Enhancing data consistency in decision matrix: Adapting Hadamard model to mitigate judgment contradiction

Cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair-wise comparison matrices (PCMs). Few of the currently-employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model, and most are heavily dependent on the method chosen for weight (priorities) derivation or the obtained closest matrix by optimization method that may change many of the original values. In this paper, we propose a Hadamard product induced bias matrix model, which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element(s) in a PCM. Through graph theory and numerical examples, we show that the adapted Hadamard model is effective in identifying and eliminating the ordinal inconsistencies. Also, for the most inconsistent element identified in the matrix, we develop innovative methods to improve the consistency of a PCM. The proposed model is only dependent on the original matrix, is independent of the methods chosen to derive the priority vectors, and preserves most of the original information in matrix A since only the most inconsistent element(s) need(s) to be modified. Our method is much easier to implement than any of the existing models, and the values it recommends for replacement outperform those derived from the literature. It significantly enhances matrix consistency and improves the reliability of PCM decision making.