A two-stage ranking method to minimize ordinal violation for pairwise comparisons

Abstract

Pairwise comparison is a powerful tool in intelligent decision making systems. Items are compared using numerical judgments that estimate the item weight ratios, which are provided by decision makers or transformed by objective data. The reliable assignment of numerical judgments is important, because judgment variety leads to significantly different results. However, it is difficult to provide exact ratios for decision makers owing to the limitations of knowledge. Although objective data provides an estimation of the ratios, a series of artificially defined rules is required to transform data into numerical judgments; however, these rules are subjective and arbitrary. Conversely, the dominance relationships between the items are obvious and reliable. Therefore, this study proposes a two-stage ranking method to minimize the ordinal violation that indicates the degree of conflict between the ranking result and the dominance. First, a 0–1 integer programming is designed and solved. Then, the second stage focuses on the topological sorting of nodes in a graph constructed using the optimal solution. To validate the effectiveness of the proposed method, we perform two experiments: a numerical example provided by participants and a real-world application involving ranking top tennis players. The results show that the proposed method not only avoids subjectivity in judgments, but also obtains the ranking that has the minimum ordinal violation among the compared methods.