs that perform well across a broad range of plausible weightings, resulting in a balanced and defensible choice.

• Product Evaluation and Design: Companies frequently design products to optimize attributes such as cost, durability, and aesthetic appeal. For example, in designing a smartphone, decision makers must weigh the importance of battery life, screen quality, and price.³ Relatively robust multicriteria optimization helps determine design specifications that ensure competitive performance across various market segments with diverse preferences, valuing the availability of different attributes with different weights.

For the application of our method, one only needs that there exists a default action, that is, a baseline alternative that performs adequately across all criteria (which can always be achieved by reindexing evaluation scales), together with the technical assumption that criteria are continuous in actions, and that the finite-dimensional action set is closed and bounded (i.e., compact).

1.2. Literature

1.2.1. Origins of Multicriteria Decision Making.

The idea of multicriteria optimization in Economics can be traced back to the distribution of resources among different individuals, leading to a set of undominated solutions such as the “contract curve” proposed by Edgeworth (1881, p. 21) for a simple exchange economy, and more generally a set of efficient outcomes as implied by Pareto (1894; 1897, sections 721–723),⁴ which cannot be improved upon for one agent without making another agent worse off. The latter avoids a direct interpersonal comparison of individuals’ utilities (or “ophelimities” in Pareto’s terminology); see also Harsanyi (1955) as well as Keeney and Raiffa (1993, chapter 10) who explore group utility functions. The drawback of such an agnostic approach to optimality with multiple criteria is that the set of Pareto-optimal allocations, because of its typically large cardinality, offers only imperfect guidance about which solution should actually be implemented. For example, in the two-agent exchange economy, the contract curve usually includes allocations that attribute all resources to any single individual, which almost completely undermines the notion of multicriteria optimization.

In Operations Research, “goal programming” refers to the notion of minimizing the weighted deviation from criterion-specific targets (Charnes and Cooper 1961). The technique was first employed in the context of executive compensation based on different “factors” (i.e., criteria) using linear programming techniques (Charnes et al. 1955). This basic approach is usually applied to a “utopian” (or “ideal”) target point, which corresponds to the (generically infeasible) vector of individually maximized criteria. Depending on the distance measure (e.g., a weighted Chebyshev distance; see, e.g., Steuer 1986, chapter 14),⁵ the corresponding solutions trade off among criteria according to the specified weights, and they are naturally Pareto-efficient.⁶ Instead of minimizing the distance to the ideal point (in the outcome space), it is also possible to maximize the distance to a (generically infeasible) “nadir” point, which contains the minimum value of each individual criterion on the Pareto-efficient set.⁷ In this approach, known as “compromise programming” (Zeleny 1974), the appropriate choice of the weights for the different criteria remains the critical point, and in our view, very little satisfying progress has been made in the assignment of weights without imposing subjectivity, which arguably amounts to picking a solution from the Pareto-efficient set. For instance, based on an exogenous ranking of the criterion importance, it is possible to apply an ordered weighted average (Yager 1988) which in turn can be related to compromise programming (Zarghami and Szidarovszky 2010, Wang and Fu 2020).⁸ Besides being subjective from the start by requiring the imposition of a preference order on criteria, this method does not provide any nontrivial performance guarantee relative to other choices of weights and/or importance rankings that might be plausible for other decision makers.⁹

1.2.2. Relative Robustness.

By contrast, we approach the selection of weights from the standpoint of relative robustness (Weber 2023), involving only comparisons between feasible points, thus avoiding fictitious targets such as the aforementioned utopian and nadir points. Rather than minimizing a fixed distance metric with specific weights, the proposed method evaluates performance in terms of the worst-case tradeoffs among criteria, ensuring a solution that remains defensible regardless of the exact weighting chosen ultimately. The underlying robustness measure, equivalent to relative regret, has been used in computer science to evaluate the relative performance of algorithms (Sleator and Tarjan 1985, Ben-David and Borodin 1994), for the scenario-based evaluation of operational decisions (Kouvelis and Yu 1997), in price discrimination (Han and Weber 2023), robust optimization (Weber 2024), and fair resource allocation (Goel et al. 2009). The idea of absolute regret (AR) is due to Savage (1951), based on the maximin robustness approach by Wald (1945) in his general treatment of sequential decision problems. The main issue with absolute regret is that it is inherently sensitive to the scale of the reference point. This sensitivity may make it impossible to derive reasonable performance guarantees, such as ensuring positive profits in a monopoly pricing problem with unknown demand (Weber 2025), because a zero-profit reference point is intrinsically small-scale. A relative robustness approach, we argue, yields more acceptable results, particularly in the context of multicriteria optimization, where changing the units of any given criterion would usually affect solutions that are based on absolute performance measures.

1.2.3. Connection to Distributionally Robust Optimization.

Because one can reinterpret a (normalized) weight vector as a probability distribution, our approach is naturally related to distributionally robust optimization (DRO), where the true probability distribution governing uncertain parameters is unknown but assumed to lie within a known ambiguity set. Important early contributions include Delage and Ye (2010), who studied DRO with moment-based sets, and Ben-Tal et al. (2013), who developed tractable reformulations for DRO with Wasserstein ambiguity sets. More recently, Blanchet and Murthy (2019) and Mohajerin Esfahani and Kuhn (2018) developed general formulations based on Wasserstein balls, offering strong out-of-sample guarantees. Whereas DRO typically focuses on expectations or risk measures over stochastic uncertainty, our approach generalizes worst-case robustness to a multicriteria setting without requiring a probabilistic model. This positions our proposed framework as a deterministic analogue to DRO, where ambiguity arises from unknown relative preferences and state-dependent criteria rather than unknown probability distributions.

1.2.4. Relative Evaluations.

The proposed approach to robust multicriteria decisions is entirely relative, in the sense that the question underlying all of our developments is “How well am I doing relative to how well I could be doing?” Indeed, the idea that the size of an object can be judged only relative to other objects goes back at least to the Taoist writings of Zhuang Zhou in the fourth century BC.¹⁰ In Economics, Cournot (1838) was among the first to note that there is no absolute value (“Il n’y a pas de valeurs absolues,” p. 22) and that inference from a social system can be likened to the observation of astronomical objects and their relative positions to each other (pp. 15–16), concluding by analogy that the concept of value is fundamentally relative (“Il y en a en ce sense que des valeurs relatives,” p. 18). In fact, human perception is inherently relative, as demonstrated by Weber (1846) and his student Fechner (1860) in extensive experiments which showed that across different senses (e.g., hearing, touch, and vision) the minimum perceptible difference is proportional to the current level of the stimulus, giving rise to the Weber-Fechner law of psychophysics. Similarly, in an economic context, it is often relative reference points such as one’s current wealth level (Kahneman and Tversky 1979) or the outcomes experienced by others (Loewenstein et al. 1989) that tend to determine human behavior. For example, when pondering whether to purchase a product from a cheaper store within walking distance, humans base decisions less on absolute gains than on the prospective relative savings in expenditure (Kahneman and Tversky 1984).

Beyond the aforementioned congruence with human perception, there are other practical arguments for relative evaluations. First, there is the insensitivity to scale, common to all relative measures such as internal rate of return (IRR), demand elasticity, profit margin, or relative regret, which allows for a direct comparison across different sizes. For example, with IRR one can readily benchmark projects of different financial magnitudes against outside investment options (of comparable risk), whereas the equivalent absolute indicator of net present value (NPV) remains silent about the required absolute investment (see, e.g., Weber 2014).¹¹ Second, normalization facilitates fairness and equity. When stakeholders have differing capacities or baselines, a relative comparison may help to ensure fairness (Goel et al. 2009). Third, relative evaluation criteria allow for comparability across contexts. For example, demand elasticity (as introduced by Marshall 1890, p. 162) is a unit-free relative measure that allows one to compare the changes of demand relative to price changes across widely differing goods, irrespective of the underlying unity of measurement (e.g., units of cars versus units of electric power). This last point is especially salient for multicriteria decision making, as the units (and inherent scale) for different criteria generally differ, so that a robustness criterion (and robust decision) should remain unaffected if the values of a given criterion are all multiplied by 10, for example. The relatively robust framework developed here uses a relative worst-case performance perspective. This methodology not only identifies solutions with reliable tradeoff characteristics, but also provides a transparent representation of the tradeoffs themselves. These tradeoffs are reflected in a robust weight vector consistent with a robust solution.

1.3. Outline

The remainder of this paper is organized as follows: Section 2 discusses the multicriteria optimization problem and associated comparative statics, together with the performance index for a robust evaluation of different decisions. Section 3 introduces pseudo-robustness and Pareto-efficiency, which together characterize relatively robust decisions. Here we also provide a computational approach for determining a relatively robust decision up to any given performance tolerance, and we allow for the possibility of close-to-arbitrary restrictions in the set of admissible weights, for example, based on a priori knowledge about physical constraints or a given priority ranking of criteria. In addition, there are extensions to criterion ambiguity and general aggregation of criteria, followed by a practical guide for how to apply the method. Section 4 focuses on discrete applications where our framework relies on virtually no assumptions, so the approach can be entirely data-driven. Section 5 concludes.

2. Basic Framework

Let $\mathcal{X}\subset {\mathbb{R}}^m$ be a nonempty, compact action set, for a given integer $m\ge 1$. Consider a decision maker who faces the multicriteria optimization problem of having to select an action (or decision, or point) $\mathit{x}\in \mathcal{X}$ so as to “simultaneously maximize” the continuous functions $f_i:\mathcal{X}\to {\mathbb{R}}_{+}$, for $i\in \mathcal{N}=\{1,\dots ,n\}$, each of which is referred to as a criterion, where $n\ge 1$ is a given integer.

Remark 1.

(i) The continuity of each criterion $f_i$ ensures that small perturbations in the action $\mathit{x}$ yield correspondingly small changes in the objective value. Notably, this assumption is trivially satisfied at any isolated point of $\mathcal{X}$.¹² In particular, this means that there is no imposed regularity requirement when the action set is finite. (ii) The requirement that each $f_i$ be nonnegative is without loss of generality: any real-valued (continuous) criterion ${\widehat{f}}_i$ can be translated as $f_i={\widehat{f}}_i-{\widehat{f}}_i^{•}\ge 0$, where ${\widehat{f}}_i^{•}={\min }_{\mathit{x}\in \mathcal{X}}{\widehat{f}}_i(\mathit{x})$ denotes the minimum value of ${\widehat{f}}_i$.¹³

2.1. Decision Problem

To evaluate goal achievement for any decision $\mathit{x}\in \mathcal{X}$, the decision maker considers a scalarization of his multicriteria optimization problem by means of a weighted objective,

where $\mathit{\lambda }=({\lambda }_1,\dots ,{\lambda }_n)\in \mathrm{\Delta }=\{\mathit{w}\in {\mathbb{R}}_{+}^n:\Vert \mathit{w}{\Vert }_1=1\}$ is a (normalized) vector of weights (with ${\lambda }_i\ge 0$ for all $i\in \mathcal{N}$, and ${\lambda }_1+\cdots +{\lambda }_n=1$). Maximizing the weighted objective yields the set of ex post optimal decisions,which by the extreme-value theorem (Rudin 1976, theorem 4.16, p. 89) is nonempty. Moreover, by the maximum theorem (Berge 1963, p. 116) the (set-valued) mapping $\mathcal{X}:\mathrm{\Delta }\rightrightarrows \mathcal{X}$ is compact-valued and upper semicontinuous.

Remark 2.

The weighted objective is homogeneous of degree one, that is, for all $\alpha >0$ and $\mathit{\lambda }\in \mathrm{\Delta }$, it is $F(·|\alpha \mathit{\lambda })=\alpha F(·|\mathit{\lambda })$, whereas the set of ex post optimal decisions is homogeneous of degree zero, in the sense that $\mathcal{X}(\alpha \mathit{\lambda })=\mathcal{X}(\mathit{\lambda })$, for all $\alpha >0$ and $\mathit{\lambda }\in \mathrm{\Delta }$. It is therefore possible to extend the definition of $F(\mathit{x}|·)$ in Equation (1) and the definition of $\mathcal{X}(·)$ in Equation (2) to a domain containing any nonzero weight vector $\mathit{w}\in {\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$, because a unique normalized weight $\mathit{\lambda }=\alpha \mathit{w}\in \mathrm{\Delta }$, with $\alpha =1/({\sum }_{i=1}^n{w}_i)>0$, is always available:¹⁴

$$\hspace{0.17em}F(\mathit{x}|\mathit{w})\triangleq (1/\alpha )F(\mathit{x}|\mathit{\lambda }),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X},$$(1′)

and

$$\mathcal{X}(\mathit{w})\triangleq \mathcal{X}(\mathit{\lambda })=\arg \underset{\mathit{x}\in \mathcal{X}}{\max }\hspace{0.17em}(1/\alpha )F(\mathit{x}|\mathit{\lambda }).$$(2′)

Remark 3.

Limited ambiguity, that is, allowing for weights in a (nonempty, compact) subset of $\mathrm{\Delta }$, is discussed in Section 3.7. Criterion ambiguity is treated in Section 3.8, and Section 3.9 investigates the use of general multicriteria objectives. All three generalizations can be treated, after suitable adjustment, within the basic framework.

To keep matters nontrivial, we assume that there exists a (feasible) default decision (${\mathit{x}}_d$) such that the decision maker’s weighted objective is positive, that is,

The nontriviality condition (N) ensures that the ex post optimal objective (or value function) is positive:

The sign-definiteness of the ex post optimal objective is critical for its role as a reference, against which the goal achievement of any feasible decision can be compared.

Remark 4.

The nontriviality condition (N) can be satisfied without affecting $\mathcal{X}(·)$, that is, without changing any set of ex post optimal decisions. It is sufficient to consider the translated criterion ${\widehat{f}}_i=f_i+s_i$ instead of $f_i$, for all $i\in \mathcal{N}$, using a suitable shift $\mathit{s}=(s_1,\dots ,s_n)\in {\mathbb{R}}_{++}^n$, in which case $\widehat{F}(\mathit{x}|\mathit{\lambda })=F(\mathit{x}|\mathit{\lambda })+c_0(\mathit{\lambda })>0$, because $c_0(\mathit{\lambda })=\mathit{\lambda }·\mathit{s}$ is strictly positive (as it is bounded from below by $\min \{s_1,\dots ,s_n\}>0$).

2.2. Comparative Statics

What happens to the criteria at the optimum when shifting weight from one criterion to another? At the optimum, one would naturally expect that a criterion which receives relatively more weight than before cannot decrease, whereas a criterion that receives relatively less weight cannot increase. The following result formalizes this intuition for a weight shift from one criterion to another.

Proposition 1.

Let ${\delta }_{ij}={\mathit{e}}_i-{\mathit{e}}_j$, where ${\mathit{e}}_i$ and ${\mathit{e}}_j$ denote the i-th and j-th Euclidean unit vectors, respectively. Consider $\mathit{\lambda },\widehat{\mathit{\lambda }}\in \mathrm{\Delta }$ such that $\widehat{\mathit{\lambda }}=\mathit{\lambda }+\epsilon {\delta }_{ij}$ for some $\epsilon >0$ and some $i,j\in \mathcal{N}$ with $i\ne j$. Then for any $(\mathit{x},\widehat{\mathit{x}})\in \mathcal{X}(\mathit{\lambda })\times \mathcal{X}(\widehat{\mathit{\lambda }})$ it is $f_i(\mathit{x})\le f_i(\widehat{\mathit{x}})$ and $f_j(\widehat{\mathit{x}})\le f_j(\mathit{x})$.

