The Need for and Possible Methods of Objective Ranking

Abstract

The classical approach in decision analysis and multiple criteria theory concentrates on subjective ranking, at most including
some aspects of intersubjective ranking (ranking understood here in a wide sense, including the selection or a classification
of decision options). Intuitive subjective ranking should be distinguished here from rational subjective ranking, based on
the data relevant for the decision situation and on an approximation of personal preferences. However, in many practical situations,
the decision maker might not want to use personal preferences, but prefers to have some objective ranking. This need of rational
objective ranking might have many reasons, some of which are discussed in this chapter. Decision theory avoided the problem
of objective ranking partly because of the general doubt in objectivity characteristic for the twentieth century; the related
issues are also discussed. While an absolute objectivity is not attainable, the concept of objectivity can be treated as a
useful ideal worth striving for; in this sense, we characterize objective ranking as an approach to ranking that is as objective
as possible. Between possible multiple criteria approaches, the reference point approach seems to be most suited for rational
objective ranking. Some of the basic assumptions and philosophy of reference point approaches are recalled in this chapter.
Several approaches to define reference points based on statistical data are outlined. Examples show that such objective ranking
can be very useful in many management situations.

KeywordRational subjective ranking-Rational objective ranking-Objectivity-Reference point approaches

Full text

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Trends in Multiple Criteria Decision Analysis

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International Series in Operations Research
& Management Science
Volume 142
Series Editor:
Frederick S. Hillier
Stanford University, CA, USA
Special Editorial Consultant:
Camille C. Price
Stephen F. Austin State University, TX, USA
For other volumes:
http://www.springer.com/series/6161

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Matthias Ehrgott
Salvatore Greco
Editors
? Jos´ e Rui Figueira
Trends in Multiple Criteria
Decision Analysis
123

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Editors
Assoc. Prof. Matthias Ehrgott
The University of Auckland
Department of Engineering Science
Auckland 1142
New Zealand
m.ehrgott@auckland.ac.nz
Assoc. Prof. Jos´ e Rui Figueira
Instituto Superior Tecnico
Departamento de Engenharia e Gestao
Tagus Park, Av. Cavaco Silva
2780-990 Porto Salvo
Portugal
figueira@ist.utl.pt
Prof. Salvatore Greco
Universit` a di Catania
Facolt` a di Economia
Corso Italia 55
95129 Catania
Italy
salgreco@unict.it
ISSN 0884-8289
ISBN 978-1-4419-5903-4
DOI 10.1007/978-1-4419-5904-1
Springer New York Dordrecht Heidelberg London
e-ISBN 978-1-4419-5904-1
Library of Congress Control Number: 2010932006
c ? Springer Science+Business Media, LLC 2010
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
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Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

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Contents
List of Figures ...................................................................... vii
List of Tables........................................................................ ix
Introduction ........................................................................ xi
Matthias Ehrgott, Jos´ e Rui Figueira, and Salvatore Greco
1Dynamic MCDM, Habitual Domains and Competence
Set Analysis for Effective Decision Making
in Changeable Spaces ........................................................
Po-Lung Yu and Yen-Chu Chen
1
2The Need for and Possible Methods of Objective Ranking .............. 37
Andrzej P. Wierzbicki
3Preference Function Modelling: The Mathematical
Foundations of Decision Theory ............................................ 57
Jonathan Barzilai
4 Robustness in Multi-criteria Decision Aiding ............................. 87
Hassene Aissi and Bernard Roy
5 Preference Modelling, a Matter of Degree .................................123
Bernard De Baets and J´ anos Fodor
6Fuzzy Sets and Fuzzy Logic-Based Methods
in Multicriteria Decision Analysis ..........................................157
Radko Mesiar and Lucia Vavr´ ıkov´ a
7Argumentation Theory and Decision Aiding ..............................177
Wassila Ouerdane, Nicolas Maudet, and Alexis Tsouki` as
v

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viContents
8Problem Structuring and Multiple Criteria Decision
Analysis ........................................................................209
Valerie Belton and Theodor Stewart
9Robust Ordinal Regression ..................................................241
Salvatore Greco, Roman Słowi´ nski, Jos´ e Rui Figueira,
and Vincent Mousseau
10Stochastic Multicriteria Acceptability Analysis (SMAA) ................285
Risto Lahdelma and Pekka Salminen
11 Multiple Criteria Approaches to Group Decision
and Negotiation ...............................................................317
D. Marc Kilgour, Ye Chen, and Keith W. Hipel
12Recent Developments in Evolutionary Multi-Objective
Optimization ..................................................................339
Kalyanmoy Deb
13Multiple Criteria Decision Analysis and Geographic
Information Systems .........................................................369
Jacek Malczewski
Contributors ........................................................................397
Index.................................................................................407

