**5. Conclusions**

The relative importance of goals in multiple criteria optimization problems may have an impact on the final solutions proposed to decision-makers. Either cardinal or ordinal information is required to establish preference among goals that, in addition, may be structured in clusters (hierarchically or by means of strict priority levels). To improve the interactive process of setting preferences in multiple criteria optimization problems, in this paper, we show that a single compact representation summarizing both AHP and lexicographical orders is possible. To this end, we introduce the concept of powerset preference rule. As the powerset derived from a set of goals includes all possible subsets of goals, a powerset preference rule allows us to compactly represent preferences of both AHP and lexicographic orders. In addition, we study the link between AHP and lexicographic orders, showing that the former is a special case of the latter.

This compact representation also facilitates the understanding of the main properties of different preference setting options. In this sense, we discuss the main relations that can be established among the goals of multiple criteria optimization problems. More precisely, we study the case of alignment of objectives that can be identified through a given powerset preference rule. We also characterize preference orders, describing a ranking of priorities that are easier to understand by decision-makers, especially when a large number of objectives are considered.

Although the compact representation described in this paper may help decision-makers to better understand the impact of preference setting on the results derived from optimization, our proposal is limited to providing a preference scheme. This scheme can be used as an input to optimization methods, and it provides a more effective way to develop the interactive process of preference eliciting and optimization. In addition, it allows to deploy mechanisms of formal reasoning that may lead to detect inconsistencies in preferences or to establish interesting theoretical properties. The search for further theoretical results derived from this compact representation is a natural extension of this work.

**Author Contributions:** Conceptualization, F.S.-M., D.P.-S., A.G.-B. and J.R.-M.; Data curation, F.S.-M., D.P.-S., A.G.-B. and J.R.-M.; Formal analysis, F.S.-M., A.G.-B. and J.R.-M.; Investigation, F.S.-M., D.P.-S., A.G.-B. and J.R.-M.; Methodology, F.S.-M., D.P.-S., A.G.-B. and J.R.-M.; Project administration, F.S.-M. and D.P.-S.; Resources, D.P.-S.; Supervision, F.S.-M., A.G.-B. and J.R.-M.; Validation, F.S.-M.; Visualization, J.R.-M.; Writing—original draft, F.S.-M. and D.P.-S.; Writing— review and editing, F.S.-M., D.P.-S., A.G.-B. and J.R.-M.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
