multiple criteria optimization state of the art annotated bibliographic surveys (international series in operations research & management science)

List of FiguresList of Tables Introduction José Figueira, Salvatore Greco, Matthias Ehrgott Human Reflection about Decision Technical Reflection about Decision: MCDA Researchers before M

MULTIPLE CRITERIA

DECISION ANALYSIS:

STATE OF THE ART SURVEYS

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./ M ATHEMATICAL 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

Print ISBN: 0-387-23067-X

Print ©2005 Springer Science + Business Media, Inc.

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

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List of Figures

List of Tables

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

Part I 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

xxi xxi xxii xxiv xxv xxxiv xxxiv

3

4 7 12 14 17 18

27

28 28 30 32 33

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 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

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

73

74 89 92 102 112 119

133

134 136 139 149 151 153 153

163

164 164 168 171 175 181 182 183 186 189

197

198 198

Pairwise Criterion Comparison Approach

One Outranking Method for Stochastic Data

Conclusions

References

Part IV 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

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

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

On the Admission of China to the World Trade Organization (WTO) 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

Two Examples of Estimating Market Share – The ANP with a Single Benefits Control Criterion

Outline of the Steps of the ANP

Complex Decisions with Dependence and Feedback

Conclusions

References

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

References

Part V Non-Classical MCDA Approaches

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

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 T OMASO Method

The Choice Problem

409

410 412 414 415 416 417 420 432 435 437 438

445

446 450 454 457 460 466 467

471

472 474 475 478 482 483 502 503 504

Decision Rule Approach

Salvatore Greco, Benedetto Matarazzo,

Fuzzy Measures and Integrals in MCDA

Michel Grabisch, Christophe Labreuche

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

Basic Definitions and Some Theory

Principles for Implementing Interactive Methods

Generating Nondominated Solutions

Solving Multiple Objective Problems

563

564 566 570 583 595 604 604

609

610 610 615 625 628 633 634

641

642 643 645 649 652 656 657

Problem Formulation and Solution Concepts

Properties of the Solution Sets

Conditions for Efficiency

Generation of the Solution Sets

Approximation of the Pareto Set

Specially Structured Problems

Current and Future Research Directions

Problem Statement and Approaches

Modality Constrained Programming Approach

Modality Goal Programming

Modal Efficiency Approach

Continuous Multicriteria Location Problems

Multicriteria Network Location Problems

Multicriteria Discrete Location Problems

Conclusions

References

Part VII Applications

Multicriteria Decision Aid/Analysis in Finance

Jaap Spronk, Ralph E Steuer, Constantin Zopounidis

Introduction

Financial Decision Making

MCDA in Portfolio Decision-Making Theory

MCDA in Discrete Financial Decision-Making Problems

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

799

800 801 819 835

Conclusions and Future Perspectives

References

MCDA and Energy Planning

Danae Diakoulaki, Carlos Henggeler Antunes, António Gomes Martins

Introduction

Multiobjective Programming Models for Energy Planning

Energy Planning Decisions with Discrete Alternatives

Conclusions

References

Multicriteria Analysis in Telecommunication Network Planning and

Design – Problems and Issues

João Clímaco, José Craveirinha

The Concept of Sustainable Development

Measuring Sustainability: The Issue of Sustainability Assessment Indexes

A Defensible Axiomatic Setting for Sustainability Composite cators

Indi-Warning! Not Always Rankings Have to Be Trusted

The Issue of the “Quality of the Social Decision Processes”

The Issue of Consistency in Multi-Criteria Evaluation of

Sustainability Policies

Conclusion

References

Part VIII MCDM Software

Multiple Criteria Decision Support Software

H Roland Weistroffer, Charles H Smith, Subhash C Narula

859

860 863 874 890 891

899

900 900 908 912 941 944

953

954

958 963 966 971

976 980 981

989

990 990 1009

Contributing Authors

Index

1011 1019

1035

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

179180181182186186187

PROMETHEE rankings, action profiles, GAIA plane.

Walking weights

Set of feasible weights

ORESTE flow chart

Outranking graph

Geometrical interpretation of preferences indices

Indifference areas

Indifference areas: rectangular

Indifference areas: rhomboidal

Indifference areas: elliptical

Aggregated semiorder structure

Aggregated pseudo-order structure

Partial profile of action

Partial profiles and partial broken lines of

Partial frequencies of

Determination of a relation between the two alternatives

on the basis of the values of global indices

Partial preorder

Choice between two lotteries

Additive Independence Criterion for Risk

Piecewise linear approximation of

Piecewise linear approximation of

The aggregation and disaggregation paradigms in MCDA [44].The disaggregation-aggregation approach [96]

The normalized marginal value function

Post-optimality analysis [43]

Ordinal regression curve (ranking versus global value)

Normalized marginal value functions

A non-monotonic partial utility function [18]

Distributional evaluation and marginal value function

Distribution of the actions and on [43]

Simplified decision support process based on

disaggre-gation approach [44]

Methodological flowchart of MARKEX [63]

