Multiple Criteria Decision Analysis (eBook)

José Rui Figueira is an Associate Professor at the Technical University of Lisbon, Portugal, and researcher at CEG-IST, Center for Management Studies of Instituto Superior Técnico and LAMSADE, University of Paris-Dauphine, France. He obtained his Ph.D. in Operations Research from University of Paris-Dauphine. Professor Figueira’s current research interests are in decision analysis, integer programming, network flows and multiple criteria decision aiding. His research has been published in such journals as European Journal of Operational Research, Computers & Operations Research, Journal of the Operational Research Society, Journal of Mathematical Modeling and Algorithms, European Business Review, Annals of Operations Research, Fuzzy Sets and Systems, 4OR, Socio-Economic Planning Sciences, Journal of Multi-Criteria Decision Analysis,and OMEGA. He is the co-editor of the book, "Multiple Criteria Decision Analysis: State of the Art Surveys, Springer Science + Business Media, Inc, 2005. He is the currently serves as Editor of the Newsletter of the European Working Group on Multiple Criteria Decision Aiding and one of the coordinators of this group. He is also member of the Executive Committee of the International Society of Multiple Criteria Decision Making.Salvatore Greco is a full professor at the Department of Economics, Catania University.  His main research interests are in the field of multicriteria decision aid, in the application of the rough set approach to decision analysis, in the axiomatic foundation of multicriteria methodology and in the fuzzy integral approach to MCDA. In these fields he cooperates with many researchers of different countries He received the Best Theoretical Paper Award, by the Decision Sciences Institute (Athens, 1999). Together with Benedetto Matarazzo, he organized the VII International Summer School on  MCDA (Catania, 2000). He is author of many articles published in important international journals and specialized books. He has been invited professor at Poznan Technical University and at the University of Paris Dauphine. He has been invited speakers in important international conferences. He is referee of the most relevant journals in the field of decision analysis.Matthias Ehrgott grew up in the Palatinate region of Germany. He studied mathematics, computer science and economics at the University of Kaiserslautern in Germany. In 2000 Matthias joined the Department of Engineering Science as a Lecturer. In 2002 he was promoted to Senior Lecturer and in 2004 to Associate Professor. From 2006 to 2008 he also held the position of directeur de recherche at Laboratoire d’Informatique de Nantes Atlantique in France. In 2011 he became Professor and the seventh Head of the Department of Engineering Science. Matthias left the University of Auckland in 2013 to take up a professorship in the Department of Management Science at the University of Lancaster.

Contents 6
List of Figures 10
List of Tables 16
Introduction 20
1 Ten Years of Success of Multiple Criteria Decision Analysis and Reasons for This New Edition 20
2 Human Reflection About Decision 21
3 Technical Reflection About Decision: MCDA Researchers Before MCDA 22
4 Reasons for This Collection of State-of-the-Art Surveys 24
5 A Guided Tour of the Book 25
5.1 Part I: The History and Current State of MCDA 25
5.2 Part II: Foundations of MCDA 26
5.3 Part III: Outranking Methods 26
5.4 Part IV: Multi-attribute Utility and Value Theories 27
5.5 Part V: Non-classical MCDA Approaches 28
5.6 Part VI: Multiobjective Optimization 29
5.7 Part VII: Applications 31
5.8 Part VIII: MCDM Software 33
References 33
Part I The History and Current State of MCDA 35
1 An Early History of Multiple Criteria Decision Making 36
1.1 Introduction 36
1.2 Early Developments 37
1.2.1 Moral Algebra 38
1.2.2 Some Early Voting Results 38
1.2.3 Pareto-Optimality 39
1.2.4 Indifference Curves and Edgeworth Box 40
1.2.5 Set Theory, Number Theory 40
1.3 Origins of Decision Analysis, Utility Theory 41
1.3.1 Economics as a Modern Science 41
1.3.2 Expected Subjective Utility 41
1.3.3 Theory of Games 42
1.3.4 Revealed Preferences 42
1.3.5 Bounded Rationality 43
1.3.6 Social Choice and Individual Values 44
1.3.7 Theory of Value 44
1.3.8 Games and Decisions 44
1.3.9 Behavioral Decision Theory 45
1.3.10 Utility Theory 46
1.3.11 The `Outranking Methods' 46
1.4 Origins of Multiple Objective Mathematical Programming 47
1.4.1 Efficient Vectors 47
1.4.2 Goal Programming 48
1.4.3 Parametric Programming 48
1.4.4 Automatic Control 49
1.4.5 Restricted Bargaining 49
1.5 Conclusion 49
2 Paradigms and Challenges 51
2.1 What Are the Expectations that Multicriteria Decision Aiding (MCDA) Responds To? 52
2.1.1 What Is Reasonable to Expect from Decision Aiding (DA)? 52
2.1.2 Why Is DA More Often Multicriteria than Monocriterion? 53
2.1.3 Can MCDA Be Always Totally Objective? 54
