Advances in Pairwise Comparisons: Detection, Evaluation and Reduction of Inconsistency 3031238834, 9783031238833

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Multiple Criteria Decision Making

Jiri Mazurek

Advances in Pairwise Comparisons Detection, Evaluation and Reduction of Inconsistency

Multiple Criteria Decision Making Series Editor Constantin Zopounidis, School of Production Engineering and Management, Technical University of Crete, Chania, Greece

This book series focuses on the publication of monographs and edited volumes of wide interest for researchers and practitioners interested in the theory of multicriteria analysis and its applications in management and engineering. The book series publishes novel works related to the foundations and the methodological aspects of multicriteria analysis, its applications in different areas in management and engineering, as well as its connections with other quantitative and analytic disciplines. In recent years, multicriteria analysis has been widely used for decision making purposes by institutions and enterprises. Research is also very active in the field, with numerous publications in a wide range of publication outlets and different domains such as operations management, environmental and energy planning, finance and economics, marketing, engineering, and healthcare. This series has been accepted by Scopus.

Jiri Mazurek

Advances in Pairwise Comparisons Detection, Evaluation and Reduction of Inconsistency

Jiri Mazurek Silesian University in Opava Orlova, Czech Republic

ISSN 2366-0023 ISSN 2366-0031 (electronic) Multiple Criteria Decision Making ISBN 978-3-031-23883-3 ISBN 978-3-031-23884-0 (eBook) https://doi.org/10.1007/978-3-031-23884-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Operations research (OR), widely regarded as a sub-field of mathematical sciences, is a discipline that deals with the development and application of advanced analytical methods to improve real-world decision making. Nowadays, pairwise comparison methods rank among the most successfully applied decision-making methods not only in OR, but in almost all areas of human activity (the Web of Science currently lists more than 22,000 papers on the analytic hierarchy process alone). To make more informed, appropriate, and precise decisions less susceptible to various biases inherent in human thinking, one important research direction aims toward the investigation of methods’ properties and, in particular, to the problem of inconsistency in human judgments (preferences). Inconsistency in decision making, and in particular in comparisons of different objects, is a ubiquitous feature that cannot be neglected and certainly requires thorough investigation. I hope this book will attract the attention of a broad community of decision makers, experts, researchers, or practitioners working in the field of operations research, and, in particular, in the field of pairwise comparison methods. The book focuses on issues and problems associated with inconsistency in pairwise comparisons from mostly theoretical perspective. Human judgments are seldom absolutely consistent, or absolutely precise; therefore, problems of measuring and handling inconsistency belong among hot topics of the current research. In this book, a reader will find my own humble contributions to the topic as well as an up-to-date, condensed state-of-the-art of the discipline that can serve as a handbook covering “almost everything important” on the problem of inconsistency in pairwise comparisons. Orlova, Czech Republic 2023

Jiri Mazurek

v

Acknowledgment

The monograph was supported by the Grant Agency of the Czech Republic, project no. 21-03085S, and I would like to thank the leader of the project, Prof. Jaroslav Ramík, for the opportunity to be the team member. Also, I would like to express my gratitude to Prof. Dominik Strzalka, who helped me with formatting the book, and Jialin Yan from the Springer production team for her kind and smooth navigation through the publishing process.

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 A Brief History of Pairwise Comparisons Methods .. . . . . . . . . . . . . . . . . . 1.2 The Goal and the Organization of the Book . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 1 3 4

2 Multiplicative Pairwise Comparisons.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Concepts and Notation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Consistency of Pairwise Comparisons . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Prioritization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 The Geometric Mean Method and the Eigenvalue Method .. . 2.3.2 Other Priority Deriving Methods . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 A Comparison of Prioritization Methods . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Scales for Pairwise Comparisons .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.6 Scale Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.7 A Note on a Possible Cause of the Inconsistency .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

7 7 8 9 9 10 12 13 15 16 18

3 Inconsistency Indices and Their Properties .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 A Review of Inconsistency Indices.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Functional and Numerical Comparisons of Inconsistency Indices . . . 3.3 System of Properties for Inconsistency Indices. . . .. . . . . . . . . . . . . . . . . . . . 3.4 Alternative System of Properties for Inconsistency Indices.. . . . . . . . . . 3.5 Software Support for Inconsistency Estimation . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

21 21 33 38 39 43 45

4 Inconsistency Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Iterative Algorithms for Inconsistency Reduction .. . . . . . . . . . . . . . . . . . . . 4.2.1 Cao et al.’s Algorithm . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Kou et al.’s Algorithm . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Mazurek et al.’s Algorithm . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

49 51 53 53 53 54

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4.2.4 Szybowski’s Algorithm .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.5 Xu and Wei’s Algorithm.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Non-iterative Approaches to Inconsistency Reduction .. . . . . . . . . . . . . . . 4.3.1 Abel et al.’s Approach . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Bozóki et al.’s Approach . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Benítez et al.’s Approach . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.4 González-Pachón and Romero’s Approach.. . . . . . . . . . . . . . . . . . . 4.4 Interactive Inconsistency Reduction . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 A Numerical Comparison of Iterative Algorithms .. . . . . . . . . . . . . . . . . . . 4.6 REDUCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5 Alternative Approaches to the Evaluation of Inconsistency in Pairwise Comparisons .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1 A Triad and Single Element Inconsistency . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 The Condition of Order Preservation . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.1 The Condition of Order Preservation for Randomly Generated Matrices . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.2 Violation Frequency Index . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.3 Preference Violation Indices . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.4 Numerical Examples.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.2.5 Properties of Preference Violation Indices . . . . . . . . . . . . . . . . . . . . 5.2.6 Monte Carlo Simulations of Preference Violation Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 Coherence Conditions for Additive Pairwise Comparisons Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

56 56 57 57 58 59 60 61 62 69 70 73 73 75 78 81 81 83 84 87 89 93

6 Inconsistency of Incomplete Pairwise Comparisons Matrices . . . . . . . . . . 6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Priority Deriving and Completion Methods . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Modified Inconsistency Indices . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Original Inconsistency Indices . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Inconsistency in the Best-Worst Method.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Numerical Comparisons of Inconsistency Indices.. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

95 96 97 99 104 106 108 111

7 Ordinal Inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Ordinal Inconsistency Indices . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Properties of Ordinal Inconsistency Indices . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 The Ordinal-Cardinal Inconsistency Thresholds . .. . . . . . . . . . . . . . . . . . . . 7.4 The Relationship Between Cardinal, Ordinal, and the COP Inconsistency .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

115 118 124 127 131

Contents

xi

7.5 Detection and Reduction of Ordinal Inconsistency.. . . . . . . . . . . . . . . . . . . 133 7.6 Numerical Simulations on Ordinal Inconsistency .. . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 136 Conclusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139

List of Abbreviations

AHP AI AIR Alo AMD AMM ANP ATI BTL BWM CC CCI CI CIH COP CR CRO CRI CSM EIM ELECTRE EV (EVM) GEM FG GCI GLSM GM (GMM) GPM GW HCI HTA

Analytic Hierarchy Process Ambiguity inconsistency index Algorithm for inconsistency reduction Abelian linearly ordered (group) Average Manhattan distance Arithmetic mean method Analytic Network Process Grzybowski’s inconsistency index Bradley-Terry-Luce model Best-Worst method Generic coherence condition Cosine consistency index Saaty’s consistency index Wu and Xu’s inconsistency index Condition of order preservation Saaty’s Consistency ratio Output-based Consistency Ratio Input-based Consistency Ratio Chi-square method Exponentially invertible measure ELimination and Choice Expressing REality Eigenvalue method Gradient eigenweight method Fedrizzi-Giove inconsistency index Geometric consistency index Geometric least squares method Geometric mean method Goal programming method Golden-Wang index Harmonic consistency index Homogeneous treatment of alternatives xiii

xiv

ICD ICI IE ILLS INSITE KI KII LDM LLS LS MACBETH MC MCDM MII MM MOO NA NJR NJV OR PAPRIKA PC PCM PD PLI (CI*) POIP POP PROMETHEE QC RI RIC RE RO SBS SCI SDR SI STJD SVD TEI T-GCI TKI

List of Abbreviations

Cavallo-d’Apuzzo index Incomplete consistency index Index exchangeability Incomplete logarithmic least squares reducINg inconSIsTency in dEcision making Koczkodaj’s inconsistency index Kendall’s inconsistency index Least distance method Least logarithmic squares Least squares Measuring Attractiveness by a Category-Based Evaluation Technique Takeda’s Inconsistency Index Multiple criteria decision making Manhattan inconsistency index Restricted max-max transitivity (of pairwise comparisons matrices) Multi-objective optimization Not available Number of judgment reversals Number of judgment violations Operations research Potentially All Pairwise RanKings of all possible Alternatives Pairwise comparisons Pairwise comparisons matrix Pairwise dominance Pelaez-Lamata inconsistency index Preservation of order of intensity preference Preservation of order preference Preference Ranking Organization Method for Enrichment Evaluations Quasi-consistency Random consistency index Row inconsistency index Relative Error Index Rank-order condition Step-by-step algorithm Sparse consistency index Standard deviation of ranks index Scale invariance Squared total judgement deviation Single value decomposition Total element inconsistency Triad based geometric inconsistency index Triad Koczkodaj’s inconsistency index