Comparisons according to volume

To choose the best hospice plan, one constructs a

hierar-chy modeling the benefits to the patient, to the institution,

and to society This is the benefits hierarchy of two

254259279280289289300301303306307313316319321

329332359

361

To choose the best hospice plan, one constructs a

hierar-chy modeling the community, institutional, and societal

costs This is the costs hierarchy of two separate hierarchies.Employee evaluation hierarchy

Hierarchies for rating benefits, costs, opportunities, and risks.Prioritizing the strategic criteria to be used in rating the BOCR.How a hierarchy compares to a network

The supermatrix of a network and detail of a component

in it

The supermatrix of a hierarchy with the resulting limit

matrix corresponding to hierarchical composition

School choice hierarchy composition

Supermatrix of school choice hierarchy gives same

re-sults as hierarchic composition

The clusters and nodes of a model to estimate the relative

market share of Walmart, Kmart and Target

The clusters and nodes of a model to estimate the relative

market share of footware

Hierarchy for rating benefits, opportunities, costs and risks

Example of sub-type b inconsistency

Example of incompatibility between (*) and (**)

Procedure for all cases of inconsistency

Suggestion of change to resolve inconsistency

Matrix of judgements and basic MACBETH scale

Representations of the MACBETH scale

“Greatest” closed intervals included in the free and

de-pendent intervals

Comparing two fuzzy intervals

Interpretation of the discriminant functions

Representation of the students problem

Classes for

Classes for

Visual representation of the classes,

Results for the neighbour algorithm,

Decision tree representing knowledge included from Table 13.1.The hierarchy of attributes and criteria for a car classifi-

cation problem

Different cases of interaction

Ternary alternatives for

385

385388

388

391

398405417424431433434435

437476493495496497499

501535

544578593

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

Two differences between fuzzy numbers and

The neo-classical view on the objective of the firm

Situations leading to MCDA in the firm

A bird’s-eye view of the framework

Feasible regions Z of (MC-Unrestr) and (MC-Bounds)

for the same eight securities

Continuous, bullet-shaped, and unbounded feasible

re-gion Z created by securities A, B and C.

Nondominated frontiers as a function of changes in the

value of upper bound parameter

An ellipsoidal feasible region projected onto two

dimen-sional risk-return space

Typical hierarchical structure of criteria used in energy

planning

MCDA methods in energy planning applications

State transition diagram

Example of priority regions

A systemic vision of sustainability issues

Impact matrix for the 4 chosen cities according to the

739740746746

748751811812817

967975

Principal t-norms and t-conorms.

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

Preference matrix for a criterion with ordinal evaluation

Preference matrix for a criterion (Max) with evaluation

on a ratio scale

Preference importance table for

Combined preferences with weights variable

Evaluation of alternatives*

Criteria and (ordinal scales)

Criterion (ordinal scale)

Criterion (ratio scale MIN)

Preference structure of weights

Pairwise comparison between and

Axiomatic system of MAPPAC basic indices

Preference indices

Table of observed stochastic dominances

Explicable concordances indices

211212212213213214214214214227228258259

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

in-stitutional 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,

Oppor-tunities, 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

377

381382

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

evalu-ations of students

Quality of classification and Shapley value for

classifi-cation 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

405406485485495498498499499500501

511

522532533536538541542551552

554616618619

Effectiveness of STEP-ZAPROS.

Summary of references

Applications of MCDA approaches in bankruptcy and

credit risk assessment

Applications of MCDA approaches in portfolio selection

Broad categories of planning problems in power systems

Key aspects in problem structuring in papers of Group A

Key aspects in problem structuring in papers of Group B

Key aspects in problem structuring in papers of Group C

Key aspects in problem structuring in papers of Group D

Example of composite indicators (Source: OECD; JRC,

2002, cited in [72, p 4])

Normalised impact matrix

Outranking matrix of the 4 cities according to the 9 indicators.Possible ranking

Weighted outranking matrix

Possible ranking with new scores

Software by problem type

845

846

847848

848864880880880881

962968968968969970992

José Figueira, Salvatore Greco, Matthias Ehrgott

Decision has inspired reflection of many thinkers since the ancient times Thegreat philosophers Aristotle, Plato, and Thomas Aquinas, to mention only afew names, discussed the capacity of humans to decide and in some mannersclaimed that this possibility is what distinguishes humans from animals Toillustrate some important aspects of decision, let us briefly quote two importantthinkers: Ignatius of Loyola (1491-1556) and Benjamin Franklin (1706-1790)

To consider, reckoning up, how many advantages and utilities follow for me from holding the proposed office or benefice [ ] , and, to consider likewise, on the contrary, the disadvantages and dangers which there are in having it Doing the same in the second part, that is, looking at the advantages and utilities there are

in not having it, and likewise, on the contrary, the disadvantages and dangers in not having the same [ ] After I have thus discussed and reckoned up on all sides about the thing proposed, to look where reason more inclines: and so, according

to the greater inclination of reason, [ ], deliberation should be made on the thing proposed.