A transfer from the j-th criterion weight ${\lambda }_j$ to the i-th criterion weight ${\lambda }_i$ augments the optimal value of criterion i and lowers the optimal value of criterion j (at least weakly). Criteria other than i and j, whose weights remain constant but whose values are affected when decisions change, may go either way as a result of the weight shift. Similarly, the value function $F^{*}(\mathit{\lambda })$ could go up or down, depending primarily on the difference between $f_i$ and $f_j$ at the optimum.

Remark 5.

The conclusion of Proposition 1 can be applied multiple times. In particular, it can also be used for shifts in nonnormalized weights $\mathit{w}=(w_j,{\mathit{w}}_{-j})\in {\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$; see Remark 2. Thus, increasing $w_j$ is equivalent to (at most) $n-1$ successive weight transfers in the normalized weight from the components of ${\mathit{\lambda }}_{-j}$ to ${\lambda }_j$, resulting in an increase of the j-th criterion at the optimum.

Remark 6.

For small shifts of weight from criterion j to criterion i, the value of the ex post optimal objective $F^{*}(\mathit{\lambda })=F({\mathit{x}}^{*}(\mathit{\lambda })|\mathit{\lambda })$ goes up (resp., down) when the corresponding score difference, $f_i({\mathit{x}}^{*}(\mathit{\lambda }))-f_j({\mathit{x}}^{*}(\mathit{\lambda }))$, is positive (resp., negative), at the selection ${\mathit{x}}^{*}(\mathit{\lambda })\in \mathcal{X}(\mathit{\lambda })$.

The following result states that for a simple nonnormalized increase of the i-th weight, the value function goes up, as long as the i-th criterion is always positive at an optimum.

Lemma 1.

Let $\mathit{w},\widehat{\mathit{w}}\in {\mathbb{R}}^n\backslash \{\mathbf{0}\}$ be nonnormalized weights, such that $\widehat{\mathit{w}}=\mathit{w}+\delta {\mathit{e}}_i$, for a given $\delta >0$ and a given $i\in \mathcal{N}$. Provided that $f_i(\mathit{x})>0$, for any $\mathit{x}\in \mathcal{X}(\mathit{w})$, it is $F^{*}(\widehat{\mathit{w}})>F^{*}(\mathit{w})$.¹⁵

2.3. Performance Index

The decision maker may a priori have no knowledge about which weight $\mathit{\lambda }\in \mathrm{\Delta }$ should be used to compute the weighted objective $F(·|\mathit{\lambda })$ in Equation (1).¹⁶ To deal with this ambiguity, the decision maker evaluates any feasible decision $\mathit{x}\in \mathcal{X}$ with respect to any given weight $\mathit{\lambda }\in \mathrm{\Delta }$ by the performance ratio,¹⁷

which is continuous on $\mathcal{X}\times \mathrm{\Delta }$ and naturally bounded from above by one. Its minimum with respect to all possible weights (cf. Endnote 13),is called the performance index (evaluated at $\mathit{x}$). The performance index measures the performance of the action $\mathit{x}$ relative to all possible weighted-sum scalarizations of the decision maker’s multicriteria decision problem. It turns out that this performance criterion depends only on the maximized individual criteria,

By the nontriviality condition (N), it is $f_i^{*}=F^{*}({\mathit{e}}_i)\ge F({\mathit{x}}_d|{\mathit{e}}_i)>0$, for all $i\in \mathcal{N}$. Thus, all maximized individual criteria are strictly positive. The next result provides an important representation of the performance index, in terms of relative goal achievement of a decision, relative to the various maximized individual criteria.

Proposition 2.

The performance index is equal to

$$\rho (\mathit{x})=\underset{i\in \mathcal{N}}{\min }{\varphi }_i(\mathit{x}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X},$$(6)

where ${\varphi }_i(\mathit{x})=f_i(\mathit{x})/f_i^{*}\in [0,1]$, for all $(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}$.

The representation in Equation (6) expresses the performance index as the minimum of the criterion-specific performance ratios ${\varphi }_i$. Hence, to compute $\rho (\mathit{x})$ as the minimum of $\phi (\mathit{x}|\mathit{\lambda })$ over all weights $\mathit{\lambda }\in \mathrm{\Delta }$, it is sufficient to restrict attention to the Euclidean unit vectors ${\mathit{e}}_i\in \mathrm{\Delta }$, as ${\varphi }_i(·)=F(·|{\mathit{e}}_i)/F^{*}({\mathit{e}}_i)$, for all $i\in \mathcal{N}$. This simplification arises because $\phi (\mathit{x}|·)$ is quasiconcave for fixed $\mathit{x}$, implying that its minimum over $\mathrm{\Delta }$ is attained at a vertex. This highlights a “perfect complementarity” among the criterion-specific performance ratios in the determination of the performance index.

Remark 7.

The idea of perfect complementarity, discussed by Cournot (1838) and Edgeworth (1897), describes elements contributing to a common goal in fixed proportions. This is equivalent to evaluating the criterion (i.e., the performance index) using the Leontief production function,¹⁸ that is, taking the minimum among the inputs ${\varphi }_i$, resulting in $\rho (\mathit{x})={\min }_{i\in \mathcal{N}}{\varphi }_i(\mathit{x})$, for all $\mathit{x}\in \mathcal{X}$, as in Proposition 2.

Example 1.

Let $\mathcal{X}\subset {\mathbb{R}}_{+}^m$ be a compact action set with $\mathcal{X}\cap {\mathbb{R}}_{++}^m\ne \varnothing$, where $m\ge 2$ is an integer. Consider a decision problem with $n=2$ criteria (containing an egalitarian and a utilitarian evaluation),

$$f_1(\mathit{x})=\min \{x_1,\dots ,x_m\}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_2(\mathit{x})=(1/m)(x_1+\cdots +x_m),$$

for all $\mathit{x}=(x_1,\dots ,x_m)$, with maximized values $f_i^{*}=\max f_i(\mathcal{X})>0$, for $i\in \mathcal{N}$. The corresponding weighted objective becomes

$$F(\mathit{x}|\mathcal{\ell })=(1-\mathcal{\ell })f_1(\mathit{x})+\mathcal{\ell }{f}_2(\mathit{x}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathcal{\ell }\in [0,1],$$

where, for simplicity, the parameter $\mathcal{\ell }$ replaces the normalized weight $\mathit{\lambda }=(1-\mathcal{\ell },\mathcal{\ell })$. The optimal value, $F^{*}(\mathcal{\ell })=\max F(\mathcal{X}|\mathcal{\ell })$, can be obtained by solving a two-stage maximization problem,

$$F^{*}(\mathcal{\ell })=\underset{t\in \mathcal{T}}{\max }\left\{(1-\mathcal{\ell })t+\mathcal{\ell }\mu (t)\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathcal{\ell }\in [0,1].$$

Here $\mathcal{T}=[t_1,t_2]$ is a compact interval, with $0<t_1\le t_2<\infty$, and the best average coordinate (subject to all coordinates being at least of size t),

is a (weakly) decreasing function on $\mathcal{T}$, independent of $\mathcal{\ell }$. Because $t<\mu (t)$, for all $t\in \mathcal{T}$, it is clear that $F^{*}(\mathcal{\ell })$ must be increasing in $\mathcal{\ell }$. The interval $\mathcal{T}$ is adapted to the problem in the sense that at $t_1$ no constraint is imposed on the computation of the mean $\mu (t)$, and at $t_2$ the maximum of the smallest coordinate of any point in $\mathcal{X}$ is reached. Accordingly, one can verify that $t_2=F^{*}(0)=f_1^{*}<f_2^{*}=F^{*}(1)=\mu (t_1)$, where $\mathcal{X}(\mathcal{\ell })=\arg {\max }_{\mathit{x}\in \mathcal{X}}F(\mathit{x}|\mathcal{\ell })$ and $0<t_1={\min }_{\mathit{\xi }\in \mathcal{X}(1)}\{{\xi }_1,\dots ,{\xi }_m\}\le t_2$, where $\mathit{\xi }=({\xi }_1,\dots ,{\xi }_m)$; see Figure 1. By means of Proposition 2 the performance index can then be written in the form

Here the criterion-specific performance ratios are ${\varphi }_1=f_1/t_2$ and ${\varphi }_2=f_2/\mu (t_1)$.

3. Robust Multicriteria Decision Making

3.1. Pseudo-Robustness

We refer to a decision $\widehat{\mathit{x}}\in \mathcal{X}$ which maximizes the performance index $\rho (·)$ as pseudo-robust. The corresponding (compact, nonempty) set of pseudo-robust decisions,¹⁹

is not necessarily a singleton, as illustrated by the following example.

Example 2.

Optimizing the performance index determined in Example 1, using the same two-stage maximization procedure, yields the optimal performance index,

$${\rho }^{*}=\underset{\mathit{x}\in \mathcal{X}}{\max }\rho (\mathit{x})=\underset{t\in \mathcal{T}}{\max }\min \left\{\frac{t}{t_2},\frac{\mu (t)}{\mu (t_1)}\right\}.$$

Here, the first term in the minimand increases continuously in t, whereas the second is (weakly) decreasing and may be discontinuous; see Figure 2. As a result, at the optimal value $\widehat{t}\in \mathcal{T}=[t_1,t_2]$ the two terms are about to cross, resulting in a “balancedness condition,”

so that ${\rho }^{*}=\widehat{t}/t_2=\widehat{t}/f_1^{*}$, as shown in Figure 2. Because for any $t\in \mathcal{T}$ it is $\mu (t)\ge t$, one obtains $\mu (\widehat{t})\ge \mu (t_2)\ge t_2=f_1^{*}$. This implies that the optimal performance index,

is always nontrivial (i.e., strictly positive). It attains its maximum possible value (of one) if and only if $\mu (·)$ is constant on $\mathcal{T}$, that is, when $\mu (t_1)=\mu (t_2)$. The set of pseudo-robust actions, $\mathrm{\Psi }=\{\mathit{x}\in \mathcal{X}:\min \{x_1,\dots ,x_m\}=\widehat{t}/{\rho }^{*}\}$, may generally contain multiple elements.

3.2. Efficiency

When $\mathrm{\Psi }$ is not a singleton, the decision maker may have good reason to prefer one pseudo-robust decision over another, based on “efficiency” (or a lack thereof). Specifically, given two feasible actions $\mathit{x},{\mathit{x}}^{\prime }\in \mathcal{X}$, we say that decision ${\mathit{x}}^{\prime }$ is more efficient than decision $\mathit{x}$ with respect to the vector of criteria $\mathit{f}=(f_1,\dots ,f_n)$, if and only if ${\mathit{x}}^{\prime }$ strictly improves on at least one criterion while weakly improving on all other criteria (over their values achieved at $\mathit{x}$). The corresponding preference relation $\succ$ on $\mathcal{X}$ is defined by²⁰

for all $\mathit{x},{\mathit{x}}^{\prime }\in \mathcal{X}$. The resulting set of efficient (or Pareto-optimal) decisions is

It is easy to see that a pseudo-robust decision $\widehat{\mathit{x}}\in \mathrm{\Psi }$ is not necessarily efficient, in the sense that it may be possible to find a different decision which strictly improves on at least one criterion-specific performance ratio while maintaining the optimal performance index achieved by $\widehat{\mathit{x}}$.

Example 3.

Following up on our analysis in Example 1 and Example 2, consider the (nonconvex, compact) action set $\mathcal{X}={[0,1]}^m\backslash {(1/2,1]}^m$, for some integer $m\ge 2$. Because $\mu (t)=1-1/(2m)$, for all $t\in \mathcal{T}=[t_1,t_2]$, where $t_1\le t_2=f_1^{*}=1/2$ and $f_2^{*}=\mu (t_1)$, the optimality condition $\widehat{t}/t_2=\mu (\widehat{t})/\mu (t_1)=1$ yields $\widehat{t}=1/2$. Hence, the optimal performance index attains its maximum possible value, ${\rho }^{*}=1$. This makes sense, because it is feasible (in $\mathcal{X}$) to attain simultaneously the highest possible minimum coordinate and the highest possible average coordinate, regardless of the weights assigned to the two objectives. Note also that the set of pseudo-robust actions,

$$\mathrm{\Psi }=\left\{\mathit{x}\in \mathcal{X}:\min \{x_1,\dots ,x_m\}=\widehat{t}/{\rho }^{*}\right\}=\left\{\mathit{x}\in {[1/2,1]}^m:\exists \hspace{0.17em}i\in \mathcal{N}\hspace{0.17em} \text{s}.\text{t}. \hspace{0.17em}{x}_i=1/2\right\},$$

has a continuum of elements. For instance, compared with the pseudo-robust action $(1,\dots ,1)/2$, all other actions in $\mathrm{\Psi }$ are more efficient, particularly those in the set of Pareto-optimal decisions, $\mathcal{P}=\{(1,\dots ,1)-({\mathit{e}}_i/2):i\in \mathcal{N}\}$, which is a subset of $\mathrm{\Psi }$. For $m=2$, we obtain that $\mathcal{P}=\{(0.5,1),(1,0.5)\}$. Figure 3 depicts the situation, including the set of Pareto-optimal decisions, for $m=3$.

Remark 8.

It is well known that for any weight vector $\mathit{\lambda }$ with strictly positive components (ensuring all criteria are considered) an ex post optimal decision must also be efficient (see, e.g., Ehrgott 2010, proposition 3.9, p. 71). That is, for any $\mathit{\lambda }\in \text{int}(\mathrm{\Delta })$, the corresponding optimal decision set satisfies $\mathcal{X}(\mathit{\lambda })\subset \mathcal{P}$.

3.3. Robust Decision Set

Requiring efficiency in addition to pseudo-robustness is important, because, by avoiding unnecessary shortfalls, it can only improve the weighted objective in Equation (1)—at least weakly. A decision ${\widehat{\mathit{x}}}^{*}\in \mathcal{X}$ is called robust if it is pseudo-robust and efficient. Thus, our goal becomes to examine the properties of the robust decision set,

which contains all available robust decisions in $\mathcal{X}$, and to then determine a direct method for the computation of a robust decision (or an arbitrarily close approximation thereof).

Example 4.

Following up on our analysis in Example 1 and Example 2, consider the (convex, compact) action set $\mathcal{X}=\{\mathit{x}\in {\mathbb{R}}^m:\Vert \mathit{x}{\Vert }_2\le 1\}$, which is equal to the unit ball in the standard Euclidean distance, where the dimension of the underlying space is given by some integer $m\ge 2$. By direct computation, $\mu (t)=1/\sqrt{m}$ for all $t\in \mathcal{T}=[t_1,t_2]$, and $t_1=f_1^{*}=1/\sqrt{m}=f_2^{*}=\mu (t_2)=t_2$. As in Example 3, the balancedness condition (8) yields $\widehat{t}=t_2$, and thus an optimal performance index of ${\rho }^{*}=1$. The set of pseudo-robust actions,

$$\mathrm{\Psi }=\left\{\mathit{x}\in \mathcal{X}:\min \{x_1,\dots ,x_m\}=\widehat{t}/{\rho }^{*}\right\}=\left\{(1/\sqrt{m},\dots ,1/\sqrt{m})\right\},$$

is a singleton. On the other hand, the set of Pareto-optimal actions,

$$\mathcal{P}=\{\mathit{x}\in {\mathbb{R}}_{+}^m:\Vert \mathit{x}{\Vert }_2=1\},$$

contains a continuum of elements, including the only pseudo-robust decision, $(1,\dots ,1)/\sqrt{m}$.