List of Abbreviations

TND TTI UT WC

Total number of deviation points Total triad inconsistency Upper triangle (of a pairwise comparisons matrix) Weakly consistent

xv

List of Figures

Fig. 2.1 A comparison of different scales. Source: author . . . . . . . . . . . . . . . . . . . . Fig. 2.2 Redundancy (.M) versus inconsistency (.RI ). Source: author . . . . . . . Fig. 3.1 Decomposition of inconsistency indices .GCI , .CI ∗ /P LI , .ICD , .KI , and .AT I . Source: [11] . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.2 Scatterplots of inconsistency indices. .K denotes Koczkodaj’s index. Source: [12] . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.3 A connection between indices from Table 3.4. Source: author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.4 A comparison of .Iχ 2 with four other indices. Source: [22] . . . . . . . . . Fig. 3.5 Values of selected inconsistency indices for a corner pairwise comparisons matrix .A, Example 3.2. Source: author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.6 Values of another set of inconsistency indices for a corner pairwise comparisons matrix .A, Example 3.2. Source: author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.7 Screenshot of PriEsT. Source: https://priority.sourceforge. io .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.8 Screenshot of PriEsT. Source: https://priority.sourceforge. io .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.1 Distribution of .d(%), .n = 4, .0.10 < CR < 0.30. Source: [32] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.2 Distribution of .D, .n = 4, .0.10 < CR < 0.30. Source: [32] . . . . . . . . Fig. 4.3 Distribution of .d(%), .n = 4, .0.30 ≤ CR < 0.80. Source: [32] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.4 Distribution of .D, .n = 4, .0.30 ≤ CR < 0.80. Source: [32] . . . . . . . . Fig. 4.5 Distribution of .d(%), .n = 8, .0.10 < CR < 0.30. Source: [32] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.6 Distribution of .D, .n = 8, .0.10 < CR < 0.30. Source: [32] . . . . . . . . Fig. 4.7 Distribution of .d(%), .n = 8, .0.30 ≤ CR < 0.80. Source: [32] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 4.8 Distribution of .D, .n = 8, .0.30 ≤ CR < 0.80. Source: [32] . . . . . . . .

15 17 30 34 35 36

37

37 44 44 65 65 66 66 67 67 68 68 xvii

xviii

List of Figures

Fig. 4.9 The output of REDUCE. Source: author . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.1 Average number of individual POP violations (EVM), .n = 7. Source: [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.2 Average number of individual POIP violations (EVM), .n = 7. Source: [26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.3 The left scale and blue dots, Alpha; the right scale and red dots, .KI . Source: author . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.4 Frequency distribution of .Alpha. Source: author .. . . . . . . . . . . . . . . . . . . Fig. 5.5 Frequency distribution of .Beta. Source: author .. . . . . . . . . . . . . . . . . . . . Fig. 5.6 Inclusion relations between coherence conditions. Source: [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 5.7 Scatterplots for the comparison of the optimization problems related with different coherence conditions. The reported plots refer to 500 randomly generated PCMs with .n = 6. Source: [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.1 The random inconsistency index. Source: [2] . . . .. . . . . . . . . . . . . . . . . . . . Fig. 6.2 Rescaled ordered distance for different inconsistency indices for incomplete PC matrices with k missing comparisons. Source: [27]; see https://www.tandfonline. com/toc/ggen20/current . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.1 Graph representation of preferences from Example 7.1. Source: author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.2 Consistent triads on the left and inconsistent triads on the right. Undirected edges correspond to ties. Source: author .. . . . . . . . . Fig. 7.3 Ordinally and cardinally consistent matrices. Source: author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.4 A graph representation of the matrix .A. Source: author . . . . . . . . . . . . . Fig. 7.5 The relationship between ordinal and cardinal inconsistency of PC matrices with respect to Koczkodaj’s inconsistency index. Source: author . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.6 The relationship between cardinal, ordinal, and the COP consistency. Source: author . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 7.7 The average values of CI and .ςg for an increasing level of disturbance. Source: [13] . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

69 79 80 85 87 88 91

92 100

110 117 119 120 123

129 133 135

List of Tables

Table 2.1 Saaty’s fundamental scale. Source: [53] . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 2.2 Scales for pairwise comparisons. Source: author . . . . . . . . . . . . . . . . . . Table 2.3 Percentage of acceptable PC matrices for all scales. Source: [56] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 3.1 The values of .RI (n). It should be noted that slightly different values were reported in other studies; see, e.g., [25] or [27]. Source: author (adapted from [3]) .. . . . . . . . . . . . . . . . . . . . Table 3.2 Inconsistency thresholds. Source: author . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 3.3 Percentiles for .CR, .KI , .P LI , and .T − GCI indices with respect to the matrix size .n; 50,000 randomly generated cases for each .n with matrix elements drawn from Saaty’s fundamental scale. .x˜ = median. Source: [43] .. . . . . . . . . . . . . . . . . . . . Table 3.4 Pearson’s correlation coefficients .ρ between inconsistency indices for 10,000 randomly generated matrices of the order .n = 6. Source: [12] . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 3.5 The lowest values of Spearman’s correlation coefficients between inconsistency indices for 10,000 randomly generated matrices and matrix order .3 ≤ n ≤ 12. Source: [15] . . . . Table 3.6 Pearson’s correlation coefficients .ρ between .Iχ 2 , .CR, ∗ .CI ≡ P LI , .GW , and .GCI , .n = 6. Source: [22] . . . . . . . . . . . . . . . . . Table 3.7 Satisfaction of properties by selected inconsistency indices. Source: author (adapted from [7, 9, 11, 42], and [52]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 3.8 Satisfaction of Axioms 1–5 and .1∗ − 4∗ by selected inconsistency indices. Source: author . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 4.1 Inconsistency reduction of the matrix .T —results by selected algorithms (best values are in bold). Source: author . . . . . . Table 4.2 AIR performance, average values for .n = 4, and initial .0.10 ≤ CR < 0.30, 630 matrices (the best values are in bold). Source: [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

13 14 14

22 31

32

34

35 36

39 41 63

63

xix

xx

List of Tables

Table 4.3 AIR performance, average values for .n = 4, and initial .0.30 ≤ CR < 0.80, 1193 matrices (the best values are in bold). Source: [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 4.4 AIR performance, average values for .n = 8, and initial .0.10 < CR < 0.30, 491 matrices (the best values are in bold). Source: [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 4.5 AIR performance, average values for .n = 8, and initial .0.30 ≤ CR < 0.80, 4082 matrices (the best values are in bold). Source: [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 5.1 Satisfaction of POP and POIP conditions (in %), .CI < 0.10. The EV method was used for the last two columns. Source: [26] .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 5.2 Satisfaction of POP and POIP conditions (in %), .CI ≥ 0.10. The EV method was used for the last two columns. Source: [26] . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 5.3 Percentile values for .Alpha and .Beta, .n = 4. Source: author .. . . . Table 6.1 Consistency index (.max ξ ) in the Best-Worst Method. Source: [39] .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 6.2 The total distance .D from the abscissae of the .D(I, k) plots for all considered indices. The smaller the value, the more robust the given inconsistency index. Source: [27]; see https://www.tandfonline.com/toc/ggen20/current . . . . . . . . . . . . . . Table 7.1 Ordinal consistency thresholds. Source: [8] . . . .. . . . . . . . . . . . . . . . . . . . Table 7.2 The thresholds under which a PC matrix cannot be ordinally inconsistent with respect to selected triad-based inconsistency indices and Saaty’s fundamental scale in the case of no ties. Source: author . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 7.3 Transitive-intransitive thresholds for selected indices; 100,000 PC matrices for each .n. Source: [1] . . .. . . . . . . . . . . . . . . . . . . .

64

64

64

78

79 88 108

111 119

131 136

Chapter 1

Introduction

Abstract Pairwise comparisons constitute a crucial part of many popular presentday multiple-criteria decision-aiding/decision-making methods such as the analytic hierarchy process (AHP). This chapter provides a brief historical overview of pairwise comparisons’ main developments as well as the organization of the presented book.

1.1 A Brief History of Pairwise Comparisons Methods The origins of pairwise comparisons can be dated back to prehistoric times, when the ascendants of Homo sapiens had to select the better stone from two specimens held in each hand, better wood for a bow, and so on. In the Medieval era, pairwise comparisons appeared for the first time in the work of the Catalan scholar and monk Ramon Llull (1232–1315); see [36]. Llull described a procedure for the selection of a prelate out of 16 candidates, where all prelates were compared pairwise and the prelate with the most wins took the position. In addition, Llull correctly stated that, in his example, a total of 120 pairwise comparisons had to be made. In the eighteenth century, the pairwise comparisons method was rediscovered by Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (1743–1794), who applied pairwise comparisons in the elections framework in what is now known as the Condorcet method [23]. The first notable work on pairwise comparisons with application in psychology can be attributed to the psychometrician L. L. Thurstone (1887–1955) and his law of comparative judgments; see [55]. His work relied on the psychophysical theory developed by E. H. Weber and G. Fechner in the nineteenth century, which postulated that the relationship between a stimulus and its perception is logarithmic. Thurstone applied pairwise comparisons to measure the perceived intensity of physical stimuli, attitudes, preferences, choices, or values. In the 1950s, Thurstone’s model was generalized into the Bradley-Terry-Luce (BTL) model [7]. During the twentieth century, individual ordinal preferences (ordinal pairwise comparisons) and their aggregation into group preferences became a focus of the social choice theory which culminated in K. J. Arrow’s famous impossibility © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mazurek, Advances in Pairwise Comparisons, Multiple Criteria Decision Making, https://doi.org/10.1007/978-3-031-23884-0_1

