Fuzzy Multiple Attribute Decision Making : Methods and Applications

This monograph is intended for an advanced undergraduate or graduate course as well as for researchers, who want a compilation of developments in this rapidly growing field of operations research. This is a sequel to our previous works: "Multiple Objective Decision Making--Methods and Applications: A state-of-the-Art Survey" (No.164 of the Lecture Notes); "Multiple Attribute Decision Making--Methods and Applications: A State-of-the-Art Survey" (No.186 of the Lecture Notes); and "Group Decision Making under Multiple Criteria--Methods and Applications" (No.281 of the Lecture Notes). In this monograph, the literature on methods of fuzzy Multiple Attribute Decision Making (MADM) has been reviewed thoroughly and critically, and classified systematically. This study provides readers with a capsule look into the existing methods, their characteristics, and applicability to the analysis of fuzzy MADM problems. The basic concepts and algorithms from the classical MADM methods have been used in the development of the fuzzy MADM methods. We give an overview of the classical MADM in Chapter II. Chapter III presents the basic concepts and mathematical operations of fuzzy set theory with simple numerical examples in a easy-to-read and easy-to-follow manner. Fuzzy MADM methods basically consist of two phases: (1) the aggregation of the performance scores with respect to all the attributes for each alternative, and (2) the rank ordering of the alternatives according to the aggregated scores.

I. Introduction.- II. Multiple Attribute Decision Making ― An Overview.- 2.1 Basics and Concepts.- 2.2 Classifications of MADM Methods.- 2.2.1 Classification by Information.- 2.2.2 Classification by Solution Aimed At.- 2.2.3 Classification by Data Type.- 2.3 Description of MADM Methods.- Method (1): DOMINANCE.- Method (2): MAXIMIN.- Method (3): MAXIMAX.- Method (4): CONJUNCTIVE METHOD.- Method (5): DISJUNCTIVE METHOD.- Method (6): LEXICOGRAPHIC METHOD.- Method (7): LEXICOGRAPHIC SEMIORDER METHOD.- Method (8): ELIMINATION BY ASPECTS (EBA).- Method (9): LINEAR ASSIGNMENT METHOD (LAM).- Method (10): SIMPLE ADDITIVE WEIGHTING METHOD (SAW).- Method (11): ELECTRE (Elimination et Choice Translating Reality).- Method (12): TOPSIS (Technique for Order Preference by Similarity to Ideal Solution).- Method (13): WEIGHTED PRODUCT METHOD.- Method (14): DISTANCE FROM TARGET METHOD.- III. Fuzzy Sets and their Operations.- 3.1 Introduction.- 3.2 Basics of Fuzzy Sets.- 3.2.1 Definition of a Fuzzy Set.- 3.2.2 Basic Concepts of Fuzzy Sets.- 3.2.2.1 Complement of a Fuzzy Set.- 3.2.2.2 Support of a Fuzzy Set.- 3.2.2.3 ?-cut of a Fuzzy Set.- 3.2.2.4 Convexity of a Fuzzy Set.- 3.2.2.5 Normality of a Fuzzy Set.- 3.2.2.6 Cardinality of a Fuzzy Set.- 3.2.2.7 The mth Power of a Fuzzy Set.- 3.3 Set-Theoretic Operations with Fuzzy Sets.- 3.3.1 No Compensation Operators.- 3.3.1.1 The Min Operator.- 3.3.2 Compensation-Min Operators.- 3.3.2.1 Algebraic Product.- 3.3.2.2 Bounded Product.- 3.3.2.3 Hamacher’s Min Operator.- 3.3.2.4 Yager’s Min Operator.- 3.3.2.5 Dubois and Prade’s Min Operator.- 3.3.3 Full Compensation Operators.- 3.3.3.1 The Max Operator.- 3.3.4 Compensation-Max Operators.- 3.3.4.1 Algebraic Sum.- 3.3.4.2 Bounded Sum.- 3.3.4.3 Hamacher’s Max Operator.- 3.3.4.4 Yager’s Max Operator.- 3.3.4.5 Dubois and Prade’s Max Operator.- 3.3.5 General Compensation Operators.- 3.3.5.1 Zimmermann and Zysno’s ? Operator.- 3.3.6 Selecting Appropriate Operators.- 3.4 The Extension Principle and Fuzzy Arithmetics.- 3.4.1 The Extension Principle.- 3.4.2 Fuzzy Arithmetics.