Representation of preferences for multiple criteria decision aiding in a new seven-valued logic
Salvatore Greco, Roman Słowiński
TL;DR
The paper tackles robustness and ambiguity in MCDA by introducing a seven-valued logic derived from rough-set theory, embedding it in a Belnap-like lattice to represent diverse truth states for preference relations. It develops two weight-perturbation based methodologies—SMAA and Robust Ordinal Regression (ROR)—to construct seven-valued preferences under both value-function and outranking aggregations, with formal propositions linking perturbation extremities to truth states. A detailed didactic example with three perspectives and four criteria demonstrates explainability, aggregation to four-valued logic, and ranking via global scores or outranking, alongside robustness analysis via SMAA and SMAA-SOR. The approach yields transparent, traceable preference models in MCDA and suggests avenues for extending to Choquet integrals, PROMETHEE-like methods, and robust multiobjective optimization.
Abstract
The seven-valued logic considered in this paper naturally arises within the rough set framework, allowing to distinguish vagueness due to imprecision from ambiguity due to coarseness. Recently, we discussed its utility for reasoning about data describing multi-attribute classification of objects. We also showed that this logic contains, as a particular case, the celebrated Belnap four-valued logic. Here, we present how the seven-valued logic, as well as the other logics that derive from it, can be used to represent preferences in the domain of Multiple Criteria Decision Aiding (MCDA). In particular, we propose new forms of outranking and value function preference models that aggregate multiple criteria taking into account imperfect preference information. We demonstrate that our approach effectively addresses common challenges in preference modeling for MCDA, such as uncertainty, imprecision, and ill-determination of performances and preferences. To this end, we present a specific procedure to construct a seven-valued preference relation and use it to define recommendations that consider robustness concerns by utilizing multiple outranking or value functions representing the decision maker s preferences. Moreover, we discuss the main properties of the proposed seven-valued preference structure and compare it with current approaches in MCDA, such as ordinal regression, robust ordinal regression, stochastic multiattribute acceptability analysis, stochastic ordinal regression, and so on. We illustrate and discuss the application of our approach using a didactic example. Finally, we propose directions for future research and potential applications of the proposed methodology.