# Incomplete Analytic Hierarchy Process with Minimum Weighted Ordinal Violations

## TL;DR

This paper introduces a novel method for decision making that effectively combines ordinal and cardinal preferences using incomplete pairwise comparison matrices, providing unique solutions and polynomial-time algorithms.

## Contribution

It proposes a new approach that blends ordinal and cardinal information, with a sufficient condition for uniqueness and an efficient algorithm for computing preferences.

## Key findings

- The method outperforms existing approaches in accuracy.
- It guarantees unique solutions under practical conditions.
- The algorithm operates in polynomial time.

## Abstract

Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually yield non-unique solutions; therefore, an approach blending ordinal and cardinal information is needed. In this work, we consider two cascading problems: first, we compute ordinal preferences, maximizing an index that combines ordinal and cardinal information; then, we obtain a cardinal ranking by enforcing ordinal constraints. Notably, we provide a sufficient condition (that is likely to be satisfied in practical cases) for the first problem to admit a unique solution and we develop a provably polynomial-time algorithm to compute it. The effectiveness of the proposed method is analyzed and compared with respect to other approaches and criteria at the state of the art.