This fragment from the “Spiritual Exercises” of St Ignatius of Loyola [14]has been taken from a paper by Fortemps and [12]

London, Sept 19, 1772

Dear Sir,

In the affair of so much importance to you, wherein you ask my advice, I cannot, for want of sufficient premises, advise you what to determine, but if you please I will tell you how [ ], my way is to divide half a sheet of paper by a line into two columns; writing over the one Pro, and over the other Con [ ] When I have thus got them all together in one view, I endeavor to estimate their respective weights; and where I find two, one on each side, that seem equal, I strike them both out If

I find a reason pro equal to some two reasons con, I strike out the three If I judge some two reasons con, equal to three reasons pro, I strike out the five; and thus proceeding I find at length where the balance lies; and if, after a day or two of

further consideration, nothing new that is of importance occurs on either side, I come to a determination accordingly [ ] I have found great advantage from this kind of equation, and what might be called moral or prudential algebra Wishing sincerely that you may determine for the best, I am ever, my dear friend, yours most affectionately.

of the considered actions and then to maximize (minimize) this objective This

is a very reductive, and in some sense also unnatural, way to look at a decisionproblem Thus, for at least thirty years, a new way to look at decision problemshas more and more gained the attention of researchers and practitioners This isthe approach considered by Loyola and Franklin, i.e., the approach of explicitlytaking into account the pros and the cons of a plurality of points of view, in otherwords the domain of Multiple Criteria Decision Analysis (MCDA) Therefore,MCDA intuition is closely related to the way humans have always been makingdecisions Consequently, despite the diversity of MCDA approaches, methodsand techniques, the basic ingredients of MCDA are very simple: a finite orinfinite set of actions (alternatives, solutions, courses of action, ), at least twocriteria, and, obviously, at least one decision-maker (DM) Given these basicelements, MCDA is an activity which helps making decisions mainly in terms

of choosing, ranking or sorting the actions

2 Technical Reflection about Decision: MCDA

Researchers before MCDA

Of course, not only philosophers reasoned about decision-making Many portant technical aspects of MCDA are linked to classic works in economics, inparticular, welfare economics, utility theory and voting oriented social choicetheory (see [28]) Aggregating the opinion or the preferences of voters or indi-viduals of a community into collective or social preferences is quite similar a

im-problem to devising comprehensive preferences of a decision-maker from a set

of conflicting criteria in MCDA [7]

Despite the importance of Ramon Llull’s (1232-1316) and Nicolaus sanus’s (1401-1464) concerns about and interests in this very topic, the origins

Cu-of voting systems are Cu-often attributed to Le Chevalier Jean-Charles de Borda(1733-1799) and Marie Jean Antoine Nicolas de Caritat (1743-1794), Le Mar-quis de Condorcet However, Ramon Llull introduced the pairwise comparisonconcept before Condorcet [13], while Nicolaus Cusanus introduced the scor-ing method about three and a half centuries before Borda [27] Furthermore, itshould be noted that a letter from Pliny the Younger AD 105) to Titus Aristoshows that he introduced the ternary approval voting strategy and was interested

in voting systems a long time before Ramon Llull and Nicolaus Cusanus [18,Chapter 2] Anyway, Borda’s scoring method [4] has some similarities withcurrent utility and value theories as has Condorcet’s method [10] with the out-ranking approach of MCDA In the same line of concerns, i.e., the aggregation

of individual preferences into collective ones, Jeremy Bentham (1748-1832)introduced the utilitarian calculus to derive the total utility for the society fromthe aggregation of the personal interests of the individuals of a community[3] Inspired by Bentham’s works, Francis Ysidro Edgeworth (1845-1926), autilitarian economist, was mainly concerned with the maximization of the util-ity of the different competing agents in economy Edgeworth tried to find thecompetitive equilibrium points for the different agents He proposed to drawindifference curves (lines of equal utility) for each agent and then derive thecontract curve, a curve that corresponds to the notion of the Pareto or efficientset [21] Not long afterwards, Vilfredo Federico Damaso Pareto (1848-1923)gave the following definition of ophelimity [utility] for the whole community[22]:

We will say that the members of a collectivity enjoy maximum ophelimity in a certain position when it is impossible to find a way of moving from that posi- tion very slightly in such a manner that the ophelimity enjoyed by each of the individuals of that collectivity increases or decreases That is to say, any small displacement in departing from that position necessarily has the effect of increas- ing the ophelimity which certain individuals enjoy, of being agreeable to some, and disagreeable to others.

From this definition it is easy to derive the concept of dominance, whichtoday is one of the fundamental concepts in MCDA

MCDA also benefits from the birth and development of game theory FélixEdouard Justin Emile Borel (1871-1956) and John von Neumann (1903-1957)are considered the founders of game theory [5, 6, 20, 19] Many concepts fromthis discipline had a strong impact on the development of MCDA

The concept of efficient point was first introduced in 1951 by Tjalling mans (1910-1985) in his paper “Analysis of production as an efficient combi-nation of activities” [15]:

Koop-A possible point in the commodity space is called efficient whenever an increase

in one of its coordinates (the net output of one good) can be achieved only at the cost of a decrease in some other coordinate (the net output of a good).

