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 Human Reflection about Decision Technical Reflection about Decision: MCDA Researchers before MCDA The Reasons for this Collection of State-of-the-Art Surveys A Guided Tour of the Book Acknowledgment to the Referees References xxi Part I xxii xxiv xxv xxxiv xxxiv An Overview of MCDA Techniques Today Paradigms and Challenges Bernard Roy What Are the Expectations that Multicriteria Decision Aiding (MCDA) Responds to? Three Basic Concepts How to Take Into Account Imperfect Knowledge? An Operational Point of View Conclusion References Part II xxi 12 14 17 18 Foundations of MCDA Preference Modelling Meltem Öztürk, Alexis Tsoukiàs, Philippe Vincke Introduction Purpose Nature of Information Notation and Basic Definitions Languages 27 28 28 30 32 33 MULTIPLE CRITERIA DECISION ANALYSIS vi Preference Structures Domains and Numerical Representations Logic of Preferences Conclusion References Conjoint measurement tools for MCDM Denis Bouyssou, Marc Pirlot Introduction and Motivation Definitions and Notation The Additive Value Model in the “Rich” Case The Additive Value Model in the “Finite” Case Extensions References Part III 39 48 56 59 60 73 74 89 92 102 112 119 Outranking Methods ELECTRE Methods JoséFigueira, Vincent Mousseau, Bernard Roy Introduction: A Brief History Main Features of ELECTRE Methods A Short Description of ELECTRE Methods Recent Developments and Future Issues Software and Applications Conclusion References PROMETHEE Methods Jean-Pierre Brans, Bertrand Mareschal History Multicriteria Problems The PROMETHEE Preference Modelling Information The PROMETHEE I and II Rankings The GAIA Visual Interactive Module The PROMETHEE VI Sensitivity Tool (The “Human Brain”) PROMETHEE V: MCDA under Constraints The PROMETHEE GDSS Procedure The DECISION LAB Software References Other Outranking Approaches Jean-Marc Martel, Benedetto Matarazzo Introduction 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 Pairwise Criterion Comparison Approach One Outranking Method for Stochastic Data Conclusions References Part IV 221 254 259 260 Multiattribute Utility and Value Theories MAUT – Multiattribute Utility Theory James S Dyer Introduction Preference Representations Under Certainty and Under Risk Ordinal Multiattribute Preference Functions for the Case of Certainty Cardinal Multiattribute Preference Functions for the Case of Risk Measurable Multiattribute Preference Functions for the Case of Certainty The Relationships Among the Multiattribute Preference Functions Concluding Remarks References UTA Methods Yannis Siskos, Evangelos Grigoroudis, Nikolaos F Matsatsinis Introduction The UTA Method Variants of the UTA Method Applications and UTA-based DSS Concluding Remarks and Future Research References The Analytic Hierarchy and Analytic Network Processes for the Measurement of Intangible Criteria and for Decision-Making Thomas L Saaty Introduction Pairwise Comparisons; Inconsistency and the Principal Eigenvector Stimulus Response and the Fundamental Scale Hospice Decision Rating Alternatives One at a Time in the AHP – Absolute Measurement Paired Comparisons Imply Dependence When is a Positive Reciprocal Matrix Consistent? In the Analytic Hierarchy Process Additive Composition is Necessary 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 Introduction Previous Research and Software Evolution Types of Preferential Information Numerical Representation of the Preferential Information Consistency – Inconsistency Consistency Test for Preferential Information Dealing with Inconsistency The MACBETH Scale 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 What is Uncertainty? Probabilistic Models and Expected Utility Pairwise Comparisons Risk Measures as Surrogate Criteria Scenario Planning and MCDA Implications for Practice References 12 Choice, Ranking and Sorting in Fuzzy Multiple Criteria Decision Aid Patrick Meyer, Marc Roubens Introduction The Data Set Valued Preference Relation and Outranking Relation Aggregation Procedures The Sorting Problem The TOMASO Method The Choice Problem 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, Introduction Dominance-based Rough Set Approach (DRSA) to