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About this book

The positive reciprocal pairwise comparison matrix (PCM) is one of the key components which is used to quantify the qualitative and/or intangible attributes into measurable quantities. This book examines six understudied issues of PCM, i.e. consistency test, inconsistent data identification and adjustment, data collection, missing or uncertain data estimation, and sensitivity analysis of rank reversal. The maximum eigenvalue threshold method is proposed as the new consistency index for the AHP/ANP. An induced bias matrix model (IBMM) is proposed to identify and adjust the inconsistent data, and estimate the missing or uncertain data. Two applications of IBMM including risk assessment and decision analysis, task scheduling and resource allocation in cloud computing environment, are introduced to illustrate the proposed IBMM.

Table of Contents

Frontmatter

Chapter 1. Introduction

Abstract
In complex decision making environment, decision making usually involves tangible and intangible multiple criteria and alternatives to choose from. To deal with such qualitative and quantitative factors in multiple criteria decision making (MCDM), in 1970s, Saaty (1978, 1979, 1980) proposed an Analytical Hierarchy Process (AHP). Since then, this method has been extensively applied into many real applications, for instance in manufacturing systems (Li and Huang 2009), quality consultants (Cebeci and Ruan 2007), software evaluation (Cebeci 2009; Peng et al. 2011a), supplier evaluation and selection (Akarte et al. 2001; Handfield et al. 2002; Chan 2003; Bayazit 2006; Chamodrakas et al. 2010; Labib 2011), strategy selection (Li and Li 2009; Chen and Wang 2010), weapon selection (Dagdeviren et al. 2009), project selection (Enea and Piazza 2004; Amiri 2010).
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 2. A New Consistency Test Index for the Data in the AHP/ANP

Abstract
The consistency test is one of the critical components both in AHP and ANP. Currently, the consistency ratio (CR) proposed by Saaty is popularly used to test the consistencies of the pairwise comparison matrices. However, when the number of comparison matrices increases, the consistency test of comparison matrices both in the AHP and ANP becomes complicated. In an attempt to simplify the consistency test, Ergu et al. (2011a) proposed a maximum eigenvalue threshold as the new consistency index for the data in the AHP and ANP, which is mathematically equivalent to the CR method. In addition, a block diagonal matrix is introduced for the comparison matrices in the AHP/ANP to conduct consistency tests simultaneously. In this Chapter, the proposed new consistency test index is comprehensively described.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 3. 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.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 4. IBMM for Missing Data Estimation

Abstract
In Chap. 3, the induced bias matrix is proposed to identify the inconsistent elements in a complete pairwise comparison matrix (PCM). Besides inconsistency, a PCM may be incomplete due to limited expertise or unwillingness to judge. For an incomplete pairwise comparison matrix (IPCM), the missing values must first be estimated in order for the IPCM to be useful. The revised PCM needs to pass the consistency test. For this purpose, we have extended the IBMM to estimate the missing values in an IPCM (Ergu et al. 2011c). The revised PCM with the estimated values by IBMM is shown to satisfy the consistency requirement. In this Chapter, the details of IBMM for missing data estimation in AHP/ANP are comprehensively addressed.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 5. IBMM for Questionnaire Design Improvement

Abstract
Questionnaire survey is a commonly used way to collect opinions and views in AHP/ANP. However, many factors such as tedious design format, redundant content, long length etc, may lead to inconsistent comparison matrix for the decision problem. Invalid or bad results of a questionnaire survey may cause the decision makers to make wrong decision. Furthermore, in the AHP/ANP, the score items for a comparison matrix in a questionnaire increase drastically if there are more comparisons, which result in longer survey.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 6. IBMM for Rank Reversal

Abstract
When a new alternative or criterion is added to the decision model or old ones are deleted from the decision matrix, the rank of the alternatives may be reversed, namely, a less preferred alternative may become more preferred. In this Chapter, the IBMM is further extended to perform the sensitivity analysis of rank reversal when a new alternative or criterion is added or old ones are deleted. Details are described below.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 7. Applications of IBMM

Abstract
The AHP and ANP are two of the widely used MCDM methods, and have been extensively applied to the real-world decision making problems. However, the inconsistency issue and missing item scores issue are still two of the major issues when AHP and ANP are used. In the previous Chapters, the IBMM is proposed to deal with the inconsistency issue, missing item scores, and rank reversal issue. In this Chapter, the IBMM is applied to two real world applications, the Task Scheduling and Resource Allocation in Cloud Computing Environment by AHP and Risk Assessment and Decision Analysis by ANP. Details are presented in Sects. 7.1 and 7.2.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Chapter 8. Induced Arithmetic Average Bias Matrix Model (IAABMM)

Abstract
In previous Chapters, IBMM and its related extensions and applications are presented. In Ergu and Kou (2012), another form of induced bias matrix model, induced arithmetic average bias matrix model (IAABMM), is proposed and proved mathematically, which is easier to be understood than the previous model. In addition, two simpler inconsistency identification processes are also analyzed and proposed. An estimating formula of inconsistency adjustment for IAABMM is derived for the first time and illustrated by two numerical examples. In this Chapter, the details of IAABMM will be described.
Gang Kou, Daji Ergu, Yi Peng, Yong Shi

Backmatter

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