2.2 Three Basic Concepts 55
2.2.1 Alternative, and More Generally, Potential Action 55
2.2.2 Criterion and Family of Criteria 56
2.2.3 Problematic as a Way in Which DA May Be Envisaged 58
2.3 How to Take into Account Imperfect Knowledge and Ill-Determination? 59
2.4 An Operational Point of View 61
2.4.1 About Multicriteria Aggregation Procedures 62
2.4.2 Approach Based on a Synthesizing Criterion 63
2.4.3 The Operational Approach Based on a Synthesizing Preference Relational System 64
2.4.4 About Other Operational Approaches 65
2.5 Conclusion 65
References 67
Part II Foundations of MCDA 72
3 Preference Modelling 73
3.1 Introduction 74
3.2 Purpose 74
3.3 Nature of Information 76
3.4 Notation and Basic Definitions 77
3.5 Languages 79
3.5.1 Classic Logic 80
3.5.2 Fuzzy Sets 80
3.5.3 Four-Valued Logics 84
3.6 Preference Structures 84
3.6.1 "426830A P, I "526930B Structures 85
3.6.2 Extended Structures 88
3.6.2.1 Preference Relations on n Ordered Points 88
3.6.2.2 Several Preference Relations 89
3.6.2.3 Incomparability 90
3.6.3 Valued Structures 91
3.7 Domains and Numerical Representations 95
3.7.1 Representation Theorems 95
3.7.2 Minimal Representation 99
3.7.2.1 Total Order, Weak Order 99
3.7.2.2 Semi-order 100
3.7.2.3 Interval Order 102
3.7.2.4 PQI Interval Order 102
3.8 Extending Preferences to Sets 103
3.8.1 Complete Uncertainty 104
3.8.2 Opportunity Sets 106
3.8.3 Sets as Final Outcomes 107
3.8.4 An Overview to Related Theories 108
3.9 Logic of Preferences 110
3.10 Conclusion 113
References 114
4 Conjoint Measurement Tools for MCDM 126
4.1 Introduction and Motivation 126
4.1.1 Conjoint Measurement Models in Decision Theory 127
4.1.2 An Aside: Measuring Length 130
4.1.3 An Example: Even Swaps 135
4.2 Definitions and Notation 141
4.2.1 Binary Relations 141
4.2.2 Binary Relations on Product Sets 142
4.2.3 Independence and Marginal Preferences 142
4.3 The Additive Value Model in the ``Rich'' Case 144
4.3.1 Outline of Theory 144
4.3.1.1 The Case of Two Attributes 144
4.3.1.2 The Case of More Than Two Attributes 149
4.3.2 Statement of Results 150
4.3.3 Implementation: Standard Sequences and Beyond 153
4.4 The Additive Value Model in the ``Finite'' Case 154
4.4.1 Outline of Theory 154
4.4.2 Implementation: LP-Based Assessment 157
4.4.2.1 UTA JacquetLagrezeSiskos82UTA 159
4.4.2.2 MACBETH BanaVansnick94MACBETH 161
4.5 Extensions 163
4.5.1 Transitive Decomposable Models 164
4.5.2 Intransitive Indifference 165
4.5.3 Nontransitive Preferences 166
References 170
Part III Outranking Methods 181
5 ELECTRE Methods 182
5.1 Introduction: A Brief History 183
5.2 Main Features of ELECTRE Methods 185
5.2.1 In What Context Are ELECTRE Methods Relevant? 185
5.2.2 Modeling Preferences Using an Outranking Relation 186
5.2.3 Structure of ELECTRE Methods 186
5.2.4 About the Relative Importance of Criteria 187
5.2.5 Discriminating Thresholds 187
5.3 A Short Description of ELECTRE Methods 188
5.3.1 Choice Problematic 188
5.3.1.1 ELECTRE I 189
5.3.1.2 ELECTRE Iv 190
5.3.1.3 ELECTRE IS 191
5.3.2 Ranking Problematic 192
5.3.2.1 ELECTRE II 193
5.3.2.2 ELECTRE III 194
5.3.2.3 ELECTRE IV 196
5.3.3 Sorting Problematic 196
5.3.3.1 ELECTRE TRI 197
5.3.3.2 ELECTRE TRI C and ELECTRE TRI nC 198
5.4 Recent Developments 199
5.4.1 Robustness Concerns 199
5.4.2 Elicitation of Parameter Values 200
5.4.2.1 Direct Elicitation Techniques 200
5.4.2.2 Indirect Elicitation Techniques 200
5.5 Software and Applications 202
5.5.1 ELECTRE Software 202
5.5.2 The Decision Deck Project 203
5.5.3 Applications 204
5.6 Conclusion 204
References 205
6 PROMETHEE Methods 213
6.1 Preamble 214
6.2 History 214
6.3 Multicriteria Problems 215
6.4 The PROMETHEE Preference Modelling Information 218
6.4.1 Information Between the Criteria 218
6.4.2 Information Within the Criteria 219
6.5 The PROMETHEE I and II Rankings 222
6.5.1 Aggregated Preference Indices 222
6.5.2 Outranking Flows 223
6.5.3 The PROMETHEE I Partial Ranking 224
6.5.4 The PROMETHEE II Complete Ranking 225
6.5.5 The Profiles of the Alternatives 225
6.6 A Few Words About Rank Reversal 226
6.7 The GAIA Visual Interactive Module 228
6.7.1 The GAIA Plane 228
6.7.2 Graphical Display of the Alternatives and of the Criteria 229
6.7.3 The PROMETHEE Decision Stickdecision stick |205. The PROMETHEE Decision Axisdecision axis |205 231
6.8 The PROMETHEE VI Sensitivity Tool (the ``Human Brain'') 233
6.9 PROMETHEE V: MCDA Under Constraints 234
6.10 FlowSort 235
6.11 The PROMETHEE GDSS Procedure 238
6.11.1 Phase I: Generation of Alternatives and Criteria 238
6.11.2 Phase II: Individual Evaluation by Each DM 239
6.11.3 Phase III: Global Evaluation by the Group 239
6.12 The D-Sight Software 240
References 243
7 Other Outranking Approaches 246
7.1 Introduction 246
7.2 Other Outranking Methods 247
7.2.1 QUALIFLEX 248
7.2.2 REGIME 251
7.2.3 ORESTE 255
7.2.4 ARGUS 258
7.2.5 EVAMIX 263
7.2.6 TACTIC 266
7.2.7 MELCHIOR 267
7.3 Pairwise Criterion Comparison Approach (PCCA) 269