Multiple-criteria Classification Variable-Consistency Dominance-Based Rough Set Approach (VC3 DRSA) Induction of Decision Rules from Rough Approximations of Upward and Downward Unions of Decision Classes Extensions of DRSA DRSA for Multiple-criteria Choice and Ranking Conclusions References 14 Fuzzy Measures and Integrals in MCDA Michel Grabisch, Christophe Labreuche Introduction Measurement Theoretic Foundations Unipolar Scales Bipolar Scales Ordinal Scales Concluding Remarks References 15 Verbal Decision Analysis Helen Moshkovich, Alexander Mechitov, David Olson Features of Unstructured Decision Problems Main Principles of Verbal Decision Analysis Decision Methods for Multicriteria Alternatives Ranking Decision Methods for Multicriteria Alternatives’ Classification Place of Verbal Decision Analysis in MCDA Conclusion 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 Introduction Basic Definitions and Some Theory Principles for Implementing Interactive Methods Generating Nondominated Solutions Solving Multiple Objective Problems Final Solution Examples of Software Systems: VIG and VIMDA 641 642 643 645 649 652 656 657 This page intentionally left blank Index solution, 676 approach, 696 approach, 683 preference structure, 40 preference structures, 43 ACADEA, 1010, 996 Achievement function, 686 Achievement function approach, 686 Achievement function increasing, 686 strictly increasing, 686 strongly increasing, 686 Achievement scalarizing function, 654 Action, ADBASE, 1004, 653, 996 Additive difference model, 116 Additive independence, 456 Additive nontransitive model, 116 Additive value function, 304–305 Additive value functions, 304 Admissible transformation, 571 AGAP, 1008, 996 AgentAllocator, 1010, 996 Aggregated index, 225 Aggregated preference index, 170, 172 Aggregation, 440, 482 Aggregation function, 568 Aggregation procedures, 14 AHP, 269, 288, 294, 650, 892 Alternative, Ambiguity, 520, 529 Ambiguous assignment, 493 Analytic Hierarchy Process, 347, 349, 377, 385, 650, 999 Analytic Network Process, 347, 349, 379, 385 Applications, 440 ARGOS, 1008–1009, 996 ARGUS, 210 Arrow K.J., 968 Aspiration level, 653–654 Attribute essential, 99 importance, 97 AutoMan, 1010, 996 Axiom Archimedean, 94, 99 restricted solvability, 99 Thomsen, 98 Axiomatic characterization, 776–777 Balance criterion, 786 Balance number, 695 apportioned, 695 Balance point, 694 Balance set approach, 694 BANKADVISOR, 1010, 996 Bankruptcy risk, 848 Banville C., 976 Barbier E.B., 958 Basic index, 226 Behavioral convergence, 660 Benefits, 347, 350, 362, 365–366, 368, 379–381, 383, 391, 402, 404 Benson approach, 684 Bi-capacity, 590 bi-capacity, 598 bi-capacity, 598 additive, 590 conjugate, 590 decomposable bi-capacity, 598 identification of bi-capacity, 598 symmetric, 590 Bicriteria problems, 775, 782, 791 Binary alternative, 578 Binary preference relation, 270 Binary relation, 32, 89, 673–674 asymmetry, 33, 38 complete, 271 completeness, 89 dominance, 118 equivalence, 89 interval order, 114 irreflexivity, 33, 38 numerical representation, 74 properties, 89 reflexive, 270 reflexivity, 33, 38 1036 MULTIPLE CRITERIA DECISION ANALYSIS semiorder, 114 symmetry, 33, 38 trace, 118 transitive, 270 transitivity, 33, 38, 41, 89 weak order, 89 Bisection, 102 Blumenthal A., 350–351 Boolean polynomial, 599 Bordley R., 292 Boundary, 524 C-K-Y-L ranking procedure, 968 Camerer C., 274 Capacity, 576 Capacity expansion, 868 Capacity 583 584 2-additive capacity, 584 additive, 576 bipolar capacity, 590 conjugate, 576 identification of capacity, 584 symmetric, 576 Capital budgeting, 806 Cardinal scale, 10 CASTART, 1011,996 Causal mapping, 995 CAUSE, 883 Cent-dian problem, 769, 776, 779, 782–784, 786, 789 Center problem, 768–769, 775, 777, 779, 782–783, 766 Cesaro sum, 354 Cesaro summability, 354, 388, 392 CGX, 1011,996 Characteristic relation, 39 Charnes A., 289–290 China, 380–381, 384, 405, 408 Choice problem, 476, 483 Choice problematic, 11, 140 Choquet integral, 485, 576, 584, 591 asymmetric, 589 symmetric, 588 Classification, 29 Clustering, 359 Co-evolutionary paradigm, 959 Coalition of criteria, 576 Commensurateness, 569, 574 Comonotonic additivity, 577 Comonotonicity, 577 Compact polyhedron, 701 Comparisons, 349, 360, 365, 371, 374, 401, 404 pairwise, 347, 349–350, 356, 385 redundant, 365 Compensability analysis, 252 Complementary criteria, 581 Complex decision, 405, 407 Complexity, 616, 632, 634, 637, 775, 778, 784, 787 Composite indicators, 964 Composition, 387–388 additive, 377 hierarchic, 386, 389 Concatenation operation, 79 Concordance, 137 Concordance condition, 218 Conditional problems, 791 Conditions for efficiency, 679 Condorcet Marquis de, 967 Cone, 673–674 polyhedral, 675 Conflict analysis, 237 Connectedness, 678 Consistency, 418 Continuous multiple criteria problem, 647 Convex hull, 771–774, 778 Convex MOP, 678 Cooper W.W., 289–290 Core, 525 Corporate financing, 806 Cost-benefit analysis, 614, 635, 976 Costs, 347, 350, 362, 379, 381, 383, 402, 404 Country risk assessment, 852 Credit risk, 848 Criterion space, 647, 825 Criterion vector, 825 Criterium Decision Plus, 996–997 Critical regime, 204 CTLite, 1009, 996 Cumulative prospect theory, 589 ordinal, 603 Customer satisfaction, 328 Cycle, 385, 404 CYCLE, 632 DAM, 996–997 Data mining, 335 Data table, 515 DEA, 790 Debreu G., 272, 276 Decision-aid, 300 Decision-making, 348–349, 355, 384, 387 Decision aiding, 28–29, 45, 52 Decision analysis, 74, 615, 617, 621, 635 Decision axis, 179–181, 186, 188 Decision Explorer, 995–996 Decision Lab, 996–997 Decision problematics, 839 Decision rules, 531, 518, 552 Decision space, 673, 825 Decision stick, 179–181 Decision support, 330 Decision support systems, 330 Decision trees, 538 Decision under uncertainty, 75–76, 546 1037 Index Decision variables, 648 Decomposable model, 570 Decomposable nontransitive model, 117 Deconvolution, 754 DEFINITE, 207 Degeneracy, 701 Degree of assignment, 493 Degrees, Deontic logic, 56 Dependence, 347, 349, 368, 374, 379, 384–386, 403, 405 Description problematic, 11 DIDASN++, 1011, 996 Difference of attractiveness, 412 DIMITRA, 1011, 996 Direct rating, 102 Disaggregation-aggregation approach, 301–302 Discrete alternatives, 878 Discrete MOP, 707 Discrete multiple criteria problem, 647 Discriminant analysis, 304, 322 Discrimination thresholds, 139 Dispatch, 873 Dispatching, 882 Distance function, 685 Distribution network planning, 871 Distributive mode, 368, 370–371, 379 Diversification, 807 Dominance-based Rough Set Approach (DRSA), 514 Dominance, 351, 353–355, 357, 365, 388, 391, 531, 540, 617, 621–623, 627, 630–633, 673 Dominance principle, 514, 517 Duality, 780, 784 Dyer J.S., 273, 275, 288, 293 Dynamic routing, 935 Ecological distribution, 959 Edwards W., 273–274, 286 Efficient, 649 Efficient basis, 702 Efficient extreme point, 702–703 Efficient face, 701 maximal, 701, 706 Efficient frontier, 649 Efficient nonbasic variable, 702 test for, 703 Efficient pivot, 703 Efficient set, 678, 826 connectedness, 678 Efficient solution, 674–675, 729 approximately, 676 condition for, 679 existence, 677 geometric characterization, 695 properly, 676, 679 strictly, 675 weakly, 675 Efficient solutions, 825 Eigenvalue, 352–356, 376 Eigenvector, 354, 377, 382, 385, 388 principal, 350–351, 354–355, 375 Ekeland’s variational principle, 780 Elastic constraint approach, 683 ELECCALC, 996, 998 ELECTRE, 1002, 1007–1008, 1013, 633, 996, 998 ELECTRE I, 140 ELECTRE II, 144 ELECTRE III, 145, 893 ELECTRE IS, 142 ELECTRE Iv, 142 ELECTRE IV, 147 ELECTRE methods, 841 history, 134 ELECTRE TRI, 148, 893 Electrical Power Districting, 1011 Elicitation, 150 Ellsberg D., 275 Employees, 372 Energy, 864 Energy planning, 865 Energy plans, 881 Energy projects, 881 Enumeration procedures, 789 Equitability approach, 693 Equity, 996, 998 Equivalence relation, 33, 38 ESY, 1011, 996 EVAMIX, 215 Even swaps, 83 Evolutionary algorithm, 940 Evolutions in telecommunications, 904 Expected utility, 455 Expert Choice, 996, 999 Expert system, 1011–1012 Exploitation procedure, 16 Extended peer-communities, 976 Extension principle, 732 Extensive facility, 784 External uncertainties, 889 Family of criteria, 10 Farquhar P.H., 273, 282 Feasibility degree, 741 Feasible Goals Method, 1004, 1013, 996 Feasible region, 647 Feasible set, 648 Feasible set in criterion space, 1004 Feedback, 347, 349, 384–385, 403, 405 Feinberg A., 275 Ferguson R.O., 