1

2

1 Introduction

theorem from 1951 [1]. In economics, ordinal preferences and their consistency were studied in the framework of consumption theory and rational choice theory; see, e.g., weak axiom of revealed preference by P. A. Samuelson [54] from 1948. In the 1960s, B. Roy developed the first modern multiple-criteria decisionmaking (MCDM) method incorporating pairwise comparisons, called ELECTRE (ELimination and Choice Expressing REality) [47]. In 1977, T. L. Saaty introduced his analytic hierarchy process (AHP), a sophisticated MCDM method (see [48– 52]), followed by the analytic network process (ANP). In the following decades, the AHP/ANP became one of the most popular multiple-criteria methods and was applied in almost all areas of human decision-making; see, e.g., [56]. Another pairwise comparisons MCDM method called PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) was introduced in 1982 by J. Brans [8, 9]. Among the latest contributions to the pairwise comparisons methods are PAPRIKA (Potentially All Pairwise RanKings of all possible Alternatives) [26], MACBETH (Measuring Attractiveness by a Category-Based Evaluation Technique) [3], or the BWM (Best-Worst Method) [46]. Over the last decades, pairwise comparisons have become one of the most popular tools in the MCDM framework. The literature on the application of pairwise comparisons amounts to thousands of papers in almost all areas of human activity; see, e.g., [5, 25], or [56]. One of the most important issues associated with pairwise comparisons is the problem of its (in)consistency. The first measure of inconsistency was introduced by Saaty in [48] in the form of the consistency index CI and consistency ratio CR. If an object A is considered two times better than an object B, and an object B is four times better than an object C, then it is expected that the object A is eight times better than C (i.e., 2 × 4 = 8); otherwise, judgments are considered inconsistent. However, in practice, humans are not able to be fully consistent, especially when the number of pairwise comparisons is large; see the experimental studies [6, 22, 34, 35, 38], or [39]. Later, many other inconsistency indices were introduced; see, e.g., [4, 24, 27, 41, 42, 45], or [53]. As the number of inconsistency indices grew, studies focusing on their comparisons and axiomatization emerged; see, e.g., [10–12, 14, 15, 28–31], or [37]. In addition, generalizations of the multiplicative pairwise comparisons, for example to Abelian linearly ordered groups (Alo-groups), have been considered; see [17, 18, 20, 33], or [43]. In 2008, the condition of order preservation (COP) in the context of pairwise comparisons was introduced in [2]. This condition can be considered an alternative approach to the evaluation of pairwise comparisons consistency. The COP states that, after comparing a set of objects pairwise, the priority vector (the vector of weights associated with every compared object) should not contradict individual judgments. That is, if an object A is directly preferred to an object B, then the weight of A should also be greater than the weight of B. Further on, if A is compared to B, and C is compared to D, with the difference between A and B being greater than the difference between C and D, then the difference between weights associated with A and B should also be greater than the difference in weights between C and D.

1.2 The Goal and the Organization of the Book

3

After its introduction, the condition of order preservation attracted the attention of several authors; see, e.g., [20, 32, 33], or [40]. Studies [32] and [33] provided a sufficient condition for the COP satisfaction with respect to inconsistency expressed by Koczkodaj’s inconsistency index for both the multiplicative case and the more general case based on Alo-groups. Finally, the study [40] proposed a new method for derivation of the priority vector based on the COP condition. More recently, alternative notions of (in)consistency based on the so-called coherence conditions, such as weak consistency, were introduced and discussed in [13, 16, 19, 21]. Finally, a summary of pairwise comparisons methods can be found in [44].

1.2 The Goal and the Organization of the Book The presented book covers both cardinal and ordinal inconsistency associated with pairwise comparisons, for both complete and incomplete pairwise comparisons matrices. The problem of inconsistency is discussed mainly in the framework of multiplicative pairwise comparisons, though additive pairwise comparisons are briefly discussed as well. The book does not cover areas of pairwise comparisons associated with uncertainty, that is, extensions toward interval pairwise comparisons, fuzzy and hesitant fuzzy pairwise comparisons, and other theoretical frameworks, which certainly deserve their own monograph. Probabilistic and group decision-making pairwise comparisons are not covered as well. As its title suggests, the book focuses on three key interconnected aspects of inconsistency: its detection, evaluation, and reduction. The book is divided into seven main chapters. Chapter 1 provides an introduction to the topic. Chapter 2 focuses on multiplicative pairwise comparisons, pairwise comparisons matrices, prioritization methods, scales used for pairwise comparisons, and the notion of inconsistency. Chapter 3 contains an overview of inconsistency indices and their properties and (dis)similarities. Chapter 4 deals with algorithmic inconsistency reduction. Chapter 5 provides alternative approaches to the inconsistency of pairwise comparisons and, specifically, deals with the condition of order preservation and coherence relations. Chapter 6 focuses on the inconsistency of incomplete pairwise comparisons, and, finally, Chap. 7 presents a brief introduction to the topic of ordinal inconsistency and its relationship with cardinal inconsistency. All the chapters include definitions of new concepts, their explanation, the most important theoretical relationships in the form of propositions and theorems, and also numerical comparisons and illustrative examples to facilitate readers’ comprehension.

4

1 Introduction

References 1. Arrow, K. J. (1951). Social Choice and Individual Values. New Haven: Yale University Press. 2. Bana e Costa, C. A., & Vansnick, J. (2008). A critical analysis of the eigenvalue method used to derive priorities in AHP. European Journal of Operational Research, 187(3), 1422–1428. 3. Bana e Costa, C. A., De Corte, J. M., & Vansnick, J. C. (2005). On the mathematical foundation of MACBETH. In Multiple Criteria Decision Analysis: State of the Art Surveys. International Series in Operations Research and Management Science (p. 78). New York: Springer. https:// doi.org/10.1007/0-387-23081-5_10. 4. Barzilai, J. (1998). Consistency measures for pairwise comparison matrices. Journal of MultiCriteria Decision Analysis, 7(3), 123–132. 5. Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198–215. https://doi.org/10.1016/j.ejor.2009.01.021. 6. Bozóki, S., Dezsö, L., Poesz, A., & Temesi, J. (2013). Analysis of pairwise comparison matrices: an empirical research. Annals of Operations Research, 211, 511–528. https://doi. org/10.1007/s10479-013-1328-1. 7. Bradley, R. A., & Milton, T. E. (1952). Rank Analysis of Incomplete Block Designs: I. The Method of Paired Comparisons. Biometrika, 39 (3/4), 324. https://doi.org/10.2307/2334029. 8. Brans, J. P. (1982). La ingenierie de la decision: elaboration de instruments de aide a la decision. La methode PROMETHEE. Québec City: Presses de l’Universite Laval. 9. Brans, J. P., & Vincke, P. (1985). A preference ranking organisation method: The PROMETHEE method for MCDM. Management Science, 31(6), 647–656. 10. Brunelli, M. (2016). Recent Advances on Inconsistency Indices for Pairwise Comparisons—A Commentary. Fundamenta Informaticae 144(3–4), 321–332. 11. Brunelli, M. (2016). On the conjoint estimation of inconsistency and intransitivity of pairwise comparisons. Operations Research Letters, 44, 672–675. 12. Brunelli, M. (2017). Studying a set of properties of inconsistency indices for pairwise comparisons. Annals of Operations Research, 248(1,2), 143–161. 13. Brunelli, M., & Cavallo, B. (2020). Incoherence measures and relations between coherence conditions for pairwise comparisons. Decisions in Economics and Finance, 43, 613–635. https://doi.org/10.1007/s10203-020-00291-x. 14. Brunelli, M., & Fedrizzi M. (2015). Axiomatic properties of inconsistency indices for pairwise comparisons. Journal of the Operational Research Society, 66(1), 1–15. 15. Brunelli, M., Canal, L., & Fedrizzi, M. (2013). Inconsistency indices for pairwise comparison matrices: a numerical study. Annals of Operations Research, 211(1), 493–509. 16. Cavallo, B. (2019). Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem. Journal of Global Optimization, 75(1), 143–161. 17. Cavallo, B., & D’Apuzzo, L. (2009). A general unified framework for pairwise comparison matrices in multicriteria methods. International Journal of Intelligent Systems, 24(4), 377–398. 18. Cavallo, B., & D’Apuzzo, L. (2015). Reciprocal transitive matrices over abelian linearly ordered groups: Characterizations and application to multi-criteria decision problems. Fuzzy Sets and Systems, 266, 33–46. https://doi.org/10.1016/j.fss.2014.07.005. 19. Cavallo, B., & D’Apuzzo, L. (2016). Ensuring reliability of the weighting vector: weak consistent pairwise comparison matrices. Fuzzy Sets and Systems, 296, 21–34. 20. Cavallo, B., & D’Apuzzo, L. (2020). Preservation of preferences intensity of an inconsistent Pairwise Comparison Matrix. International Journal of Approximate Reasoning, 116, 33–42. 21. Cavallo, B., D’Apuzzo, L., & Basile, L. (2016). Weak Consistency for Ensuring Priority Vectors Reliability. Journal of Multi-criteria Decision Analysis, 23, 126–138. 22. Cavallo, B., Ishizaka, A., Olivieri, M. G., & Squillante, M. (2019). Comparing inconsistency of pairwise comparison matrices depending on entries. Journal of the Operational Research Society, 70(5), 842–850. https://doi.org/10.1080/01605682.2018.1464427.