In the same year (1951) Harold William Kuhn (born 1925) and Albert WilliamTucker (1905-1995) introduced the concept of vector maximum problem [16]

In the sixties, basic MCDA concepts were explicitly considered for the firsttime As two examples we mention Charnes’ and Cooper’s works on goal pro-gramming [8] and the proposition of ELECTRE methods by Roy [23] Theseventies saw what is conventionally considered the “official” starting point ofMCDA, the conference on “Multiple Criteria Decision Making” organised in

1972 by Cochrane and Zeleny at Columbia University in South Carolina [9].Since then MCDA has seen a tremendous growth which continues today

3 The Reasons for this Collection of State-of-the-Art Surveys

The idea of MCDA is so natural and attractive that thousands of articles anddozens of books have been devoted to the subject, with many scientific journalsregularly publishing articles about MCDA To propose a new collection of state-of-the-art surveys of MCDA in so rich a context may seem a rash enterprise.Indeed, some objections come to mind There are many and good handbooksand reviews on the subject (to give an idea consider [1,11, 25, 26, 29]) The mainideas are well established for some years and one may question the contributionsthis volume can provide Moreover, the field is so large and comprises devel-opments so heterogeneous that it is almost hopeless to think that an exhaustivevision of the research and practice of MCDA can be given

We must confess that at the end of the work of editing this volume we agreewith the above remarks However, we believe that a new and comprehensivecollection of state-of-the-art surveys on MCDA can be very useful The mainreasons which, despite our original resistance, brought us to propose this bookare the following:

Many of the existing handbooks and reviews are not too recent SinceMCDA is a field which is developing very quickly this is an importantreason

Even though the field of research and application of MCDA is so large,there are some main central themes around which MCDA research andapplications have been developed Therefore our approach was to try topresent the – at least in our opinion – most important of these ideas

1

2

With reference to the first point, we can say that we observed many theoreticaldevelopments which changed MCDA over the last ten years We tried to consider

these changes as much as possible and in this perspective strong points of thebook are the following:

It presents the most up-to-date discussions on well established ologies and theories such as outranking based methods and MAUT.The book also contains surveys of new, recently emerged fields such asconjoint measurement, fuzzy preferences, fuzzy integrals, rough sets andothers

method-1

2

Following these points we drafted a list of topics and asked well knownresearchers to present them We encouraged the authors to cooperate with theaim to present different perspectives if topics had some overlap We asked theauthors to present a comprehensive presentation of the most important aspects

of the field covered by their chapters, a simple yet concise style of exposition,and considerable space devoted to bibliography and survey of relevant literature

We also requested a sufficiently didactic presentation and a text that is useful forresearchers in MCDA as well as for people interested in real life applications.The importance of these requirements is related also to the specific waythe MCDA community looks at its research field It can be summarized in theobservation that there is a very strong and vital link between theoretical andmethodological developments on the one hand and real applications on theother hand Thus, the validity of theoretical and methodological developmentscan only be measured in terms of the progress given to real world practice.Moreover, interest of MCDA to deal with concrete problems is related to theconsideration of a sound theoretical basis which ensures the correct application

of the methodologies taken into account

In fact, not only the chapters of our book but rather all MCDA contributionsshould satisfy the requirements stated out above, because they should be not too

“esoteric” and therefore understandable for students, theoretically well founded,and applicable to some advantage in reality

Of course, this book can be read from the first to the last page However, we thinkthat this is not the only possibility and it may not even be the most interestingpossibility In the following we propose a guided tour of the book suggestingsome reference points that are hopefully useful for the reader

This part is important because MCDA is not just a collection of theories, ologies, and techniques, but a specific perspective to deal with decision prob-lems Losing this perspective, even the most rigorous theoretical developmentsand applications of the most refined methodologies are at risk of being meaning-

method-less, because they miss an adequate consideration of the aims and of the role ofMCDA We share this conviction with most MCDA researchers Bernard Roydiscusses these “pre-theoretical” assumptions of MCDA and gives an overview

of the field Bernard Roy, besides giving many important theoretical tions, engaged himself in thorough reflections on the meaning and the value ofMCDA, proposing some basic key concepts that are accepted throughout theMCDA community

This part of the book is related to a fundamental problem of MCDA, the sentation of preferences Classically, for example in economics, it is supposedthat preference can be represented by a utility function assigning a numeri-cal value to each action such that the more preferable an action, the larger itsnumerical value Moreover, it is very often assumed that the comprehensiveevaluation of an action can be seen as the sum of its numerical values for theconsidered criteria Let us call this the classical model It is very simple but nottoo realistic Indeed, there is a lot of research studying under which conditionsthe classical model holds These conditions are very often quite strict and it isnot reasonable to assume that they are satisfied in all real world situations Thus,other models relaxing the conditions underlying the classical model have beenproposed This is a very rich field of research, which is first of all importantfor those interested in the theoretical aspects of MCDA However, it is also ofinterest to readers engaged in applications of MCDA In fact, when we adopt aformal model it is necessary to know what conditions are supposed to be sat-isfied by the preferences of the DM In the two chapters of this part problemsrelated to the representations of preferences are discussed

repre-Meltem Öztürk, Alexis Tsoukiàs, and Philippe Vincke present a very tive review of preference modelling, starting from classical results but arriving

exhaus-at the frontier of some challenging issues of scientific activity relexhaus-ated to fuzzylogic and non-classical logic

