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