In Example 3, we examined a problem where $\mathcal{P}⊊\mathrm{\Psi }$ (so $\mathcal{R}=\mathcal{P}$), whereas in Example 4, for the same multicriteria decision problem with a different action set, it was $\mathrm{\Psi }⊊\mathcal{P}$ (so $\mathcal{R}=\mathrm{\Psi }$). Neither of these two extremes might apply, in which case $\mathcal{R}\notin \{\mathrm{\Psi },\mathcal{P}\}$, as illustrated next.

Example 5.

Mixing and matching features from Example 3 and Example 4, let us consider the (nonconvex, compact) action set $\mathcal{X}=\{\mathit{x}\in {\mathbb{R}}^2:\Vert \mathit{x}{\Vert }_2\le 1\}\backslash {(1/2,1]}^2$ in the Euclidean plane. The corresponding set of Pareto-optimal actions is given by the intersection of the unit circle with both the action set $\mathcal{X}$ and the positive quadrant ${\mathbb{R}}_{+}^2$, so $\mathcal{P}=\{\mathit{x}\in \mathcal{X}\cap {\mathrm{ℝ}}_{+}^2:\Vert \mathit{x}{\Vert }_2=1\}$. Meanwhile, the set of pseudo-robust actions is $\mathrm{\Psi }=\{\mathit{x}\in \mathcal{X}:\min \{x_1,x_2\}=1/2\}$. Thus, by Equation (11) it is $\mathcal{R}=\mathrm{\Psi }\cap \mathcal{P}=\{(1/2,\sqrt{3}/2),(\sqrt{3}/2,1/2)\}$, which in this setting means $\mathcal{R}\notin \{\mathrm{\Psi },\mathcal{P}\}$.

The existence of robust actions is ensured by the next result, together with the fact that the robust decision set $\mathcal{R}$ must be closed and bounded (i.e., compact).

Lemma 2.

The robust decision set $\mathcal{R}$ is nonempty and compact.

The proof of Lemma 2 starts by noting that the set $\mathcal{P}$ of efficient actions is compact because it must be bounded (by the boundedness of the encompassing action set $\mathcal{X}$) and closed (by the continuity of the criteria). One can then construct a sequence of pseudo-robust actions in the (nonempty) compact set $\mathrm{\Psi }$ which might be inefficient (or else $\mathcal{R}\ne \varnothing$ holds true immediately). But each inefficient pseudo-robust action suggests the existence of a more efficient action, which incidentally must also be pseudo-robust. Given that $\mathrm{\Psi }$ is compact, the Bolzano-Weierstrass theorem (Berge 1963, p. 67) then guarantees the existence of a converging subsequence of pseudo-robust actions, with a limit that must be a feasible decision which is both pseudo-robust and efficient, so $\mathcal{R}\ne \varnothing$. Compactness of the robust decision set then follows, as it has to be both closed and bounded.

Lemma 3.

The optimal performance index ρ* is such that $\max \rho (\mathcal{P})={\rho }^{*}=\max \rho (\mathcal{X})$ and $\rho (\mathcal{R})=\rho (\mathrm{\Psi })=\{{\rho }^{*}\}$.

The preceding result (re)states the fact that the decision maker can restrict attention to efficient actions when maximizing the performance index, meaning that there always exists an efficient action which attains the optimal performance index ${\rho }^{*}$; this action is by definition robust. Conversely, any robust action necessarily achieves a performance index of ${\rho }^{*}$, which is quite straightforward in light of both Lemma 2 and the definition of the robust decision set in Equation (11).

3.4. Robust Decisions: Computation

How can one determine a (relatively) robust decision? By Lemma 3 we can limit attention to maximizing the performance index over all efficient decisions in our search for robust decisions. Thus, to be guaranteed an efficient decision which is also approximately pseudo-robust, by virtue of Proposition 2 we introduce the $\epsilon$-augmented performance index,

so that ${\mathrm{\Phi }}_0=\rho ={\min }_{i\in \mathcal{N}}{\varphi }_i$, and ${\mathrm{\Phi }}_1=(1/n){\sum }_{i\in \mathcal{N}}{\varphi }_i$. Because ${\mathrm{\Phi }}_{\epsilon }(·)$ is a continuous function, its maximizer, referred to as the set of $\epsilon$-robust actions,is nonempty and compact, again by virtue of the extreme-value theorem and the maximum theorem. It is useful, for our further analysis, to shift the perspective from the available decisions in $\mathcal{X}$ to their respective consequences (or “outcomes”) in terms of their robustness performance, given the prevailing weight ambiguity.

Remark 9.

The (nonempty, compact) outcome space,

$$\mathcal{Y}=\left\{\mathit{y}\in {[0,1]}^n:y_i={\varphi }_i(\mathit{x}),\hspace{0.17em}(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}\right\},$$(14)

contains the vectors $\mathit{y}$ of criterion-specific performance ratios, achieved by the available decisions $\mathit{x}$, so $\mathcal{Y}=\varphi (\mathcal{X})$, where $\varphi =({\varphi }_1,\dots ,{\varphi }_n)$. As several feasible decisions might result in the same score vector, the mapping $\varphi :\mathcal{X}\to \mathcal{Y}$ may not be one-to-one. Consider now the $\epsilon$-augmented performance index in the outcome space,

$${\widehat{\mathrm{\Phi }}}_{\epsilon }(\mathit{y})=(1-\epsilon )\underset{i\in \mathcal{N}}{\min }{y}_i+(\epsilon /n){\displaystyle \sum _{i\in \mathcal{N}}{y}_i},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(\mathit{y},\epsilon )\in \mathcal{Y}\times {\mathbb{R}}_{+}.$$

Using ideas from Example 1, maximization of this weighted objective yields

where $\mathcal{T}=[t_1,t_2]$ is a suitable compact interval, anddenotes the average (criterion-specific) performance ratio. The interval boundaries $t_1,t_2$ of $\mathcal{T}$, with $0<t_1\le t_2\le 1$, are given by

In addition, one can easily verify that

Proposition 1 implies that any selection $t_{\epsilon }\in {\mathcal{T}}_{\epsilon }=\arg {\max }_{t\in \mathcal{T}}\{(1-\epsilon )t+\epsilon \widehat{\mu }(t)\}$ must be decreasing in $\epsilon$, and similarly, $\widehat{\mu }(t_{\epsilon })$ must be increasing in $\epsilon$, at least weakly. The latter also follows directly from the monotonicity of $t_{\epsilon }$, because $\widehat{\mu }(t)$ must be nonincreasing in t.

Lemma 4.

Let $\epsilon \in (0,1]$. (i) ${ℛ}_0=\mathrm{\Psi }$. (ii) ${ℛ}_{\epsilon }\subset \mathcal{P}$. (iii) If $\mathit{x}\in {ℛ}_{\epsilon }\backslash \mathrm{\Psi }$ and $\widehat{\mathit{x}}\in \mathrm{\Psi }$, then there exists ${\epsilon }^{\prime }\in (0,\epsilon )$ such that ${\mathrm{\Phi }}_{\widehat{\epsilon }}(\mathit{x})<{\mathrm{\Phi }}_{\widehat{\epsilon }}(\widehat{\mathit{x}})$, for all $\widehat{\epsilon }\in [0,{\epsilon }^{\prime }]$.

Part (i) of the preceding result notes that without $\epsilon$-augmentation (i.e., for $\epsilon =0$) the weighted objective ${\mathrm{\Phi }}_{\epsilon }$ specializes to the performance index (via Proposition 2), the maximization of which produces the set of pseudo-robust decisions, as in Equation (7). Part (ii) stipulates that whenever the $\epsilon$-augmentation is nontrivial (i.e., for $\epsilon >0$), maximization yields efficient actions. Finally, part (iii) means that if a decision $\mathit{x}$ maximizes the $\epsilon$-augmented performance index in Equation (12), for some nontrivial $\epsilon \in (0,1]$, without being pseudo-robust, then any pseudo-robust decision $\widehat{\mathit{x}}\in \mathrm{\Psi }$ would strictly improve upon $\mathit{x}$ in terms of any $\widehat{\epsilon }$-augmented performance index, as long as $\widehat{\epsilon }$ (smaller than $\epsilon$) lies in a sufficiently small right-neighborhood of the origin.

We are now able to establish a cornerstone property for the practice of robust multicriteria optimization, in the sense that a robust decision (i.e., an element of $\mathcal{R}$) can be obtained as a lower limit of the set of $\epsilon$-robust actions, for $\epsilon \to 0^{+}$.²¹ In Section 3.5, we then show that $\epsilon >0$ can be chosen so as to guarantee an approximation of the optimal performance ${\rho }^{*}$ up to any desired precision.

Proposition 3.

Let $\mathcal{Q}={\underset{¯}{\text{Lim}}}_{\epsilon \to 0^{+}}{ℛ}_{\epsilon }$. Then $\mathcal{Q}\subset \mathcal{R}$ and $\mathcal{Q}\ne \varnothing$.

The following example shows that it is possible that $\mathcal{Q}⊊\mathcal{R}$. In other words, some points in the robust decision set might not be reached using the proposed approximation procedure.

Example 6.

Consider a (nonconvex, compact) action set as shown in Figure 4(a), specified by

$$\mathcal{X}=\left\{\mathit{x}\in [0,1]\times {\mathbb{R}}_{+}:x_2\le 2+\sqrt{1-x_1}\right\}\cup \{(2,\frac{3}{2})\}.$$

There are $n=2$ criteria that simply measure the coordinate achievement, so $f_i(\mathit{x})=x_i$, for all $(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}$. The set of efficient actions is

whereas the set of pseudo-robust actions is given by $\mathrm{\Psi }=\{(1,x_2):1\le x_2\le 2\}\cup \{(2,\frac{3}{2})\}$. By its definition in Equation (11) the set of robust actions can be obtained as the intersection of the preceding sets:

At this point, let us consider the $\epsilon$-augmented performance index in Equation (12). By Lemma 4 we have ${ℛ}_0=\mathrm{\Psi }$, and for $\epsilon \in (0,1]$ the maximizer of ${\mathrm{\Phi }}_{\epsilon }$ is efficient, so

where ${\varphi }_i=f_i/f_i^{*}$, for $i\in \mathcal{N}$, with $(f_1^{*},f_2^{*})=(2,3)$, and ${\mathrm{\Phi }}_{\epsilon }(2,\frac{3}{2})=(2+\epsilon )/4$. By direct computation one finds that the maximizer of the (nontrivially) $\epsilon$-augmented performance index,²²is a singleton. Proposition 3 then guarantees that the lower limit of this maximizer is robust:

The fact that $(2,\frac{3}{2})$ is an isolated point (cf. Endnote 12) is not important, because one could easily connect it to $\mathcal{X}$ by adding points to the action set that are always suboptimal.²³

3.5. Approximation Error of $\epsilon$-Robust Decisions

Maximizing the $\epsilon$-augmented performance index ${\mathrm{\Phi }}_{\epsilon }$ in Equation (12) enables us to approximate a robust decision to an arbitrary prespecified precision, as a function of $\epsilon \in (0,1]$. The quality of any approximate decision ${\mathit{x}}_{\epsilon }\in {ℛ}_{\epsilon }$ in Equation (11) is gauged by its approximation error,

that is, the difference between the optimal performance index, ${\rho }^{*}=\max \rho (\mathcal{X})$, and the achieved performance index, ${\rho }_{\epsilon }=\rho ({\mathit{x}}_{\epsilon })$. This quantity measures the loss in robustness from using the $\epsilon$-approximation rather than the true robust decision. For the remainder of this discussion, we keep the selection ${\mathit{x}}_{\epsilon }\in {ℛ}_{\epsilon }$ fixed, for all $\epsilon \in [0,1]$. If we denote by ${\mu }_{\epsilon }=\mu ({\mathit{x}}_{\epsilon })=(1/n){\sum }_{i\in \mathcal{N}}{\varphi }_i({\mathit{x}}_{\epsilon })$ the average performance ratio achieved by ${\mathit{x}}_{\epsilon }$, the optimal $\epsilon$-augmented performance index becomes

The following result guarantees continuity, as well as first- and second-order monotonicity of the optimal $\epsilon$-augmented performance index.

Lemma 5.

${\mathrm{\Phi }}_{\epsilon }^{*}$ is continuous, increasing, and convex in $\epsilon \in [0,1]$.

The proof of the first-order monotonicity uses the fact that ${\mu }_{\epsilon }\ge {\rho }_{\epsilon }$ (for all $\epsilon \in [0,1]$), together with natural properties of an optimal solution to Equation (13) in order to establish that ${\mathrm{\Phi }}_{\epsilon }^{*}$ in Equation (16) must increase in $\epsilon$ (at least weakly). That the optimal $\epsilon$-augmented performance index also exhibits a second-order monotonicity means that tightening the approximation parameter further and further leads to progressively slower decreases of ${\mathrm{\Phi }}_{\epsilon }^{*}$ toward ${\mathrm{\Phi }}_0^{*}={\rho }^{*}$ (as $\epsilon \to 0^{+}$). The convexity of the optimal $\epsilon$-augmented performance index also implies its smoothness (almost everywhere), as pointed out next.

Remark 10.

By the Rademacher theorem (Villani 2008, theorem 10.8, p. 222), the convexity of ${\mathrm{\Phi }}_{\epsilon }^{*}$ in $\epsilon \in [0,1]$, which implies Lipschitz continuity, guarantees that the optimal $\epsilon$-augmented performance index is differentiable almost everywhere (a.e.) on [0, 1]. The Alexandrov theorem (Villani 2008, theorem 14.25, p. 402) goes even further by establishing its second-order differentiability a.e., in the sense of having a Taylor expansion with a smaller-than-quadratic local error at almost every point $\epsilon \in [0,1]$. By the envelope theorem (see, e.g., Mas-Colell et al. 1995, theorem M.L.1, pp. 965–966), at points of differentiability one therefore obtains:

$$\frac{d{\mathrm{\Phi }}_{\epsilon }^{*}}{d\epsilon }={\mu }_{\epsilon }-{\rho }_{\epsilon }\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\dot{\forall }\epsilon \in [0,1].$$

In other words, the gradient of the optimal $\epsilon$-augmented performance index is (a.e.) equal to the difference between the average performance ratio (${\mu }_{\epsilon }$) and the minimum performance ratio (${\rho }_{\epsilon }$) attained by the approximately robust decision ${\mathit{x}}_{\epsilon }$.

The maximized weighted objective in Equation (16) is a convex combination of ${\rho }_{\epsilon }$ and ${\mu }_{\epsilon }$; the latter both exhibit “natural” comparative statics as implied by the reweighting result in Proposition 1.

Lemma 6.

(i) ${\rho }_{\epsilon }$ is decreasing in $\epsilon \in [0,1]$. (ii) ${\rho }^{*}\ge {\rho }_{\epsilon }$, for all $\epsilon \in [0,1]$. (iii) ${\lim }_{\epsilon \to 0^{+}}{\rho }_{\epsilon }={\rho }^{*}$. (iv) ${\mu }_{\epsilon }$ is increasing in $\epsilon \in [0,1]$. (v) ${\mu }_{\epsilon }\ge {\rho }^{*}$, for all $\epsilon \in [0,1]$.

Parts (i) and (iv) of Lemma 6 assert that decreasing the augmentation parameter $\epsilon$ can only decrease the average criterion-specific performance ratio ${\mu }_{\epsilon }$ and at the same time increase the performance index ${\rho }_{\epsilon }$, where both are achieved at a given $\epsilon$-robust selection ${\mathit{x}}_{\epsilon }$. Meanwhile, by parts (ii) and (v) the optimal performance ratio ${\rho }^{*}=\max \rho (\mathcal{X})$ is bracketed by these two values, in the sense that ${\rho }_{\epsilon }\le {\rho }^{*}\le {\mu }_{\epsilon }$, for all $\epsilon \in [0,1]$. Finally, parts (i) and (iii) establish the monotone convergence of ${\rho }_{\epsilon }$ to ${\rho }^{*}$, as $\epsilon \to 0^{+}$.

The following result provides a performance guarantee for any $\epsilon$-approximation of our robust multicriteria decision problem, alluded to at the outset of our discussion.

Proposition 4.

Fix any $\delta >0$. If $\epsilon \le \delta /{\mu }_1$, then ${\psi }_{\epsilon }\le \delta$.

A special case of the preceding result is that for any $\delta >0$ the approximation error ${\psi }_{\delta /{\mu }_1}$ cannot exceed $\delta$. In other words, any $\epsilon$-robust decision ${\mathit{x}}_{\epsilon }$, for $\epsilon =\delta /{\mu }_1$, attains a performance index ${\rho }_{\epsilon }\in [{\rho }^{*}-\delta ,{\rho }^{*}]$, where ${\rho }^{*}$ is the optimal performance index. The following example applies this result in an already familiar context.

Example 7.

Somewhat similar to Example 6, we consider the (nonconvex, compact) action set

$$\mathcal{X}=\left\{\mathit{x}\in [0,1]\times {\mathbb{R}}_{+}:x_2\le 2+\sqrt{1-x_1}\right\}\cup \{(3,\frac{1}{2})\},$$

together with the same $n=2$ criteria $f_i(\mathit{x})=x_i$, for all $(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}$. Consequently, the set of efficient actions is $\mathcal{P}=\left\{\mathit{x}\in [0,1]\times {\mathbb{R}}_{+}:x_2=2+\sqrt{1-x_1}\right\}\cup \{(3,\frac{1}{2})\}$, and the set of pseudo-robust actions is $\mathrm{\Psi }=\{(1,x_2):1\le x_2\le 2\}$. Thus, by Equation (11) the set of robust actions amounts to $\mathcal{R}=\mathrm{\Psi }\cap \mathcal{P}=\{(1,2)\}$, which is a singleton. By Lemma 4 the $\epsilon$-augmented performance index ${\mathrm{\Phi }}_{\epsilon }$ in Equation (12) is efficient (i.e., ${ℛ}_{\epsilon }\subset \mathcal{P}$), for all $\epsilon \in (0,1]$, and ${ℛ}_0=\mathrm{\Psi }$. In particular, the optimal value of the $\epsilon$-augmented performance index is

$${\mathrm{\Phi }}_{\epsilon }^{*}=\underset{\mathit{x}\in \mathcal{P}}{\max }{\mathrm{\Phi }}_{\epsilon }(\mathit{x})=\max \left\{\underset{x_1\in [0,1]}{\max }\left\{\frac{(1-\epsilon )x_1}{3}+\frac{\epsilon }{2}\left(\frac{x_1}{3}+\frac{2+\sqrt{1-x_1}}{3}\right)\right\},{\mathrm{\Phi }}_{\epsilon }(3,\frac{1}{2})\right\},$$

where ${\varphi }_i=f_i/f_i^{*}$, for $i\in \mathcal{N}$, with $(f_1^{*},f_2^{*})=(3,3)$, and ${\mathrm{\Phi }}_{\epsilon }(3,\frac{1}{2})=(2+5\epsilon )/12$. For sufficiently small $\epsilon$, the maximizer,

$${ℛ}_{\epsilon }=\left\{\left(1-{\left(\frac{\epsilon /2}{2-\epsilon }\right)}^2,2+\frac{\epsilon /2}{2-\epsilon }\right)\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<\epsilon <\frac{4}{7}\left(2-\frac{1}{\sqrt{2}}\right)\approx 0.7388,$$

is single-valued, achieving ${\mathrm{\Phi }}_{\epsilon }^{*}=(1/24)(16-3{\epsilon }^2)/(2-\epsilon )$.²⁴ By Proposition 3 the lower limit of this maximizer for $\epsilon \to 0^{+}$ is robust, and in our case:

$$\mathcal{Q}={\underset{¯}{\text{Lim}}}_{\epsilon \to 0^{+}}{ℛ}_{\epsilon }=\{(1,2)\}=\mathcal{R};$$

see Figure 4(b). Note also that ${ℛ}_1=\{(3,\frac{1}{2})\}$, which in turn yields the average criterion-specific performance ratio achieved by the decision ${\mathit{x}}_{\epsilon }\in {ℛ}_{\epsilon }$, for $\epsilon =1$:

$${\mu }_1=\frac{1}{2}{\left(\frac{x_1}{f_1^{*}}+\frac{x_2}{f_2^{*}}\right)|}_{(x_1,x_2)=(3,\frac{1}{2})}=\frac{7}{12}\approx 0.5833.$$

Thus, to guarantee that the approximation error ${\psi }_{\epsilon }$ in Equation (15) cannot exceed $\delta =5\%$, it is by Proposition 4 enough to find an action ${\mathit{x}}_{\epsilon }$ that maximizes the $\epsilon$-augmented performance index ${\mathrm{\Phi }}_{\epsilon }$ in Equation (12) for some $\epsilon \in (0,\delta /{\mu }_1]\approx (0,8.6\%]$. Incidentally, for $\epsilon =8\%$, we find ${\rho }_{\epsilon }\approx 33.32\%$, so that the realized approximation error of ${\psi }_{\epsilon }\approx 0.01\%$ (with respect to ${\rho }^{*}=1/3$) is actually much smaller than the prespecified 5% approximation-error bound.

It is also possible to derive a priori performance estimates without knowledge of the optimal performance index (${\rho }^{*}$) just by solving the $\epsilon$-approximation problem in Equation (13) for some admissible $\epsilon$. Indeed, Lemma 6 (ii) and Lemma 5 together imply that

For any $\epsilon \in [0,1]$, let us now consider the midpoint estimator,

The latter means that ${\widehat{\rho }}_{\epsilon }$ effectively approximates ${\rho }^{*}$, for $\epsilon \to 0^{+}$.

Lemma 7.

$|{\widehat{\rho }}_{\epsilon }-{\rho }^{*}|\le \epsilon ({\mu }_{\epsilon }-{\rho }_{\epsilon })/2$, for all $\epsilon \in [0,1]$.

Because ${\mu }_{\epsilon }-{\rho }_{\epsilon }\in [0,1]$, a somewhat simpler (though generally less precise) approximation inequality than the one given in the preceding Lemma 7 is $|{\widehat{\rho }}_{\epsilon }-{\rho }^{*}|\le \epsilon /2$, for all $\epsilon \in [0,1]$.

Remark 11.

For a robust decision $\widehat{\mathit{x}}\in \mathcal{R}$ let $\widehat{\mathit{y}}=\left(f_1\right(\widehat{\mathit{x}}),\dots ,f_n(\widehat{\mathit{x}}\left)\right)$; in addition, let ${\mathit{y}}^{*}=(f_1^{*},\dots ,f_n^{*})$ be the utopian point in the outcome space. Because by construction ${\widehat{y}}_i\ge {\rho }^{*}{y}_i^{*}$, for all $i\in \mathcal{N}$, it is

$${\rho }^{*}\ge \underset{¯}{\rho }=\sup \left\{r\in [0,1]:r{\mathit{y}}^{*}\in \mathcal{Y}\right\},$$

where $\sup \hspace{0.17em}\varnothing =0$. In the balanced case, where $f_i(\widehat{\mathit{x}})={\rho }^{*}{f}_i^{*}$, for all $i\in \mathcal{N}$, the lower bound becomes tight (i.e., ${\rho }^{*}=\underset{¯}{\rho }$). However, in general $\underset{¯}{\rho }$ may not be very reliable (e.g., generically worse than the performance index ${\rho }_d=\rho ({\mathit{x}}_d)$, attained by the default decision ${\mathit{x}}_d$).

3.6. Robust Weights

3.6.1. General Case.

An interesting and useful byproduct of a robust decision $\widehat{\mathit{x}}\in \mathcal{R}$ is its associated robust (normalized) weight,

where the positive constantdenotes the harmonic mean of the criterion scores at the robust decision. By construction one obtains that the contribution to the robustly weighted objective $F(·|\widehat{\mathit{\lambda }})$, evaluated at the robust decision $\widehat{\mathit{x}}$, is equal for all criteria, in the sense that

When the action set is convex, then the chosen robust action naturally also maximizes the robustly weighted criterion, that is, $\widehat{\mathit{x}}\in \mathcal{X}(\widehat{\mathit{\lambda }})$. One can think of the robust weight as an endogenous belief that can be used to evaluate the expected criterion achievement. It defines the tradeoffs compatible with the robust decision, where the latter was found while remaining agnostic over all possible weights (in $\mathrm{\Delta }$).

3.6.2. Balanced Case.

In the case where $f_i(\widehat{\mathit{x}})={\rho }^{*}{f}_i^{*}$, for all $i\in \mathcal{N}$ (cf. Remark 11), we have

whereis the harmonic mean of the maximum criterion scores. The balancedness condition then becomes

In the balanced case, the robust weight can therefore be determined based on the maximized individual criteria alone. If all maximized individual criteria are equal, the robust weight becomes uniform. Indeed, $f_i^{*}=c>0$, for all $i\in \mathcal{N}$, implies that $h^{*}=c/n$, so $\widehat{\mathit{\lambda }}=(1/n,\dots ,1/n)$.²⁵

Example 8.

In Example 1, we determined the value function $F^{*}(\mathcal{\ell })=(1-\mathcal{\ell })\widehat{t}+\mathcal{\ell }\mu (\widehat{t})$, so that by Equations (19′) and (20′) in this balanced case (as established in Example 2):

$$\widehat{\mathit{\lambda }}=\left(1-\widehat{\mathcal{\ell }},\widehat{\mathcal{\ell }}\right)=\left(\frac{\mu (\widehat{t})}{\widehat{t}+\mu (\widehat{t})},\frac{\widehat{t}}{\widehat{t}+\mu (\widehat{t})}\right)=\left(\frac{f_2^{*}}{f_1^{*}+f_2^{*}},\frac{f_1^{*}}{f_1^{*}+f_2^{*}}\right).$$

Generically, it is $F^{*}(\widehat{\mathcal{\ell }})\ne F(\widehat{\mathit{x}}|\widehat{\mathcal{\ell }})$, even when the decision set is convex. To see this, let $m=n=2$ and consider the convex domain, $\mathcal{X}=\{\mathit{x}\in {\mathbb{R}}_2^{+}:{\nu }_1{x}_1+{\nu }_2{x}_2\le 1\}$, where the constants ${\nu }_1,{\nu }_2$ are such that $0<{\nu }_1<{\nu }_2<\infty$. Then $\mu (t)=(1-{\nu }_{2}t)/{\nu }_1+t$, so that ${\mu }^{\prime }(t)=1-({\nu }_2/{\nu }_1)<0$. Meanwhile, $F(t|\mathcal{\ell })=(1-\mathcal{\ell })t+\mathcal{\ell }\mu (t)$, for $t\in [0,f_1^{*}]=[0,1/({\nu }_1+{\nu }_2)]$, and

Therefore, $F^{*}(\mathcal{\ell })=F(t^{*}(\mathcal{\ell })|\mathcal{\ell })=(\mathcal{\ell }/{\nu }_1)+\max \{0,1-(\mathcal{\ell }{\nu }_2/{\nu }_1)\}{f}_1^{*}$, where

and $f_1^{*}=1/({\nu }_1+{\nu }_2)$. Because $f_2^{*}=1/{\nu }_1$, we find ${\nu }_1/{\nu }_2=f_1^{*}/(f_2^{*}-f_1^{*})$. By virtue of the fact that $\widehat{\mathcal{\ell }}=f_1^{*}/(f_1^{*}+f_2^{*})<{\nu }_1/{\nu }_2$, it is $t^{*}(\widehat{\mathcal{\ell }})=f_1^{*}$. The balancedness condition,is equivalent to $\widehat{t}/f_1^{*}=1-(\widehat{t}/f_1^{*})$, so $\widehat{t}=f_1^{*}/2$. Hence, we find that $\widehat{t}\ne t^{*}(\widehat{\mathcal{\ell }})$. Let us check the corresponding performance index. Indeed,

Note also,

That is, the robust weight puts more than twice as much emphasis on the Rawlsian (or egalitarian) objective as on the utilitarian objective. One can conclude that in general there is no “robustness equivalence principle,” in the sense that substituting the robust parameter into the original scalarization (via maximization of the corresponding weighted objective) might not lead to a robust decision.²⁶

Example 9.

Consider the robust allocation of two resources to $n=2$ agents. The total amount of each resource has been normalized to one. Allocations are determined as decisions $\mathit{x}=(x_1,x_2)\in {[0,1]}^2$, under which agent 1 obtains $x_1$ of resource 1 and $x_2$ of resource 2, whereas agent 2 obtains $1-x_1$ of resource 1 and $1-x_2$ of resource 2. Agent 1’s utility is $f_1(\mathit{x})=x_1^{\alpha }{x}_2^{1-\alpha }$, and agent 2’s utility is $f_2(\mathit{x})={(1-x_1)}^{\beta }{(1-x_2)}^{1-\beta }$, where $\alpha ,\beta \in (0,1)$ are given scalars. Figure 5 provides an illustration in the corresponding Edgeworth box (see, e.g., Pareto 1906, p. 187). Because a robust allocation is necessarily Pareto-optimal, both agents’ marginal rates of substitution for the two goods must be equal, so $\mathit{x}\in \mathcal{P}$ if and only if

$$\frac{\partial f_1(\mathit{x})/\partial x_1}{\partial f_1(\mathit{x})/\partial x_2}=\frac{\alpha }{1-\alpha }\frac{x_2}{x_1}=\frac{\beta }{1-\beta }\frac{1-x_2}{1-x_1}=\frac{\partial f_2(\mathit{x})/\partial x_1}{\partial f_2(\mathit{x})/\partial x_2}.$$(22)

In addition, a robust allocation must maximize the performance ratio, and one can verify that this requires the boundary performance ratios to be equal (i.e., a balancedness condition), so

because $f_1^{*}=f_2^{*}=1$. In the interesting special case where $\alpha +\beta =1$, Equations (22) and (23) together yield the unique robust solution $\widehat{\mathit{x}}=(\alpha ,\beta )$, resulting in an optimal performance index of ${\rho }^{*}={\alpha }^{\alpha }{\beta }^{\beta }={\alpha }^{\alpha }{(1-\alpha )}^{1-\alpha }$, with the last expression being a strictly convex function in $\alpha \in (0,1)$ achieving its minimum (of $1/2$) at $\alpha =1/2$ and ${\rho }^{*}$ going to one as $\alpha \to \{0^{+},1^{-}\}$.²⁷

3.7. Limited Ambiguity

Consider the (nonempty, compact) subset $\mathrm{\Omega }\subset {\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$ which may reflect the decision maker’s a priori knowledge about the relevant weights for the problem at hand, and let

be the corresponding $\mathrm{\Omega }$-conditioned performance index. The different weight combinations considered feasible may not necessarily be normalized, especially at an initial stage in practice when information about reasonable weight ranges is being compiled. The function $\pi :{\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}\to \mathrm{\Delta }$, with $\pi (\mathit{w})=\mathit{w}/\Vert \mathit{w}{\Vert }_1$, describes the radial projection of any nonzero weight $\mathit{w}$ in ${\mathbb{R}}_{+}^n$ onto the unit simplex $\mathrm{\Delta }$. It allows for the normalization under weight ambiguity; see Remark 2. The radial projection is a nonlinear function which is homogeneous of degree zero (i.e., $\pi (\alpha \mathit{w})=\pi (\mathit{w})$, for all $\alpha >0$ and all $\mathit{w}\in {\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$). It can be represented using the conical closure (see, e.g., Berge 1963, p. 14), $\mathcal{C}(\mathrm{\Omega })=\{\widehat{\mathit{w}}\in {\mathbb{R}}_{+}^n:\widehat{\mathit{w}}=\alpha \mathit{w},\hspace{0.17em}\mathit{w}\in \mathrm{\Omega },\hspace{0.17em}\alpha \in {\mathbb{R}}_{+}\}$, as described, among other useful properties of $\pi (·)$, in the following auxiliary result.