Denis Bouyssou and Marc Pirlot discuss the axiomatic basis of the differentmodels to aggregate multiple criteria preferences We believe that this chapter

is very important for the future of MCDA Initially, the emphasis of MCDAresearch was on proposal of new methods But gradually the necessity to un-derstand the basic conditions underlying each method and its specific axioma-tization became more and more apparent This is the first book on MCDA with

so much space dedicated to the subject of foundations of MCDA

In this part of the book the class of outranking based multiple criteria decisionmethods is presented Given what is known about the decision-maker’s prefer-

ences and given the quality of the performances of the actions and the nature

of the problem, an outranking relation is a binary relation S defined on the set

of potential actions A such that if there are enough arguments to decidethat is at least as good as whereas there is no essential argument to refutethat statement [24] Methods which strictly apply this definition of outrankingrelation are the ELECTRE methods They are very important in many respects,not least historically, since ELECTRE I was the first outranking method [2].However, within the class of outranking methods we generally consider allmethods which are based on pairwise comparison of actions Thus, anotherclass of very well known multiple criteria methods, PROMETHEE methods,are considered in this part of the book Besides ELECTRE and PROMETHEEmethods, many other interesting MCDA methods are based on the pairwise com-parison of actions José Figueira, Vincent Mousseau and Bernard Roy presentthe ELECTRE methods; Jean-Pierre Brans and Bertrand Mareschal presentthe PROMETHEE methods and Jean-Marc Martel and Benedetto Matarazzoreview the rich literature of other outranking methods

4.4 Part IV: Multiattribute Utility and Value Theories

In this part of the book we consider multiple attribute utility theory (MAUT).This MCDA approach tries to assign a utility value to each action This utility is

a real number representing the preferability of the considered action Very oftenthe utility is the sum of the marginal utilities that each criterion assigns to theconsidered action Thus, this approach very often coincides with what we calledthe classical approach before As we noted in commenting Part I, this approach

is very simple at first glance It is often applied in real life, e.g., every time

we aggregate some indices by means of a weighted sum we are applying thisapproach Despite its simplicity the approach presents some technical problems.The first are related to the axiomatic basis and to the construction of marginalutility functions (i.e., the utility functions relative to each single criterion),both in case of decision under certainty and uncertainty These problems areconsidered by James Dyer in a comprehensive chapter about the fundamentals

of this approach

Yannis Siskos, Vangelis Grigoroudis and Nikolaos Matsatsinis present thevery well known UTA methods, which on the basis of the philosophy of theaggregation-disaggregation approach and using linear programming, build aMAUT model that is as consistent as possible with the DM’s preferences ex-pressed in actual previous decisions or on a “training sample” The philosophy

of aggregation-disaggregation can be summarized as follows: How is it ble to assess the decision-maker’s preference model leading to exactly the samedecision as the actual one or at least the most “similar” decision?

possi-Thomas Saaty presents a very well known methodology to build utility tions, the AHP (Analytic Hierarchy Process) and its more recent extension,the ANP (Analytic Network Process) AHP is a theory of measurement thatuses pairwise comparisons along with expert judgments to deal with the mea-surement of qualitative or intangible criteria The ANP is a general theory ofrelative measurement used to derive composite priority ratio scales from in-dividual ratio scales that represent relative measurements of the influence ofelements that interact with respect to control criteria The ANP captures theoutcome of dependence and feedback within and between clusters of elements.Therefore AHP with its dependence assumptions on clusters and elements is aspecial case of the ANP.

func-Carlos Bana e Costa, Jean-Claude Vansnick, and Jean-Marie De Corte presentanother MCDA methodology based on the additive utility model This method-ology is MACBETH (Measuring Attractiveness by a Categorical Based Evalu-ation Technique) It is an MCDA approach that requires only qualitative judge-ments about differences of values of attractiveness of one action over anotheraction to help an individual or a group to quantify the relative preferability ofdifferent actions In simple words, the MACBETH approach tries to answer thefollowing questions: How can we build an interval scale of preferences on a set

of actions without forcing evaluators to produce direct numerical tions of their preferences? How can we coherently aggregate these qualitativeevaluations using an additive utility model?