- 3.4.2.1 Fuzzy Number.- 3.4.2.2 Addition of Fuzzy Numbers.- 3.4.2.3 Subtraction of Fuzzy Numbers.- 3.4.2.4 Multiplication of Fuzzy Numbers.- 3.4.2.5 Division of Fuzzy Numbers.- 3.4.2.6 Fuzzy Max and Fuzzy Min.- 3.4.3 Special Fuzzy Numbers.- 3.4.3.1 L-R Fuzzy Number.- 3.4.3.2 Triangular (or Trapezoidal) Fuzzy Number.- 3.4.3.3 Proof of Formulas.- 3.4.3.3.1 The Image of Fuzzy Number N.- 3.4.3.3.2 The Inverse of Fuzzy Number N.- 3.4.3.3.3 Addition and Subtraction.- 3.4.3.3.4 Multiplication and Division.- 3.5 Conclusions.- IV. Fuzzy Ranking Methods.- 4.1 Introduction.- 4.2 Ranking Using Degree of Optimality.- 4.2.1 Baas and Kwakernaak’s Approach.- 4.2.2 Watson et al.’s Approach.- 4.2.3 Baldwin and Guild’s Approach.- 4.3 Ranking Using Hamming Distance.- 4.3.1 Yager’s Approach.- 4.3.2 Kerre’s Approach.- 4.3.3 Nakamura’s Approach.- 4.3.4 Kolodziejczyk’s Approach.- 4.4 Ranking Using ?-Cuts.- 4.4.1 Adamo’s Approach.- 4.4.2 Buckley and Chanas’ Approach.- 4.4.3 Mabuchi’s Approach.- 4.5 Ranking Using Comparison Function.- 4.5.1 Dubois and Prade’s Approach.- 4.5.2 Tsukamoto et al.’s Approach.- 4.5.3 Delgado et al.’s Approach.- 4.6 Ranking Using Fuzzy Mean and Spread.- 4.6.1 Lee and Li’s Approach.- 4.7 Ranking Using Proportion to The Ideal.- 4.7.1 McCahone’s Approach.- 4.8 Ranking Using Left and Right Scores.- 4.8.1 Jain’s Approach.- 4.8.2 Chen’s Approach.- 4.8.3 Chen and Hwang’s Approach.- 4.9 Ranking with Centroid Index.- 4.9.1 Yager’s Centroid Index.- 4.9.2 Murakami et al.’s Approach.- 4.10 Ranking Using Area Measurement.- 4.10.1 Yager’s Approach.- 4.11 Linguistic Ranking Methods.- 4.11.1 Efstathiou and Tong’s Approach.- 4.11.2 Tong and Bonissone’s Approach.- V. Fuzzy Multiple Attribute Decision Making Methods.- 5.1 Introduction.- 5.2 Fuzzy Simple Additive Weighting Methods.- 5.2.1 Baas and Kwakernaak’s Approach.- 5.2.2 Kwakernaak’s Approach.- 5.2.3 Dubois and Prade’s Approach.- 5.2.4 Cheng and McInnis’s Approach.- 5.2.5 Bonissone’s Approach.- 5.3 Analytic Hierarchical Process (AHP) Methods.- 5.3.1 Saaty’s AHP Approach.- 5.3.2 Laarhoven and Pedrycz’s Approach.- 5.3.3 Buckley’s Approach.- 5.4 Fuzzy Conjunctive/Disjunctive Method.- 5.4.1 Dubois, Prade, and Testemale’s Approach.- 5.5 Heuristic MAUF Approach.- 5.6 Negi’s Approach.- 5.7 Fuzzy Outranking Methods.- 5.7.1 Roy’s Approach.- 5.7.2 Siskos et al.’s Approach.- 5.7.3 Brans et al.’s Approach.- 5.7.4 Takeda’s Approach.- 5.8 Maximin Methods.- 5.8.1 Gellman and Zadeh’s Approach.- 5.8.2 Yager’s Approach.- 5.9 A New Approach to Fuzzy MADM Problems.- 5.9.1 Converting Linguistic Terms to Fuzzy Numbers.- 5.9.2 Converting Fuzzy Numbers to Crisp Scores.- 5.9.3 The Algorithm.- VI. Concluding Remarks.- 6.1 MADM Problems and Fuzzy Sets.- 6.2 On Existing MADM Solution Methods.- 6.2.1 Classical Methods for MADM Problems.- 6.2.2 Fuzzy Methods for MADM Problems.- 6.2.2.1 Fuzzy Ranking Methods.- 6.2.2.2 Fuzzy MADM Methods.- 6.3 Critiques of the Existing Fuzzy Methods.- 6.3.1 Size of Problem.- 6.3.2 Fuzzy vs. Crisp Data.- 6.4 A New Approach to Fuzzy MADM Problem Solving.- 6.4.1 Semantic Modeling of Linguistic Terms.- 6.4.2 Fuzzy Scoring System.- 6.4.3 The Solution.- 6.4.4 The Advantages of the New Approach.- 6.5 Other Multiple Criteria Decision Making Methods.- 6.5.1 Multiple Objective Decision Making Methods.- 6.5.2 Methods of Group Decision Making under Multiple Criteria.- 6.5.2.1 Social Choice Theory.- 6.5.2.2 Experts Judgement/Group Participation.- 6.5.2.3 Game Theory.- 6.6 On Future Studies.- 6.6.1 Semantics of Linguistic Terms.- 6.6.2 Fuzzy Ranking Methods.- 6.6.3 Fuzzy MADM Methods.- 6.6.4 MADM Expert Decision Support Systems.- VII. Bibliography.