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List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
2.1
8.1
8.2
The behavior mechanism ..................................................
The interrelationships among four elements of competence set ......... 21
Two domains of competence set analysis ................................. 22
Innovation dynamics ....................................................... 25
Decision blinds ............................................................. 31
Decision blind is reducing as we move from A to B then to C........... 32
The general OEAM spiral of evolutionary knowledge creation.......... 41
The process of MCDA (from [8]) .........................................211
An illustration of the process of “dynamic decision
problem structuring” suggested by Corner et al. [24] ....................215
Combining problem structuring and multicriteria modelling ............219
Causal map from Hawston workshops ....................................223
Overall value tree for fisheries rights allocation ..........................224
Synthesized map which provided the starting point for the
visual thinking workshop ..................................................226
SSM process and tools .....................................................228
Rich picture for King Communications and Security Ltd ................229
Multicriteria model for King Communications and
Security Ltd.................................................................231
NRF value tree..............................................................232
Example of the use of scenarios to define performance levels ...........233
Necessary partial ranking at the first iteration ............................278
Necessary partial ranking at the second iteration .........................278
Necessary partial ranking at the third iteration............................279
Traditional approach: decision model determines
the “best” solution based on criteria measurements and
DMs’ preferences...........................................................286
Inverse approach: identify preferences that are favourable
for each alternative solution................................................287
Favourable weights and acceptability indices in
deterministic 2-criterion case with linear utility function ................291
Favourable rank weights and rank acceptability indices in
deterministic 2-criterion case with linear utility function ................292
5
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
9.1
9.2
9.3
10.1
10.2
10.3
10.4
vii

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viiiList of Figures
10.5
10.6
10.7
10.8
Rank acceptability indices in deterministic 2-criterion problem .........293
Examples of simulated cardinal values for four ranks....................299
Feasible weight space in the 3-criterion case .............................300
Uniform distribution in the weight space in the 3-criterion
case (projected into two dimensions)......................................301
Uniformly distributed weights with interval constraints for weights ....302
Uniformly distributed weights with two constraints for
trade-off ratios ..............................................................303
Uniformly distributed weights with importance order....................303
With an additive utility/value function holistic preference
statements correspond to general linear constraints in the
weight space ................................................................304
DMs and stakeholders in Ralgreen BR....................................329
Feasible states in the Elmira conflict model...............................331
Option prioritizing for MoE ...............................................332
Evolution of the Elmira conflict ...........................................332
A set of points and the first non-dominated front are shown .............341
Schematic of a two-step multi-criteria optimization and
decision-making procedure ................................................343
A posteriori MCDM methodology
employing independent single-objective
optimizations ...............................................................344
Schematic of the NSGA-II procedure .....................................346
The crowding distance calculation ........................................347
NSGA-II on ZDT2 .........................................................348
NSGA-II on KUR ..........................................................348
Non-constrained-dominationfronts .......................................349
Obtained nondominated solutions using NSGA ..........................351
Four trade-off trajectories ..................................................351
Vector and raster data models in GIS......................................371
Cumulative numbers of the GIS-MCDA articles published
in refereed journals, 1990–2004[996] ....................................375
10.9
10.10
10.11
10.12
11.1
11.2
11.3
11.4
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
13.1
13.2

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List of Tables
1.1
1.2
A structure of goal functions ..............................................
Eight methods for expanding and enriching habitual
domains .................................................................... 17
Nine principles of deep knowledge ....................................... 18
Data for an example on international business management
(Empl. D employees) ...................................................... 51
An example of objective ranking and classification for the
data from Table 2.1 ........................................................ 52
Four different types of reference profile distributions .................... 54
Performance matrix of potential actions ..................................118
Efficient set of actions for b D 190........................................118
Efficient set of actions for b D 180........................................118
Parametric families of continuous distributions...........................152
Cycle-transitivity for the continuous distributions in Table 5.1 ..........152
Fox and Parsons’ argument scheme .......................................195
Atkinson’s argument scheme ..............................................196
Problem structuring methods and the link to MCDA.....................218
Consideration of MCDA in the light of Rosenhead and
Minger’s characteristics of problem structuring methods ................235
The set A of Pareto optimal solutions for the illustrative
MOO problem ..............................................................277
Problem with four alternatives and two criteria to be maximized........291
SMAA applications ........................................................307
Condorcet paradox: Preference orderings of 1, 2, and 3
over A, B, and C............................................................321
Satisfaction of screening criteria...........................................330
Classification of the GIS-MCDA articles according to the
GIS data model, the spatial dimension of the evaluation
criteria (EC), and the spatial definition of decision
alternatives (DA) ...........................................................377
8
1.3
2.1
2.2
2.3
4.1
4.2
4.3
5.1
5.2
7.1
7.2
8.1
8.2
9.1
10.1
10.2
11.1
11.2
13.1
ix

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x List of Tables
13.2Classification of the GIS-MCDA articles according to the
multicriteria decision rule; multiattribute decision analysis
(MADA) and multiobjective decision analysis (MODA).
Some articles presented more than one combination rule ................378
Classification of GIS-MCDA papers according to the type
of multicriteria decision method for individual decision maker..........379
Classification of GIS-MCDA papers according to the type
of multicriteria decision methods for group decision making............379
13.3
13.4