7.3.1 MAPPAC 273
7.3.2 PRAGMA 286
7.3.3 IDRA 293
7.3.4 PACMAN 296
7.4 One Outranking Method for Stochastic Data 300
7.4.1 Martel and Zaras' Method 300
7.5 Conclusions 305
References 305
Part IV Multiattribute Utility and Value Theories 308
8 Multiattribute Utility Theory (MAUT) 309
8.1 Introduction 309
8.2 Preference Representations Under Certainty and Under Risk 311
8.2.1 Preference Functions for Certainty (Value Functions) 313
8.2.2 Preference Functions for Risky Choice (Utility Functions) 315
8.2.3 Comment 316
8.3 Ordinal Multiattribute Preference Functions for the Case of Certainty 317
8.3.1 Preference Independence 317
8.3.2 Assessment Methodologies 319
8.4 Cardinal Multiattribute Preference Functions for the Case of Risk 321
8.4.1 Utility Independence 322
8.4.2 Additive Independence 323
8.4.3 Assessment Methodologies 324
8.5 Measurable Multiattribute Preference Functions for the Case of Certainty 324
8.5.1 Weak Difference Independence 325
8.5.2 Difference Independence 326
8.5.3 Assessment Methodologies 328
8.5.3.1 Verification of the Independence Conditions 328
8.5.3.2 Assessment of the Measurable Value Functions 329
8.5.4 Goal Programming and Measurable Multiattribute Value Functions 330
8.5.4.1 Goal Programming as an Approximation to Multiattribute Preferences 331
8.6 The Relationships among the Multiattribute Preference Functions 334
8.6.1 The Additive Functions 334
8.6.2 The Multiplicative Functions 335
8.7 Concluding Remarks 336
References 337
9 UTA Methods 339
9.1 Introduction 340
9.1.1 General Philosophy 340
9.1.2 The Disaggregation-Aggregation Paradigm 341
9.1.3 Historical Background 342
9.2 The UTA Method 344
9.2.1 Principles and Notation 344
9.2.2 Development of the UTA Method 345
9.2.3 The UTASTAR Algorithm 348
9.2.4 Robustness Analysis 351
9.2.5 A Numerical Example 352
9.3 Variants of the UTA Method 356
9.3.1 Alternative Optimality Criteria 356
9.3.2 Meta-UTA Techniques 360
9.3.3 Stochastic UTA Method 361
9.3.4 UTA-Type Sorting Methods 363
9.3.5 Other Variants and Extensions 366
9.3.6 Other Disaggregation Methods 367
9.4 Applications and UTA-Based DSS 370
9.5 Concluding Remarks and Future Research 378
References 379
10 The Analytic Hierarchy and Analytic Network Processes for the Measurement of Intangible Criteria and for Decision-Making 387
10.1 Introduction 388
10.2 Pairwise Comparisons Inconsistency and the Principal Eigenvector
10.3 Stimulus Response and the Fundamental Scale 395
10.3.1 Validation Example 398
10.3.2 Clustering and Homogeneity Using Pivots to Extend the Scale from 1–9 to 1–?
10.4 Hospice Decision 400
10.5 Rating Alternatives One at a Time in the AHP: Absolute/ Measurement 409
10.5.1 Evaluating Employees for Salary Raises 410
10.6 Paired Comparisons Imply Dependence 411
10.7 When Is a Positive Reciprocal Matrix Consistent? 413
10.8 In the Analytic Hierarchy Process Additive Composition Is Necessary 415
10.9 Benefits, Opportunities, Costs and Risks 416
10.10 On the Admission of China to the World Trade Organization 417
10.11 The Analytic Network Process 421
10.11.1 The Classic AHP School Example as an ANP Model 426
10.11.2 Criteria Weights Automatically Derived from Supermatrix 427
10.12 Two Examples of Estimating Market Share: The ANP with a Single Benefits Control Criterion 428
10.12.1 Example 1: Estimating the Relative Market Share of Walmart, Kmart and Target 429
10.12.1.1 The Unweighted Supermatrix 429
10.12.1.2 The Cluster Matrix 430
10.12.1.3 The Weighted Supermatrix 431
10.12.1.4 Synthesized Results from the Limit Supermatrix 433
10.12.2 Example 2: US Athletic Footwear Market in 2000 434
10.12.2.1 Clusters and Elements (Nodes) 434
10.13 Outline of the Steps of the ANP 436
10.14 Complex Decisions with Dependence and Feedback 439
10.14.1 The National Missile Defense Example 439
10.15 Synthesis of Individual Judgments into a Representative Group Judgment 441
10.16 Conclusions 442
References 443
11 On the Mathematical Foundations of MACBETH 444
11.1 Introduction 444
11.2 Previous Research and Software Evolution 447
11.3 Types of Preferential Information 448
11.3.1 Type 1 Information 448
11.3.2 Type 1+2 Information 449
11.4 Numerical Representation of the Preferential Information 449
11.4.1 Type 1 Scale 449
11.4.2 Type 1+2 Scale 450
11.5 Consistency: Inconsistency 450
11.6 Consistency Test for Preferential Information 452
11.6.1 Testing Procedures 452
11.6.2 Pre-test of the Preferential Information 452
11.6.3 Consistency Test for Type 1 Information 453
11.6.3.1 Consistency Test for Incomplete (? = ?) Type 1 Information 453
11.6.3.2 Consistency Test for Complete (? = ?) Type 1 Information 453
11.6.4 Consistency Test for Type 1+2 Information 454
11.7 Dealing with Inconsistency 454
11.7.1 Systems of Incompatible Constraints 455
11.7.2 Example 1 456
11.7.3 Identifying Constraints which Cause Inconsistency 458
11.7.4 Augmentation: Reduction in a Judgement with p Categories 461
11.7.4.1 Preliminaries 461