Lemma 8.

Let $\mathrm{\Omega }$ be a (nonempty, compact) subset of ${\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$. Then (i) (a) $\pi (\mathrm{\Omega })=\mathcal{C}(\mathrm{\Omega })\cap \mathrm{\Delta }$ is nonempty and compact, and (b) $\pi (\mathrm{\Omega })=\pi (\partial \mathrm{\Omega })$. (ii) If ${\mathrm{\Omega }}^{\prime }\subset \mathrm{\Omega }$ is open (in ${\mathbb{R}}_{+}^n$), then $\pi ({\mathrm{\Omega }}^{\prime })$ is open (in $\mathrm{\Delta }$). (iii) If ${\mathrm{\Omega }}^{\prime }\subseteq \mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\prime }\ne \varnothing$, then $\partial \pi ({\mathrm{\Omega }}^{\prime })=\partial \pi (\partial {\mathrm{\Omega }}^{\prime })$ is nonempty and compact. (iv) If $\mathrm{\Omega }$ is convex, then $\pi (\mathrm{\Omega })$ is convex. (v) If ${\mathrm{\Omega }}^{\prime }\subset \mathrm{\Omega }$ is a straight line segment, then $\pi ({\mathrm{\Omega }}^{\prime })$ is a straight line segment (or a point). (vi) $\pi (\text{co}(\mathrm{\Omega }))=\text{co}(\pi (\partial \mathrm{\Omega }))$. (vii) If $\mathrm{\Omega }={\mathrm{\Omega }}^{\prime }\cup {\mathrm{\Omega }}^{″}$ for some ${\mathrm{\Omega }}^{\prime },{\mathrm{\Omega }}^{″}\subset {\mathbb{R}}_{+}^n$, then $\pi (\mathrm{\Omega })=\pi ({\mathrm{\Omega }}^{\prime })\cup \pi ({\mathrm{\Omega }}^{″})$. (viii) If $\mathrm{\Omega }={\mathrm{\Omega }}^{\prime }\cap {\mathrm{\Omega }}^{″}$ for some ${\mathrm{\Omega }}^{\prime },{\mathrm{\Omega }}^{″}\subset {\mathbb{R}}_{+}^n$, then $\pi (\mathrm{\Omega })=\mathcal{C}({\mathrm{\Omega }}^{\prime }\cap {\mathrm{\Omega }}^{″})\cap \mathrm{\Delta }$.

Part (i) of Lemma 8 notes that (a) the radial projection of $\mathrm{\Omega }$ onto $\mathrm{\Delta }$ can be obtained by simply intersecting the conical closure of $\mathrm{\Omega }$ with $\mathrm{\Delta }$, and (b) one can limit attention to the boundary $\partial \mathrm{\Omega }$ (ignoring interior points of $\mathrm{\Omega }$). Although a continuous function generally does not map open sets to open sets (e.g., a constant function would not), part (ii) guarantees that $\pi (·)$ does exactly that, provided the “interiority” of a point in $\mathrm{\Omega }$ is assessed in ${\mathbb{R}}_{+}^n$ and then, after its projection, in the lower-dimensional $\mathrm{\Delta }$. Part (iii) ensures that weight ambiguity in any (nonempty) subset ${\mathrm{\Omega }}^{\prime }$ of $\mathrm{\Omega }$ leads to a compact domain $\partial \pi (\partial {\mathrm{\Omega }}^{\prime })$ for a robust decision according to Proposition 5 below. Part (iv) establishes that convex sets are projected to convex sets.²⁸ Following (v) and (vi), the radial projection leaves straight-line geometries intact, thus, for example, converting (bounded) polyhedra in ${\mathbb{R}}_n^{+}$ to polytopes in $\mathrm{\Delta }$. Finally, parts (vii) and (viii) note that the radial projection of a union of sets is the union of the corresponding single-set projections, but the same does generally not apply to an intersection.²⁹

Example 10.

Assume the (nonnormalized) weights $\mathit{w}$ considered reasonable by the decision maker are such that each of its components $w_i$, for $i\in \mathcal{N}$, is known to lie in some interval, resulting in a rectangular ambiguity set,

$$\mathrm{\Omega }=[\underset{¯}{\mathit{w}},\overline{\mathit{w}}]={\displaystyle \prod _{i\in \mathcal{N}}}[{\underset{¯}{w}}_i,{\overline{w}}_i],$$

with bounds $\underset{¯}{\mathit{w}}=({\underset{¯}{w}}_1,\dots ,{\underset{¯}{w}}_n)\ne \mathbf{0}$ and $\overline{\mathit{w}}=({\overline{w}}_1,\dots ,{\overline{w}}_n)\ge \underset{¯}{\mathit{w}}$. Using Lemma 8, leveraging the preservation of straight-line geometries, it can be verified that the radial projection of $\mathrm{\Omega }$ onto $\mathrm{\Delta }$ has the form

$$\pi (\mathrm{\Omega })=\text{co}\left(\left\{\frac{\underset{¯}{\mathit{w}}+{\delta }_1{\mathit{e}}_1}{\Vert \underset{¯}{\mathit{w}}{\Vert }_1+{\delta }_1},\dots ,\frac{\underset{¯}{\mathit{w}}+{\delta }_n{\mathit{e}}_n}{\Vert \underset{¯}{\mathit{w}}{\Vert }_1+{\delta }_n}\right\}\right),$$(25)

where $\mathit{\delta }=({\delta }_1,\dots ,{\delta }_n)=\overline{\mathit{w}}-\underset{¯}{\mathit{w}}$ and $\Vert \underset{¯}{\mathit{w}}{\Vert }_1={\sum }_{l=1}^n{\underset{¯}{w}}_l$. As an example, Figure 6 illustrates that the radial projection of a box in $n=3$ dimensions is a triangle in $\mathrm{\Delta }$. Thus, the shape of $\pi (\mathrm{\Omega })$ depends on (at most) n of the original $2^n$ vertices of $\mathrm{\Omega }$. Moreover, if the modulus of the lower bound of the (nontrivial) box-shaped ambiguity set $\mathrm{\Omega }$ becomes very small, then the situation tends to become equivalent to full ambiguity. To see this, consider a $\overline{\mathit{w}}$ with only positive components and let $\underset{¯}{\mathit{w}}=\epsilon {\mathit{e}}_1$, for a sufficiently small $\epsilon >0$. Then, by Equation (25) it is $\pi (\mathrm{\Omega })=\text{co}(\{{\mathit{\lambda }}^{(1)},\dots ,{\mathit{\lambda }}^{(n)}\})$, where ${\mathit{\lambda }}^{(1)}={\mathit{e}}_1$ and ${\mathit{\lambda }}^{(i)}=(\epsilon {\mathit{e}}_1+{\overline{w}}_i{\mathit{e}}_i)/(\epsilon +{\overline{w}}_i)$, for $i\in \{2,\dots ,n\}$. Hence, ${\lim }_{\epsilon \to 0^{+}}{\mathit{\lambda }}^{(i)}={\mathit{e}}_i$, for all $i\in \mathcal{N}$, implying full ambiguity in the limit, as $\mathrm{\Delta }=\text{co}(\{{\mathit{e}}_1,\dots ,{\mathit{e}}_n\})=\pi ({\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\})$.

Under limited ambiguity, a coordinate-wise decomposition is no longer available. However, because $\phi (\mathit{x}|·)$ is quasiconcave, for any given $\mathit{x}\in \mathcal{X}$, as established in the proof of Proposition 2, the upper contour sets of the performance ratio in the space of weights are necessarily convex. This in turn allows restricting attention, for the representation of the performance index, to the extreme points of the convex hull of the ambiguity set, which yields a representation much in the same spirit as before.

Proposition 5.

The $\mathrm{\Omega }$-conditioned performance index in Equation (24) is such that

$$\rho (\mathit{x}|\mathrm{\Omega })=\underset{\mathit{\lambda }\in \partial \pi (\partial \mathrm{\Omega })}{\min }\phi (\mathit{x}|\mathit{\lambda })=\underset{\mathit{\lambda }\in \mathrm{\Lambda }}{\min }\phi (\mathit{x}|\mathit{\lambda }),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X},$$

for some “extremal base” $\mathrm{\Lambda }$ which is a (compact) subset of $\partial \pi (\partial \mathrm{\Omega })$, such that $\text{co}(\mathrm{\Lambda })=\text{co}(\pi (\mathrm{\Omega }))$.

Although choosing $\mathrm{\Lambda }=\partial \pi (\mathrm{\Omega })=\partial \pi (\partial \mathrm{\Omega })$ (cf. Lemma 8 (i) (b)) is always possible, it is usually advantageous to opt for the smallest possible extremal base $\mathrm{\Lambda }$, so it consists only of the “extreme points” of $\text{co}(\pi (\mathrm{\Omega }))$, where the latter cannot be represented as convex combinations of other points in $\mathrm{\Lambda }$ (see, e.g., Rockafellar 1970, section 18, p. 162).³⁰ If $\mathrm{\Omega }$ is finite or a (finite) polytope, then the smallest extremal base $\mathrm{\Lambda }$ of extreme points of $\text{co}(\pi (\mathrm{\Omega }))$ is also finite.³¹ Because any bounded convex set can be approximated by a finite polytope (Bronstein 2008), a finite extremal base can be used to represent the convex hull of the given ambiguity set (or its radial projection onto $\mathrm{\Delta }$) up to any desired precision. That $\pi (\mathrm{\Omega })$ may be nonconvex is not important because the level sets of the performance ratio are convex, so that the minima of $\phi (\mathit{x}|·)$ on $\pi (\mathrm{\Omega })$ are attained on the boundary of its convexification, $\text{co}(\pi (\mathrm{\Omega }))$.

Example 11.

Consider the robust allocation of resources to $n=2$ agents as in Example 9, given the box-shaped ambiguity set $\mathrm{\Omega }$ as in Example 11, with its radial projection specified by Equation (25),

$$\pi (\mathrm{\Omega })=\text{co}(\mathrm{\Lambda })=\text{co}\left(\left\{\frac{({\overline{w}}_1,{\underset{¯}{w}}_2)}{{\overline{w}}_1+{\underset{¯}{w}}_2},\frac{({\underset{¯}{w}}_1,{\overline{w}}_2)}{{\underset{¯}{w}}_1+{\overline{w}}_2}\right\}\right)⊊\mathrm{\Delta },$$

where $\mathrm{\Lambda }=\{{\mathit{\lambda }}^{(1)},{\mathit{\lambda }}^{(2)}\}$. Thus, by virtue of Proposition 5 the $\mathrm{\Omega }$-conditioned performance index becomes

$$\rho (\mathit{x}|\mathrm{\Omega })=\min \left\{\frac{{\overline{w}}_1{f}_1(\mathit{x})+{\underset{¯}{w}}_2{f}_2(\mathit{x})}{{\max }_{\widehat{\mathbf{x}}\in \mathcal{X}}\left\{{\overline{w}}_1{f}_1(\widehat{\mathit{x}})+{\underset{¯}{w}}_2{f}_2(\widehat{\mathit{x}})\right\}},\frac{{\underset{¯}{w}}_1{f}_1(\mathit{x})+{\overline{w}}_2{f}_2(\mathit{x})}{{\max }_{\widehat{\mathbf{x}}\in \mathcal{X}}\left\{{\underset{¯}{w}}_1{f}_1(\widehat{\mathit{x}})+{\overline{w}}_2{f}_2(\widehat{\mathit{x}})\right\}}\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X}.$$

For a pseudo-robust decision $\mathit{x}$ (in $\mathrm{\Psi }$), balancedness must hold, so that, by substituting $f_1,f_2$ from Example 9,

Meanwhile, a feasible allocation decision $\mathit{x}=(x_1,x_2)$ is Pareto-optimal (in $\mathcal{P}$) if and only if it satisfies Equation (22), as before. When the box-shaped ambiguity set becomes smaller, the elements of the extremal base $\mathrm{\Lambda }$ become more similar and coincide in the limit (i.e., ${\mathit{\lambda }}^{(1)}-{\mathit{\lambda }}^{(2)}\to \mathbf{0}$ as $\Vert \overline{\mathit{w}}-\underset{¯}{\mathit{w}}{\Vert }_1\to 0^{+}$). Figure 7 shows the robust allocation $\widehat{\mathit{x}}=\widehat{\mathit{x}}(\mathrm{\Omega })$ on the contract curve between the Pareto-optimal allocations ${\mathit{x}}^{*}({\mathit{\lambda }}^{(i)})$ that maximize the weighted objective $F(\mathit{x}|{\mathit{\lambda }}^{(i)})$ for $i\in \mathcal{N}$. Under full ambiguity (i.e., for $\mathrm{\Omega }=\mathrm{\Delta }$), it is ${\mathit{\lambda }}^{(i)}\equiv {\mathit{e}}_i$; see Example 9.

Example 12.

