Decision Making Under Uncertainty: A Z-Number-Based Regret Principle

Decision-making theory has developed over many decades at the intersection of economics, mathematics, psychology, and engineering. Its classical foundations include Bernoulli’s expected utility theory, von Neumann and Morgenstern’s rational choice theory, and the criteria proposed by Savage, Wald, Hurwicz, and others. However, in real-world contexts, decisions are made under uncertainty, incompleteness, and unreliability of information, which classical approaches do not adequately address. To overcome these limitations, modern multi-criteria decision-making methods such as Analytic Hierarchy Process (AHP), Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I Kompromisno Resenje (Compromise solution approach) (VIKOR), and ELimination Et Choix Traduisant la REalité (Elimination and Choice Expressing Reality) (ELECTRE), as well as their fuzzy and Z-number extensions, are widely applied to the modeling and evaluation of complex systems. These Z-number extensions are based on the concept of Z-numbers introduced by Lotfi Zadeh in 2011 to formalize higher-order uncertainty. This study introduces the Z-Regret principle, which extends Savage’s regret criterion through the use of Z-numbers. Supported by Rafik Aliev’s mathematical justifications concerning arithmetic operations on Z-numbers, the model evaluates regret not only as a loss relative to the best alternative but also by incorporating the degree of confidence and reliability of this evaluation. Calculations for the selection of digital advertising platforms in terms of performance assessment under various scenarios demonstrate that the Z-Regret principle enables more stable and well-founded decision-making under uncertainty.