11.7.4.2 Exploitation of the Constraints of SEI 462
11.7.4.3 Search for Suggestions 463
11.7.5 Example 2 465
11.8 The MACBETH Scale 466
11.8.1 Definition of the MACBETH Scale 466
11.8.2 Discussing the Uniqueness of the Basic MACBETH Scale 467
11.8.3 Presentation of the MACBETH Scale 468
11.8.4 Determining by Hand the Basic MACBETH Scale 469
11.9 Discussion About a Scale 473
11.10 MACBETH and MCDA 474
References 476
Part V Non-classical MCDA Approaches 487
12 Dealing with Uncertainties in MCDA 488
12.1 What is Uncertainty? 489
12.1.1 Internal Uncertainty 490
12.1.2 External Uncertainty 490
12.2 Sensitivity Analysis and Related Methods 493
12.3 Probabilistic Models and Expected Utility 495
12.4 Pairwise Comparisons 500
12.5 Risk Measures as Surrogate Criteria 503
12.6 Scenario Planning and MCDA 506
12.7 Implications for Practice 512
References 513
13 Decision Rule Approach 518
13.1 Introduction 519
13.2 Dominance-Based Rough Set Approach (DRSA) 522
13.2.1 Data Table 522
13.2.2 Dominance Principle 524
13.2.3 Decision Rules 525
13.2.4 Rough Approximations 526
13.2.5 Properties of Rough Approximations 529
13.2.6 Quality of Approximation, Reducts and Core 532
13.2.7 Importance and Interaction Among Criteria 534
13.3 Variable Consistency Dominance-Based Rough Set Approach (VC-DRSA) 535
13.4 Induction of Decision Rules from Rough Approximations of Upward and Downward Unions of Decision Classes 537
13.4.1 A Syntax of Decision Rules Involving Dominance with Respect to Partial Profiles 537
13.4.2 Different Strategies of Decision Rule Induction 541
13.4.3 Application of Decision Rules 541
13.4.4 Decision Trees: An Alternative to Decision Rules 544
13.5 Extensions of DRSA 545
13.5.1 DRSA with Joint Consideration of Dominance, Indiscernibility and Similarity Relations 545
13.5.2 DRSA and Interval Orders 547
13.5.3 Fuzzy DRSA: Rough Approximations by Means of Fuzzy Dominance Relations 548
13.5.4 DRSA with Missing Values: Multiple-Criteria Classification Problem with Missing Values 550
13.5.5 DRSA for Decision Under Uncertainty 551
13.5.6 DRSA for Hierarchical Structure of Attributes and Criteria 552
13.6 DRSA for Multiple-Criteria Choice and Ranking 553
13.6.1 Pairwise Comparison Table (PCT) as a Preference Information and a Learning Sample 554
13.6.2 Multigraded Dominance 555
13.6.3 Induction of Decision Rules from Rough Approximations of Graded Preference Relations 557
13.6.4 Use of Decision Rules for Decision Support 558
13.6.5 Illustrative Example 559
13.6.6 Fuzzy Preferences 562
13.6.7 Preferences Without Degree of Preferences 563
13.7 DRSA and Operations Research Problems 563
13.7.1 DRSA to Interactive Multiobjective Optimization (IMO-DRSA) 564
13.7.2 DRSA to Interactive Evolutionary Multiobjective Optimization 564
13.7.3 DRSA to Decision Under Uncertainty and Time Preference 565
13.8 Conclusions 566
References 568
14 Fuzzy Measures and Integrals in MCDA 574
14.1 Introduction 574
14.2 Measurement Theoretic Foundations 577
14.2.1 Basic Notions of Measurement, Scales 577
14.2.2 Bipolar and Unipolar Scales 579
14.2.3 Construction of the Measurement Scales and Absolute References Levels 580
14.3 Unipolar Scales 581
14.3.1 Notion of Interaction: A Motivating Example 581
14.3.2 Capacities and Choquet Integral 582
14.3.3 Construction of Utility Functions 584
14.3.3.1 Difficulty of the Construction of Utility Functions 584
14.3.3.2 General Method for Building Utility Functions 585
14.3.3.3 Construction of Utility Functions Without any Commensurability Assumption 586
14.3.4 Justification of the Use of the Choquet Integral 586
14.3.4.1 Justification Through Information on the Binary Alternatives 586
14.3.4.2 Axiomatization of the Choquet Integral as an Aggregation Function 588
14.3.4.3 Axiomatizations of the Choquet Integral with Utility Functions 589
14.3.5 Shapley Value and Interaction Index 590
14.3.6 k-Additive Measures 593
14.3.6.1 Definition of the k-Additive Measures 593
14.3.6.2 Axiomatic Characterization of the Choquet Integral w.r.t. 2-Additive Measures 594
14.3.7 Final Recommendation and Identification of Capacities 595
14.3.7.1 Preferential Information 595
14.3.7.2 Identification of a Capacity, and Associated Recommendation 596
14.3.7.3 Robust Preference Relations 596
14.3.7.4 Explanation of the Recommendation 597
14.4 Bipolar Scales 598
14.4.1 A Motivating Example 598
14.4.2 The Symmetric Choquet Integral and Cumulative Prospect Theory 599
14.4.2.1 Definitions 599
14.4.2.2 Application to the Example 600
14.4.3 Bi-capacities and the Corresponding Integral 601
14.4.4 Representation of the Motivating Example 603
14.4.5 General Method for Building Utility Functions 605
14.4.6 Justification of the Use of the Generalized Choquet Integral 606
14.4.6.1 Required Information 606
14.4.6.2 Measurement Conditions 607
14.4.7 Shapley Value and Interaction Index 608
14.4.8 Particular Models 609
14.4.9 Identification of Bi-capacities 610
14.5 Ordinal Scales 610
14.5.1 Introduction 610