References

5

23. marquis de Condorcet, M. J. A. (1785). Essai sur l’application de l’analyse a la probabilite des decisions rendues a la pluralite des voix, Paris, France. 24. Golden, B., & Wang, Q. (1989). An alternate measure of consistency. In B. Golden, E. Wasil, & P. T. Harker (Eds.), The analytic hierarchy process, applications and studies (pp. 68–81). Berlin: Springer. 25. Govindan, K., & Jepsen, M. B. (2015). ELECTRE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 250(1), 1–29. https://doi.org/10.1016/j.ejor.2015.07.019. 26. Hansen, P., & Ombler, F. (2008). A new method for scoring additive multi-attribute value models using pairwise rankings of alternatives. Journal of Multi-Criteria Decision Analysis, 15(3–4), 87–107. https://doi.org/10.1002/mcda.428. 27. Koczkodaj, W. W. (1993). A new definition of consistency of pairwise comparisons. Mathematical and Computer Modeling, 18(7), 79–84. 28. Koczkodaj, W. W., & Magnot, J.-P. (2017). Axiomatization of Inconsistency Indicators for Pairwise Comparisons. ArXiv:1509.03781v2. 29. Koczkodaj, W. W., & Szwarc, R. (2014). On axiomatization of inconsistency indicators for pairwise comparisons. Fundamenta Informaticae, 132(4), 485–500. 30. Koczkodaj, W. W., & Urban, R. (2018). Axiomatization of inconsistency indicators for pairwise comparisons. International Journal of Approximate Reasoning, 94, 18–29. https://doi.org/10. 1016/j.ijar.2017.12.001. 31. Koczkodaj, W. W., Magnot, J.-P., Mazurek, J., Peters, J. F., Rakhshani, H., Soltys, M., Strzalka, D., Szybowski, J., & Tozzi, A. (2017). On normalization of inconsistency indicators in pairwise comparisons. International Journal of Approximate Reasoning, 86, 73–79. 32. Kulakowski, K. (2015). Notes on Order Preservation and Consistency in AHP. European Journal of Operational Research, 245, 333–337. 33. Kulakowski, K., Mazurek, J., Ramík, J., & Soltys, M. (2019). When is the condition of preservation met? European Journal of Operational Research, 277, 248–254. 34. Linares, P. (2009). Are inconsistent decisions better? An experiment with pairwise comparisons. European Journal of Operational Research, 193, 492–498. 35. Linares, P., Lumbreras, S., Santamaría, A., & Veiga, A. (2016). How relevant is the lack of reciprocity in pairwise comparisons? An experiment with AHP. Annals of Operations Research, 245, 227–244. https://doi.org/10.1007/s10479-014-1767-3. 36. Lull, R. (1274–1283). Artifitium electionis personarum (The method for the elections of persons). https://www.math.uni-augsburg.de/htdocs/emeriti/pukelsheim/llull/. 37. Mazurek, J. (2018). Some notes on the properties of inconsistency indices in pairwise comparisons. Operations Research and Decisions, 1, 27–42. 38. Mazurek, J., & Nenickova, Z. (2020). Occurrence and Violation of Transitivity of Preferences in Pairwise Comparisons. In S. Kapounek, & H. Vranova (Eds.), 38th International Conference on Mathematical Methods in Economics (pp. 371–376), Czech Republic: Mendel University Brno. 39. Mazurek, J., & Perzina, R. (2017). On the inconsistency of pairwise comparisons: an experimental study. Scientific papers of the University of Pardubice—Series D3, 41, 102–109. 40. Mazurek, J., & Ramík, J. (2019). Some new properties of inconsistent pairwise comparison matrices. International Journal of Approximate Reasoning, 113, 119–132. 41. Obata, T., Shiraishi, S., Daigo, M., & Nakajima, N. (1999). Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: Computational experiments. In ISAHP. Japan: Kobe. 42. Peláez, J. I., & Lamata, M. T. (2003). A new measure of inconsistency for positive reciprocal matrices. Computer and Mathematics with Applications, 46(12), 1839–1845. 43. Ramík, J. (2015). Pairwise comparison matrix with fuzzy elements on alo-groups. Information Sciences, 297, 236–253. 44. Ramík, J. (2020). Pairwise Comparisons Method: Theory and Applications in Decision Making. Lecture Notes in Economics and Mathematical Systems. Berlin: Springer.

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45. Ramík, J., & Korviny, P. (2010). Inconsistency of pairwise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161, 1604–1613. 46. Rezaei, J. (2015). Best-worst multi-criteria decision-making method. Omega, 53, 40–57. 47. Roy, B. (1968). Classement et choix en presence de points de vue multiples (la methode ELECTRE). La Revue de Informatique et de Recherche Operationelle (RIRO), 8, 57–75. 48. Saaty, T. L. (1977). A Scaling Method for Priorities in Hierarchical Structures. Journal of Mathematical Psychology, 15, 234–281. 49. Saaty, T. L. (1980). Analytic Hierarchy Process. New York: McGraw-Hill. 50. Saaty, T. L. (1994). Fundamentals of Decision Making. Pittsburgh, USA: RWS Publications. 51. Saaty, T. L. (2004). Decision making—The analytic hierarchy and network processes (AHP/ANP). Journal of Systems Science and Systems Engineering, 13(1), 1–34. 52. Saaty, T. L. (2008). Decision making with the analytic hierarchy process. International Journal of Services Sciences, 1, 83–98. 53. Salo, A. A., & Hämäläinen, R. (1995). Preference Programming through Approximate Ratio Comparisons. European Journal of Operational Research, 82(3), 458–475. 54. Samuelson, P. A. (1948). Consumption theory in terms of revealed preferencem. Economica, 15(60), 243–253. 55. Thurstone, L. L. (1927). A law of comparative judgments. Psychological Reviews, 34, 273– 286. 56. Vaidya, O. S., & Kumar, S. (2006). Analytic hierarchy process: An overview of applications. European Journal of Operational Research, 169, 1–29.

Chapter 2

Multiplicative Pairwise Comparisons

Abstract This chapter begins with a brief introduction into the multiplicative pairwise comparisons framework. It introduces concepts such as a pairwise comparisons matrix, a priority vector, a priority deriving (prioritization) method, or a definition of inconsistency. Numerical examples, comparisons of prioritization methods via Monte Carlo experiments, and a discussion of scales used for pairwise comparisons are provided as well.

This chapter provides a brief introduction into multiplicative pairwise comparisons framework. It includes definitions of basic concepts, applied notation, a definition of (cardinal) consistency, and an overview of prioritization methods and scales used for pairwise comparisons. It should be noted that, although pairwise comparisons in the multiplicative setting are the most common both in practice and in the literature, there are also other systems, such as additive or fuzzy; see, e.g., [7, 19, 29, 34, 47, 49, 50] or [51].

2.1 Concepts and Notation Let C = {c1 , c2 , . . . , cn }, n ∈ N, n > 1 be a non-empty and finite set of compared objects (alternatives, criteria, sub-criteria, etc.). Let S be a pairwise comparisons scale (see Chap. 2.5). Let aij ∈ S, aij > 0 denote the relative importance of object i over object j , where i, j ∈ {1, . . . , n}. For instance, aij = 3 means that object i is three times more important, or more preferred, than object j . Definition 2.1 Pairwise comparisons of n objects form a n × n square matrix A = [aij ] called a pairwise comparisons matrix (PCM). ⎡

1 ⎢ a21 A=⎢ ⎣ ... an1

a12 1 ... ...

⎤ . . . a1n ... ... ⎥ ⎥, 1 ... ⎦ ... 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mazurek, Advances in Pairwise Comparisons, Multiple Criteria Decision Making, https://doi.org/10.1007/978-3-031-23884-0_2

7

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2 Multiplicative Pairwise Comparisons

Definition 2.2 A pairwise comparisons matrix An×n = [aij ] is reciprocal if aij = 1/aj i ∀i, j ∈ {1, . . . , n}

(2.1)

Hereinafter, it is assumed that all pairwise comparisons matrices are reciprocal. Let’s call a pairwise comparisons method any decision-making method that involves pairwise comparisons, and let prioritization method (a priority deriving method) be any procedure that derives a priority vector w = (w1 , . . . , wn ) (vector of weights of all n compared objects) from a n × n PC matrix. We say that the priority vector w is associated with the PC matrix A or that the priority vector w is derived by a priority generating method based on the PC matrix A. In addition, the priority vector w is usually normalized, i.e., n 

wi = 1

(2.2)

i=1

Based on the values w1 , . . . , wn , compared objects can be ranked from the best to the worst (with possible ties), which is the goal of pairwise comparisons methods.

2.2 Consistency of Pairwise Comparisons Formally, the (cardinal) consistency of pairwise comparisons is defined as follows: Definition 2.3 The matrix An×n = [aij ] is said to be consistent if aik = aij · aj k ∀i, j, k ∈ {1, . . . , n}

(2.3)

Note that a consistent matrix is also reciprocal, but the converse statement is not true in general. Example 2.1 Consider the following PC matrix A = [aij ] of the order n = 4. It can be seen that A is reciprocal, but it is not consistent, since, for instance, a13 · a34 = a14 (2 × 5 = 5). ⎡

⎤ 1 12 2 5 ⎢ 2 1 4 4⎥ ⎥ A=⎢ ⎣ 1 1 1 5⎦, 2 4 1 1 1 5 4 5 1 The following proposition provides a relationship between entries of a consistent PC matrix and its priority vector.

2.3 Prioritization Methods

9

Proposition 2.1 Let An×n = [aij ] be a consistent pairwise comparisons matrix. Then there is a unique (up to a multiplication constant) priority vector w = (w1 , . . . , wn ) associated with A satisfying aij =

wi , ∀i, j. wj

(2.4)

Proof See [52].

 

In addition, if a PC matrix is consistent, then all its rows (or columns) are collinear (multiples of each other). The standard notion of consistency in pairwise comparisons requires that if, for example, an object A is two times better than an object B, and the object B is three times better than an object C, then the object A should be six times better than C. Though the concept of consistency is obvious and logical, when solving real-world problems, decision-makers are seldom consistent. One problem associated with inconsistency is that it requires absolute precision, which is not natural for human experts. For example, if an expert considers object A to be 50% (1.5 times) better than object B and object B to be 50% (1.5 times) better than object C, then A should be precisely 125% (2.25 times) better than C, which would arguably be rarely heard from an expert. The second problem associated with inconsistency stems from the fact that as the number n of compared objects increases, the number of pairwise comparisons grows as n(n−1) (not counting reciprocals); hence, for n = 4, there are 6 pairwise 2 comparisons; for n = 5, there are 10 pairwise comparisons; for n = 6, we get 15 pairwise comparisons; and so forth. With the growing number of comparisons, it is more and more difficult to remain consistent, since the capabilities of the human working memory are limited to 7 ± 2 items, according to the classic study of Miller [46]. A study [44] investigated how often decision-makers (undergraduate university students) are consistent in their judgments expressed by pairwise comparisons matrices of the order 3 ≤ n ≤ 7. In the case of smaller matrices (n = 3 and n = 4), more than 90% of decision-makers were inconsistent. For n = 7, the frequency of inconsistent matrices reached 100%. This result indicates that the problem of inconsistency in pairwise comparisons is omnipresent and cannot be neglected.

