Normalization of Multidimensional Data for Multi-Criteria Decision Making Problems: Inversion, Displacement, Asymmetry (International Series in Operations Research & Management Science, 348)

✍ Scribed by Irik Z. Mukhametzyanov

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 314
• Category: Library

✦ Synopsis

This book presents a systematic review of multidimensional normalization methods and addresses problems frequently encountered when using various methods and ways to eliminate them.

The invariant properties of the linear normalization methods presented here can be used to eliminate simple problems and avoid obvious errors when choosing a normalization method. The book introduces valuable, novel techniques for the multistep normalization of multidimensional data. One of these methods involves inverting the normalized values of cost attributes into profit attributes based on the reverse sorting algorithm (ReS algorithm). Another approach presented is the IZ method, which addresses the issue of shift in normalized attribute values. Additionally, a new method for normalizing the decision matrix is proposed, called the MS method, which ensures the equalization of average values and variances of attributes.

Featuring numerous illustrative examples throughout, the book helps readers to understand what difficulties can arise in multidimensional normalization, what to expect from such problems, and how to solve them. It is intended for academics and professionals in various areas of data science, computing in mathematics, and statistics, as well as decision-making and operations.

✦ Table of Contents

Preface
Contents
About the Author
List of Abbreviations
List of Figures
List of Tables
Chapter 1: Introduction
1.1 The Problem of Multi-criteria Decision-Making
1.2 Multidimensional Normalization in the Context of Decision Problems
References
Chapter 2: The MCDM Rank Model
2.1 MCDM Rank Model
2.2 The Target Value of Attributes
2.3 Significance of Criteria: Multivariate Assessment
2.3.1 Subjective Weighting Methods: Pairwise Comparisons and AHP Process
2.3.2 Subjective Weighting Methods: Best-Worst Method
2.3.3 Objective Weighting Methods: Entropy, CRITIC, SD
Entropy Weighting Method (EWM) [26, 27, 37]
CRiteria Importance Through Inter-criteria Correlation (CRITIC) [28]
Standard Deviation (SD)
2.4 Aggregation of the Attributes: An Overview of Some Methods
2.4.1 Value Measurement Methods
Simple Additive Weighting (SAW) or Weighted Sum Method (WSM) [1]
Weighted Product Method (WPM) [39]
Weighted Aggregated Sum Product Assessment (WASPAS) [39]
Multi-Attributive Border Approximation Area Comparison (MABAC) [45]
Complex Proportional Assessment (COPRAS) Method [46]
2.4.2 Goal or Reference Level Models
Distance Metric
Reference Point (RP) Method [47]
COmbinative Distance-based ASsessment (CODAS)
Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [1]
VIsekriterijumsko KOmpromisno Rangiranje (VIKOR) [40]
Gray Relation Analysis (GRA) [49, 50]
2.4.3 Outranking Techniques
Preference Ranking Organization METHod for Enrichment Evaluations (PROMETHEE) [41]
Organisazion, RangEment ot SynTEze de donnecs relationnelles (ORESTE) [43, 44]
2.4.4 Rank Reversal Problem
2.4.5 Distinguishability of the Performance Indicator of Alternatives
2.5 Design of the MCDM Model
2.6 Conclusions
References
Chapter 3: Normalization and MCDM Rank Model
3.1 General Principles for Normalizing Multidimensional Data
3.1.1 Preserving the Ordering Values of Attributes
3.1.2 Scale Invariance of Normalized Values of Attributes
3.1.3 Principle of Additive Significance of Attributes
3.1.4 Interpretation of Normalized Values of Attributes
3.2 Linear Multivariate Normalization Methods
3.2.1 How Is the Shift Factor Determined?
3.2.2 How Is Scaling Determined?
3.2.3 Disadvantages of Data Standardization
3.3 Asymmetry in the Distribution of Features
3.3.1 Measures of Asymmetry
3.4 The Outlier Detection
3.5 Non-linear Normalization: General Principles
3.6 Target Inversion in Multivariate Normalization
3.7 Isotropy of Scales of Normalized Values
3.8 Impact of the Choice of Normalization Method on the Rating
3.9 Conclusions
References
Chapter 4: Linear Methods for Multivariate Normalization
4.1 Basic Linear Methods for Multivariate Normalization
4.2 Scaling Factor Ratios
4.3 Invariant Properties of Linear Normalization Methods
4.3.1 Invariance of the Dispositions of Alternatives
4.3.2 Isotropic of Scaling: Invariance of Rating
4.3.3 Invariants of Numerical Characteristics of the Sample
4.4 Re-normalization
4.4.1 Invariant Re-normalization Properties for Linear Methods
4.5 Meaningful Interpretation of Linear Scales
4.6 Some Features of Individual Linear Normalization Methods
4.6.1 Max-method of Normalization
4.6.2 The Displacement of Normalized Values in Domains for the Sum and Vec Methods
4.6.3 Loss of Contribution to the Performance Indicator in the Max-Min Method
4.6.4 dSum Method of Normalization
4.6.5 Z-score Method of Normalization
4.6.6 mIQR Method of Normalization