14.5.2 The Sugeno Integral 612
14.5.3 Symmetric Ordinal Scales and the Symmetric Sugeno Integral 613
14.5.4 A Model of Decision Based on the Sugeno Integral 615
14.5.5 The Lexicographic Sugeno Integral 616
14.5.6 Identification of Capacities 618
14.6 Concluding Remarks 618
References 619
15 Verbal Decision Analysis 625
15.1 Introduction 625
15.1.1 Features of Unstructured Decision Problems 626
15.2 Main Principles of Verbal Decision Analysis 627
15.2.1 Natural Language of a Problem Description 627
15.2.2 Psychological Basis for Decision Rules Elaboration 628
15.2.3 Theoretical Basis for Decision Rules Elaboration 629
15.2.4 Consistency Check of Decision Maker's Information 630
15.2.5 Explanation of the Analysis Result 631
15.3 Decision Methods for Multiple Criteria Alternatives' Ranking 631
15.3.1 Problem Formulation 631
15.3.2 The Joint Ordinal Scale: Method ZAPROS-LM 633
15.3.2.1 Verification of the Structure of the Decision Maker's Preferences 636
15.3.3 Joint Scale for Quality Variation: ZAPROS III 637
15.3.4 Goal Oriented Process for Quality Variations: STEP-ZAPROS 639
15.3.5 Working in the Space of Real Alternatives: UniCombos 641
15.4 Decision Methods for Multiple Criteria Alternatives' Classification 642
15.4.1 Problem Formulation 643
15.4.2 An Ordinal Classification Approach: ORCLASS 643
15.4.3 Effectiveness of Preference Elicitation 644
15.4.4 Class Boundaries 645
15.4.5 Real Alternatives Classification: SAC and CLARA 646
15.4.6 Hierarchical Ordinal Classification 647
15.5 Place of Verbal Decision Analysis in MCDA 648
15.5.1 Multi Attribute Utility Theory and Verbal Decision Analysis Methods 648
15.5.2 Practical Value of the Verbal Decision Analysis Approach 652
15.6 Conclusion 653
References 653
16 A Review of Fuzzy Sets in Decision Sciences: Achievements, Limitations and Perspectives 657
16.1 Introduction 657
16.2 Membership Functions in Decision-Making 660
16.2.1 Membership Functions and Truth Sets in Decision Analysis 660
16.2.2 Truth-Sets as Value Scales: The Meaning of End-Points 661
16.2.3 Truth-Sets as Value Scales: Quantitative or Qualitative? 662
16.2.4 From Numerical to Fuzzy Value Scales 664
16.2.4.1 Evaluations by Pairs of Values 664
16.2.4.2 Linguistic vs. Numerical Scales 665
16.3 The Two Meanings of Fuzzy Preference Relations 667
16.3.1 Unipolar vs. Bipolar Fuzzy Relations 667
16.3.2 Fuzzy Strict Preference Relations 668
16.3.3 Fuzzy Preference Relations Expressing Uncertainty 669
16.3.4 Transitivity and Arrow's Theorem 670
16.4 Fuzzy Outranking Relations in Multicriteria Decision Problems 672
16.4.1 Fuzzy Concordance Relations 673
16.4.2 Fuzzy Discordance Relations and the Veto Principle 675
16.4.3 Choosing, Ranking and Sorting with Fuzzy Preference Relations 677
16.5 Fuzzy Connectives for Decision Evaluation in the Qualitative Setting 680
16.5.1 Aggregation Operations: Qualitative or Quantitative 681
16.5.2 Refinements of Qualitative Aggregation Operations 682
16.5.3 Numerical Encoding of Qualitative Aggregation Functions 686
16.5.4 Bipolarity in Qualitative Evaluation Processes 688
16.6 Uncertainty Handling in Decision Evaluation Using Fuzzy Intervals 690
16.6.1 Fuzzy Weighted Averages 690
16.6.2 Fuzzy Extensions of the Analytical Hierarchy Process 695
16.7 Comparing Fuzzy Intervals: A Constructive Setting 698
16.7.1 Four Views of Fuzzy Intervals 699
16.7.2 Constructing Fuzzy Interval Ranking Methods 701
16.7.2.1 Metric Approach 702
16.7.2.2 Random Interval Approach 703
16.7.2.3 Imprecise Probability Approach 703
16.7.2.4 Gradual Number Approach 704
16.8 Conclusion 705
References 706
Part VI Multiobjective Optimization 712
17 Vector and Set Optimization 713
17.1 Introduction 713
17.2 Pre- and Partial Orders 715
17.3 Vector Optimization 718
17.3.1 Optimality Concepts 718
17.3.2 Existence Results 723
17.3.3 Application: Field Design of a Magnetic Resonance System 732
17.3.4 Vector Optimization with a Variable Ordering Structure 735
17.4 Set Optimization 746
17.4.1 Vector Approach 746
17.4.2 Set Approach 748
References 753
18 Continuous Multiobjective Programming 756
18.1 Introduction 756
18.2 Problem Formulation and Solution Concepts 758
18.2.1 Partial Orders and Pareto Optimality 758
18.2.2 Cones and Nondominated Outcomes 759
18.2.3 Domination Sets and Variable Cones 761
18.2.4 Local, Proper, and Approximate Solutions 762
18.3 Properties of the Solution Sets 763
18.4 Conditions for Efficiency 765
18.4.1 First Order Conditions 765
18.4.2 Second Order Conditions 767
18.5 Generation of the Solution Sets 767
18.5.1 Scalarization Methods 767
18.5.1.1 The Weighted-Sum Approach 768
18.5.1.2 The Weighted t-th Power Approach 768
18.5.1.3 The Weighted Quadratic Approach 769
18.5.1.4 The Guddat et al. Approach 769
18.5.1.5 The -Constraint Approach 770
18.5.1.6 The Improved -Constraint Approach 770
18.5.1.7 The Penalty Function Approach 771
18.5.1.8 The Benson Approach 772
18.5.1.9 Reference Point Approaches 772