289 Finance, 806 Financial decision making, 805 Financial investment, 807 Financial management, 335 FINCLAS, 1011, 996 1038 MULTIPLE CRITERIA DECISION ANALYSIS FINEVA, 1012, 996 Finite dominating set, 783 Fishburn P.C., 271, 273–274,289 Flexible programming problem, 730 Four-valued outranking, 554 Fractile optimization model, 744 Free search, 659 Fritz-John necessary conditions, 679 Future trends, 945 Fuzzy dominance, 542 Fuzzy equality relation, 730 Fuzzy inequality relation, 730 Fuzzy integral Choquet, 114 Sugeno, 114 Fuzzy integral, 330 Fuzzy interval, 478 Fuzzy linear function, 732 Fuzzy max, 749 Fuzzy measure, 576 fuzzy measure, 583 fuzzy measure, 584 2-additive fuzzy measure, 584 decomposable fuzzy measure, 583 non-monotonic fuzzy measure, 576 Fuzzy multiobjective programming, 463 Fuzzy number, 731 Fuzzy orders, 45 fuzzy interval order, 45 fuzzy partial order, 45 fuzzy partial preorder, 46 fuzzy semi-order, 45 fuzzy total order, 45 fuzzy weak order, 45 Fuzzy ordinal value, 478 Fuzzy preferences, 558 Fuzzy sets, 35, 451, 453, 888 De Morgan triplets, 37–38, 46 t-conorm, 36, 45 t-norm, 36, 45 Fuzzy unceratinty, 982 GAIA, 997 GAIA plane, 176–181, 185–186, 188 Game, 576 bi-cooperative game, 590 ternary voting game, 589 Gauge-based approach, 690 inner, 690 outer, 690 Gauge, 771, 773, 778-779 GDF-method, 651 Generalised criterion-preference function, 170, 176, 185 Generalized center problem, 782 Genetic algorithm, 1005–1006, 1011 Geoffrion A., 275, 292 Geometrical descriptions, 774, 776–777, 779, 789 Global frequency, 244 Global preference, 302, 315, 318, 332 Global preference index, 229 Global value function, 305 GMCR, 1009, 996 Goal-attainment approach, 688 Goal, 350, 353, 369, 381, 384, 386, 389 Goal programming, 289, 463, 650, 687, 788 Archimedean, 650 Gorman W.M., 276 Gradient method, 377 Graphical representation, 33, 54 Greedy algorithm, 699 Green accounting, 963 Guddat et al approach, 682 R.P., 273 R.P., 288 Hadamard product, 392, 399 Hard constraint, 729 Harker P., 377 Heuristics, 698, 700 Hierarchical decision problem, 547 Hierarchy, 359, 362, 366–367, 371, 379, 382, 384–386, 388–389, 404 HiPriority, 1008, 996 HIVIEW, 996, 999 Homogeneous elements, 350, 359 Hospice decision, 361 Human judgment, 614–616 Ideal mode, 368, 370–371, 373, 382 Ideal point, 676 Identification of capacities, 606 IDRA, 248 Ignizio J.P., 289 Ijiri J., 289 Imperfect knowledge, 12 Implicit value function, 651 Importance, 527 Importance index, 580, 596 Shapley, 581 Imprecision, 451 Incomparability, 29, 42–44, 46, 165–167, 174 Incompatibility, 426 Inconsistency, 350–351, 354–355, 365–366, 377, 415, 418, 426 Independence, 90 weak, 90, 113 Indifference, 29, 31, 40, 45–46, 49, 165, 170, 174 Indifference area, 233 Indifference nontransitive, 114 Indiscernibility, 540 Indiscernibility principle, 513 Indiscernibility relation, 847 Induction, 512, 535 Inequality measures, 76 1039 Index Information, 28, 30, 38, 43–44, 59 Intangible, 347, 349–350, 362, 370 Integer programming, 787 Intelligent agents, 334 Intensity, 351, 353, 372, 384 Intensity level, 352 Interaction, 527 Interaction between criteria, 575, 569, 581–582, 596 Interaction index, 582, 596 Interaction transform, 583 Interactive, 656 Interactive algorithms, 788 Interactive multicriteria, 920 Interior point methods, 705 Internal uncertainties, 887 Intertemporal decision, 76 Interval criterion weights, 651 Interval orders, 542 Interval scale, 10 Intervals, 414, 438 Investment appraisal, 851 INVEX, 1012, 996 IRIS, 1007, 996 Jia J., 275 Joint Gains, 1009, 996 Joint Ordinal Scale (JOS), 619, 622–625, 627–628, 637 Judgement reciprocal, 350 Judgement, 414 Judgment, 347, 349, 351, 353, 356–357, 365, 372, 375, 377, 382, 384, 389, 402, 405 absolute, 350–351 comparative, 350–351 inconsistent, 355, 365 relative, 351 Judgment, 353 K-shortest path, 931 Kahneman D., 274 Kamenetzky R.D., 273, 288 Keeney R.L., 268–269, 274, 277, 280, 282, 286, 288, 295 Keller L.R., 273, 275 Kemeny J., 967 Kendall’s 316 Kenney R.L., 277 Kirkwood C.W., 273, 282, 286, 292 Knowledge, 517 certain, 533 doubtful, 533 possible, 533 Krantz D.H., 280 Kreps D.M., 271 Krzysztofowicz R., 275 Kuhn-Tucker constraint qualification, 679–680 Kuhn-Tucker necessary conditions, 679 Kuhn-Tucker sufficient conditions, 680 L-good choice, 506 L-R fuzzy number, 732 LBS, 652 Least squares, 355 logarithmic, 355 Level curve, 695 Level set, 695 Level set approach, 694 Level set strict, 695 Levenglick A., 967 Lexicographic approach, 691 Lexicographic max-ordering approach, 693 Lexicographic order, 289 Lexicographical, 617 Light beam search, 652 Limit supermatrix, 388–389, 391–392, 399 Limits, 438 Linear programming, 917 Load management, 875 Local search, 699 Location problems, 791, 765, 768–769, 780 continuous, 771 discrete, 787–788 network, 781 networks, 787 Logarithmic response, 356 Logic of preferences, 56 Logical Decisions, 996, 999 Logics, 33 classical logic, 34 four-valued logics, 39 fuzzy sets, 35, 45 multivalued logics, 31, 46, 39 nonclassical logics, 31, 39 Lottery, 273 Lower approximation, 521–522, 551 Luce R.D., 276 M&P, 1000 MACBETH, 1000, 413, 440, 573, 650, 996, 999 MACBETH method, 110 MACBETH scale, 435 MACBETH software, 414 MacModel, 1000, 996 MAPPAC, 226 Marginal analysis, 371 Marginal value functions, 306 Market portfolio, 837 Market share, 391–392, 396, 399, 402, 405–406 MARKEX, 1012, 996 Martel and Zaras’ method, 256 Martinez-Alier J., 959 Matrix representation, 33 MAUT, 633–636, 650 Max-ordering approach, 692–693 1040 MCDM, 75 Measuarble multiattribute value function, 283 Measurable multiattribute preference function additive, 284 multiplicative, 284 Measurable multiattribute value function difference consistent, 283 weak difference independence, 283 Measureable multiattribute preference function difference independence, 285 Measureable multiattribute value function assessment, 286 Measurement, 347–349, 379, 570 absolute, 371 conjoint, 73 difference measurement, 571 extensive, 82 inequality, 76 interval scale, 97, 101 length, 77 ordinal, 79 ordinal measurement, 570 relative, 349, 359, 375 scale of, 370 uniqueness, 76 Medi-center problem, 782 Median problem, 766, 768, 772–773, 776–779, 782, 786, 789 MEDIATOR, 1009–1010, 996 MEDICS, 1012, 996 MELCHIOR, 220 Mergers and acquisitions, 852 Merits, 350, 379, 384,404–406 Metaheuristics, 699–700 local-search-based, 699 population-based, 700 Method outranking, 118 MHDIS method, 846 Michnik J., 289 MICROQUALIFLEX, 201 MIIDAS, 1000, 996 MINORA, 1000, 996 Missing values, 545 Modality constrained programming, 735 Modality goal programming, 753 Modality optimization model, 746 Modernization planning, 916 Modified Pascoletti and Serafini approach, 689 Modulus, 389 MOFP, 710 MOIP, 708 MOIRA, 1012, 996 MOLP, 649 MOMHLib++, 1005, 996 Monotonicity, 576 Monotonicity constraints, 309 MULTIPLE CRITERIA DECISION ANALYSIS Monotonicity of preferences, 307, 317 MOP, 673 convex, 678 discrete, 707 fractional, 710 linear, 700 piecewise linear, 710 quadratic, 710 stability, 677 Morgenstern O., 269, 273 Most preferred solution, 649 MOZOP, 708 MPS, 649 Multiattribute, 944 Multiattribute preference function, 275 conjoint measurement, 277 mutual preference independence, 276 ordinal additive value function, 277 assessment, 277 preference independence, 276 additive ordinal multiattribute value function, 276 Multiattribute utility function additive, 281 additive independence, 282 assessment, 282 multplicative, 281 utility independence, 280 Multiattribute utility theory, 455, 843 Multicriteria analysis, 912 Multicriteria decision support systems, 853 Multicriteria routing, 926 Multicriteria Simplex methods, 702–703 Multifacility, 766, 773, 776, 791 MultiGen, 1005, 996 Multigraded dominance, 550 Multiobjective linear programming, 649, 700 Multiobjective optimization, 326 Multiobjective programming, 672, 867 Multiple-criteria choice, 548 Multiple-criteria classification, 515, 545 Multiple attribute utility theory, 650 Multiple Criteria Decision Making, 646 Multiple Criteria Decision Making Support Systems (MCDSS), 646 Multiple criteria design problem, 647 Multiple criteria evaluation problem, 647 Multiple criteria optimization, 825 Multiple objective linear programming, 729 Multistat Optimizer, 1005, 996 Musgrave A., 980 MUSTARD, 1000, 996 Nadir point, 677 NAIADE, 1001, 996 National missile defense, 405 Nearness, 29–30 Necessarily efficient solution, 759 Negative