# As simple as possible but not simpler in Multiple Criteria Decision Aiding: the ROR-SMAA level dependent Choquet integral approach

## TL;DR

This paper introduces a novel level dependent Choquet integral approach for multiple criteria decision aiding, incorporating robust and probabilistic analyses to handle multiple compatible models and decision maker preferences.

## Contribution

It proposes two formulations of level dependent capacities and integrates ROR and SMAA to analyze all compatible models in decision making.

## Key findings

- The methodology effectively ranks universities based on multiple criteria.
- It accounts for uncertainty by considering all compatible capacities.
- Probabilistic rankings provide nuanced decision support.

## Abstract

The level dependent Choquet integral has been proposed to handle decision making problems in which the importance and the interaction of criteria may depend on the level of the alternatives' evaluations. This integral is based on a level dependent capacity, which is a family of single capacities associated to each level of evaluation for the considered criteria. We present two possible formulations of the level dependent capacity where importance and interaction of criteria are constant inside each one of the subintervals in which the interval of evaluations for considered criteria is split or vary with continuity inside the whole interval of evaluations. Since, in general, there is not only one but many level dependent capacities compatible with the preference information provided by the Decision Maker, we propose to take into account all of them by using the Robust Ordinal Regression (ROR) and the Stochastic Multicriteria Acceptability Analysis (SMAA). On one hand, ROR defines a necessary preference relation (if an alternative a is at least as good as an alternative b for all compatible level dependent capacities), and a possible preference relation (if a is at least as good as b for at least one compatible level dependent capacity). On the other hand, considering a random sampling of compatible level dependent capacities, SMAA gives the probability that each alternative reaches a certain ranking position as well as the probability that an alternative is preferred to another. A real-world decision problem on rankings of universities is provided to illustrate the proposed methodology.