18.5.1.10 Direction-Based Approaches 776
18.5.1.11 Gauge-Based Approaches 778
18.5.1.12 Composite and Other Approaches 779
18.5.2 Approaches Based on Non-Pareto Optimality 779
18.5.2.1 The Lexicographic Approach 779
18.5.2.2 The Max-Ordering Approach 780
18.5.2.3 The Lexicographic Max-Ordering Approach 781
18.5.2.4 The Equitability Approach 782
18.5.3 Descent Methods 783
18.5.4 Set-Oriented Methods 783
18.5.4.1 The Balance and Level Set Approaches 784
18.5.4.2 The ?-Efficiency Approach 785
18.5.4.3 Continuation Methods 785
18.5.4.4 Covering Methods 786
18.6 Approximation of the Pareto Set 787
18.6.1 Quality Measures for Representations 788
18.6.1.1 Measures of Cardinality 788
18.6.1.2 Measures of Coverage 788
18.6.1.3 Measures of Spacing 789
18.6.1.4 Hybrid Measures 790
18.6.2 Representation and Approximation Approaches 791
18.6.2.1 Representation for BOPs 791
18.6.2.2 Representation for MOPs 792
18.6.2.3 Polyhedral Approximation 794
18.6.2.4 Nonlinear Approximation 795
18.7 Specially Structured Problems 796
18.7.1 Multiobjective Linear Programming 796
18.7.1.1 Multicriteria Simplex Methods 798
18.7.1.2 Interior Point Methods 800
18.7.1.3 Objective Space Methods 801
18.7.2 Nonlinear MOPs 803
18.7.2.1 MOPs with Piecewise Linear Objectives 804
18.7.2.2 Quadratic MOPs 804
18.7.2.3 Polynomial MOPs 805
18.7.2.4 Fractional MOPs 805
18.7.3 Parametric Multiobjective Programming 806
18.7.3.1 Parametric MOPs 806
18.7.3.2 Parametrization of the Scalarized MOP 807
18.7.4 Bilevel Multiobjective Programming 808
18.7.4.1 Relationships between Bilevel Single Objective and Multiobjective Programming 808
18.7.4.2 Theory of Bilevel Multiobjective Programming 809
18.7.4.3 Methodology for Bilevel Multiobjective Programming 811
18.8 Current and Future Research Directions 812
18.8.1 Research on Set-Oriented Methods 813
18.8.2 Theoretical and Methodological Studies Motivated by Mathematical and Real-Life Applications 813
18.8.3 Applications in New Areas 814
18.8.4 Integration of Multiobjective Programming with Multicriteria Decision Analysis (MCDA) 814
18.9 Conclusion 814
References 815
19 Exact Methods for Multi-Objective CombinatorialOptimisation 833
19.1 Introduction 833
19.1.1 Definitions 834
19.1.2 Computational Complexity 838
19.1.3 Connectedness of Efficient Solutions 840
19.1.4 Bounds and Bound Sets 840
19.1.5 Outlook 842
19.2 Extending Single Objective Algorithms 843
19.2.1 Labelling Algorithms 843
19.2.2 Greedy Algorithms 844
19.3 Scalarisation 845
19.3.1 Scalarisation Algorithms from the Literature 850
19.4 The Two-Phase Method 850
19.4.1 The Two Phase Method for Two Objectives 851
19.4.2 The Two Phase Method for Three Objectives 855
19.4.3 Two-Phase Algorithms from the Literature 856
19.5 Multi-Objective Branch and Bound 857
19.5.1 Branching and Node Selection 857
19.5.2 Bounding and Fathoming Nodes 858
19.5.3 Multi-Objective Branch and Bound Algorithms from the Literature 860
19.6 Conclusion 860
References 862
20 Fuzzy Multi-Criteria Optimization: Possibilistic and Fuzzy/Stochastic Approaches 867
20.1 Introduction 868
20.2 Problem Statement and Preliminaries 869
20.3 Single Objective Function Case 873
20.3.1 Optimization of Upper and Lower Bounds 873
20.3.2 Possibly and Necessarily Optimal Solutions 876
20.3.3 Minimax Regret Solutions and the Related Solution Concepts 881
20.4 Multiple Objective Function Case 884
20.4.1 Possibly and Necessarily Efficient Solutions 884
20.4.2 Efficiency Test and Possible Efficiency Test 891
20.4.3 Necessary Efficiency Test 893
Implicit Enumeration Algorithm Bitran 895
20.5 Interactive Fuzzy Stochastic Multiple Objective Programming 898
20.5.1 Fuzzy Random Variable 899
20.5.2 Brief Survey of Fuzzy Random Multiple Objective Programming 901
20.5.3 Problem Formulation 901
20.5.4 Possibilistic Expectation Model 905
20.5.4.1 Interactive Satisficing Method for the Possibilistic Expectation Model 907
20.5.5 Possibilistic Variance Model 908
20.5.5.1 Extended Dinkelbach-Type Algorithm for Solving (20.114) 912
20.5.5.2 Interactive Satisficing Method for the Possibilistic Variance Model 913
20.5.6 Recent Topics: Random Fuzzy Multiple Objective Programming 913
References 914
21 A Review of Goal Programming 919
21.1 Introduction 919
21.2 Goal Programming Variants 920
21.2.1 Lexicographic Goal Programming 920
21.2.2 Weighted Goal Programming 921
21.2.3 Chebyshev Goal Programming 922
21.2.4 Extended Goal Programming 923
21.2.5 Meta Goal Programming 924
21.2.6 Multi-Choice Goal Programming 924
21.2.7 Fuzzy Goal Programming 925
21.2.8 Goal Programming with Non-standard Preferences 925
21.2.9 Integer and Binary Goal Programming 926
21.2.10 Non-linear and Fractional Goal Programming 926
21.3 Goal Programming as Part of a Mixed-Modelling Framework 927
21.3.1 Goal Programming as a Statistical Tool 927
21.3.2 Goal Programming and Other Distance-Metric Based Approaches 928
21.3.3 Goal Programming and Pairwise Comparison Techniques 930