In some applications, the weights are naturally ranked by importance (see, e.g., Wang and Fu 2020). The ordered set of weights, ${\mathrm{\Omega }}^{\prime }=\{(w_1,\dots ,w_n)\in {\mathbb{R}}_{+}^n:w_1\ge w_2\ge \cdots \ge w_n\}=\mathcal{C}({\mathrm{\Omega }}^{\prime })$, has a radial projection onto $\mathrm{\Delta }$ of the form $\pi ({\mathrm{\Omega }}^{\prime })=\text{co}(\{{\mathit{\lambda }}^{\prime (i)}:i\in \mathcal{N}\})$, where ${\mathit{\lambda }}^{\prime (i)}=\frac{1}{i}{\sum }_{j=1}^i{\mathit{e}}_j$, for all $i\in \mathcal{N}$. Combining this importance-ranking with a box-shaped ambiguity set ${\mathrm{\Omega }}^{″}=[\underset{¯}{\mathit{w}},\underset{¯}{\mathit{w}}+\mathit{\delta }]$ as in Example 10, with $\pi ({\mathrm{\Omega }}^{″})=\text{co}(\{{\mathit{\lambda }}^{″(i)}:i\in \mathcal{N}\})$, where ${\mathit{\lambda }}^{″(i)}=(\underset{¯}{\mathit{w}}+{\delta }_i{\mathit{e}}_i)/(\Vert \underset{¯}{\mathit{w}}{\Vert }_1+{\delta }_i)$, for all $i\in \mathcal{N}$, leads to

$$\mathrm{\Lambda }=\left\{{𝟙}_{\left\{\underset{¯}{\mathit{w}}+{\delta }_i{\mathit{e}}_i\notin {\mathrm{\Omega }}^{\prime }\right\}}\hspace{0.17em}{\mathit{\lambda }}^{\prime (i)}+{𝟙}_{\left\{\underset{¯}{\mathit{w}}+{\delta }_i{\mathit{e}}_i\in {\mathrm{\Omega }}^{\prime }\right\}}\hspace{0.17em}{\mathit{\lambda }}^{″(i)}:i\in \mathcal{N}\right\}.$$

This extremal base (of cardinality n) suits practical applications where a decision maker disposes of plausible ranges for the weights to be placed on the criteria, together with a ranking of their importance (cf. Section 4.3.2).

Remark 12.

(i) Limiting ambiguity can only increase robustness performance. That is, if $\mathrm{\Omega },\widehat{\mathrm{\Omega }}\subset {\mathbb{R}}_{+}^n\backslash \{\mathbf{0}\}$ are nonempty and compact and satisfy $\pi (\widehat{\mathrm{\Omega }})\subseteq \pi (\mathrm{\Omega })$, then $\rho (·|\mathrm{\Omega })\le \rho (·|\widehat{\mathrm{\Omega }})$, which follows directly from the definition of the $\mathrm{\Omega }$-conditioned performance index in Equation (24). (ii) In the absence of ambiguity, the optimal robustness performance has to be maximal. That is, if $\widehat{\mathrm{\Omega }}=\{\mathit{w}\}$, then ${\rho }^{*}(\widehat{\mathrm{\Omega }})={\max }_{\mathit{x}\in \mathcal{X}}\rho (\mathit{x}|\{\mathit{w}\})=F^{*}(\mathit{w})/F^{*}(\mathit{w})=1$.

3.8. Criterion Ambiguity

Given a (nonempty, compact) action set $\mathcal{X}\subset {\mathbb{R}}^m$ as in Section 2, we now allow for ambiguity in each of the $n^{\prime }$ criteria, given by the continuous functions $g_i:\mathcal{X}\times \mathrm{\Theta }\to {\mathbb{R}}_{+}$, for $i\in {\mathcal{N}}^{\prime }=\{1,\dots ,n^{\prime }\}$, which map $(\mathit{x},\mathit{\theta })$ to real numbers, where $\mathit{x}\in \mathcal{X}$ is a feasible action and $\mathit{\theta }$ is an ex ante unknown state in the (nonempty, compact) state space $\mathrm{\Theta }\subset {\mathbb{R}}^l$, and where $l,m,n^{\prime }\ge 1$ are given integers. The unknown state $\mathit{\theta }$ represents the decision maker’s uncertainty about the exact value that each of his $n^{\prime }$ objectives may attain under a chosen action $\mathit{x}$.

Remark 13.

(i) Continuity of the criteria in the state implies that a small perturbation in the state can have only a small impact on the decision maker’s objective. This continuity is automatically satisfied when the state space $\mathrm{\Theta }$ is finite.³² (ii) The unknown state may introduce dependencies between the different $g_i$. Indeed, even if $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_{n^{\prime }})$ (for $l=n^{\prime }$) and criteria are such that $g_i(\mathit{x},\mathit{\theta })\equiv g_i(\mathit{x},{\theta }_i)$, for all $i\in {\mathcal{N}}^{\prime }$, then the different criteria’s ambiguity may still be linked via common constraints in $\mathrm{\Theta }$.³³ But if in addition the state space is a Cartesian product, so $\mathrm{\Theta }={\mathrm{\Theta }}_1\times \cdots {\mathrm{\Theta }}_{n^{\prime }}$, and ${\theta }_i\in {\mathrm{\Theta }}_i$, for all $i\in \mathcal{N}$, then the decoupling of ambiguity across the different criteria is complete, in the sense that observing the part of the state that determines one criterion does not reduce ambiguity for any other criterion.

To assess goal achievement of an action in a given state, the decision maker considers a scalarization of his multicriteria optimization problem by means of a weighted objective,

where ${\mathrm{\Delta }}^{\prime }=\{{\mathit{w}}^{\prime }=(w_1,\dots ,w_{n^{\prime }})\in {\mathbb{R}}_{+}^{n^{\prime }}:w_1+\cdots +w_{n^{\prime }}=1\}$ denotes the set of all (normalized) weights. As in the nontriviality condition (N) (cf. Section 2), we assume that there exists a default decision (${\mathit{x}}_d$) such that the decision maker’s weighted objective is positive across all states, that is,

The modified nontriviality condition (N′) ensures that the ex post optimal objective is always positive, so

for all $(\mathit{\theta },{\mathit{\lambda }}^{\prime })\in \mathrm{\Theta }\times {\mathrm{\Delta }}^{\prime }.$

Remark 14.

The modified nontriviality condition (N′) can always be satisfied, without altering the set of ex post optimal decisions, by using a (positive, translated) criterion ${\widehat{g}}_i=g_i+\epsilon$, for some $\epsilon >0$; see also Remark 4.

Given a (nonempty, compact) ambiguity set ${\mathrm{\Omega }}^{\prime }\subset {\mathbb{R}}_{+}^{n^{\prime }}$, the decision maker’s robustness objective, as in Section 3.7, is to maximize the $({\mathrm{\Omega }}^{\prime },\mathrm{\Theta })$-conditioned performance index,

The following result recasts the robustness objective into a by-now-familiar representation.

Proposition 6.

Let ${\mathrm{\Lambda }}^{\prime }\subset {\mathrm{\Delta }}^{\prime }$ be an extremal base of $\text{co}(\pi ({\mathrm{\Omega }}^{\prime }))$. Then

$$\rho (\mathit{x}|{\mathrm{\Omega }}^{\prime },\mathrm{\Theta })=\underset{{\mathit{\lambda }}^{\prime }\in {\mathrm{\Lambda }}^{\prime }}{\min }\left\{\underset{\mathit{\theta }\in \mathrm{\Theta }}{\min }\frac{G(\mathit{x},\mathit{\theta }|{\mathit{\lambda }}^{\prime })}{G^{*}(\mathit{\theta }|{\mathit{\lambda }}^{\prime })}\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X}.$$(27)

Based on Proposition 6, if the (smallest) extremal base is finite, we can reduce the general multicriteria decision problem to our basic framework in Section 2, with full weight ambiguity and no criterion ambiguity.

Corollary 1.

Assume that there exists a (finite, smallest) integer $n\ge 1$ so that ${\mathrm{\Lambda }}^{\prime }=\{{\mathit{\lambda }}^{\prime (1)},\dots ,{\mathit{\lambda }}^{\prime (n)}\}$ and $\text{co}({\mathrm{\Lambda }}^{\prime })=\text{co}(\pi ({\mathrm{\Omega }}^{\prime }))$, and let $f_i(\mathit{x})={\min }_{\mathit{\theta }\in \mathrm{\Theta }}\{G(\mathit{x},\mathit{\theta }|{\mathit{\lambda }}^{\prime (i)})/G^{*}(\mathit{\theta }|{\mathit{\lambda }}^{\prime (i)})\}$, for all $(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}$. Then

$$\rho (\mathit{x})=\underset{i\in \mathcal{N}}{\min }\left\{\frac{f_i(\mathit{x})}{f_i^{*}}\right\},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X},$$(28)

where $f_i^{*}={\max }_{\mathit{x}\in \mathcal{X}}{f}_i(\mathit{x})=1$, for all $i\in \mathcal{N}=\{1,\dots ,n\}$, represents $\rho (·|{\mathrm{\Omega }}^{\prime },\mathrm{\Theta })$ in Equation (27).

Example 13.

(i) In the case of full ambiguity, it is ${\mathrm{\Lambda }}^{\prime }={\mathrm{\Delta }}^{\prime }$, so that $n=n^{\prime }$ and ${\mathrm{\Delta }}^{\prime }=\text{co}(\{{\mathit{e}}_1,\dots ,{\mathit{e}}_n\})=\mathrm{\Delta }$, with $f_i(\mathit{x})={\min }_{\mathit{\theta }\in \mathrm{\Theta }}\{g_i(\mathit{x},\mathit{\theta })/G^{*}(\mathit{\theta }|{\mathit{e}}_i)\}$, for all $(\mathit{x},i)\in \mathcal{X}\times \mathcal{N}$. (ii) Consider now a multicriteria decision problem under full ambiguity, with criteria of the form $g_i(\mathit{x},\mathit{\theta })={({\widehat{g}}_i(\mathit{x}))}^{{\theta }_i}$, for all $i\in \mathcal{N}$, where the state $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_n)$ is only known to lie in the (nonempty, compact) set $\mathrm{\Theta }\subset {\mathbb{R}}_{+}^n$ (with $\mathrm{\Theta }\ne \{\mathbf{0}\}$ to avoid trivialities), and where the functions ${\widehat{g}}_i:\mathcal{X}\to {\mathbb{R}}_{++}$ are continuous. Let ${\widehat{g}}_i^{*}=\max {\widehat{g}}_i(\mathcal{X})>0$ and ${\overline{\theta }}_i={\max }_{\mathit{\theta }\in \mathrm{\Theta }}{\theta }_i$; furthermore, set

$$f_i(\mathit{x})={\left(\frac{{\widehat{g}}_i(\mathit{x})}{{\widehat{g}}_i^{*}}\right)}^{{\overline{\theta }}_i}=\underset{\mathit{\theta }\in \mathrm{\Theta }}{\min }\left(\frac{{\widehat{g}}_i^{{\theta }_i}(\mathit{x})}{\underset{\mathit{x}\in \mathcal{X}}{\max }{\widehat{g}}_i^{{\theta }_i}(\mathit{x})}\right),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{x}\in \mathcal{X},$$

for all $i\in \mathcal{N}$, taking into account that ${\xi }^{\theta }$ decreases in $\theta$, for all $\xi \in [0,1]$. The extreme state $\overline{\mathit{\theta }}=({\overline{\theta }}_1,\dots ,{\overline{\theta }}_n)$ determines the robust choice, as it leads ceteris paribus to the smallest criterion values—thus following a precautionary principle. Then, by Corollary 1 we follow our basic framework, maximizing the performance index in Equation (28) with respect to all Pareto-optimal decisions in Equation (10). For instance, if $\mathrm{\Theta }=\mathrm{\Delta }$, the reduction to the weighted objective in Equation (1) obtains with $f_i={\widehat{g}}_i/{\widehat{g}}_i^{*}$, for all $i\in \mathcal{N}$.

Remark 15.

(i) The minimization over the state space in Example 13 reflects a robust, precautionary stance: the decision maker evaluates actions under the most adverse plausible realization of the state with respect to the achievable performance ratio, consistent with a notion of relative worst-case robustness. (ii) The treatment of criterion ambiguity in this section also bears a conceptual resemblance to models of DRO, where decisions are evaluated against worst-case distributions within a specified ambiguity set. In our framework, the uncertainty is not over probability distributions but over states $\mathit{\theta }\in \mathrm{\Theta }$, with performance assessed under the worst-case realization of both the state and the weights. This can be viewed as a nonprobabilistic analogue of DRO, where the ambiguity set ${\mathrm{\Omega }}^{\prime }$ over the weights plays a role similar to ambiguity sets over distributions in DRO models. In particular, the two-layer minimization in Equation (27)—over weights and states—mirrors the inner DRO minimization over distributions, highlighting a parallel structure between worst-case evaluation across distributional and multicriteria settings.

3.9. General Multicriteria Objectives

In certain practical applications, it may seem appealing to consider alternatives to the arithmetic mean in Equation (1), such as a harmonic or geometric mean, with suitable weights. One could even think of using a (weighted) power mean which would accommodate each of the earlier options as a special case; see Example 14 below, which illustrates aggregation ambiguity. Rather than commit to a particular functional form for aggregating multiple criteria, however, we propose a general class of multicriteria objectives that adhere to a set of reasonable axioms. These axioms reduce (in the case of uniform weights) to those known to characterize the quasiarithmetic mean. A key finding, notably, is that maximizing this general weighted objective is entirely equivalent to maximizing the arithmetic mean objective in Equation (1), provided the criteria are suitably transformed.

Let $H:\mathcal{Y}\times \mathrm{\Delta }\to \mathbb{R}$ denote a (continuous) multicriteria objective, where $\mathcal{Y}\subseteq {\mathbb{R}}^n$ is a (nonempty) domain such that $\widehat{\mathit{f}}(\mathcal{X})\subset \mathcal{Y}$, with the vector of criteria $\widehat{\mathit{f}}=({\widehat{f}}_1,\dots ,{\widehat{f}}_n)$, and where $\mathrm{\Delta }\subset {\mathbb{R}}_{+}^n$ is the unit simplex; see Section 2.1. For any decision $\mathit{x}\in \mathcal{X}$ and weight $\mathit{\lambda }\in \mathrm{\Delta }$, the multicriteria objective produces an overall score $\widehat{H}(\mathit{x}|\mathit{\lambda })=H(\widehat{\mathit{f}}(\mathit{x})|\mathit{\lambda })$, which the decision maker would like to maximize by choosing an appropriate decision—in the presence of (possibly limited) ambiguity about $\mathit{\lambda }$ (and possibly also about $\widehat{\mathit{f}}$) as discussed in Section 3.7 (and Section 3.8). To guide the form of a sufficiently flexible and interpretable objective, we posit five axioms (Axioms 1–5), which nest those proposed by Kolmogorov (1930) in his seminal work “On the notion of mean.”

Axiom 1

(Monotonicity). For all $\mathit{y}=(y_1,\dots ,y_n)\in \mathcal{Y}$,  $\widehat{\mathit{y}}=({\widehat{y}}_1,\dots ,\widehat{y})\in \mathcal{Y}$,  $\mathit{\lambda }=({\lambda }_1,\dots ,{\lambda }_n)\in \mathrm{\Delta }$, and $i\in \mathcal{N}$:

$$\widehat{\mathit{y}}-\mathit{y}=({\widehat{y}}_i-y_i){\mathit{e}}_i,\hspace{0.17em}{\widehat{y}}_i>y_i\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\{\begin{array}{ll}H(\widehat{\mathit{y}}|\mathit{\lambda })>H(\mathit{y}|\mathit{\lambda }),\hfill & \text{if} {\lambda }_i>0,\hfill \\ H(\widehat{\mathit{y}}|\mathit{\lambda })=H(\mathit{y}|\mathit{\lambda }),\hfill & \text{if} {\lambda }_i=0.\hfill \end{array}$$

Axiom 2

(Symmetry). For all $\mathit{y}=(y_1,\dots ,y_n)\in \mathcal{Y}$,  $\mathit{\lambda }=({\lambda }_1,\dots ,{\lambda }_n)\in \mathrm{\Delta }$, and $i,j\in \mathcal{N}$:

$$(\widehat{\mathit{y}},\widehat{\mathit{\lambda }})=(\mathit{y},\mathit{\lambda })+(y_i-y_j,{\lambda }_i-{\lambda }_j)({\mathit{e}}_j-{\mathit{e}}_i)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}H(\widehat{\mathit{y}}|\widehat{\mathit{\lambda }})=H(\mathit{y}|\mathit{\lambda }).$$

Axiom 3

(Reflexivity). $H(\alpha {\mathbf{1}}_n|\mathit{\lambda })=\alpha$, for all $\alpha >0$ (with $\alpha {\mathbf{1}}_n\in \mathcal{Y}$) and $\mathit{\lambda }\in \mathrm{\Delta }$.³⁴

Axiom 4

(Associativity). For all $\mathit{y}=(y_1,\dots ,y_n)\in \mathcal{Y}$,  $\mathit{\lambda }=({\lambda }_1,\dots ,{\lambda }_n)\in \mathrm{\Delta }$, and $m\in \mathcal{N}\backslash \{n\}$:³⁵

$$({\lambda }_1,\dots ,{\lambda }_m)\ne \mathbf{0},\hspace{0.17em}H\left({\displaystyle \sum _{i=1}^m{y}_i}{\mathit{e}}_i|\frac{\sum _{i=1}^m{\lambda }_i{\mathit{e}}_i}{\sum _{i=1}^m{\lambda }_i}\right)=\alpha \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}H(\alpha {\mathbf{1}}_m,y_{m+1},\dots ,y_n|\mathit{\lambda })=H(\mathit{y}|\mathit{\lambda }).$$

Axiom 5

(Coordinate Filter). $H(\mathit{y}|{\mathit{e}}_i)=y_i$, for all $\mathit{y}=(y_1,\dots ,y_n)\in \mathcal{Y}$ and $i\in \mathcal{N}$.

The significance of these five basic requirements is as follows: Monotonicity (Axiom 1) means that as long as the weight component ${\lambda }_i$ is positive, increasing the value $y_i$ of criterion $i\in \mathcal{N}$ must also increase the weighted objective, whereas for ${\lambda }_i=0$ the weighted objective becomes insensitive to criterion i; symmetry (Axiom 2) requires that the weighted objective is invariant with respect to any joint permutation of indices belonging to criteria and their associated weights; reflexivity (Axiom 3) imposes score-consistency in the sense that if all criterion scores are identical, then that should also be the value of the weighted objective, no matter what (normalized) weights are applied; in a similar vein, associativity (Axiom 4) postulates that replacing a group of inputs with their internal weighted average leaves the overall aggregation unchanged; finally, in order to guarantee that the weights provide a homotopic relation between all criteria in isolation, the weighted objective is a coordinate filter (Axiom 5) if it yields the i-th component of $\mathit{y}$ when putting all weight on the i-th coordinate (i.e., for $\mathit{\lambda }={\mathit{e}}_i$).

Based on Kolmogorov’s result,³⁶ we define the general h-mean,

where the kernel $h:\mathcal{D}\to \mathbb{R}$ is a continuous strictly monotonic function, defined on a suitable domain $\mathcal{D}\subset \mathbb{R}$.

Lemma 9.

The general h-mean $H:\mathcal{Y}\times \mathrm{\Delta }\to \mathbb{R}$ in Equation (29) satisfies Axioms 1–5.

The following example illustrates the flexibility afforded by the general h-mean.

Example 14.

Consider two well-known averages.

1. Let $\mathcal{Y}={\mathbb{R}}_{++}^n$. For any weight $\mathit{\lambda }\in \mathrm{\Delta }$ and power parameter $p\in \mathbb{R}$, define the power mean:

   $$M_p(\mathit{y}|\mathit{\lambda })=\{\begin{array}{ll}{\left({\sum }_{i=1}^n{\lambda }_i{y}_i^p\right)}^{1/p},\hfill & p\ne 0,\hfill \\ {\prod }_{i=1}^n{y}_i^{{\lambda }_i},\hfill & p=0,\hfill \end{array}$$
   which corresponds to Equation (29) with $h(y)=y^p$ for $p\ne 0$, and $h(y)=\log y$ for $p=0$. This recovers the harmonic ($p=-1$), geometric ($p=0$), and arithmetic ($p=1$) means, among others. In the limit, the power mean approaches the minimum (for $p\to -\infty$) or maximum (for $p\to +\infty$) of all $y_i$ with positive weights. Importantly, $M_p$ is increasing in p (see, e.g., Hardy et al. 1934, theorem 16, p. 26) and uniquely satisfies homogeneity: $M_p(\alpha \mathit{y}|\mathit{\lambda })=\alpha M_p(\mathit{y}|\mathit{\lambda })$ for all $\alpha >0$ (see, e.g., Hardy et al. 1934, theorem 84, p. 68).³⁷

2. For $h(·)=\exp (·)$, the general h-mean in Equation (29) becomes a weighted mean in the log-semiring, related to the LogSumExp (LSE) function, because $H(\mathit{y}|\mathit{\lambda })=\text{LSE}(y_1+\log {\lambda }_1,\dots ,y_n+\log {\lambda }_n)$, for all $(\mathit{y},\mathit{\lambda })\in {\mathbb{R}}^n\times \text{int}(\mathrm{\Delta })$, where $\text{LSE}(\mathit{y})=\log ({\sum }_{i=1}^n\exp (y_i))$, for all $\mathit{y}\in {\mathbb{R}}^n$. Such a formulation carries fruit in probabilistic modeling and neural networks, where LSE and softmax appear naturally because of their smoothness properties.³⁸

3.9.1. Reduction to Basic Framework.

Crucially, the general h-mean in Equation (29) reduces to our standard weighted objective in Equation (1) under a transformation of the criteria, by setting $f_i=h\circ {\widehat{f}}_i$, so that

If the kernel h is increasing (resp., decreasing), then maximizing $\widehat{H}(·|\mathit{\lambda })$ is equivalent to maximizing (resp., minimizing) $F(·|\mathit{\lambda })$. Thus, our earlier results carry over directly—with minor adjustments in the decreasing case, as detailed in Section 3.9.2.

Example 15.

Consider Example 13 (ii) for the “diagonal” state space $\mathrm{\Theta }=\{\mathit{\theta }\in {\mathbb{R}}_{+}^n:{\theta }_1=\cdots ={\theta }_n\in [\underset{¯}{p},\overline{p}]\}$, where the constants $\underset{¯}{p},\overline{p}$ are such that $0<\underset{¯}{p}<\overline{p}<\infty$. By Example 14 (i), this represents a situation in which the decision maker is uncertain about the appropriate aggregation method, except that a (homogeneous) power mean $M_p$ should be used, for some $p\in [\underset{¯}{p},\overline{p}]$. The result in Example 13 (ii) suggests as robust choice the largest available option: the power mean $M_{\overline{p}}$.

Remark 16.

(i) As Example 15 suggests, the insights about criterion ambiguity in Section 3.8 may sometimes be combined with the general representation of multicriteria objectives in Section 3.9 to handle aggregation ambiguity, that is, uncertainty over which aggregation rule or kernel should be used. (ii) For the general h-mean in Equation (29), we define $H=h^{-1}\circ F_h$ with $F_h={\sum }_{i=1}^n{\lambda }_i(h\circ f_i)$. However, in general, $h^{-1}\circ (F_h/F_h^{*})\ne (h^{-1}\circ F_h)/(h^{-1}\circ F_h^{*})$ with $F_h^{*}(·)=\max F_h(\mathcal{X}|·)$, so our optimization focuses on $F_h$ directly. (iii) For the practically very important power mean in Example 14 (i) and Example 15 (as long as $p\ne 0$), the robust optimization of $F_h$ is equivalent to the robust optimization of $M_p$, because $h(·)={(·)}^p$ and $h^{-1}(·)={(·)}^{1/p}$ both feature multiplicative separability.

3.9.2. Special Case: Minimization Under Decreasing Kernel.

For a decreasing kernel, $f_i=h\circ {\widehat{f}}_i$ transforms larger ${\widehat{f}}_i$ into smaller $f_i$, so that the objective reflects a smaller-is-better interpretation (akin to optimizing a loss function). Assuming ${\mathbb{R}}_{++}^n\subset \mathcal{Y}$, by Axiom 3 it is ${\lim }_{\alpha \to 0^{+}}H(\alpha {\mathbf{1}}_n|\mathit{\lambda })=H({\mathbf{0}}^{+}|\mathit{\lambda })=0$. Because in most practical applications (e.g., for logarithmic or inverse transformations in multiobjective loss minimization) it is $h(0^{+})=\infty$, we obtain $h^{-1}(\infty )=0^{+}$, so that on the compact action set $\mathcal{X}$ it is $f_i^{•}=\min f_i(\mathcal{X})=h({\widehat{f}}_i^{*})>0$, for all $i\in \mathcal{N}$, as ${\widehat{f}}_i^{*}=\max {\widehat{f}}_i(\mathcal{X})$ is necessarily finite. Hence, we can consider the adjusted performance ratio,

where $F^{•}(\mathit{\lambda })=\min F(\mathcal{X}|\mathit{\lambda })$, for all $\mathit{\lambda }\in \mathrm{\Delta }$. A representation of the performance index $\rho (\mathit{x})$ in Equation (4) then obtains, analogous to Equation (6), for all $\mathit{x}\in \mathcal{X}$, as the minimum of the (adjusted) criterion-specific performance ratios ${\varphi }_i(\mathit{x})=f_i^{•}/f_i(\mathit{x})$ with respect to $i\in \mathcal{N}$. This defines the (nonempty, compact) set of pseudo-robust actions,

The set of Pareto-optimal actions $\mathcal{P}$ needs to be based on the original vector of criteria $\widehat{\mathit{f}}$, so that

analogous to Equation (10). The robust decision set is then $\mathcal{R}=\mathrm{\Psi }\cap \mathcal{P}$ as before, in Equation (11).

3.10. Applying the Method

“What do I do? What do I get? How do I adapt it to my case?”—We now address these practitioner questions by outlining the method’s core, the interpretation of its results, and important extensions for customization, as a navigation device for approaching the multicriteria decision problem introduced in Section 2.1 and its generalizations.

1. (Core) To determine an approximately robust decision ${\widehat{\mathit{x}}}_{\epsilon }$ (at any prespecified precision $\epsilon$) with respect to n criteria, one needs to solve just $n+1$ optimization problems: one to maximize the $\epsilon$-augmented performance index ${\mathrm{\Phi }}_{\epsilon }$ in Equations (12) and (13) and n to compute the normalization constants $f_i^{*}$ in Equation (5) for the criterion-specific performance ratios ${\varphi }_i$ in Equation (6). The associated normalized robust weight $\widehat{\mathit{\lambda }}\in \mathrm{\Delta }$, which encapsulates the tradeoffs between the criteria embedded in both the action set and the shape of the criterion functions (independent of any scaling), is then obtained (approximately) from Equations (19) and (20) by setting $\widehat{\mathit{x}}={\widehat{\mathit{x}}}_{\epsilon }$.

2. (Interpretation) The performance index $\rho ({\widehat{\mathit{x}}}_{\epsilon })\in [0,1]$, with $\rho (·)$ given in Equation (6), guarantees a minimum percentage that the robust solution ${\widehat{\mathit{x}}}_{\epsilon }$ achieves of the optimal weighted objective in Equation (1), for any weight in $\mathrm{\Delta }$. This means that the performance of ${\widehat{\mathit{x}}}_{\epsilon }$ is guaranteed to be no worse than $\rho ({\widehat{\mathit{x}}}_{\epsilon })$ times the best achievable performance under any possible weight vector, ensuring robustness to unknown or contested preferences. In this manner, the weight-induced subjectivity in multicriteria optimization can be removed. The robust weight rationalizes the robust decision as an optimum of the weighted objective in Equation (1) for $\mathit{\lambda }=\widehat{\mathit{\lambda }}$. The vector $\widehat{\mathit{\lambda }}$ can be interpreted as the revealed weight structure that best justifies the robust solution, based on the problem’s internal tradeoffs.

3. (Customization) The method accommodates several practically important extensions:

   • – Partial weight information: Prior knowledge or constraints on weights can be imposed by requiring weight vectors (not necessarily normalized) to belong to a suitable subset (cf. Section 3.7).

   • – State-dependent criteria: Uncertain or context-dependent criteria $f_i$ can be captured by worst-case performance ratios, effectively reducing the problem to the core framework (cf. Section 3.8).

   • – Generalized aggregation: Rather than relying on the arithmetic mean in Equation (1), one may adopt alternative scalarization functions consistent with an axiomatic foundation (cf. Section 3.9).

Overall, the relatively robust methodology provides a computationally tractable and conceptually transparent toolkit for balancing multiple, possibly ambiguous objectives in diverse real-world settings (cf. Section 4.3).

4. Discrete Applications

4.1. Finite Action Set

4.1.1. Relative Robustness Criterion.

Consider a finite set of alternatives $\mathcal{X}=\{1,\dots ,J\}$, evaluated according to n positive criteria $f_1,\dots ,f_n:\mathcal{X}\to {\mathbb{R}}_{++}$. Each alternative $j\in \mathcal{X}$ receives a score $f_i(j)=s_{ij}>0$, for all $i\in \mathcal{N}$. If we set ${\widehat{s}}_i={\max }_{j\in \mathcal{X}}{s}_{ij}$, then the performance index for the j-th option becomes

By Proposition 3, a robust decision $j^{*}\in \mathcal{R}$ can be found by computing the lower limit, as follows:

This robust decision achieves the optimal performance index,

As discussed in Section 3, simply maximizing the performance index ${\rho }_j$ yields pseudo-robust solutions, which are generically inefficient; see, for instance, Example 6. We now compare the proposed approach with several alternative robustness criteria.

4.1.2. Alternative Robustness Criteria.

The following three robustness criteria, defined in the present context of discrete action sets, are frequently used in the literature for dealing with parameter ambiguity.

1. The Laplace criterion corresponds to a weighted objective $F(·|{\mathit{\lambda }}_{\text{Laplace}})$ in Equation (1) with a uniform weight, ${\mathit{\lambda }}_{\text{Laplace}}=(1,\dots ,1)/n\in \mathrm{\Delta }$, leading to the “Laplace solution,”

   $$j_{\text{Laplace}}\in \arg \underset{j\in \mathcal{X}}{\max }F(j|{\mathit{\lambda }}_{\text{Laplace}})=\arg \underset{j\in \mathcal{X}}{\max }{\displaystyle \sum _{i=1}^n{s}_{ij}}.$$(34)

2. The worst-case criterion is defined as the minimum payoff across the admissible weights, which yields a so-called “maximin solution,”

   $$j_{\text{WC}}\in \arg \underset{j\in \mathcal{X}}{\max }\underset{\mathit{\lambda }\in \mathrm{\Delta }}{\min }F(j|\mathit{\lambda })=\arg \underset{j\in \mathcal{X}}{\max }\underset{i\in \mathcal{N}}{\min }{s}_{ij}.$$(35)

3. The absolute-regret criterion evaluates absolute regret, that is, the maximum difference ex post between what is and the best that could have been, resulting in the “(minimax) absolute-regret solution,”

   $$j_{\text{AR}}\in \arg \underset{j\in \mathcal{X}}{\min }\underset{\mathit{\lambda }\in \mathrm{\Delta }}{\max }\left\{F^{*}(\mathit{\lambda })-F(j|\mathit{\lambda })\right\}=\arg \underset{j\in \mathcal{X}}{\min }\underset{i\in \mathcal{N}}{\max }\left\{{\widehat{s}}_i-s_{ij}\right\}.$$(36)

4.1.3. Comparison.

The following example, which features a finite action set, illustrates the proposed robust solution in Equation (32) against decisions recommended by alternative robustness criteria, notably the Laplace criterion in Equation (34), the WC (or maximin) solution in Equation (35), and the solution minimizing the (maximum) AR in Equation (36).