4.6.7 mMAD-Method of Normalization
4.7 Conclusions
References
Chapter 5: Inversion of Normalized Values: ReS-Algorithm
5.1 Optimization Goal Inversion
5.2 Permissible Pairs of Transformations to the Benefit and Cost Criteria
5.3 Overview of Inverse Transforms and Compliance with Multidimensional Data Normalization Requirements
5.3.1 Max Method of Normalization
5.3.2 Sum Method of Normalization
5.3.3 Vec Method of Normalization
5.3.4 Max-Min Method of Normalization
5.3.5 dSum Method of Normalization
5.3.6 Z-score Method of Normalization
5.4 Universal Goal Inversion Algorithm: ReS-Algorithm
5.4.1 Reverse Sorting Algorithm
5.4.2 ReS-Algorithm
5.4.3 Basic Properties of the ReS-Algorithm
5.5 Conclusions
References
Chapter 6: Rank Reversal in MCDM Models: Contribution of the Normalization
6.1 Main Factors Determining Rank Reversal in MCDM Problems
6.2 Relative Preference for Different Normalizations
6.3 Assessing the Contribution of an Individual Attribute to the Performance Indicator of an Alternative
6.4 Rank Reversal Due to Normalization
6.5 Conclusions
References
Chapter 7: Coordination of Scales of Normalized Values: IZ-Method
7.1 Ratio of Feature Scales
7.2 The Domains Displacement of Normalized Values of Various Attributes
7.3 Attribute Equalizer
7.3.1 Transformation of Normalized Values Using Fixed Point Technique
7.4 Elimination of Displacement in the Domains of Normalized Values: IZ-Method
7.5 Choice of Conditionally General Scale [I, Z] Normalized Values
7.6 Invariant Properties of the IZ-Method
7.7 Generalization of the IZ-Method
7.8 IZ Transformation for Non-linear Aggregation Methods: Example for COPRAS, WPM, and WASPAS Methods
7.9 Conclusions
References
Chapter 8: MS-Transformation of Z-Score
8.1 Standardized Scoring
8.2 MS-Transformation of Z-Score
8.3 Selecting a Conditionally Common Scale [I, Z] for MS-Transformation
8.4 Invariant Properties of MS-Transformation
8.5 MS-Transformations for Non-linear Aggregation Methods: Example for WPM and WASPAS Methods
8.6 Conclusions
References
Chapter 9: Non-linear Multivariate Normalization Methods
9.1 Non-linear Data Transformation as a Way to Eliminate Asymmetry in the Distribution of Features
9.2 Non-linear Data Pre-processing Procedures. Transition to the Non-linear Scales
9.3 Transformation of Normalized Data: Post-processing of Data
9.3.1 Post-processing with Max-Min Normalization
9.3.2 Post-processing with Z-Score Normalization
9.3.3 Weighted Product Model and Post-processing of Normalized Values
9.4 Inversion of Normalized Values and Matching the Areas of Normalized Values of Different Criteria
9.5 Numerical Example of Data Pre-processing
9.6 Numerical Example of Data Post-processing
9.7 Conclusions
References
Chapter 10: Normalization for the Case Nominal Value the Best´´ 10.1 Target Criteria and Target-Based Normalization 10.2 Review of Target Normalization Methods 10.3 Generalization of Normalization Methods of Target Criteria for Linear Case 10.4 Comparative Normalization of Target Criteria Using Linear Methods 10.5 Normalization of Target Criteria: Non-linear Methods-Concept of Harrington´s Desirability Function 10.5.1 One-Sided DF for LTB and STB Criteria 10.5.2 Two-Sided DF for the NTB Criteria 10.5.3 Consistent DF-Normalization for LTB, STB, and NTB Criteria 10.5.4 The Desirability Function: Power Form 10.5.5 The Desirability Function: Gaussian Form 10.6 Conclusions References Chapter 11: Comparative Results of Ranking of Alternatives Using Different Normalization Methods: Computational Experiment 11.1 Methodology of Computational Experiment 11.2 Normalization Methods 11.3 A Decision Matrix Generation with High Sensitivity of Rank to the Normalization Methods 11.4 Graphical Illustration of Normalized Values 11.5 Results of Ranking of Alternatives for Decision Matrix D0 11.5.1 Borda Voting Principles 11.5.2 Distinguishability of Ratings 11.6 Results of Ranking of Alternatives for Decision Matrix D1 11.6.1 Borda Count 11.6.2 Distinguishability of Ratings 11.7 Conclusions References Chapter 12: Significant Difference of the Performance Indicator of Alternatives 12.1 Relative Difference in the Performance Indicator of Alternatives 12.2 Ranking Algorithm Using Distinguishability Criteria 12.3 Numerical Example of the Ranking of Alternatives, Taking into Account the Criterion of Distinguishability 12.4 Assessing the Significance of the Difference in the Ratings of Alternatives in the VIKOR Method 12.5 Evaluation of the Distinguishability of the Rating When the Decision Matrix Is Varied 12.6 Statistics of the Performance Indicator of Alternatives When Varying the Decision Matrix 12.6.1 Statistical Experiment 12.6.2 Distribution of the Performance Indicator of Alternatives 12.7 Ranking Alternatives Based on Simple Comparison of the Rating 12.8 Evaluation of the Criterion Value of the Performance Indicator Based on the Error in the Evaluation of the Decision Matrix 12.9 Conclusions References Conclusion Appendix: Program CodeNormalization of Multidimensional Data´´ for MatLab System