Interval-valued Value Functions from Uncertain Preferences
Diego García-Zamora, José Rui Figueira
TL;DR
The work addresses deriving interval-valued value scales when decision-makers provide interval judgments due to uncertainty. It introduces interval-valued preference relations (IPR) with a consistency notion relative to a neutral interval $u$, and proves a representation theorem linking the IPR to monotonic interval value functions $v_i$ via $z_{ij}+u=v_i-v_j$. A normalization procedure yields interval-valued scales around 1, and the framework unifies fuzzy, Saaty, and Deck of Cards approaches through transformations, offering optimization models to diagnose and repair inconsistencies with analytic solution expressions. The methodology enables robust, interpretable interval scales suitable for interactive decision support. It lays groundwork for extending classical decision models to interval-valued inputs and for applying interval arithmetic in decision analysis under uncertainty.
Abstract
The construction of numerical value scales (or priority values) is a recurrent topic in decision-aiding research. However, in real contexts, uncertainty and limited cognitive precision often lead decision-makers to provide interval judgments rather than exact values. In this scenario, even though obtaining a numerical value scale from interval preferences could be feasible, it implies a loss of information and an oversimplification of the input information. This paper proposes a general framework for deriving interval-valued value scales from interval-valued pairwise comparisons. This implies addressing fundamental challenges regarding the elicitation, the determination of representative value functions, and the preservation of properties such as monotonicity and interpretability. We start by presenting a new definition for the consistency of an interval-valued preference relation. We show that when consistency holds, interval-valued value functions can be determined, and we establish conditions for their uniqueness. Furthermore, we show that our proposal extends and unifies classical models such as fuzzy preference relations, Saaty's preference relations, and the Deck of Cards Method. Methodologically, we develop optimization models that provide procedures to guide decision-makers towards consistency when contradictions arise. The framework provides a coherent and interpretable foundation for constructing monotonic interval value functions, bridging classical and interval-valued preference models, and enhancing robustness in decision-making.