2.3 Prioritization Methods 2.3.1 The Geometric Mean Method and the Eigenvalue Method The geometric mean method (also known as the least logarithmic squares method) and the eigenvalue (eigenvector) method are the two main methods for the derivation of a priority vector from a pairwise comparisons matrix. The eigenvalue (EV)

10

2 Multiplicative Pairwise Comparisons

method was proposed by Saaty [52], and the geometric mean (GM) method was introduced by Crawford and Williams [12, 13]. In the EV method, the priority vector is equated with the right eigenvector corresponding to the largest eigenvalue of a PC matrix An×n = [aij ]: A · w = λmax · w,

(2.5)

where λmax ≥ n is a positive eigenvalue (the existence of λmax is guaranteed by the Perron-Frobenius theorem) and w is the corresponding right eigenvector of A. Usually, it is assumed that w is normalized: w = 1. In the GM method, the priority vector w is derived as the geometric mean of all rows of A: wi =

n

1/n aik

/

k=1

n n 

j =1

1/n aj k

, ∀i

(2.6)

k=1

This formula is equivalent to finding a solution of the following non-linear programming problem:

n  n  wi 2 min lnaij − ln wj

(2.7)

i=1 j =1

s.t.

n 

wi = 1, wi ≥ 0, ∀i

(2.8)

i=1

Again, the priority vector w is normalized.

2.3.2 Other Priority Deriving Methods Apart from the two main methods mentioned in the previous chapter, other methods have been proposed in the literature. In the arithmetic mean method (AMM), also called the mean of normalized values method [27], the priority vector w is derived from the pairwise comparisons matrix A = [aij ] as follows: n

wi =

1  aij , n cj j =1

where cj is the sum of the j -th column of A.

(2.9)

2.3 Prioritization Methods

11

Hence, according to the AMM, each column of the matrix A is divided by its sum (a column is normalized), and then the weight of the i-th object is given as the arithmetic mean of the i-th row. Again, w = 1. In the least squares method (LSM) (it is unclear who should be credited for its introduction; see, e.g., [4, 10], or [54]), the vector of weights w is determined by a solution of the following problem:

n  n  wi 2 min : ; ∀i, j aij − wj

(2.10)

i=1 j =1

Recently, an interesting method was proposed by Siraj et al. [55]. This method is based on spanning trees of a graph representation of a given PC matrix, and the final priority vector is the arithmetic mean of all spanning trees’ priority vectors: η 1 w= w(τs ), η

(2.11)

s=1

where Cayley’s formula η = nn−2 denotes the number of spanning trees for a complete PC matrix (a full graph) and w(τs ) denotes the priority vectors of individual spanning trees. An obvious drawback of this method is the large number of spanning trees; therefore, the authors suggest the application of a proper sampling. The spanning tree prioritization method was later elaborated in [22, 43]. Further on, in [40], it was shown that the spanning tree method is equivalent with the (row) geometric mean method for complete PC matrices, while the study [5] proved the equivalence for incomplete PC matrices as well. Simple row sum/average priority deriving methods were proposed by Ra and Saaty; see [48, 53]. In these methods, a priority vector corresponds to the sums or averages over rows of a PC matrix. Mazurek and Ramík [45] introduced a new prioritization method in the context of the condition of order preservation; see Chap. 5. Other less frequently applied approaches to the derivation of a priority vector, such as the linear programming method (LPM), chi-square method (CSM), gradient eigenweight method (GEM), least distance method (LDM), geometric least squares method (GLSM), or goal programming method (GPM), can be found, e.g., in [2, 6, 9, 15, 17, 23, 30, 38], or [39], where, in particular, studies [17, 26] deal with nonreciprocal PC matrices. Example 2.2 Consider the following PC matrix A: ⎡

⎤ 1 12 2 5 ⎢ 2 1 4 4⎥ ⎥ A=⎢ ⎣ 1 1 1 5⎦, 2 4 1 1 1 5 4 5 1

12

2 Multiplicative Pairwise Comparisons

We will find the corresponding priority vectors w. From relation (2.5), the maximum eigenvalue λmax = 4.254; thus, wEV = (0.282, 0.474, 0.179, 0.065). Via relation (2.6), we obtain wGM = (0.294, 0.468, 0.175, 0.062). And, finally, via relation (2.9), we get wAM = (0.283, 0.466, 0.183, 0.068). Though the priority vectors slightly differ, all three methods provide the same ranking of objects in the following order: (2, 1, 3, 4).

2.4 A Comparison of Prioritization Methods Example 2.2 shows that priority vectors obtained by different methods are not the same when the PC matrix is inconsistent. In general, priority vectors derived by the EV and GM methods are identical in the case of a consistent pairwise comparisons matrix [13]. Additionally, in the case of n = 3, priority vectors for both methods coincide even when the PC matrix is inconsistent; see, e.g., [13] or [35]. The study [25] established that for not-soinconsistent matrices, both the EV and GM methods produce very similar results, with differences “small beyond human perception.” For numerical comparisons of prioritization methods, see, e.g., [11, 27, 37], or [39]. Ishizaka and Lusti [27] found that the priority deriving methods that differ the most are the left eigenvalue method and the right eigenvalue method and that the rankings’ contradictions among different methods (the mean normalized value method, left eigenvalue method, right eigenvalue method, and geometric mean method were examined) increase with increasing matrix size and the value of the consistency ratio CR almost linearly. However, the differences were rather small, so the authors concluded their study by saying [27]: There is a high level of agreement between the different . . . techniques. . . We do not think that one method is superior to another. We advise decision makers also to consider other criteria like ‘easy to use’ while selecting their derivation method

From a theoretical point of view, Kulakowski et al. [36] proved the following theorem about the distance between priority vectors obtained via the EV and GM methods. Theorem 2.1 For every n × n PC matrix A, n > 3, and two priority vectors wEV and wGM , wEV = 1, wGM = 1, obtained by the eigenvalue and geometric mean method, respectively, it holds that κ 2 − 1 ≤ MD(wEV , wGM ) ≤ ( κ12 − 1), where MD denotes Manhattan distance, κ = 1 − KI (A), and KI is Koczkodaj’s inconsistency index (see the next chapter). Proof See [36].

 

Hence, Theorem 2.1 postulates the lower and upper bound of the Manhattan distance between wEV and wGM . Consequently, if a PC matrix is inconsistent (KI > 0), then wEV and wGM cannot be the same.

2.5 Scales for Pairwise Comparisons

13

Further on, several studies compared properties of the GM and EV methods. The study [3] discussed the topic of inefficiency of the priority vector obtained by the EVM, while efficiency of 18 prioritization methods was investigated in [11]. In [14], it was shown that, unlike the GMM, the EVM violates rank and weight monotonicity. Numerical studies [1, 11, 42], or [60] found that the GMM is more favorable than the EVM as well, and therefore, the authors recommended it as the most appropriate prioritization method.

2.5 Scales for Pairwise Comparisons The most frequently applied scale S for pairwise comparisons is the so-called Saaty’s fundamental scale Sf from 1 to 9 (with reciprocals); see Table 2.1. This scale is a discrete, linear 9-point scale. Other discrete linear scales proposed in the literature use 10 points, 5 points, or even only 3 points (the study of Fülöp et al. [21] showed that the reduction of a linear discrete scale to a scale with only 3 points did not result in a significant loss of information). Among continuous scales, the scale R+ is probably most popular. Apart from linear scales, non-linear scales, such as logarithmic, exponential, or geometric, have also been suggested for pairwise comparisons; see Table 2.2. The existence of many scales leads to a natural question as to which scale is the most suitable for pairwise comparisons. Dong et al. [16] provided comparisons on several scales for pairwise comparisons and concluded that the best scale was the geometric one. Elliott [18] experimentally compared three different scales with the result that none of the scales accurately captured the preferences of all individuals. Triantaphyllou et al. [58] compared 78 scales to conclude that no single scale could outperform all the other scales. Starczewski [57] examined the effect of a scale (he compared the fundamental scale, extension scale, and geometric scale) on a priority vector and found that scales with more options lead to a better (more precise) evaluation of a priority vector. Franek and Kresta [20] compared Saaty’s scale to other scales for both consistent Table 2.1 Saaty’s fundamental scale. Source: [53]

Intensity of importance 1 2 3 4 5 6 7 8 9

Definition Equal importance Weak or slight importance Moderate importance Moderate plus importance Strong importance Strong importance plus Very strong importance Very, very strong importance Extreme importance

14

2 Multiplicative Pairwise Comparisons

Table 2.2 Scales for pairwise comparisons. Source: author Scale Linear Power Geometric Root square Logarithmic Inverse linear

Definition S = ax S = xa S = a x−1 S = (x)1/2 S = loga (x + 1) 9 S = 10−x

Domain a > 0, x ∈ {1, 2, . . . , 9}, or x > 0 a > 1 , x ∈ {1, 2, . . . , 9}, or x > 0 a > 1 , x ∈ {1, 2, . . . , 9}, or x > 0 x ∈ {1, 2, . . . , 9}, or x > 0 a > 1 , x ∈ {1, 2, . . . , 9}, or x > 0 x ∈ {1, 2, . . . , 9}

Balanced

S=

x ∈ {0.5, 0.55, . . . , 0.9}

x 1−x

S=9

Balanced power

S=

Asymptotical

x−1 n−1

tanh−1 {

√ 3(x−1) } 14

x ∈ {1, 2, . . . , n} x ∈ {1, 2, . . . , 9}

Table 2.3 Percentage of acceptable PC matrices for all scales. Source: [56] Matrix size n 3 4 5

