An extension of Ordered Weighted Averaging over intervals with application to optimization under risk

The Ordered Weighted Averaging (OWA) operator is a traditional and commonly used criterion for aggregating discrete values of uncertain quantities. In this paper, it is shown that the discrete OWA naturally extends to the continuous case by using the concept of a distortion risk measure. It is shown how to apply the distortion risk measure to optimization problems with a linear objective function, whose coefficients are random variables with continuous distribution functions supported on intervals. The case where these coefficients are independent, uniformly distributed random variables is explored in more detail. The computational complexity of the resulting optimization problem is analyzed, and solution methods with approximation guarantees are proposed. These methods are also verified through computational experiments.

💡 Research Summary

The paper presents a novel framework that extends the classic Ordered Weighted Averaging (OWA) operator from its traditional discrete formulation to a continuous setting by leveraging the concept of a distortion risk measure. The authors begin by recalling the discrete OWA, defined as a weighted sum of ordered components of a vector, and show how it can be interpreted in a stochastic context using empirical quantiles (Value‑at‑Risk, VaR). They then introduce a Basic Unit Monotone (BUM) function Q: