On the Derivation of Weights from Incomplete Pairwise Comparisons Matrices via Spanning Trees with Crisp and Fuzzy Confidence Levels

TL;DR

This paper introduces a novel method for deriving priority vectors from incomplete pairwise comparison matrices by utilizing spanning trees and confidence levels, including fuzzy logic, with computational complexity analysis.

Contribution

It presents a new approach combining spanning trees and confidence levels to derive priorities from incomplete PC matrices, including fuzzy confidence modeling.

Findings

Method effectively handles incomplete data with confidence levels.

Numerical examples demonstrate the approach's advantages.

Provides a formula for the upper bound of spanning trees for complexity assessment.

Abstract

In this paper, we propose a new method for the derivation of a priority vector from an incomplete pairwise comparisons (PC) matrix. We assume that each entry of a PC matrix provided by an expert is also evaluated in terms of the expert's confidence in a particular judgment. Then, from corresponding graph representations of a given PC matrix, all spanning trees are found. For each spanning tree, a unique priority vector is obtained with the weight corresponding to the confidence levels of entries that constitute this tree. At the end, the final priority vector is obtained through an aggregation of priority vectors achieved from all spanning trees. Confidence levels are modeled by real (crisp) numbers and triangular fuzzy numbers. Numerical examples and comparisons with other methods are also provided. Last, but not least, we introduce a new formula for an upper bound of the number of spanning trees, so that a decision maker gains knowledge (in advance) on how computationally demanding the proposed method is for a given PC matrix.