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MULTIPLE CRITERIA DECISION ANALYSIS: STATE OF THE ART SURVEYS Recent titles in the INTERNATIONAL SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE Frederick S. Hillier, Series Editor, Stanford University Zhu, J. / QUANTITATIVE MODELS FOR PERFORMANCE EVALUATION AND BENCHMARKING Ehrgott, M. & Gandibleux, X. / MULTIPLE CRITERIA OPTIMIZATION: State of the Art Annotated Bibliographical Surveys Bienstock, D. / Potential Function Methods for Approx. Solving Linear Programming Problems Matsatsinis, N.F. & Siskos, Y. / INTELLIGENT SUPPORT SYSTEMS FOR MARKETING DECISIONS Alpern, S. & Gal, S. / THE THEORY OF SEARCH GAMES AND RENDEZVOUS Hall, R.W./ HANDBOOK OF TRANSPORTATION SCIENCE Ed. Glover, F. & Kochenberger, G.A. / HANDBOOK OF METAHEURISTICS Graves, S.B. & Ringuest, J.L. / MODELS AND METHODS FOR PROJECT SELECTION: Concepts from Management Science, Finance and Information Technology Hassin, R. & Haviv, M./ TO QUEUE OR NOT TO QUEUE: Equilibrium Behavior in Queueing Systems Gershwin, S.B. et al/ANALYSIS & MODELING OF MANUFACTURING SYSTEMS Maros, I./ COMPUTATIONAL TECHNIQUES OF THE SIMPLEX METHOD Harrison, T., Lee, H. & Neale, J./ THE PRACTICE OF SUPPLY CHAIN MANAGEMENT: Where Theory And Application Converge Shanthikumar, J.G., Yao, D. & Zijm, W.H./ STOCHASTIC MODELING AND OPTIMIZATION OF MANUFACTURING SYSTEMS AND SUPPLY CHAINS Nabrzyski, J., Schopf, J.M., GRID RESOURCE MANAGEMENT: State of the Art and Future Trends Thissen, W.A.H. & Herder, P.M./ CRITICAL INFRASTRUCTURES: State of the Art in Research and Application Carlsson, C., Fedrizzi, M., & Fullér, R./ FUZZY LOGIC IN MANAGEMENT Soyer, R., Mazzuchi, T.A., & Singpurwalla, N.D./ MATHEMATICAL RELIABILITY: An Expository Perspective Chakravarty, A.K. & Eliashberg, J./ MANAGING BUSINESS INTERFACES: Marketing, Engineering, and Manufacturing Perspectives Talluri, K. & van Ryzin, G./ THE THEORY AND PRACTICE OF REVENUE MANAGEMENT Kavadias, S. & Loch, C.H./ PROJECT SELECTION UNDER UNCERTAINTY: Dynamically Allocating Resources to Maximize Value Brandeau, M.L., Sainfort, F., Pierskalla, W.P./ OPERATIONS RESEARCH AND HEALTH CARE: A Handbook of Methods and Applications Cooper, W.W., Seiford, L.M., Zhu, J./ HANDBOOK OF DATA ENVELOPMENT ANALYSIS: Models and Methods Luenberger, D.G./ LINEAR AND NONLINEAR PROGRAMMING, Ed. Sherbrooke, C.C./ OPTIMAL INVENTORY MODELING OF SYSTEMS: Multi-Echelon Techniques, Second Edition Chu, S.-C., Leung, L.C., Hui, Y. V., Cheung, W./ 4th PARTY CYBER LOGISTICS FOR AIR CARGO Simchi-Levi, Wu, Shen/ HANDBOOK OF QUANTITATIVE SUPPLY CHAIN ANALYSIS: Modeling in the E-Business Era Gass, S.I. & Assad, A.A./ AN ANNOTATED TIMELINE OF OPERATIONS RESEARCH: An Informal History Greenberg, H.J./ TUTORIALS ON EMERGING METHODOLOGIES AND APPLICATIONS IN OPERATIONS RESEARCH Weber, C./ UNCERTAINTY IN THE ELECTRIC POWER INDUSTRY: Methods and Models for Decision Support * A list of the early publications in the series is at the end of the book * MULTIPLE CRITERIA DECISION ANALYSIS: STATE OF THE ART SURVEYS Edited by JOSÉ FIGUEIRA University of Coimbra SALVATORE GRECO University of Catania MATTHIAS EHRGOTT University of Auckland Springer eBook ISBN: Print ISBN: 0-387-23081-5 0-387-23067-X ©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.kluweronline.com http://www.springeronline.com Contents List of Figures xiii List of Tables xvii Introduction José Figueira, Salvatore Greco, Matthias Ehrgott 1. Human Reflection about Decision 2. Technical Reflection about Decision: MCDA Researchers before MCDA 3. The Reasons for this Collection of State-of-the-Art Surveys 4. A Guided Tour of the Book 5. Acknowledgment to the Referees References xxi Part I xxii xxiv xxv xxxiv xxxiv An Overview of MCDA Techniques Today 1 Paradigms and Challenges Bernard Roy 1. What Are the Expectations that Multicriteria Decision Aiding (MCDA) Responds to? 2. Three Basic Concepts 3. How to Take Into Account Imperfect Knowledge? 