Example 16.

Consider a discrete-choice situation for $J=5$ options, for which the scores $s_{ij}$ across $n=3$ criteria are recorded in Table 1. In the context of relative robustness, options 2 and 5 are tied for the highest performance index in Equation (31) and are thus both pseudo-robust, so $\mathrm{\Psi }=\{2,5\}$. At the same time, option 5 Pareto-dominates option 2 (with $\mathcal{P}=\{1,3,4,5\}$), so that $j^{*}=5\in \mathcal{R}=\mathrm{\Psi }\cap \mathcal{P}$ is the unique robust choice. This solution can also be obtained directly from Equation (32), written in the form

$$j^{*}\in {\underset{¯}{\text{Lim}}}_{\epsilon \to 0^{+}}\arg \underset{j\in \mathcal{X}}{\max }{\mathrm{\Phi }}_{\epsilon }(j),$$

Table 1. Discrete Decision Options in Example 16, Evaluated with Different Robustness Criteria

Table 1. Discrete Decision Options in Example 16, Evaluated with Different Robustness Criteria

Option (j)	Criteria			Performance ratios/index				Alternative evaluations
	$s_{1j}$	$s_{2j}$	$s_{3j}$	$s_{1j}/{\widehat{s}}_1$	$s_{2j}/{\widehat{s}}_2$	$s_{3j}/{\widehat{s}}_3$	${\rho }_j$	Laplace	WC	AR
1	24	20	16	1	10/11	4/15	4/15	20	16	44
2	7	6	32	7/24	3/11	8/15	3/11	15	6	28
3	1	2	60	1/24	1/11	1	1/24	21	1	23
4	22	22	16	11/12	1	4/15	4/15	20	16	44
5	12	6	36	1/2	3/11	3/5	3/11	18	6	24

---

Note. Values in bold indicate optimality for the corresponding robustness criterion.

where ${\mathrm{\Phi }}_{\epsilon }(j)=(1-\epsilon ){\rho }_j+(\epsilon /n){\sum }_{i=1}^n(s_{ij}/{\widehat{s}}_i)$, for all $j\in \mathcal{X}$ and all $\epsilon \in [0,1]$. Indeed, because ${\rho }_2={\rho }_5=3/11>{\rho }_1={\rho }_4=4/15>{\rho }_3=1/24$, for sufficiently small $\epsilon \in (0,1]$ the $\epsilon$-augmented performance index is

$$\underset{j\in \{2,5\}}{\max }{\mathrm{\Phi }}_{\epsilon }(j)>{\mathrm{\Phi }}_{\epsilon }(4)>{\mathrm{\Phi }}_{\epsilon }(3).$$

Meanwhile, it is

Therefore, ${ℛ}_{\epsilon }=\{5\}$, for all sufficiently small $\epsilon >0$, so $j^{*}\in {\underset{¯}{\text{Lim}}}_{\epsilon \to 0^{+}}{ℛ}_{\epsilon }=\{5\}$, and one finds $j^{*}=5$ as the unique robust option. Regarding alternative robustness evaluation, both the Laplace criterion in Equation (34) and the absolute-regret criterion in Equation (36) produce the solution $j_{\text{Laplace}}=j_{\text{AR}}=3$ with very poor performance in at least one criterion. In fact, changing the payoff $s_{13}$ from one to zero would produce even a zero performance index and zero worst-case performance guarantee for that same decision (still optimal under these criteria),³⁹ which may be rather difficult to justify in any real-world scenario. Finally, maximizing the worst-criterion performance in Equation (35) leads to indifference between options 1 and 4, at a suboptimal performance index and largest absolute regret. By contrast, the proposed robust solution $j^{*}=5$ provides (by construction) the best performance index and a reasonable compromise solution in terms of the other criteria. It guarantees a strictly positive performance across all criteria. From Equations (19) and (20) we find that the corresponding (normalized) robust weight vector is

Consistent with Equation (21), it is such that

where 3.6 corresponds to the harmonic mean of the criterion scores for the robust option $j^{*}$. Consider now the robustly reweighted scores ${\sigma }_{ij}={\widehat{\lambda }}_i{s}_{ij}$, for all $(i,j)\in \mathcal{N}\times \mathcal{X}$, displayed in Table 2, together with the associated robustness evaluations. It is interesting that after reweighting, option 3 goes from lowest to largest absolute regret, which highlights the sensitivity of the AR criterion to uncertainty about the relative importance of the criteria. The maximin payoff (WC) solution shifts from options 1 and 4 to option 5, whereas the Laplace solution shifts from option 3 to option 4. By contrast, the performance index remains unaffected by the robust reweighting (or any other reweighting), so option 5 is still the unique robust solution.

Table 2. Discrete Decision Options in Example 16, Evaluated after Robust Reweighting

4.2. Data-Driven Approach

Consider any (nonempty, compact) action set $\mathcal{X}\subset {\mathbb{R}}^m$ for some integer $m\ge 1$, as in Section 2. Assume further that the decision maker does not know the functional form of the multicriterion $\mathit{f}=(f_1,\dots ,f_n)$, but is still able to observe its value ${\mathit{s}}^k=(s_1^k,\dots ,s_n^k)$ for different decisions ${\mathit{x}}^k\in \mathcal{X}$ over the course of $K\ge 1$ experiments, where $k\in \mathcal{K}=\{1,\dots ,K\}$. Indeed, given the realized score set $\widehat{\mathcal{S}}=\{{\mathit{s}}^k:k\in \mathcal{K}\}$ and the realized action set $\widehat{\mathcal{X}}=\{{\mathit{x}}^k:k\in \mathcal{K}\}\subset \mathcal{X}$, the average criterion scores are⁴⁰

for all $(\mathit{x},i)\in \widehat{\mathcal{X}}\times \mathcal{N}$. With this, the (data-driven) criterion-specific performance ratios are given bywhere ${\widehat{f}}_i^{*}=\max {\widehat{f}}_i(\widehat{\mathcal{X}})$ is the i-th maximum average criterion score. The (data-driven) performance index,is defined for all observed decisions. A (data-driven) robust decision ${\widehat{\mathit{x}}}^{*}$ (among all sampled actions in $\widehat{\mathcal{X}}$) can then be determined in the standard way by applying Proposition 3 to the observational analogues of our standard approach:

In other words, $\widehat{\rho }(\mathit{x})$ captures the worst-case relative performance of action $\mathit{x}$ across all observed criteria, and ${\widehat{\mathit{x}}}^{*}$ identifies a robust decision, which is efficient and has the best-possible worst-case performance. Note that instead of going through the motions of actually taking the limit, it is sufficient to maximize the $\epsilon$-augmented performance index, that is, the maximand in Equation (37), for a sufficiently small $\epsilon \in (0,1]$. This approximates the optimal robustness performance, captured by the optimal performance index ${\rho }^{*}$, arbitrarily closely by virtue of the $\epsilon$-performance guarantee provided in Proposition 4. Finally, as in Equations (19) and (20), the (data-driven) robust weight is

Based on the observed data, this vector provides a robust estimate of the tradeoffs between the different criteria from the vantage point of relative robustness.

Example 17.

Consider the joint evaluation of human performance on a certain task (such as giving a university lecture) by K evaluators who score J individuals in N different performance dimensions (or criteria) on a Likert scale (from one to seven),⁴¹ so $\mathcal{K}=\{1,\dots ,K\}$, $\mathcal{N}=\{1,\dots ,N\}$, and $\mathcal{X}=\{1,\dots ,J\}$. Figure 8 shows a spider plot comparing $J=4$ different individuals. Assuming that all evaluators scored all the individuals using a seven-point Likert scale, the realized score set $\widehat{\mathcal{S}}$ has elements ${\mathit{s}}^k\in {\{1,\dots ,7\}}^N$, for all $k\in \mathcal{K}$, whereas the realized action set $\widehat{\mathcal{X}}$ is equal to $\mathcal{X}$. Individual 3, although never achieving a highest criterion-specific average score, exhibits the best data-driven performance index (${\widehat{\rho }}^{*}=\widehat{\rho }(3)=76.19\%={\max }_{j\in \{1,2,3,4\}}\widehat{\rho }(j)$). Because in a noisy observation environment ties in the performance index are fairly unlikely, the set of pseudo-robust alternatives $\mathrm{\Psi }$ tends to be a singleton, thus also resulting in a singleton set of robust options (e.g., $\mathcal{R}=\{3\}$). This example illustrates how the data-driven approach identifies robust performance through balanced tradeoffs rather than peak performance in any one dimension, thus favoring individual 3, whose weakest dimension is relatively strong.

4.3. Real-World Applications

Relatively robust multicriteria optimization provides a flexible and transparent framework for decision-making under uncertainty. As developed in Sections 2 and 3, it is especially well suited for complex problems where the relative importance of multiple criteria is ambiguous or hard to justify. The method systematically balances competing objectives using only minimal assumptions on the decision maker’s preferences, and delivers solutions that are both Pareto-efficient and robust (cf. Proposition 3). We illustrate its practical relevance in three diverse contexts: the energy trilemma, quality-adjusted life year (QALY)-based health evaluations, and corporate resource allocation. In each case, the relatively robust methodology facilitates structured tradeoff management and data-informed robustness, whether operating in an abstract policy space or on a finite empirical action set (cf. Sections 4.1 and 4.2).

4.3.1. Energy Trilemma: Balancing Energy Security, Equity, and Sustainability.

The World Energy Council’s Energy Trilemma framework evaluates nations on three key dimensions, namely “energy security” (i.e., the reliability and resilience of energy supply), “energy equity” (i.e., the accessibility and affordability of energy for all citizens), and “environmental sustainability” (i.e., the reduction of greenhouse gas emissions and environmental impact). Nations face challenges in improving one dimension without compromising others (World Energy Council 2024).⁴² For instance, increasing energy equity by subsidizing fossil fuels may undermine sustainability, while prioritizing environmental sustainability through renewable energy investment might initially reduce energy security or equity. An optimal strategy depends on the relative importance attributed to each of the three criteria. In this setting, a relatively robust approach can

• Identify Robust Policies: By modeling each nation’s energy policies and outcomes as feasible decisions, the method identifies those policies that achieve strong tradeoffs across all three dimensions. For example, a relatively robust policy might balance investment in renewable energy, grid modernization, and subsidies for low-income households, ensuring consistent performance regardless of variations of the relative importance across dimensions, which might still be importance-ranked (cf. Section 3.7).

• Quantify Tradeoffs: The approach provides a clear representation of tradeoffs, such as how much equity might need to be sacrificed for a given improvement in sustainability under different weighting scenarios.

• “Robustify” Strategic Goals: Policymakers can develop long-term strategies that are resilient to evolving societal priorities, such as shifts toward greater emphasis on the precautionary principle (cf. Section 3.9).

4.3.2. QALY Impact of Diseases: Evaluating Quality-Adjusted Life Years.

QALYs measure the impact of diseases and medical treatments by combining longevity and quality of life into a single index (Pliskin 1974, Pliskin and Beck 1976, Miyamoto et al. 1998).⁴³ The impact of different impairments—such as mobility loss, chronic pain, or cognitive decline—depends on how these are weighted relative to each other in calculating overall health outcomes. Subject to restrictions on weights (e.g., a priority ranking; cf. Example 12) and with a flexible aggregation of criteria (cf. Section 3.9) the method can

• Handle Uncertain Weightings: When precise utility-weightings for different impairments are unavailable (as one would generally assume), the method identifies health interventions or treatment plans that remain effective across a wide range of subjective assessments. For example, it might suggest treatments that optimize outcomes for both pain relief and mobility restoration, ensuring robust quality-of-life improvements.

• Optimize Resource Allocation: Health authorities can use the method to prioritize interventions that deliver the highest QALY improvements per dollar spent, even when the relative importance of various health dimensions (e.g., physical versus mental health) is uncertain.⁴⁴

• Support Evidence-Based Policy: The method provides a performance guarantee for proposed policies, demonstrating their effectiveness regardless of the relative emphasis placed on specific impairments.

4.3.3. Resource Allocation in Companies: Balancing Risk, Resources, and Rival Assets.

In organizations, resource allocation involves deciding how to distribute limited resources—financial, human, or physical—across competing projects. Each project is evaluated based on multiple criteria, such as “risk of noncompletion” (i.e., the probability that a project fails because of delays or unforeseen issues), “use of human resources” (i.e., the availability and workload of skilled personnel required for the project), and “use of rival assets” (i.e., the occupation of assets that cannot be used by multiple projects simultaneously, for example, specialized machinery). Applying the framework of relatively robust decision making allows organizations to⁴⁵

• Identify Balanced Portfolios: By modeling projects and their criteria as feasible decisions $\mathit{x}\in \mathcal{X}$, with performance criteria $f_1(\mathit{x}),\dots ,f_n(\mathit{x})$, the method selects a portfolio of projects that balances competing objectives, ensuring efficient resource use even when preferences over criteria are unclear.

• Manage Tradeoffs: By aggregating criterion-specific performance ratios ${\varphi }_i(\mathit{x})$ into a performance index $\rho (\mathit{x})$, the method reveals, for example, how prioritizing low-risk projects impacts resource utilization and asset deployment, informing decisions about a robust balance between risk and resource efficiency.

• Adapt to Uncertainty: As organizational priorities shift (e.g., toward innovation or risk aversion), the method ensures that allocation strategies can remain robust across changing criteria (cf. Section 3.8) and weight configurations for the criteria, and possibly across different aggregation methods (cf. Section 3.9).

In settings with discrete project sets and empirical observations, a data-driven approach can be applied (cf. Section 4.2).

5. Conclusion

Multicriteria optimization aims to resolve conflicts between competing objectives by finding Pareto-efficient decisions for which improving one criterion necessarily degrades at least one other. Although Pareto-efficiency sets a minimum standard of nonwastefulness, practical decision making often requires selecting a single compromise solution from the available Pareto frontier. Scalarization, such as assigning weights to criteria, is a common method for operationalizing this selection, but it assumes knowledge of the weights, which may not be available or easily justifiable, and it may also exclude Pareto-efficient solutions when the action set is nonconvex.

Here, we seek to mitigate the dependence on precise weight specifications by introducing a performance index that evaluates the worst-case weighted performance of a decision relative to its maximum potential. A Pareto-efficient decision that maximizes this index is viewed as relatively robust, balancing competing criteria in a way that offers resilience to weight uncertainty. A critical feature of the method is its computational simplicity, relying only on the compactness of the feasible set and the continuity of criterion functions to guarantee the existence and basic regularity of the solutions. This ensures broad applicability, even for complex, nonconvex problems. Criterion ambiguity and more general aggregation methods can be accommodated. In the case of finite sets, this method is completely general and can be operationalized through a data-driven approach under virtually no assumptions.