Linear 22.21 3.36 0.26

Balanced 94.59 98.88 99.93

Geometric 12.97 0.47 0.01

Inverse 41.78 18.37 7.34

Log 67.61 58.71 58.15

Power 10.10 0.29 0.00

Root 51.72 29.91 14.64

and inconsistent pairwise comparisons matrices. According to the authors, Saaty’s scale is still favorable, but if a decision-maker demands higher consistency, root square or logarithmic scales should be used. Cavallo and Ishizaka [8] performed an opinion survey dealing with the estimation of cities’ distances to evaluate the suitability (accuracy) of eight pairwise numerical scales. They found that the power and geometric scales were the least accurate, while the inverse linear scale was the most accurate, followed by the root square and balanced scale. Ishizaka et al. [28] compared seven different scales and applied the additive and the multiplicative aggregation techniques. They found that with the geometric and power scales, a compromise is never selected when aggregation is additive and rarely when aggregation is multiplicative, while the logarithmic scale used with the multiplicative aggregation most often selects the compromise that is desirable by consumer choice theory. Siraj et al. [56] performed Monte Carlo simulations to investigate how well is inconsistency of a PC matrix preserved for different scales (balanced, geometric, inverse, logarithmic, power, and root). They generated 50,000 PC matrices of the order 3 ≤ n ≤ 5 and evaluated cardinal consistency by Saaty’s consistency ratio CR (see the next chapter) and ordinal inconsistency by the number of circular triads (see the last chapter). As can be seen from Table 2.3, the percentage of acceptably inconsistent PC matrices decreased with the matrix order with the exception of the balanced scale, which provided more than 90% of acceptable PC matrices. On the other hand, in the case of the linear, geometric, and power scales, less than 1% of acceptable PC matrices were found for n = 5. Due to large differences, the authors conclude that the possibility of having an acceptable PC matrix is highly sensitive to

2.6 Scale Normalization

15

the selected scale or, in other words, the value of CR is sensitive to the measurement scale chosen for generating PC matrices. As for ordinal inconsistency, the authors found that for linear, geometric, and power scales, the percentage of intransitive, yet acceptably inconsistent, matrices is low, below 1%. Other scales produced more than 20% of intransitive acceptable PC matrices. All aforementioned scales were comprised of natural numbers (integers) or positive real numbers. Wajch [59] considered more general scales such as negative real numbers or complex numbers; however, Koczkodaj et al. [33] argued that the most general meaningful scale for pairwise comparisons (i.e., ratios) is the scale of positive real numbers. In addition, other study of Koczkodaj et al. [32] argues from a theoretical point of view that the best scale for pairwise comparisons is (0, ∞).

2.6 Scale Normalization When at least two different scales are used for pairwise comparisons, the following problem emerges: Let’s consider two pairwise comparisons scales, a discrete scale from 1 to 3 (with reciprocals), and a discrete scale from 1 to 100 (with reciprocals). In the former case, the preference aij = 2 means a medium preference of object i to object j, while in the latter case, the medium preference is aij = 50 (see Fig. 2.1), and the value aij = 2 has a different meaning. This problem was first recognized in [31]. The scale for pairwise comparisons cannot be simply neglected and has to be taken into account, especially in situations when different scales are used simultaneously or when the results of pairwise comparisons of the same set of objects with different scales need to be compared. In such cases, scale normalization is necessary. In [41], a following simple power transform (with several desirable properties) for the scale normalization was proposed. Definition 2.4 Let the scale for the multiplicative pairwise comparisons (without reciprocals) be given as S = [1, m], m > 1, m ∈ R. The normalization f is a transformation that converts the scale S onto the unit interval [1, 2] so that the

Fig. 2.1 A comparison of different scales. Source: author

16

2 Multiplicative Pairwise Comparisons

following conditions hold: (a) f : [1, m] → [1, 2], (b) f is strictly increasing, and (c) f (1) = 1 and f (m) = 2. Proposition 2.2 Let A = [aij ] be a pairwise comparisons matrix, aij = [1/m, m]. Let f be the power transform as follows: f (aij ) = aijk , k =

ln 2 ln m

(2.12)

Then the transform satisfies conditions from Definition 2.4. Proof [41]

 

Proposition 2.3 Let A = [aij ] be a pairwise comparisons matrix. Then, the power transform preserves the most inconsistent triad with respect to Koczkodaj’s inconsistency index KI . Proof [41]

 

Proposition 2.4 Let A = [aij ] be a pairwise comparisons matrix. Then, the power transform does not change the ranking of all alternatives if the weights of alternatives are determined by the geometric mean method. Proof [41]

 

2.7 A Note on a Possible Cause of the Inconsistency Consider an expert who provides only (n − 1) pairwise comparisons of n objects, the minimum amount necessary to find a priority vector (comparisons must form a strongly connected graph, e.g., the expert may provide all pairwise comparisons above the main diagonal). Then, the defining relation for consistency (2.3) is not violated since there is no triplet of indices (i, j, k), 1 ≤ i < j < k ≤ n such that all three matrix elements aij , aj k , aik exist; thus, relation (2.3) is never violated. However, it does not mean that all comparisons are accurate, without mistakes. In fact, all comparisons might be erroneous, but altogether they form a system of (n − 1) judgments, which is contradiction-free. According to Harker [24], the redundancy in pairwise comparisons plays a useful role in that a decision-maker can incorrectly answer one pairwise comparison, but the final weights will not be greatly affected due to the redundancy and the averaging effect of the eigenvalue method. Therefore, one would not want to make only (n−1) pairwise comparison since a certain amount of redundancy is necessary to “correct” any errors in the judgments. However, the completion of all n(n − 1)/2 judgments might be a laborious task. The reason for the inconsistency of pairwise comparisons likely rests in redundancy of pairwise comparisons with respect to the minimal amount of pairwise comparisons necessary to find a priority vector. Only (n − 1) pairwise comparisons

2.7 A Note on a Possible Cause of the Inconsistency

17

suffice to find a priority vector; however, a complete PC matrix requires n(n−1) 2 pairwise comparisons (above the main diagonal). And these extra comparisons may contain contradicting information. To quantify the redundancy, the following measure M is introduced: Definition 2.5 Let An×n = [aij ] be a pairwise comparisons matrix. Then the measure of redundancy M of pairwise comparisons is defined as follows:  M = ln

 n n(n − 1) /(n − 1) = ln 2 2

(2.13)

The redundancy measure M is simply the logarithm of the ratio of the number of all pairwise comparisons to the number of minimum pairwise comparisons necessary to find a priority vector. Figure 2.2 provides the relationship between Saaty’s random consistency index RI and the variable M for PC matrices of the increasing order n. As can be seen, this relationship is monotonically increasing (and almost perfectly quadratic), which means the greater the redundancy, the greater the inconsistency as well.

Fig. 2.2 Redundancy (M) versus inconsistency (RI ). Source: author

18

2 Multiplicative Pairwise Comparisons

References 1. Bajwa, G., Choo, E. U., & Wedley, W. C. (2008). Effectiveness of deriving priority vectors from reciprocal pairwise comparison matrices. Asia-Pacific Journal of Operational Research, 25(3), 279–299. https://doi.org/10.1142/S0217595908001754. 2. Basile, L., & d’Apuzzo, L. (2002). Weak consistency and quasi-linear means imply the actual ranking. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(3), 227–239. 3. Bozóki, S. (2014). Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency. Optimization, 63(12), 1893–1901. https://doi.org/10.1080/02331934. 2014.903399. 4. Bozóki, S., & Rapcsák, T. (2008). On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization, 42(2), 157–175. 5. Bozóki, S., & Tsyganok, V. (2019). The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices. International Journal of General Systems, 48(4), 362–381. https://doi.org/10.1080/03081079.2019.1585432. 6. Bryson, N. (1995). A Goal Programming Method for Generating Priority Vectors. The Journal of the Operational Research Society, 46(5), 641–648. https://doi.org/10.2307/2584536. 7. Cavallo, B., & D’Apuzzo, L. (2009). A general unified framework for pairwise comparison matrices in multicriteria methods. International Journal of Intelligent Systems, 24(4), 377–398. 8. Cavallo, B., & Ishizaka, A. (2022). Evaluating scales for pairwise comparisons. Annals of Operations Research, 1–15. https://doi.org/10.1007/s10479-022-04682-8. 9. Chandran, B., Golden, B., & Wasil, E. (2005). Linear programming models for estimating weights in the analytic hierarchy process. Computers and Operations Research, 32(9), 2235– 2254. https://doi.org/10.1016/j.cor.2004.02.010. 10. Chu, A. T. W., Kalaba, R. E., & Spingarn, K. (1979). A comparison of two methods for determining the weights of belonging to fuzzy sets. Journal of Optimization Theory and Applications, 27(4), 531–538. 11. Choo, E. U., & Wedley, W. C. (2004). A common framework for deriving preference values from pairwise comparison matrices. Computers and Operations Research, 31(6), 893–908. https://doi.org/10.1016/S0305-0548(03)00042-X. 12. Crawford, G. B. (1987). The geometric mean procedure for estimating the scale of a judgement matrix. Mathematical Modelling, 9(3), 327–334. https://doi.org/10.1016/02700255(87)90489-1. 13. Crawford, G., & Williams, C. (1985). A Note on the Analysis of Subjective Judgment Matrices. Journal of Mathematical Psychology, 29(4), 387–405. 14. Csató, L., & Petróczy, D. G. (2021). On the monotonicity of the eigenvector method. European Journal of Operational Research, 292(1), 230–237. https://doi.org/10.1016/j.ejor.2020.10.020. 15. Dijkstra, T. K. (2013). On the extraction of weights from pairwise comparison matrices. Central European Journal of Operational Research, 21, 103–123. https://doi.org/10.1007/ s10100-011-0212-9. 16. Dong, Y., Xu, Y., Li, H., & Dai, M. (2008). A comparative study of the numerical scales and the prioritization methods in AHP. European Journal of Operational Research, 186, 229–242. https://doi.org/10.1016/j.ejor.2007.01.044. 17. Dopazo, E., & González-Pachón, J. (2003). Consistency-driven approximation of a pairwise comparison matrix. Kybernetika, 39(5), 561–568. 18. Elliott, M. A. (2010). Selecting numerical scales for pairwise comparisons. Reliability Engineering and System Safety, 95(7), 750–763. 19. Fedrizzi, M., Brunelli, M., & Caprila, A. (2020). The linear algebra of pairwise comparisons. International Journal of Approximate Reasoning, 118, 190–207. https://doi.org/10.1016/j.ijar. 2019.12.009.