4. An Operational Point of View 5. Conclusion References Part II xxi 3 4 7 12 14 17 18 Foundations of MCDA 2 Preference Modelling Meltem Öztürk, Alexis Tsoukiàs, Philippe Vincke 1. Introduction 2. Purpose Nature of Information 3. Notation and Basic Definitions 4. Languages 5. 27 28 28 30 32 33 MULTIPLE CRITERIA DECISION ANALYSIS vi 6. Preference Structures Domains and Numerical Representations 7. 8. Logic of Preferences 9. Conclusion References 3 Conjoint measurement tools for MCDM Denis Bouyssou, Marc Pirlot Introduction and Motivation 1. Definitions and Notation 2. The Additive Value Model in the “Rich” Case 3. The Additive Value Model in the “Finite” Case 4. Extensions 5. References Part III 39 48 56 59 60 73 74 89 92 102 112 119 Outranking Methods 4 ELECTRE Methods JoséFigueira, Vincent Mousseau, Bernard Roy Introduction: A Brief History 1. Main Features of ELECTRE Methods 2. A Short Description of ELECTRE Methods 3. 4. Recent Developments and Future Issues 5. Software and Applications 6. Conclusion References 5 PROMETHEE Methods Jean-Pierre Brans, Bertrand Mareschal 1. History 2. Multicriteria Problems 3. The PROMETHEE Preference Modelling Information 4. The PROMETHEE I and II Rankings 5. The GAIA Visual Interactive Module 6. The PROMETHEE VI Sensitivity Tool (The “Human Brain”) PROMETHEE V: MCDA under Constraints 7. 8. The PROMETHEE GDSS Procedure The DECISION LAB Software 9. References 6 Other Outranking Approaches Jean-Marc Martel, Benedetto Matarazzo 1. Introduction 2. Other Outranking Methods 133 134 136 139 149 151 153 153 163 164 164 168 171 175 181 182 183 186 189 197 198 198 vii Contents 3. Pairwise Criterion Comparison Approach 4. One Outranking Method for Stochastic Data Conclusions 5. References Part IV 221 254 259 260 Multiattribute Utility and Value Theories 7 MAUT – Multiattribute Utility Theory James S. Dyer 1. Introduction 2. Preference Representations Under Certainty and Under Risk 3. Ordinal Multiattribute Preference Functions for the Case of Certainty 4. Cardinal Multiattribute Preference Functions for the Case of Risk 5. Measurable Multiattribute Preference Functions for the Case of Certainty 6. The Relationships Among the Multiattribute Preference Functions 7. Concluding Remarks References 8 UTA Methods Yannis Siskos, Evangelos Grigoroudis, Nikolaos F. Matsatsinis Introduction 1. 2. The UTA Method Variants of the UTA Method 3. 4. Applications and UTA-based DSS Concluding Remarks and Future Research 5. References 9 The Analytic Hierarchy and Analytic Network Processes for the Measurement of Intangible Criteria and for Decision-Making Thomas L. Saaty 1. Introduction 2. Pairwise Comparisons; Inconsistency and the Principal Eigenvector 3. Stimulus Response and the Fundamental Scale 4. Hospice Decision 5. Rating Alternatives One at a Time in the AHP – Absolute Measurement 6. Paired Comparisons Imply Dependence 7. When is a Positive Reciprocal Matrix Consistent? 8. In the Analytic Hierarchy Process Additive Composition is Necessary 9. Benefits, Opportunities, Costs and Risks 10. On the Admission of China to the World Trade Organization (WTO) 11. The Analytic Network Process (ANP) 265 266 267 273 278 281 290 292 294 297 298 302 313 328 334 335 345 346 348 354 359 369 372 373 375 377 378 382 viii MULTIPLE CRITERIA DECISION ANALYSIS Two Examples of Estimating Market Share – The ANP with a Single Benefits Control Criterion 13. Outline of the Steps of the ANP Complex Decisions with Dependence and Feedback 14. Conclusions 15. References 12. 10 On the Mathematical Foundation of MACBETH Carlos A. Bana e Costa, Jean-Marie De Corte, Jean-Claude Vansnick 1. Introduction Previous Research and Software Evolution 2. Types of Preferential Information 3. Numerical Representation of the Preferential Information 4. Consistency – Inconsistency 5. Consistency Test for Preferential Information 6. Dealing with Inconsistency 7. 8. The MACBETH Scale 9. Discussion About a Scale MACBETH and MCDA 10. References Part V 389 400 403 405 406 409 410 412 414 415 416 417 420 432 435 437 438 Non-Classical MCDA Approaches 11 Dealing with Uncertainties in MCDA Theodor J Stewart 1. What is Uncertainty? 2. Probabilistic Models and Expected Utility Pairwise Comparisons 3. 