Many approaches have been proposed in MCDA besides outranking methodsand multiattribute utility theory In this part of the book we try to collect in-formation about some of the most interesting proposals First, the question ofuncertainty in MCDA is considered Theo Stewart discusses risk and uncertainty

in MCDA It is necessary to distinguish between internal uncertainties (related

to decision-maker values and judgements) and external uncertainties (related

to imperfect knowledge concerning consequences of actions) The latter, responding to the most accepted interpretation of uncertainty in the specializedliterature, has been considered in the chapter Four broad approaches for deal-ing with external uncertainties are discussed These are multiattribute utilitytheory and some extensions; stochastic dominance concepts, primarily in thecontext of pairwise comparisons of alternatives; the use of surrogate risk mea-sures such as additional decision criteria; and the integration of MCDA andscenario planning

cor-The second consideration is the fuzzy set approach to MCDA Most realworld decision problems take place in a complex environment where conflict-ing systems of logic, uncertain and imprecise knowledge, and possibly vague

preferences have to be considered To face such complexity, preference ing requires the use of specific tools, techniques, and concepts which allow theavailable information to be represented with the appropriate granularity In thisperspective, fuzzy set theory has received a lot of attention in MCDA for a longtime Patrick Meyer and Marc Roubens present the fuzzy set approach to MCDAfor choice, ranking, and sorting problems In this chapter, several MCDA ap-proaches based on fuzzy evaluations are reviewed The authors give details on

model-a sorting procedure for the model-assignment of model-alternmodel-atives to grmodel-aded clmodel-asses whenthe available information is given by interacting points of view and a subset

of prototypic alternatives whose assignment is given beforehand A softwarededicated to that approach (TOMASO) is briefly presented Finally they recallthe concepts of good and bad choices based on dominant and absorbent kernels

in the valued digraph that corresponds to an ordinal valued outranking relation.Salvatore Greco, Benedetto Matarazzo and present thedecision rule approach to MCDA This approach represents the preferences interms of “if , then .” decision rules such as, for example, “if the maximumspeed of car is at least 175 km/h and its price is at most $12000, then car

is comprehensively at least medium” This approach is related to rough settheory and to artificial intelligence Its main advantages are the following The

DM gives information in the form of examples of decisions, which requiresrelatively low cognitive effort and which is quite natural The decision model isalso expressed in a very natural way by decision rules This permits an absolutetransparency of the methodology for the DM Another interesting feature ofthe decision rule approach is its flexibility, since any decision model can beexpressed in terms of decision rules and, even better, the decision rule modelcan be much more general than all other existing decision models used inMCDA

Michel Grabisch and Christophe Labreuche present the fuzzy integral proach that is known in MCDA for the last two decades In very simple wordsthis methodology permits a flexible modeling of the importance of criteria In-deed, fuzzy integrals are based on a capacity which assigns an importance toeach subset of criteria and not only to each single criterion Thus, the importance

ap-of a given set ap-of criteria is not necessarily equal to the sum ap-of the importance

of the criteria from the considered subset Consequently, if the importance ofthe whole subset of criteria is smaller than the sum of the importances of itsindividual criteria, then we observe a redundancy between criteria, which insome way represents overlapping points of view On the other hand, if the im-portance 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 evaluations ofwhich reinforce one another On the basis of the importance of criteria measured

by means of a capacity, the criteria are aggregated by means of specific fuzzy

integrals, the most important of which are the Choquet integral (for cardinalevaluations) and the Sugeno integral (for ordinal evaluations).

Finally, Helen Moshkovich, Alexander Mechitov and David Olson presentthe verbal decision methods MCDA This is a class of methods originated fromthe work of one of the MCDA pioneers, the late Oleg Larichev The idea ofverbal decision analysis is to build a decision model using mostly qualitativeinformation expressed in terms of a language that is natural for the DM More-over, measurement of criteria and preference elicitation should be psycholog-ically valid The methods, besides being mathematically sound, should checkthe DM’s consistency and provide transparent recommendations

The classical formulation of an Operations Research model is based on the imization or minimization of an objective function subject to some constraints

max-A very rich and powerful arsenal of methodologies and techniques has beendeveloped and continues to be developed within Operations Research How-ever, it is very difficult to summarize all the points of view related to the desiredresults of the decision at hand in only one objective function Thus, it seemsnatural to consider a very general formulation of decision problems where a set

of objective functions representing different criteria have to be “optimized” Todeal with these types of problems requires not only to generalize the method-ologies developed for classical single objective optimization problems, but also

to introduce new methodologies and techniques permitting to compare differentobjectives according to the preferences of the DM In this part of the book wetried to give adequate space to these two sides of multiobjective programmingproblems

Emphasis on the side of gathering information from the decision-maker andconsequent preference representation is given in the first chapter of this part, inwhich Pekka Korhonen introduces the main concepts and basic ideas of inter-active methods dealing with multiobjective programming problems The basicobservation is that, since the DM tries to “maximize” a set of criteria in con-flict with each other and an increment of one criterion can only be reached byaccepting a decrement of at one or more other criteria, we need to compare theadvantages coming from increments with respect to some criteria with the dis-advantages coming from corresponding decrements of other criteria A utility

or value function representing DM preferences would seem the most priate for this aim, but the key assumption in multiple objective programming

appro-is that thappro-is utility function appro-is unknown Therefore many methodologies havebeen proposed with the aim of developing a fruitful dialogue with the DM per-mitting, on the one hand, to provide the DM with relevant information aboutnon-dominated solutions and, on the other hand, to obtain useful information

about the preferences of the DM This dialogue is generally assisted by cific software, very often employing graphical representations of the results Itpermits to define a solution which the DM can accept as a good compromise.