21.3.3.1 Using the AHP to Determine Goal Programming Preferential Weights 930
21.3.3.2 Using Goal Programming as a Technique to Derive the Weighting Vector in AHP 930
21.3.4 Goal Programming and Other Multi-Criteria Decision Analysis Techniques 931
21.3.4.1 Goal Programming and Interactive Methods 931
21.3.4.2 Goal Programming and A Posteriori Techniques 931
21.3.4.3 Goal Programming and Discrete Choice/Outranking Methods 932
21.3.5 Goal Programming and Computing/Artificial Intelligence Techniques 932
21.3.5.1 Goal Programming and Pattern Recognition 932
21.3.5.2 Goal Programming and Fuzzy Logic 933
21.3.5.3 Goal Programming and Meta Heuristic Methods 933
21.3.6 Goal Programming and Data Envelopment Analysis 934
21.4 Application of Goal Programming 935
21.5 Conclusions 936
References 936
22 Interactive Nonlinear Multiobjective Optimization Methods 943
22.1 Introduction 943
22.2 Concepts 945
22.3 Introduction to Interactive Methods 947
22.4 Methods Using Aspiration Levels 950
22.4.1 Reference Point Method 950
22.4.2 GUESS Method 952
22.4.3 Light Beam Search 953
22.4.4 Other Methods Using Aspiration Levels 955
22.5 Methods Using Classification 955
22.5.1 Step Method 955
22.5.2 Satisficing Trade-Off Method 957
22.5.3 Reference Direction Method 959
22.5.4 NIMBUS Method 960
22.5.5 Other Methods Using Classification 963
22.6 Methods Where Solutions Are Compared 964
22.6.1 Chebyshev Method 964
22.6.2 NAUTILUS Method 965
22.6.3 Other Methods Where Solutions Are Compared 969
22.7 Methods Using Marginal Rates of Substitution 969
22.7.1 Interactive Surrogate Worth Trade-Off Method 970
22.7.2 Geoffrion-Dyer-Feinberg Method 971
22.7.3 Other Methods Using Marginal Rates of Substitution 973
22.8 Navigation Methods 973
22.8.1 Reference Direction Approach 973
22.8.2 Pareto Navigator Method 975
22.8.3 Pareto Navigation Method 977
22.8.4 Other Navigation Methods 978
22.9 Other Interactive Methods 978
22.10 Comparing the Methods 979
22.11 Conclusions 979
References 980
23 MCDA and Multiobjective Evolutionary Algorithms 993
23.1 Introduction 993
23.2 Multiobjective Evolutionary Algorithms 994
23.2.1 Non-dominated Sorting Genetic Algorithm (NSGA-II) 996
23.2.2 Indicator-Based MOEAs 997
23.2.3 Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D) 998
23.2.4 Interactive Evolutionary Algorithms 999
23.3 MCDM to Support the Selection from a Set of Solutions Generated by an MOEA 999
23.4 Integrating User Preferences in MOEA 999
23.4.1 Scaling 1001
23.4.2 Constraints 1002
23.4.3 Providing a Reference Point 1003
23.4.4 Limiting Possible Trade-Offs 1007
23.4.5 Weighting the Objective Space 1009
23.4.6 Specifying a Distribution over Utility Functions 1009
23.4.7 Approaches Based on Outranking Relations 1011
23.4.8 Approaches Based on Solution Comparison 1011
23.4.8.1 Determining a Representative Value Function 1012
23.4.8.2 Determining a Set of Compatible Value Functions 1015
23.4.8.3 Other Algorithmic Principles 1017
23.5 Summary and Open Research Questions 1019
References 1020
Part VII Applications 1025
24 Multicriteria Decision Aid/Analysis in Finance 1026
24.1 Introduction 1026
24.2 Financial Decision Making 1028
24.2.1 Issues, Concepts, and Principles 1028
24.2.2 Focus of Financial Research 1030
24.2.3 Descriptive vs. Conditional-Normative Modelling 1032
24.2.4 Decision Support for Financial Decisions 1035
24.2.5 Relevance of MCDA for Financial Decisions 1036
24.2.6 A Multicriteria Framework for Financial Decisions 1039
24.2.6.1 Principles 1040
24.2.6.2 Allocation Decisions 1041
24.2.6.3 Uncertainty and Risk 1041
24.2.6.4 A Bird's-Eye View of the Framework 1042
24.2.6.5 The Framework and Modern Financial Theory 1044
24.3 MCDA in Portfolio Decision-Making Theory 1044
24.3.1 Portfolio Selection Problem 1045
24.3.2 Background on Multicriteria Optimization 1047
24.3.3 Two Model Variants 1048
24.3.4 Bullet-Shaped Feasible Regions 1049
24.3.5 Assumptions and Nondominated Sensitivities 1052
24.3.6 Expanded Formulations and New Assumptions 1055
24.3.7 Nondominated Surfaces 1056
24.3.8 Idea of a Projection 1057
24.3.9 Further Research in MCDA in Portfolio Analysis 1058
24.4 MCDA in Discrete Financial Decision-Making Problems 1059
24.4.1 Outranking Relations 1060
24.4.2 Utility Functions-Based Approaches 1062
24.4.3 Decision Rule Models: Rough Set Theory 1065
24.4.4 Applications in Financial Decisions 1066
24.5 Conclusions and Future Perspectives 1071
References 1071
25 Multi-Objective Optimization and Multi-Criteria Analysis Models and Methods for Problems in the Energy Sector 1081
25.1 Introduction 1082
25.2 Multi-Objective Optimization Models and Methods for Energy Planning 1085
25.2.1 Power Generation Expansion Planning and Operation Planning 1086
25.2.2 Transmission and Distribution Network Planning 1093
25.2.3 Reactive Power Planning and Voltage Regulation 1102
25.2.4 Unit Commitment and Dispatch Problems 1105
25.2.5 Load Management 1110
25.2.6 Energy-Economy Planning Models 1113
25.2.7 Energy Markets 1115