4. Risk Measures as Surrogate Criteria 5. Scenario Planning and MCDA 6. Implications for Practice References 12 Choice, Ranking and Sorting in Fuzzy Multiple Criteria Decision Aid Patrick Meyer, Marc Roubens 1. Introduction 2. The Data Set 3. Valued Preference Relation and Outranking Relation 4. Aggregation Procedures 5. The Sorting Problem 6. The TOMASO Method 7. The Choice Problem 8. Conclusion References 445 446 450 454 457 460 466 467 471 472 474 475 478 482 483 502 503 504 Contents ix 13 Decision Rule Approach Salvatore Greco, Benedetto Matarazzo, 1. Introduction 2. Dominance-based Rough Set Approach (DRSA) to Multiple-criteria Classification Variable-Consistency Dominance-Based Rough Set Approach (VC3. DRSA) 4. Induction of Decision Rules from Rough Approximations of Upward and Downward Unions of Decision Classes Extensions of DRSA 5. 6. DRSA for Multiple-criteria Choice and Ranking 7. Conclusions References 14 Fuzzy Measures and Integrals in MCDA Michel Grabisch, Christophe Labreuche Introduction 1. Measurement Theoretic Foundations 2. Unipolar Scales 3. Bipolar Scales 4. Ordinal Scales 5. Concluding Remarks 6. References 15 Verbal Decision Analysis Helen Moshkovich, Alexander Mechitov, David Olson 1. Features of Unstructured Decision Problems 2. Main Principles of Verbal Decision Analysis Decision Methods for Multicriteria Alternatives Ranking 3. Decision Methods for Multicriteria Alternatives’ Classification 4. Place of Verbal Decision Analysis in MCDA 5. Conclusion 6. References Part VI 507 508 511 525 527 536 544 555 557 563 564 566 570 583 595 604 604 609 610 610 615 625 628 633 634 Multiobjective Mathematical Programming 16 Interactive Methods Pekka Korhonen 1. Introduction Basic Definitions and Some Theory 2. Principles for Implementing Interactive Methods 3. 4. Generating Nondominated Solutions Solving Multiple Objective Problems 5. Final Solution 6. Examples of Software Systems: VIG and VIMDA 7. 641 642 643 645 649 652 656 657 MULTIPLE CRITERIA DECISION ANALYSIS x Concluding Remarks 8. References 17 Multiobjective Programming Matthias Ehrgott, Margaret M. Wiecek Introduction 1. Problem Formulation and Solution Concepts 2. Properties of the Solution Sets 3. Conditions for Efficiency 4. Generation of the Solution Sets 5. 6. Approximation of the Pareto Set Specially Structured Problems 7. Current and Future Research Directions 8. Conclusions 9. References 18 Multiple Objective Linear Programming with Fuzzy Coefficients Masahiro Inuiguchi 1. Introduction 2. Problem Statement and Approaches Modality Constrained Programming Approach 3. 4. Modality Goal Programming 5. Modal Efficiency Approach Concluding Remarks 6. References 19 MCDM Location Problems Stefan Nickel, Justo Puerto, Antonio M. Rodríguez-Chía 1. Introduction Location Problems 2. Continuous Multicriteria Location Problems 3. Multicriteria Network Location Problems 4. Multicriteria Discrete Location Problems 5. Conclusions 6. References Part VII 661 662 667 668 669 673 675 676 692 696 707 708 708 723 724 725 731 749 754 757 757 761 762 764 767 776 783 787 787 Applications 20 Multicriteria Decision Aid/Analysis in Finance Jaap Spronk, Ralph E. Steuer, Constantin Zopounidis Introduction 1. Financial Decision Making 2. MCDA in Portfolio Decision-Making Theory 3. MCDA in Discrete Financial Decision-Making Problems 4. 799 800 801 819 835 Contents xi 5. Conclusions and Future Perspectives References 21 MCDA and Energy Planning Danae Diakoulaki, Carlos Henggeler Antunes, António Gomes Martins 1. Introduction 2. Multiobjective Programming Models for Energy Planning Energy Planning Decisions with Discrete Alternatives 3. 4. Conclusions References 22 Multicriteria Analysis in Telecommunication Network Planning and Design – Problems and Issues João Clímaco, José Craveirinha 1. Motivation 2. Overview of Current Evolutions in Telecommunication Networks and Services Multicriteria Analysis in Telecommunication Network Planning and 3. Design Review and Discussion of Applications of MA to Telecommunication 4. Network Planning Future Trends 5. References 23 Multiple Criteria Decision Analysis and Sustainable Development Giuseppe Munda The Concept of Sustainable Development 1. Measuring Sustainability: The Issue of Sustainability Assessment 2. Indexes A Defensible Axiomatic Setting for Sustainability Composite Indi3. cators Warning! Not Always Rankings Have to Be Trusted ... 