Robust Multivariate Quantiles in Ranking Problems

Multi-criteria decision-making is a valuable tool for evaluating and ranking alternatives with multiple conflicting criteria. Traditional methods assume precise inputs, which is rarely the case in practice. A recent approach tackles this using cumulative distribution functions and quantiles for multivariate random vectors, relying on cone-induced partial orders to build a conservative ranking procedure that reflects multiple expert opinions. Yet, expert weights and evaluations are often uncertain due to preferences, limited data, or subjective judgment. This paper extends the cone-based method to address both types of uncertainty. By modeling weights and evaluations as uncertain sets, we develop a dual-uncertainty framework to enhance decision robustness. We introduce robust versions of cone distribution functions and set-valued quantiles. Numerical examples illustrate how uncertainty affects rankings, offering a flexible tool for robust analysis in finance, sustainability planning, and public policy.