Mean-based incomplete pairwise comparisons method with the reference values

TL;DR

This paper introduces two new methods for deriving weight vectors from incomplete pairwise comparison matrices using reference values, extending existing heuristic estimation techniques with proven optimality and feasibility.

Contribution

It presents novel geometric and arithmetic HRE-based methods for incomplete data, including proofs of optimality, feasibility, and conditions for solution existence.

Findings

The geometric HRE method is proven to be optimal.

Existence of feasible solutions for the geometric method is established.

Sufficient conditions for solutions in the arithmetic variant are provided.

Abstract

In this article, we propose two quantitative methods for calculating weight vectors for incomplete pairwise comparison matrices using reference values. Both procedures are extensions of arithmetic and geometric heuristic estimation (HRE) methods. The proposed solutions allow flexible selection of the number of reference alternatives and the range of comparisons, from the acceptable minimum to a complete set. In this paper, we prove that the newly introduced geometric HRE method for incomplete data is optimal. For this method, we also prove the existence of a feasible solution. In the paper, we also provide sufficient conditions for the existence of a solution for the second arithmetic variant of the HRE method. We illustrate the presented methods with numerical examples.