Further Discussions on Induced Bias Matrix Model for the Pair-Wise Comparison Matrix

• Published in: Journal of optimization theory and applications Vol. 161; no. 3; pp. 980 - 993
• Main Authors: Ergu, Daji, Kou, Gang, Fülöp, János, Shi, Yong
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.06.2014; Springer Nature B.V
• Subjects: Applications of Mathematics; Bias; Calculus of Variations and Optimal Control; Optimization; Eigenvalues; Engineering; Hierarchies; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix; Operations Research/Decision Theory; Optimization; Science; Studies; Theorems; Theory of Computation; Analytic network process (ANP); Inconsistency identification; The induced bias matrix model (IBMM); Reciprocal pairwise comparison matrix (RPCM)
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-012-0223-2

The inconsistency issue of pairwise comparison matrices has been an important subject in the study of the analytical network process. Most inconsistent elements can efficiently be identified by inducing a bias matrix only based on the original matrix. This paper further discusses the induced bias matrix and integrates all related theorems and corollaries into the induced bias matrix model. The theorem of inconsistency identification is proved mathematically using the maximum eigenvalue method and the contradiction method. In addition, a fast inconsistency identification method for one pair of inconsistent elements is proposed and proved mathematically. Two examples are used to illustrate the proposed fast identification method. The results show that the proposed new method is easier and faster than the existing method for the special case with only one pair of inconsistent elements in the original comparison matrix.