outranking flow, 173–175 1041 Index Negative transitivity, 270 Net outranking flow, 174 Net outranking flows, 179 Network, 349, 384–386, 388, 391–393, 396, 403, 405–406 Neurath O., 962 Neurons, 351, 356 Neutral level, 572 NIMBUS, 1006 No stable value function, 652 Non-additive utility functions, 330 Non-discordance, 137 Non-monotonic preferences, 317 Nonconvex problems, 791 Nondominated, 648 Nondominated criterion vector, 825 Nondominated frontier, 649 Nondominated outcome, 674 Nondominated point existence, 677 Nondominated set, 674, 678, 826 connectedness, 678 properties of, 675 Nondominated surfaces, 837 Nonscalarizing methods, 680, 691, 696 approach, 696 balance set approach, 694 equitability approach, 693 level set approach, 694 lexicographic approach, 691 lexicographic max-ordering approach, 693 max-ordering approach, 692 Norgaard R.B., 958 Norm, 685 monotonically increasing, 685 strictly monotonically increasing, 685 Normal boundary intersection approach, 690 Normalization, 378 Numerical representation, 30, 48, 53, 568, 570 Numerical scale, O’Connor M., 959 Objective function vector-valued, 673 Objective function, 648 Objective space, 673 Objective space methods, 705 OnBalance, 1001, 996 Operational planning, 942 Opportunities, 347, 350, 379, 381, 383, 402, 404 Option pricing theory, 811 ORCLASS, 629 Ordered median problem, 779 Ordered sets, 40 complete order, 48 complete preorder, 53 quasi-order, 44 total order, 40 complete order, 40, 53 complete preorder, 41, 49, 53 double threshold orders, 43, 51 interval order, 28, 31, 35, 42, 50, 55 linear order, 40, 48, 53 partial order, 44, 52 partial preorder, 44 partial weak order, 52 PQI interval order, 51 pseudo-orders, 43–45, 51 quasi-order, 44 semi-order, 41 semiorder, 54 simple order, 40, 48, 53 total order, 48, 53 total preorder, 41, 49, 53 weak order, 41, 49, 53 Ordered sorting, 476 Ordering problem, 476, 484 Ordinal classification, 619, 629–631, 637 Ordinal judgments, 616–619, 621, 634 Ordinal scale, ORESTE, 207 Outcome set-based algorithm, 707 Outranking, 460, 893 Outranking method, 166 Outranking methods, 16 Outranking relation, 137, 479 Outranking relations, 840 PACMAN, 251 Pairwise comparison, 166–167,169–170 Pairwise comparison table (PCT), 549 Pairwise comparisons, 315, 458 Pairwise Criterion Comparison Approach (PCCA), 222 Parameter space investigation, 698 Parametric analysis, 772 Pareto condition strong Pareto condition, 601 weak Pareto condition, 601 Pareto face maximal, 706 Pareto preference, 675 Pareto Principle, 617 Pareto Race, 662 Pareto relation, 675 Pareto set approximation, 696 exact approximation, 696 heuristic approaches, 698 heuristic methods, 700 nonlinear approximation, 698 piece-wise linear approximation, 697 point-wise approaximation, 697 Partial frequency, 242 Partial order, 52, 674 Partial profile, 521,531 1042 Participative multi-criteria evaluation, 976 Pascoletti and Serafini approach, 688 Performance, Perron eigenvalue, 377 Perron theory, 354 Perturbation, 353–354, 377 Piecewise linear objective, 710 Point-objective location problem, 766, 771, 789 Policies, 881 Portfolio selection, 849 Portfolio theory, 823 Positive outranking flow, 173, 175 Positive reciprocal matrix, 375–377 Possibly efficient solution, 759 Post-optimality analysis, 307 Posterior articulation of preferences, 653 Potential action, Potential index, 226 Power generation, 865, 880 PRAGMA, 240 Pratt J.W., 274 Pre-aggregation, 482 Pre-scoring, 484 PREFCALC, 1001, 996, 998 PREFDIS, 1007, 996 Preference, 165, 170, 172, 174, 673, 675 Preference asymmetry, 270 Preference difference, 102 Preference differences, 75 Preference disaggregation, 300, 844 Preference elicitation, 615–617, 621–622, 624, 630–631, 633 Preference function, 169–171, 175, 186, 188 Preference model, 300–301, 306, 327, 330, 332, 336 Preference modelling, 52 Preference relation monotonicity, 601 symmetric, 270 Preference structure, 29, 39, 44, 48 Preference Pareto, 675 independence, 274 indifference, 270 interval, 88 nontransitive, 115 ratio, 273 strict, 270 weak, 271 weak order, 270 Preferences, 412 Preferential independence, 617–618, 622, 624 Preferential independence, 628 Preferential independence, 633, 636 Preferential