spe-In the next chapter, Matthias Ehrgott and Margaret Wiecek introduce ematical methods to solve multiobjective programming (MOP) problems Intheir survey, they present solution concepts of MOP, properties of efficient andnondominated sets, optimality conditions, solution techniques, approximation

math-of efficient and nondominated sets, and specially-structured problems includinglinear and discrete MOPs as well as selected nonlinear MOPs The contents ofthe chapter have been selected on the idea that the primary (although not nec-essarily the ultimate) goal of multiobjective programming is to seek solutions

of MOPs and therefore a special attention was paid to methods suitable forfinding these solutions Since the ultimate goal of MOP problem is selection of

a preferred solution, for which an adequate representation of DM preferences

is necessary, this chapter is well complemented by the previous one

Masahiro Inuiguchi deals with multiple objective programming problemswith fuzzy coefficients The introduction of fuzziness in multiple objectiveprogramming is due to the observation that in real world problems imprecisespecifications of parameters fluctuating in certain ranges are very usual Forexample, let us consider an activity for which the acceptable expense is 100million dollars However, the DM may accept the expense of 100.1 million dol-lars if the objective functions take much better values by this small violation ofthe constraint Due to their specific nature, fuzzy multiobjective programmingproblems need an interpretation which leads to specific approaches to the prob-lem Since fuzzy programming has a relatively long history, many approachesrelated to different interpretations of the fuzzy MOP have been proposed In thischapter the approach based on necessity and possibility is considered, as many

of the approaches proposed in the specialized literature are of this type Thedifference to other approaches often lies solely in the measures employed forthe evaluation of a fuzzy event Thus, describing the approaches based on pos-sibility and necessity measures would be sufficient to acknowledge the essence

of multiple objective programming problems with fuzzy coefficients

Finally, this part is concluded by a chapter that deals with an area of erations Research in which multiobjective programming has been used quitefrequently Stefan Nickel, Justo Puerto and Antonio Rodríguez-Chía presentthe multiple criteria approach to locational analysis An important characteris-tic of location models is their intrinsic multiple criteria nature In this contextdifferent criteria are related to one or several new facilities and depend on thedistances of these facilities to the set of fixed or demand facilities There are atleast two natural ways of deriving the different criteria First, a decision about anew facility to be located is typically a group decision and each decision makerwill have his own preferences, which may be expressed by a corresponding

Op-criterion Secondly, the functions may represent different evaluation criteriafor the new facility to be located, like cost, reachability, risk, etc The chapterprovides a broad overview of the most representative multiple criteria locationproblems which have been divided into the three classes of continuous, network,and discrete problems.

It is apparent that the validity and success of all the developments of MCDAresearch are measured by the number and quality of the decisions supported byMCDA methodologies Applications in this case discriminate between resultsthat are really interesting for MCDA and results that, even though beautifuland interesting for economics, mathematics, psychology, or other scientificfields, are not interesting for MCDA The applications of MCDA in real worldproblems are very numerous and in very different fields Therefore, it was clearfrom the outset that it would be impossible to cover all the fields of application

of MCDA We decided to select some of the most significant areas

Jaap Spronk, Ralph Steuer and Constantin Zopounidis discuss the tions of MCDA in finance A very valuable feature of their chapter is the focus

contribu-on justificaticontribu-on of the multidimensicontribu-onal character of financial decisicontribu-ons and theuse of different MCDA methodologies to support them The presentation ofthe contributions of MCDA in finance permits to structure complex evaluationproblems in a scientific context and in a transparent and flexible way, with theintroduction of both quantitative (i.e., financial ratios) and qualitative criteria

in the evaluation process

Danae Diakoulaki, Carlos Henggeler Antunes and António Gomes Martinspresent applications of MCDA in energy planning problems In modern tech-nologically developed societies, decisions concerning energy planning must bemade in complex and sometimes ill-structured contexts, characterized by tech-nological evolution, changes in market structures, and new societal concerns.Decisions to be made by different agents (at utility companies, regulatory bod-ies, and governments) must take into account several aspects of evaluation such

as technical, socio-economic, and environmental ones, at various levels of cision making (ranging from the operational to the strategic level) and withdifferent time frames Thus, energy planning problems inherently involve mul-tiple, conflicting and incommensurate axes of evaluation The chapter aims atexamining to which extent the use of MCDA in energy planning applicationshas been influenced by those changes currently underway in the energy sector,

de-in the overall socio-economic context, and de-in particular to which extent it isadapted to the new needs and structuring and modelling requirements.João Clímaco and José Craveirinha present multiple criteria decision analysis

in telecommunication network planning and design Decision making processes

in this field take place in an increasingly complex and turbulent environmentinvolving multiple and potentially conflicting options Telecommunication net-works is an area where different socio-economic decisions involving commu-nication issues have to be made, but it is also an area where technologicalissues are of paramount importance This interaction between a complex socio-economic environment and the extremely fast development of new telecommu-nication technologies and services justifies the interest in using multiple criteriaevaluation in decision making processes The chapter presents a review of con-tributions in these areas, with particular emphasis on network modernisationplanning and routing problems and outlines an agenda of current and futureresearch trends and issues for MCDA in this area.