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40. Lundy, M., Siraj, S., & Greco, S. (2017). The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis. European Journal of Operational Research, 257(1), 197–208. 41. Mazurek, J. (2019). On the problem of different pairwise comparison scales in the multiplicative AHP framework. Scientific papers of the University of Pardubice, 46(2), 124–133. 42. Mazurek, J., & Kulakowski, K. (2020). Satisfaction of the condition of order preservation. A simulation study. Operations Research and Decisions, 2, 77–89. https://doi.org/10.37190/ ord200205. 43. Mazurek, J., & Kulakowski, K. (2022). On the derivation of weights from incomplete pairwise comparisons matrices via spanning trees with crisp and fuzzy confidence levels. International Journal of Approximate Reasoning, 150, 242–257. https://doi.org/10.1016/j.ijar.2022.08.014. 44. Mazurek, J., & Perzina, R. (2017). On the inconsistency of pairwise comparisons: an experimental study. Scientific papers of the University of Pardubice—Series D3, 41, 102–109. 45. Mazurek, J., & Ramík, J. (2019). Some new properties of inconsistent pairwise comparison matrices. International Journal of Approximate Reasoning, 113, 119–132. 46. Miller, G. A. (1956). The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychological Review, 63(2), 81–97. https://doi.org/10. 1037/h0043158. 47. Orlowski, S. A. (1978). Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems, 1, 155–167. 48. Ra, J. W. (1987). Analysis of the column-row sums approach for pairwise comparisons, unpublished (Masters thesis). Pittsburgh: University of Pittsburgh. 49. Ramík, J. (2015). Pairwise comparison matrix with fuzzy elements on alo-group. Information Sciences, 297, 236–253. https://doi.org/10.1016/j.ins.2014.11.010. 50. Ramík, J. (2020). Pairwise Comparisons Method. Lecture Notes in Economics and Mathematical Systems. Berlin: Springer. 51. Ramík, J., & Korviny, P. (2010). Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161(11), 1604–1613. https://doi. org/10.1016/j.fss.2009.10.011. 52. Saaty, T. L. (1977) A Scaling Method for Priorities in Hierarchical Structures. Journal of Mathematical Psychology, 15, 234–281. 53. Saaty, T. L. (1980). Analytic Hierarchy Process. New York: McGraw-Hill. 54. Saaty, T. L., & Vargas, L. G. (1984). Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Mathematical Modelling, 5(5), 309–324. 55. Siraj, S., Mikhailov, L., & Keane, J. (2012). Enumerating all spanning trees for pairwise comparisons. Computers and Operations Research, 39(2), 191–199. https://doi.org/10.1016/ j.cor.2011.03.010. 56. Siraj, S., Mikhailov, L., & Keane, J. A. (2015). Contribution of individual judgments toward inconsistency in pairwise comparisons. European Journal of Operational Research, 242(2), 557–567. https://doi.org/10.1016/j.ejor.2014.10.024. 57. Starczewski, T. (2017). Remarks on the impact of the adopted scale on the priority estimation quality. Journal of Applied Mathematics and Computational Mechanics, 16(3), 105–116. https://doi.org/10.17512/jamcm.2017.3.10. 58. Triantaphyllou, E., Lootsma, F. A., Pardalos, P. M., & Mann, S. H. (1994). On the Evaluation and Application of Different Scales For Quantifying Pairwise Comparisons in Fuzzy Sets. Journal of Multi-Criteria Decision Analysis, 3(3), 133–155. 59. Wajch, E. (2019). From pairwise comparisons to consistency with respect to a group operation and Koczkodaj’s metric. International Journal of Approximate Reasoning, 106, 51–62. 60. Wedley, W. C., Choo, E. U., & Wijnmalen, D. J. D. (2016). Efficacy analysis of ratios from pairwise comparisons. Fundamenta Informaticae, 146(3), 321–338. https://doi.org/10.3233/ FI-2016-1389.

Chapter 3

Inconsistency Indices and Their Properties

Abstract This chapter provides a review of (cardinal) inconsistency indices, two systems of indices’ (desirable) properties, and an overview of satisfaction of these properties by selected indices. Further on, numerical studies on indices’ comparisons are presented and discussed as well.

In this chapter, a review of inconsistency indices is provided along with a discussion of their properties. Inconsistency indices can be divided into cardinal and ordinal ones regarding the type of PC matrix to be evaluated. Here, we deal with cardinal inconsistency and cardinal inconsistency indices, while the less frequently used ordinal inconsistency indices are discussed in Chap. 7.

3.1 A Review of Inconsistency Indices Formally, a (cardinal) inconsistency index is a function I : A → R+ 0 , where A denotes the set of PC matrices. Usually, it is assumed that for a consistent PC matrix, an inconsistency index is equal to 0, and the higher the value of the inconsistency index, the higher the inconsistency of the corresponding PC matrix. Various inconsistency indices measuring the extent of inconsistency of pairwise comparisons have been proposed since T. L. Saaty introduced the analytic hierarchy process (AHP) in 1977; see, e.g., [8] or [46]. Perhaps the most popular is Saaty’s consistency index CI and consistency ratio CR; see [49, 50]. Definition 3.1 The consistency index CI of a PC matrix A of the order n is defined as follows: CI (A) =

λmax − n . n−1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Mazurek, Advances in Pairwise Comparisons, Multiple Criteria Decision Making, https://doi.org/10.1007/978-3-031-23884-0_3

(3.1)

21

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3 Inconsistency Indices and Their Properties

Table 3.1 The values of RI (n). It should be noted that slightly different values were reported in other studies; see, e.g., [25] or [27]. Source: author (adapted from [3]) Matrix size n 3 4 5 6

RI 0.5245 0.8815 1.1086 1.2479

Matrix size n 7 8 9 10

RI 1.3417 1.4056 1.4499 1.4854

Matrix size n 11 12 13 14

RI 1.5140 1.5365 1.5551 1.5713

The consistency ratio CR is defined as CR(A) =

CI (A) , RI (n)

(3.2)

where RI (n) in the definition above denotes the random consistency index (the arithmetic mean of randomly generated PC matrices of a given order with Saaty’s scale) dependent on n (see Table 3.1) and λmax is the largest (positive) eigenvalue of A. The value λmax ≥ n, and λmax = n only if a pairwise comparisons matrix is consistent [49]. Another popular measure of inconsistency is Koczkodaj’s inconsistency index KI [21, 31]: Definition 3.2 Koczkodaj’s inconsistency index KI of an n × n (n > 2) reciprocal matrix A = [aij ] is given as       aik akj  aij    , 1− min 1 − KI (A) = max i,j,k∈{1,...,n} aik akj   aij 

(3.3)

Obviously, 0 ≤ KI < 1. The main difference between CI (CR) and KI is that KI is a local and “worstcase” index, since it provides the value of inconsistency of the most inconsistent triad (the triplet aij , aj k , aik ) contained in a PC matrix, while CI (CR) is a global and average index. Example 3.1 Consider the PC matrix A from Example 2.2. We will estimate its inconsistency. ⎡

⎤ 1 12 2 5 ⎢ 2 1 4 4⎥ ⎥ A=⎢ ⎣ 1 1 1 5⎦, 2 4 1 1 1 5 4 5 1

3.1 A Review of Inconsistency Indices

23

The inconsistency of the matrix A is as follows: −n The principal right eigenvalue λmax = 4.254; hence, CI (A) = λmax = n−1 4.254−4 = 0.085. 3 0.085 The random consistency index RI = 0.8815; hence, CR = CI RI = 0.8815 = 0.096. And, finally, Koczkodaj’s inconsistency index KI (A): there are four triads in the matrix A. The most inconsistent triad is (a23, a34 , a24 ), with its inconsistency being 4 1 − min( 4·5 4 , 4·5 ) = 1 − 0.2 = 0.8. Other inconsistency indices are listed below. The least squares (LS) index by Chu et al. [17] is defined as follows: Definition 3.3 Least squares (LS) index [17]. Let An×n = [aij ] be a pairwise comparisons matrix; let w = (w1 , . . . , wn ) be a priority vector associated with A. Then the LS index is defined as follows: LS(A) = min



n n n    wi 2 aij − , s.t. wi = 1, wi > 0 wj i=1 j =1,j =i

(3.4)

i=1

If the PC matrix A is consistent, then LS(A) = 0; otherwise, LS(A) > 0. Definition 3.4 Golden-Wang (GW) index [27]. Let A∗ be the normalized n × n matrix obtained from the matrix An×n = [aij ] by dividing each column of A by the sum of all the elements in that column. Further on, let w∗ be the normalized priority vector obtained from A∗ by the EM or GM method. Then the GW index is defined as n n 1  ∗ GW (A) = · |aij − wi∗ | n

(3.5)

i=1 j =1

If the PC matrix A is consistent, then GW (A) = 0; otherwise, GW (A) > 0. The following index of Peláez and Lamata is based on (sub)determinants of a PC matrix of the order n = 3. Definition 3.5 Peláez-Lamata index (PLI/ CI ∗ ) [47]. Let An×n = [aij ] be a pairwise comparisons matrix. Then the PLI index is defined as follows: n−1 n−2   6 · P LI (A) = n(n − 1)(n − 2)

n  aij aj k aik + −2 aij aj k aik

(3.6)

i=1 j =i+1 k=j +1

Again, if the PC matrix A is consistent, then P LI (A) = 0; otherwise, P LI (A) > 0. The study [48] proposed a refined modification of the P LI index denoted as C + , which is normalized to the interval [0, 1] via a suitable scaling function.