information, 416, 512, 549 Preference preference difference, 271 MULTIPLE CRITERIA DECISION ANALYSIS PRIAM, 1001, 996 PRIME Decisions, 1001, 1004, 996 Priorities, 347, 349–352, 354–355, 357, 362, 366, 368–369, 372, 375, 377, 379, 381–382, 385, 388–389, 392, 396, 399, 402, 405–406 global, 368, 402 local, 368 PROAFTN, 1007, 996 Probability distribution, 75 Problem formulation, 90 Problem statement, 300–301 Problem structure, 614, 618, 627, 629, 634, 636–637 Problem structuring, 883 Problematic, 11 Product set, 75, 90 Profiles of alternatives, 175–176, 185–186, 188 PROMETHEE, 1011–1012, 1014, 633, 893, 997 Promethee GDSS group decision support system, 183 PROMETHEE I partial ranking, 173 PROMETHEE II complete ranking, 174–175, 188 PROMETHEE methods, 842 PROMETHEE V, 182–183 PROMETHEE VI, 181 Properly efficient solution, 676 Psychological validity, 615 QoS issues, 910 QoS routing, 926 Quadratic function, 710 QUALIFLEX, 199 Qualitative judgements, 414 Qualitative scales, 888 Quality of approximation, 525, 552 Quantitative scale, Questioning procedure, 416 Raiffa H., 277, 268–269, 274, 277, 280, 282, 286, 288, 295 Ranking, 548 Ranking problematic, 12, 143 Ratings, 372–373, 405–406 Ratio scale, 10, 357, 372, 399 Raynaud H., 968 Reactive power compensation, 872 Reciprocal matrix, 355 Reduced cost matrix, 702 Reduct, 525 Reducts, 848 Redundancy, 496 Reference point|688 reference actions, 302 Reference actions, 302, 306, 309, 320, 322 Reference direction approach, 689 Reference direction method, 661 Reference direction vector, 661 Reference level, 653 Reference point, 654, 684 Reference point approach, 788 Index Reference set, 845 Reference set approach, 688 REGIME, 202 Regime matrix, 203 Relative importance of criteria, 138 REMBRANDT, 1002, 996 Reservation level, 653 RGDB, 1002, 996 Risk-return trade-off, 807 Risk, 450 Risk aversion, 807 Risk measures, 461 Risks, 347, 350, 379, 381, 383, 402, 404 Robust programming, 749 Robustness, 779 Robustness concerns, 149 Rough approximations, 519 Rough set theory, 847 Rough sets, 451, 453 Routing models, 925 Roy approach, 688 Roy B., 976 Salary raises, 372 Salo A.A., 273, 288 SANEX, 1012, 996 SANNA, 1002, 996 Sarin R.A., 293 Sarin R.K., 275, 288 Satisfactory level, 573 Satisficing bounded rationality, 573 Savage L.J., 274 Scalarization, 680, 691 approach, 683 achievement function approach, 686 Benson approach, 684 direction-based approaches, 688 distance function, 684 elastic constraint approach, 683 gauge-based approach, 690 goal-attainment approach, 688 Guddat et al approach, 682 modified Pascoletti and Serafini approach, 689 normal boundary intersection approach, 690 Pascoletti and Serafini approach, 688 reference direction approach, 689 reference point approaches, 684 reference set approach, 688 Roy aproach, 688 weighted-sum approach, 681 weighted power approach, 681 weighted geometric mean approach, 687 weighted quadratic approach, 682 Scalarizing function, 653 Scale, 9, 360, 418, 435,570 absolute, 347, 349, 371, 399 bipolar scale, 572, 587 bounded scale, 573 1043 fundamental, 347, 349, 356–357, 367, 372 interval, 351, 356 interval scale, 571 ordinal, 75, 351 ordinal scale, 571, 599 ratio scale, 571 relative, 349 unidimensional, 350 unipolar scale, 572, 574 Scaling, 888 SCDAS, 1010, 996 Scenario planning, 464 Scenarios, 887, 889 Scores, Scott D., 272 Search algorithms, 943 Second-order conditions, 680 Selection problem, 647 Semi-obnoxious, 777, 782, 787 Sensitivity walking weights, 188–189 Separability weak, 113 Set of actions, 28 Set of alternatives, 28 Shapley importance index, 596 Shapley value, 494, 528, 581 Similarity, 29, 46, 540 Simpson decisions, 775 Simulated annealing, 1005, 699 Single facility, 766, 773, 784, 786 Sink node, 385 Šipoš integral, 588 Siting, 882 Skills Evaluator, 1012, 996 Social choice, 76 Social incommensurability, 961 Social multi-criteria evaluation, 977 Socio-economic issues, 944 Soft constraint, 729 Software, 414 Software Decision Lab, 168, 170, 181, 186, 188–189 Promcalc, 168, 170, 181, 183, 186 Solutions efficient, 767 lexicographic, 767 max-ordering, 767 Pareto, 767 properly efficient, 767 strictly efficient, 767 weakly efficient, 767 SOLVEX, 1006, 996 Sorting, 