4. The Issue of the “Quality of the Social Decision Processes” 5. The Issue of Consistency in Multi-Criteria Evaluation of 6. Sustainability Policies Conclusion 7. References Part VIII 848 849 859 860 863 874 890 891 899 900 900 908 912 941 944 953 954 958 963 966 971 976 980 981 MCDM Software 24 Multiple Criteria Decision Support Software H. Roland Weistroffer, Charles H. Smith, Subhash C. Narula Introduction 1. Software Overview 2. Concluding Remarks 3. 989 990 990 1009 MULTIPLE CRITERIA DECISION ANALYSIS xii References 1011 Contributing Authors 1019 Index 1035 List of Figures 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 Graphical representation of R. Matrix representation of R. Graphical representation of the semiorder. Comparing the length of two rods. Comparing the length of composite rods. Using standard sequences. Building a standard sequence on Building a standard sequence on The grid. The entire grid. The Thomsen condition. Restricted solvability on Value function when is discrete. Value function when is continuous. Inferring parameter values for ELECTRE TRI. Preference function. Valued outranking graph. The PROMETHEE outranking flows. Profile of an alternative. Projection on the GAIA plane. Alternatives and criteria in the GAIA plane. PROMETHEE II ranking. PROMETHEE decision axis and stick. Piloting the PROMETHEE decision stick. “Human Brain”. Two types of decision problems. Conflict between DM’s. Overview PROMETHEE GDSS procedure. Main window. 34 34 55 78 79 81 94 95 96 97 99 100 106 107 151 169 172 173 175 176 177 179 180 181 182 186 186 187 xiv 5.14 5.15 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 MULTIPLE CRITERIA DECISION ANALYSIS 188 PROMETHEE rankings, action profiles, GAIA plane. 188 Walking weights. 202 Set of feasible weights. 210 ORESTE flow chart. 215 Outranking graph. 228 Geometrical interpretation of preferences indices. 232 Indifference areas. 233 Indifference areas: rectangular. 234 Indifference areas: rhomboidal. 234 Indifference areas: elliptical. 236 Aggregated semiorder structure. Aggregated pseudo-order structure. 236 241 Partial profile of action 242 Partial profiles and partial broken lines of Partial frequencies of 243 Determination of a relation between the two alternatives on the basis of the values of global indices. 254 259 6.15 Partial preorder. Choice between two lotteries. 279 7.1 Additive Independence Criterion for Risk. 280 7.2 Piecewise linear approximation of 289 7.3 Piecewise linear approximation of 289 7.4 The aggregation and disaggregation paradigms in MCDA [44]. 300 8.1 301 8.2 The disaggregation-aggregation approach [96]. The normalized marginal value function. 8.3 303 8.4 Post-optimality analysis [43]. 306 Ordinal regression curve (ranking versus global value). 307 8.5 Normalized marginal value functions. 8.6 313 8.7 A non-monotonic partial utility function [18]. 316 8.8 Distributional evaluation and marginal value function. 319 Distribution of the actions and on [43]. 8.9 321 8.10 Simplified decision support process based on disaggregation approach [44]. 329 8.11 Methodological flowchart of MARKEX [63]. 332 Comparisons according to volume. 9.1 359 9.2 To choose the best hospice plan, one constructs a hierarchy modeling the benefits to the patient, to the institution, and to society. This is the benefits hierarchy of two separate hierarchies. 361 List of Figures xv To choose the best hospice plan, one constructs a hierarchy modeling the community, institutional, and societal costs. This is the costs hierarchy of two separate hierarchies. 362 Employee evaluation hierarchy. 371 9.4 Hierarchies for rating benefits, costs, opportunities, and risks. 