24

3 Inconsistency Indices and Their Properties

Definition 3.6 The geometric consistency index (GCI) [1, 18]. Let An×n = [aij ] be a pairwise comparisons matrix, and let w = (w1 , . . . , wn ) be a priority vector derived from A by the geometric mean method. The geometric consistency index GCI is defined as follows:  n−1  n   wj 2 2 GCI (A) = ) ln(aij (n − 1)(n − 2) wi

(3.7)

i=1 j =i+1

Definition 3.7 The triads geometric consistency index (T-GCI) [2]. Let An×n = [aij ] be a pairwise comparisons matrix. The triads geometric consistency index T − GCI is defined as follows: T − GCI (A) =

2



i 1 for inconsistent matrices. Definition 3.12 Cavallo-d’Apuzzo index (ICD ) [16]. Let An×n = [aij ] be a pairwise comparisons matrix. Then the inconsistency index ICD is defined as follows: ICD (A) =

n−2  n−1 

6   n   n(n−1)(n−2) aik aij aj k max , aij aj k aik

(3.13)

i=1 j =i+1 k=j +1

Shiraishi, Obata, and Daigo [53] proposed the coefficient c3 of the characteristic polynomial of a PC matrix as an inconsistency index. Definition 3.13 The c3 index [53]. Let An×n = [aij ] be a pairwise comparisons matrix. Then the inconsistency index c3 is defined as follows: c3 (A) =

n−2  n−1 

 n   aij aj k aik 2− − aij aj k aik

(3.14)

i=1 j =i+1 k=j +1

Unlike any other index, the index c3 attains negative values for inconsistent matrices, while for consistent matrices, c3= 0. In [13], it is shown that the index c3 is proportional to the index P LI : c3 = − n3 P LI .

26

3 Inconsistency Indices and Their Properties

Mazurek [42] applied row vectors of a PC matrix to define the row inconsistency index. Definition 3.14 The row inconsistency index (RIC) [42]. Let An×n = [aij ] be a pairwise comparisons matrix, and let ri , rj denote row vectors of A, i, j ∈ {1, l . . . , n}. Then the RI C is defined as follows:  2 cos(ϕi ) n(n − 1) n−1 n

RI C(A) = 1 −

(3.15)

i=1 j >i

where cos(ϕi ) =

ri ·rj ri · rj .

The RI C = 0 for consistent matrices and 0 < RI C < 1 for inconsistent ones. Fedrizzi and Ferrari [22] interpreted a pairwise comparisons matrix as a contingency table and proposed a chi-square-based inconsistency index. Definition 3.15 Chi-square-based index (χ 2 ) [22]. Let An×n = [aij ] be a pairwise comparisons matrix, and let E = [eij ] be a chi-square-based consistent approximation of A. Then the χ 2 inconsistency index is defined as follows: χ 2 (A) =

n n   (aij − eij )2 eij

(3.16)

i=1 j =1

For a consistent matrix A, χ 2 (A) = 0. In concordance with Saaty’s consistency ratio CR, the authors introduced a variant of the previous inconsistency index given as follows: Iχ 2 (A) =

χ 2 (A) , RIn (χ 2 )

(3.17)

where RIn (χ 2 ) is a random index—a mean value of χ 2 (A) obtained from a large set of randomly generated PC matrices of the order n. The cosine consistency index CCI was proposed by Kou and Lin [36]. The index is based on collinearity (or the lack of it) among columns of a PC matrix. Definition 3.16 Cosine consistency index CCI [36]. Let An×n = [aij ] be a pairwise comparisons matrix of the order n. Then the CCI inconsistency index is defined as follows: ⎧ ⎛ ⎞2 ⎫1/2 ⎪ ⎪ n n ⎨ ⎬   1 ⎝ ⎠ CCI (A) = bij ⎪ n⎪ ⎩ i=1 j =1 ⎭

(3.18)

3.1 A Review of Inconsistency Indices

27

where aij bij = n

2 k=1 akj

(3.19)

The CCI = 1 for consistent matrices and CCI > 1 for inconsistent matrices. Kulakowski [38] proposed the local inconsistency index E. Definition 3.17 Index (E) [38]. Let An×n = [aij ] be a pairwise comparisons matrix. Then the inconsistency index E is defined as follows: 

 wj E = max aij −1 . ij wi

(3.20)

For consistent matrices, the index E = 0 and E > 0 otherwise. Gass and Rapcsak [26] proposed an inconsistency index based on the singular value decomposition (SVD) of the PC matrix A and its rank 1 approximation. Definition 3.18 SVD index [26]. Let An×n = [aij ] be a pairwise comparisons matrix. Let A = U DV T , where D is a diagonal matrix with diagonal elements αi . Further on, let A1 be an approximation of A such that A1 = uα1 v T , where u and v are the first columns of U and V , respectively. The SV D inconsistency index is given as follows: SV D(A) = A − A1 .

(3.21)

The SV D index is equal to 0 for consistent matrices. Dixit [20] suggested using the following inconsistency index S derived in the context of non-equilibrium entropy production in the induced maximum path entropy random walks: Definition 3.19 S index [20]. Let An×n = [aij ] be a pairwise comparisons matrix, and let u and w be the left and right Perron-Frobenius eigenvectors, respectively. Then the inconsistency index S is defined as follows: S=

2 λmax

n  n 

ui wj aij log aij

(3.22)

i=1 j =1

Like the majority of the indices, the index S = 0 also for consistent matrices and is positive otherwise. Wu and Xu [60] proposed the following inconsistency index CIH .

28

3 Inconsistency Indices and Their Properties

Definition 3.20 CIH index ! [60]. Let An×n = [aij ] be a pairwise comparisons matrix, and let gij = ( nk=1 aik akj )1/n . Then the inconsistency index CIH is defined as follows: n n 1  CIH (A) = 2 aij gj i n

(3.23)

i=1 j =1

The matrix G = [gij ] can be considered a consistent approximation of A. If the matrix A itself is consistent, then CIH = 1; otherwise, CIH > 1. Kulakowski and Szybowski [39] proposed a triad-based inconsistency index analogous to Koczkodaj’s inconsistency index, but instead of an extreme case (extreme triad), their two indices I1 and I2 provide a mean value of inconsistency of all triads. Definition 3.21 I1 , I2 indices [39]. Let An×n = [aij ] be a pairwise comparisons a a matrix. Let K(t) = min{|1 − aijaikajk |, |1 − ijaikjk |} denote the inconsistency of the triad t ≡ (aij , aj k , aik ). Then the inconsistency index Ii is defined as follows:  6 t K(t) I1 = (3.24) n(n − 1)(n − 2) and the index I2 is defined as follows: I2 =

 6( t K2 (t))1/2 n(n − 1)(n − 2)

(3.25)

Both indices are equal for consistent matrices. Moreover, the authors prove that for a given PC matrix, the following inequalities hold: 0 ≤ I2 ≤ I1 ≤ KI ≤ 1. Independently, the mean Koczkodaj inconsistency index AT I was suggested by Grzybowski [29]. Also, Grzybowski proposed an extension to Saaty’s consistency index CI for nonreciprocal matrices [28]. Fedrizzi et al. [24] considered a norm-based inconsistency index Id for skewsymmetric PC matrices (matrices obtained from multiplicative PC matrices by log transformation). Definition 3.22 Id index [24]. Let A be a norm in Rn×n , and let d(A, B) be the distance between the matrices A and B. Let L be the linear space of all skewsymmetric matrices of the order n, and let L∗ be the linear subspace of L of consistent matrices. Then the norm-based inconsistency index Id (A) is given as follows: Id (A) = d(A, L∗ ) = min{d(A, B); B ∈ L∗ } = min{ A − B ; B ∈ L∗ } = A − A∗ , where A∗ is the solution of the minimization problem.

(3.26)

3.1 A Review of Inconsistency Indices

29

Obviously, for consistent matrices, the index is equal to 0; otherwise, it is positive. Further on, Fedrizzi and Giove [23] introduced a new inconsistency index F G (originally called ρ) in the context of reciprocal relations (also called fuzzy preference relations) represented by a PC matrix R = [rij ], rij ∈ [0, 1], rij + rj i = 1. A matrix R is consistent if rij − rik − rkj + 0.5 = 0, ∀i, j, k [57]. Definition 3.23 F G index [23]. Let Rn×n = [rij ] be a PC matrix of the order n. Then the FG inconsistency index is given as follows: F G(R) =

n−1 n−2  

n 

n (rij − rik − rkj + 0.5) / 3 2

i=1 j =i+1 k=j +1

(3.27)

In [13], it was proved that F G = 4ln32 (9) GCI under assumption of the log transformation between reciprocal relations PC matrix and multiplicative PC matrix, where the entries of the latter matrix are drawn from the Saaty scale. Osei-Bryson proposed the following consistency indicator (index) ρ [45]: Definition 3.24 Consistency indicator ρ [45]. Let A = [aij ] be a n × n PC matrix, wi a c let C = [cij ] = [ ] be its consistent approximation, and let sij = min{ cijij , aijij }. wj Further on, let 0 < τ ≤ 1 be a tolerance parameter. Then the consistency indicator ρ is defined as follows: ρ(A) = 2



(i, j )/n(n − 1),

(3.28)

1≤i