29 Sorting problem, 476, 484, 486 Sorting problematic, 12, 147 Source node, 385 1044 MULTIPLE CRITERIA DECISION ANALYSIS Stability of MOP, 677 Stability external, 677 Stakeholder multi-criteria decision aid, 976 Stakeholders, 889 Standard sequence, 81, 93, 98, 102 States of nature, 453 Steel Mill Scheduling, 996 Steps of ANP, 402 Stimulus, 350, 356 Stimulus response, 356 Stochastic, 888 Stochastic approaches, 889 Stochastic dominance, 459 Stochastic matrix, 389, 396 Stochastic programming, 463 Strength of preference, 271 Strength Pareto Evolutionary Algorithm, 1005 Strictly efficient solution, 675 Strong preference, 29, 45–46, 58 Strong sustainability, 981 Structuration of objectives, 90 Subdifferentials, 774, 778 Substitutability, 572 Substitutive criteria, 581 Sugeno integral, 599, 601 symmetric, 603 Suggestions, 415, 432 Supermatrix, 386, 388–390, 392, 396, 404 unweighted, 392, 396 weighted, 396, 399 Suppes P., 272 Support, 656 Sustainable development, 866, 957 Swing procedure, 617 Sylvester J.J., 388 Symmetric maximum, 602 Symmetric minimum, 602 Symmetrical importance, 965 Synergy, 496 Synthesis, 350, 360, 370–371, 377–379, 389, 404 Tabu search, 700 TACTIC, 218 Tchebycheff criterion, 319 Tchebychev norm, 771, 783 Technical incommensurability, 961 Technological evolutions, 907 TELOS, 1013, 996 Ternary alternative, 590, 594 Threshold, 43, 47, 49–50 Thresholds, 166, 170, 186 Tolerant decision maker, 569 TOMASO, 1008, 487, 996 TOMMIX, 1006, 996 Topological properties, 774, 780 TOPSIS, 1002, 996 Tradeoff, 83, 349, 380 Transitivity, 355 Transitivity of preferences, 618, 622, 624, 626 Transitivity of preferences, 628 Transitivity of preferences, 633 Transmission, 871 Trapezoidal number, 478 Tree networks, 783–784, 786 TRIMAP, 1006, 996 Trzaskalik T., 289 Tukey J.W., 276 Tversky A., 274 Uncapacitated facility location, 790 Uncertainty, 30, 35, 42–43, 45, 450, 887 external, 451 internal, 451 Unfuzzy non dominated solution, 484 Unit commitment, 873 Upper approximation, 521–522, 551 UTA method, 108,844 UTA Plus, 1001–1002, 996 UTADIS method, 845 Utility function, 270, 320, 568, 648 Utility independence, 456 Utility theory, 273 continuity, 274 independence, 273 Utility expected, 75 Utopia point, 676, 685 V.I.S.A, 1003 Validation, 359 Value, 438 Value and utility theory, 891 Value function, 74, 270, 648 additive, 76, 100 finite case, 102 Linear Programming, 106 rich case, 92 cardinal, 272 decomposable, 112 interval scale, 272 measurable, 271 multiplicative, 77 ordinal, 271 pseudoconcave, 661 Value space, 648 Valued preference relation, 479 Valued relation, 37, 45 Valued structures, 45 Variable-consistency dominance-based rough set approach (VC-DRSA), 529 Variable space, 648 Vector-maximum approach, 651 Vector of ordered outcomes, 694 Vectorial best approximation, 779 Venture capital, 852 1045 Index Verbal scale, Verification, 624 Veto criterion, 569 VIG, 1006, 659, 661, 996 VIMDA, 1003, 661, 996 VIP Analysis, 1003, 996 Von Neumann J., 269, 273 Von Winterfeldt D., 273–274, 286 Voting theory, 775 Wakker P., 274 Water Quality Plan, 996 Weak comparability, 960 Weak order, 306 Weak preference, 29, 43, 58 Weak separability, 572 directional weak separability, 600 Weak sustainability, 981 Weakly efficient, 649 Weakly efficient solution, 674–675 Weakly nondominated, 648 Weakly nondominated outcome, 674 Weber-Fechner law, 356 WebHIPRE, 996 Weight space decomposition, 702 Weighted-sum approach, 681 Weighted power aproach, 681 Weighted geometric mean approach, 687 Weighted quadratic approach, 682 Weighting, 888 Weights, 167–168, 170, 175–176, 179–181, 185–186, 188 WINGDSS, 1010, 996 WINPRE, 1004, 996 World Trade Organization, 380 WWW-NIMBUS, 1006, 996 Young H.P., 967 Zadeh L.A., 982 ZAPROS-STEP, 625 ZAPROS, 619, 622, 624–630, 636–637 ZAPROS III, 625–627 This page intentionally left blank Early Titles in the INTERNATIONAL SERIES IN OPERATIONS 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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. .. overlapping points of view On the other hand, if the importance of the whole subset of criteria is larger than the sum of the importances of its members, then we observe a synergy between criteria, the