380 9.5 9.6 Prioritizing the strategic criteria to be used in rating the BOCR. 381 384 9.7 How a hierarchy compares to a network. 9.8 The supermatrix of a network and detail of a component 385 in it. The supermatrix of a hierarchy with the resulting limit 9.9 matrix corresponding to hierarchical composition. 385 388 9.10a School choice hierarchy composition. 9.10b Supermatrix of school choice hierarchy gives same re388 sults as hierarchic composition. 9.11 The clusters and nodes of a model to estimate the relative 391 market share of Walmart, Kmart and Target. 9.12 The clusters and nodes of a model to estimate the relative market share of footware. 398 405 9.13 Hierarchy for rating benefits, opportunities, costs and risks. 417 10.1 Example of sub-type b inconsistency. 424 10.2 Example of incompatibility between (*) and (**). 431 10.3 Procedure for all cases of inconsistency. 433 10.4 Suggestion of change to resolve inconsistency. 434 10.5 Matrix of judgements and basic MACBETH scale. 435 10.6 Representations of the MACBETH scale. 10.7 “Greatest” closed intervals included in the free and de437 pendent intervals. 476 Comparing two fuzzy intervals. 12.1 493 12.2 Interpretation of the discriminant functions. 495 12.3 Representation of the students problem. 496 12.4 Classes for 497 12.5 Classes for 499 12.6 Visual representation of the classes, neighbour algorithm, 12.7 Results for the 501 13.1 Decision tree representing knowledge included from Table 13.1. 535 13.2 The hierarchy of attributes and criteria for a car classifi544 cation problem. 578 14.1 Different cases of interaction. 593 14.2 Ternary alternatives for 9.3 xvi 16.1 16.2 16.3 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 MULTIPLE CRITERIA DECISION ANALYSIS Illustrating the projection of a feasible and an infeasible aspiration level point onto the nondominated surface. An example of the Pareto Race screen. The screen of the computer-graphics interface in VIMDA. Fuzzy inequality and equality relations. L-R fuzzy number Possibility and necessity measures. Differences among six extended fuzzy relations. Relations among VWF, WF1, WF2, MF1, MF2, SF1, SF2 and VSF. Relations among VWF, MF1, MF2 and VSF. Fuzzy max Symmetric triangular fuzzy number defined by and 18.10 Two differences between fuzzy numbers and 20.1 The neo-classical view on the objective of the firm. 20.2 Situations leading to MCDA in the firm. 20.3 A bird’s-eye view of the framework. 20.4 Feasible regions Z of (MC-Unrestr) and (MC-Bounds) for the same eight securities. 20.5 Continuous, bullet-shaped, and unbounded feasible region Z created by securities A, B and C. 20.6 Nondominated frontiers as a function of changes in the value of upper bound parameter 20.7 An ellipsoidal feasible region projected onto two dimensional risk-return space 21.1 Typical hierarchical structure of criteria used in energy planning. 21.2 MCDA methods in energy planning applications. 22.1 State transition diagram. 22.2 Example of priority regions. 23.1 A systemic vision of sustainability issues. 23.2 Impact matrix for the 4 chosen cities according to the selected indicators. 23.3 The ideal problem structuring in SMCE. 651 660 661 726 728 731 736 739 740 746 746 748 751 811 812 817 826 827 830 834 882 888 914 930 956 967 975 List of Tables 2.1 2.2 3.1 5.1 5.2 5.3 5.4 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Principal t-norms and t-conorms. Various with Evaluation of the 5 offices on the 5 attributes. Evaluation table. Weights of relative importance. Types of generalised criteria Preference function). Single criterion net flows. Rank evaluation of alternatives (impact matrix). The concordance/discordance indices. Concordance matrix. Rank evaluation of alternatives (impact matrix). Regime matrix. Position-matrix. City-block. Preference matrix for a criterion with ordinal evaluation. Preference matrix for a criterion (Max) with evaluation on a ratio scale. 6.10 Preference importance table for 6.11 Combined preferences with weights variable. 6.12 Evaluation of alternatives*. 6.13 Criteria and (ordinal scales). 6.14 Criterion (ordinal scale). 