Finally, Giuseppe Munda addresses applications of MCDA in problems cerning sustainable development Sustainable development is strongly related

con-to environmental questions, i.e., sustainable development generalizes mental management taking into account not only an ecological but also socio-economic, technical and ethical perspectives Ecological problems were amongthe first to be dealt with by MCDA Therefore, there is a strong tradition in thisfield and many interesting stimuli for MCDA research came from there Theextensive perspective of sustainable development is very significant because itimproves the quality of decisions concerning the environment taking into ac-count other criteria, which are not strictly environmental but which stronglyinteract with it In making sustainability policies operational, basic questions

environ-to be answered are sustainability of what and whom? As a consequence, tainability issues are characterised by a high degree of conflict Therefore, inthis context MCDA appears as an adequate approach

Application of an MCDA method requires such a considerable amount of putation that even the development of many MCDA methodologies withoutthe use of a specialized software is hardly imaginable While software is aneven more important element in the application of MCDA methodologies, thisdoes not mean that to have a good software is sufficient to apply an MCDAmethodology correctly Clearly, software is a tool and it should be used as atool Before using a software, it is necessary to have a sound knowledge of theadopted methodology and of the decision problem at hand

com-After these remarks about cautious use of software, the problem is: Whatsoftware is available for MCDA? Heinz Roland Weistroffer, Subhash Narulaand Charles H Smith present well known MCDA software packages Whilethere is certainly some MCDA software available that is not present in thechapter, it can help the reader She may get suggestions of well known software,

but also information about aspects to be taken into account when evaluating asoftware for adoption in an application.

The editors are very grateful to Euro Beinat, Nabil Belacel, Denis sou, John Buchanan, João Clímaco, Danae Diakoulaki, Luís Dias, MichaelDoumpos, Ernest Forman, Philippe Fortemps, Lorraine Gardiner, ChristopheGonzales, Michel Grabisch, Winfried Hallerbach, Raimo P Hämäläinen, Car-los Henggeler Antunes, Masahiro Inuiguchi, Robin Keller, Pekka Korhonen,Masahiro Inuiguchi, Christophe Labreuche, Risto Lahdelma, Thierry Marchant,Benedetto Matarazzo, Manuel Matos, Nikolaos Matsatsinis, Kaisa Miettinen,Maria Franca Norese, Wlodzimierz Ogryczak, Patrice Perny, Jacques Pictet,Marc Pirlot, Jean-Charles Pomerol, Justo Puerto, Marc Roubens,

Bouys-Jerzy Stefanowsky, Ralph Steuer, Theo Stewart, Christianne Tammer,Jean-Claude Vansnick, Luis Vargas, Philippe Vincke, Peter Wakker, MargaretWiecek, Szymon Wilk who served as referees for this volume

The editors would also like to express their gratitude to people who supportedthem with very valuable advice along all the preparation of the book: BernardRoy, Denis Bouyssou, Benedetto Matarazzo, Gary Folvenand Frederick S Hillier

Acknowledgments

José Figueira was supported by the grant SFRH/BDP/6800/2001 (Fundaçãopara a Ciência e Tecnologia, Portugal) and gratefully acknowledges DIMACSResearch Center at Rutgers University and LAMSADE at University Paris-Dauphine for the welcome during his sabbatical leave and the short visits tothe Catania University, Auckland University and London School of Economics.His research has partially benefited also from MONET research grant (POCTI/GES/37707) and the luso-french scientifique collaborations ICCTI/Embassy

of France in Lisbon (500B4) and Program Pessoa 2004 Matthias Ehrgott waspartially supported by University of Auckland grant 3602178/9275 and by theDeutsche Forschungsgemeinschaft grant Ka 477/27-1

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AN OVERVIEW OF MCDA TECHNIQUES TODAY

PARADIGMS AND CHALLENGES

Abstract The purpose of this introductory part is to present an overall view of what MCDA

is today In Section 1, I will attempt to bring answers to questions such as: what

is it reasonable to expect from MCDA? Why decision aiding is more often criteria than monocriterion? What are the main limitations to objectivity? Section

multi-2 will be devoted to a presentation of the conceptual architecture that constitutes the main keys for analyzing and structuring problem situations Decision aiding cannot and must not be envisaged jointly with a hypothesis of perfect knowledge Different ways for apprehending the various sources of imperfect knowledge will

be introduced in Section 3 A robustness analysis is necessary in most cases The crucial question of how can we take into account all criteria comprehensively in order to compare potential actions to one another will be tackled in Section 4 In this introductory part, I will only present a general framework for positioning the main operational approaches that exist today In Section 5, I will discuss some more philosophical aspects of MCDA For providing some aid in a decision con- text, we have to choose among different paths which one seems to be the most appropriate, or how to combine some of them: the path of realism which leads

to the quest for a discussion for discovering, the axiomatic path which is often associated with the quest of norms for prescribing, or the path of constructivism which goes hand in hand with the quest of working hypothesis for recommending.

Keywords: Multiple criteria decision aiding, imperfect knowledge, aggregation procedures.