25.3 Energy Planning Decisions with Discrete Alternatives 1117
25.3.1 Comparison of Power Generation Technologies 1119
25.3.2 Energy Plans and Policies 1119
25.3.3 Selection of Energy Projects 1135
25.3.4 Siting Decisions 1135
25.3.5 Energy Efficiency 1135
25.3.6 Miscellaneous 1135
25.3.7 The Choice of Criteria 1135
25.3.8 Technical Criteria 1145
25.3.9 Economic Criteria 1149
25.3.9.1 Costs 1149
25.3.9.2 Economic Performance 1151
25.3.10 Environmental Criteria 1153
25.3.10.1 Local Impacts 1153
25.3.10.2 Global Impacts 1155
25.3.11 Social Criteria 1155
25.3.11.1 Health Impacts 1155
25.3.11.2 Risks 1156
25.3.11.3 Development 1157
25.3.11.4 Acceptability 1157
25.3.12 MCDA Methods 1159
25.3.13 Uncertainty Treatment 1162
25.4 Conclusions 1164
References 1165
26 Multicriteria Analysis in Telecommunication Network Planning and Design: A Survey 1180
26.1 Motivation 1180
26.2 Overview of Current Evolutions in Telecommunication Networks and Services 1181
26.2.1 Major Technological Evolutions 1181
26.2.2 Increasing Relevance of QoS Issues in the New Technological Platforms 1185
26.3 Multicriteria Analysis in Telecommunication Network Planning and Design 1186
26.4 Review and Discussion of Applications of MA to Telecommunication Network Planning 1191
26.4.1 Routing Models 1191
26.4.1.1 Background Concepts 1191
26.4.1.2 Review of Multiple Criteria Routing Approaches 1192
26.4.2 Network Planning and Design 1217
26.4.3 Models Studying Interactions Between Telecommunication Evolution and Socio-Economic Issues 1224
26.5 Future Trends 1226
26.5.1 Routing Models 1227
26.5.2 Network Planning and Design and Models Studying Interactions Between Telecommunication Evolution and Socio-Economic Issues 1230
References 1231
27 Multiple Criteria Decision Analysis and SustainableDevelopment 1247
27.1 The Concept of Sustainable Development and the Incommensurability Principle 1247
27.2 Measuring Sustainability: The Issue of Sustainability Assessment Indexes 1253
27.3 A Defensible Setting for Sustainability Composite Indicators 1257
27.4 Warning! Not Always Rankings Have to Be Trusted … 1261
27.5 The Issue of the “Quality of the Social Decision Processes” 1265
27.6 The Issue of Consistency in Multi-Criteria Evaluation of Sustainability Policies 1270
27.7 Conclusion 1273
References 1274
28 Multicriteria Portfolio Decision Analysis for Project Selection 1280
28.1 Introduction 1280
28.2 A Formal Framework for MCPDA 1282
28.3 Modelling Challenges 1285
28.3.1 Structuring 1285
28.3.2 Exploring 1288
28.4 Application Domains 1291
28.4.1 R& D Project Selection
28.4.2 Military Planning and Procurement 1294
28.4.3 Commissioning Health Services 1296
28.4.4 Environment and Energy Planning 1297
28.5 Conclusion and Directions for Future Research 1300
References 1301
Part VIII MCDM Software 1310
29 Multiple Criteria Decision Analysis Software 1311
29.1 Introduction 1311
29.2 General Overview of Available MCDA Software 1312
29.2.1 MADA Versus MOO Software 1312
29.2.2 MCDA Methods Implemented 1313
29.2.3 Group Decision Support 1318
29.2.4 Platform Supported 1318
29.3 Software Review 1319
29.3.1 1000Minds 1319
29.3.2 4eMka2/jMAF 1322
29.3.3 ACADEA 1322
29.3.4 Accord 1322
29.3.5 Analytica Optimizer 1323
29.3.6 APOGEE 1323
29.3.7 BENSOLVE 1324
29.3.8 Criterium Decision Plus (CDP) 1324
29.3.9 DecideIT 1324
29.3.10 Decision Explorer® 1325
29.3.11 Decision Desktop Software (d2)/Diviz 1325
29.3.12 Decision Lab 2000/Visual PROMETHEE 1326
29.3.13 DPL 8 1326
29.3.14 D-Sight 1327
29.3.15 ELECTRE III-IV 1327
29.3.16 ELECTRE IS 1328
29.3.17 ELECTRE TRI 1328
29.3.18 Equity3 1328
29.3.19 ESY 1329
29.3.20 Expert Choice 1329
29.3.21 FGM 1329
29.3.22 FuzzME 1330
29.3.23 GeNIe & SMILE
29.3.24 GUIMOO 1330
29.3.25 HIPRE 3+ 1331
29.3.26 HiPriority 1331
29.3.27 HIVIEW3 1331
29.3.28 IDS Multicriteria Assessor (IDS Version 2.1) 1332
29.3.29 IND-NIMBUS 1332
29.3.30 INPRE and ComPAIRS 1333
29.3.31 IRIS 1333
29.3.32 iMOLPe 1334
29.3.33 interalg 1334
29.3.34 iSight 1334
29.3.35 JAMM 1335
29.3.36 Logical Decisions 1335
29.3.37 MakeItRational 1335
29.3.38 Markex (Market Expert) 1336
29.3.39 MindDecider 1336
29.3.40 MINORA 1336
29.3.41 M-MACBETH and WISED 1337
29.3.42 modeFrontier 1337
29.3.43 MOIRA and MOIRA Plus 1338
29.3.44 NAIADE 1338
29.3.45 OnBalance 1338
29.3.46 Optimus 1339
29.3.47 ParadisEO-MOEO 1339
29.3.48 Pareto Front Viewer 1339
29.3.49 Prime Decisions 1340
29.3.50 Priority Mapper 1340
29.3.51 Prism's Group Decision Support System 1340
29.3.52 PROBE 1341
29.3.53 RGDB 1341
29.3.54 RICH Decisions 1342
29.3.55 Rubis (Plug-in) 1342
29.3.56 SANNA 2009 1342
29.3.57 MC-SDSS for ArcGIS 1343
29.3.58 SOLVEX 1343
29.3.59 TransparentChoice 1343
29.3.60 Triptych 1343
29.3.61 TRIMAP 1344
29.3.62 UTA Plus 1344
29.3.63 Very Good Choice 1344
29.3.64 VIP Analysis 1345
29.3.65 Visual Market/2 1345
29.3.66 VISA 1345
29.3.67 VisualUTA 1346
29.3.68 WINGDSS 1346
29.3.69 WINPRE 1346
29.4 Concluding Remarks 1347
References 1348
Index 1352