6.15 Criterion (ratio scale MIN). 6.16 Preference structure of weights. and 6.17 Pairwise comparison between 6.18 Axiomatic system of MAPPAC basic indices. 6.19 Preference indices. 6.20 Table of observed stochastic dominances. 6.21 Explicable concordances indices. 37 55 84 165 168 170 176 201 201 203 204 204 208 208 211 211 212 212 213 213 214 214 214 214 227 228 258 259 xviii 8.1 8.2 8.3 8.4 8.5 8.6 8.7 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 MULTIPLE CRITERIA DECISION ANALYSIS Criteria values and ranking of the DM. Marginal value functions (initial solution). Linear programming formulation (post-optimality analysis). Post-optimality analysis and final solution. Marginal value functions (final solution). LP size of UTA models. Indicative applications of the UTA methods. The fundamental scale of absolute numbers. Which drink is consumed more in the U.S.? An example of estimation using judgments. The entries in this matrix respond to the question: Which criterion is more important with respect to choosing the best hospice alternative and how strongly? The entries in this matrix respond to the question: Which subcriterion yields the greater benefit with respect to institutional benefits and how strongly? The entries in this matrix respond to the question: Which model yields the greater benefit with respect to direct care and how strongly? The entries in this matrix respond to the question: Which criterion is a greater determinant of cost with respect to the care method and how strongly? The entries in this matrix respond to the question: Which criterion incurs greater institutional costs and how strongly? The entries in this matrix respond to the question: Which model incurs greater cost with respect to institutional costs for recruiting staff and how strongly? Synthesis (P=Priorities, M=Model). (continued) Ranking intensities. Ranking alternatives. Random index. Calculating returns arithmetically. Normalized criteria weights and normalized alternative weights from measurements in the same scale (additive synthesis). Priority Ratings for the Merits: Benefits, Costs, Opportunities, and Risks. Intensities: Very High (0.42), High (0.26), Medium (0.16), Low (0.1), Very Low (0.06). Four methods of synthesizing BOCR using the ideal mode. 309 311 312 312 313 317 335 356 358 364 365 365 365 366 366 367 368 371 372 374 376 377 381 382 List of Tables xix 9.17 9.18 9.19 9.19 9.20 9.21 9.21 9.22 9.23 389 389 392 393 394 395 396 397 9.24 9.25 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 15.1 15.2 15.3 The supermatrix. The limit supermatrix. The unweighted supermatrix. (continued) The cluster matrix. The weighted supermatrix. (continued) The synthesized results for the alternatives. Footwear actual statistics and model results along with the compatibility index. Priority ratings for the merits: Benefits, opportunities, costs and risks. Overall syntheses of the alternatives. Number of alternatives per evaluation level. Score of each of the evaluation levels. Profiles of the students. Importance indexes for the students problem. Global accuracy. Per class accuracy: Precise assignments. Assignments of the elements to intervals of classes. Importance indexes. Global and weighted accuracy in %. Data table presenting examples of comprehensive evaluations of students. Quality of classification and Shapley value for classification Cl and set of criteria P. Evaluations of new students. Evaluations of new students. Information table of the illustrative example. Students with interval evaluations. Example of missing values in the evaluation of students. Substitution of missing values in the evaluation of students. Decision table with reference objects. A fragment of Ranking of warehouses for sale by decision rules and the Net Flow Score procedure. Criteria for applicant evaluation. Comparison of hypothetical alternatives. An Example of a joint ordinal scale. 401 405 406 485 485 495 498 498 499 499 500 501 511 522 532 533 536 538 541 542 551 552 554 616 618 619
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