IBMM for Inconsistent Data Identification and Adjustment in the AHP/ANP

Abstract

As stated previously, the inconsistent elements should be identified if the pairwise comparison matrix (PCM) failed to the consistency test, therefore, the methods for identifying and adjusting the inconsistent elements in the PCM have been extensively studied since the AHP/ANP were developed by Saaty. However, existing methods are either too complicated to be applied in the revising process of the inconsistent comparison matrix or are difficult to preserve most of the original comparison information due to the use of a new pairwise comparison matrix. Therefore, Ergu et al. (2011b) developed a simple method for improving the consistency ratio of the pairwise comparison matrix in ANP, namely, an induced bias matrix (IBM) was developed to identify and adjust the inconsistent data in the ANP/AHP. The proposed method was further extended to estimate the missing item scores, optimize the questionnaire design and analyze the risk in decision making as well as task scheduling and resource allocation (Ergu et al. 2011c, 2011d, 2011e; Ergu and Kou 2011). To make the proposed model more comprehensive and robust, Ergu et al. (2011f) integrated the fundamental theorems and corollaries into one model, the induced bias matrix model (IBMM), and the related theorems and corollaries were also proved mathematically in Ergu et al. (2011b, 2011c). In this Chapter, all theorems and corollaries related to IBMM and their proofs are discussed systematically in order to understand the proposed IBMM explicitly.