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Introduction
Matthias Ehrgott, Jos´ e Rui Figueira, and Salvatore Greco
1 Introduction
When 5 years ago we edited the book “Multiple Criteria Decision Analysis: State
of the Art Surveys” with 24 chapters written by 49 international leading experts, we
believed that the book would cover the research field for several years. But over the
last 5 years Multiple Criteria Decision Analysis (MCDA) has received an increas-
ing interest and has experienceda developmentfaster than we expected. Thus, what
looked like a comprehensive collection of state-of-the-art surveys appears clearly
partial and incomplete a few years later. New approaches and new methodologies
have been developed which even contribute to change the paradigm of MCDA. A
researcher who does not take into account the new contributed risks to be discon-
nected from the main trends of the discipline and to have a misleading conception
of it. These thoughts convinced us to explore the map of the new trends in MCDA
in order to recognize the most promising new contributions. This book comprises
13 chapters, once again written by leading international experts, that summarize
trends in MCDA that were not covered in our previous book and that describe the
development of rapidly evolving sub-fields of MCDA.
Po-Lung Yu and Yen-Chu Chen present the theory of dynamic multiple criteria
decision analysis, habitual domains, and competence set analysis. In real life, most
decisions are dynamic with multiple criteria. Even though most of the MCDA lit-
erature assumes that the parameters involved in decision problems – such as the set
of alternatives, the set of criteria, the preference structures of the decision makers
– are more or less fixed and steady, in reality – for most nontrivial decision prob-
lems – these parameters can change dynamically. In fact, satisfactory solutions are
obtained only when those parameters are properly structured. To analyze the deci-
sion process in a dynamic context the concepts of habitual domain and competence
set are of fundamental importance. A habitual domain is the set of ideas and con-
cepts which we encode and store in our brain, gradually stabilized over a period
of time. The competence set is a collection of ideas, knowledge, resources, skills,
and effort for the effective solution of a decision problem. Competence set analy-
sis and habitual domain theory suggest how to expand and enrich our competence
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xiiM. Ehrgott et al.
set and how to maximize the value of our competence set. In this perspective, any
decision problem can be dealt with by restructuring its elements and environmental
facets in order to gain a broader and richer perception permitting to derive effective
solutions.
Andrzej P. Wierzbicki discusses the need for and possible methods of objective
ranking after observing that the classical approach in decision analysis and multiple
criteria theory concentrates on subjective ranking. However, in many practical situ-
ations, the decision maker might not want to use personal preferences, but prefers
to have some objective ranking. One reason for objectivity is that decisions of a
given class might influence other people, e.g., some decision situations dominating
in technology creation, such as constructing a safe bridge or a safe car. Thus, tech-
nologists stress objectivity but real managers also know well that there are many
managerial situations where stressing objectivity is necessary. Therefore, even if it
canbeagreedthatanabsoluteobjectivityisnotattainable,itis reasonabletotreatthe
conceptofobjectivityas a usefulideal worth strivingfor,lookingforobjectiverank-
ing interpreted as an approach to ranking that is as objective as possible. Between
many possible multiple criteria approaches, the reference point approach (already
introduced in the literature to deal with interactive multiple criteria optimization) is
mentioned as the best suited methodology for rational objective ranking, because
reference levels needed in this approach can be established to some extent objec-
tively – statistically from the given data set.
Jonathan Barzilai in his provocative chapter discusses preference function mod-
elling, i.e., the mathematical foundations of decision theory. He formulates the
conditions that must be satisfied for the mathematical operations of linear alge-
bra and calculus to be applicable and claims that the mathematical foundations of
decision theory and related theories depend on these conditions, which have not
been correctly identified in the classical literature. He argues that Operations Re-
search and Decision Analysis Societies should act to correct fundamental errors in
the mathematical foundations of measurement theory, utility theory, game theory,
mathematical economics, decision theory, mathematical psychology, and related
disciplines. Consequences of this approach to some MCDA methodologies such
as AHP or value theory are also discussed.
Hassene Aissi and Bernard Roy discuss robustness in MCDA. The term robust
refers to a capacity for withstanding“vagueapproximations”and/or “zones of igno-
rance” in order to prevent undesirable impacts. Robustness concerns are related to
theobservationthatanactionismade,executed,andjudgedinareal-lifecontextthat
may not correspond exactly to the model on which the decision analysis is based.
The gap between formal representationand real-life context originates frailty points
against which the robustness concern attempts to protect. Robustness concerns can
be dealt with using approaches involving a single robustness criterion, completing
a preference system that has been defined previously, or using several criteria. Ro-
bustness can be consideredother thanby usingone orseveral criteria to comparethe
solutions in approaches that involve one or several properties designed to character-
ize the robust solution or to draw robust conclusions. The considerations developed

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Introductionxiii
in this chapter show that the use of multiple criteria for apprehending robustness
in MCDA is a field of research open to future development, both theoretically and
practically.
Bernard De Baets and J´ anos Fodor consider preferences expressed in a gradual
way. The key concept is that the application of two-valued (yes or-no) preferences,
regardless of their sound mathematical theory, is not satisfactory in everyday situ-
ations. Therefore, it is desirable to consider a degree of preference. There are two
main frameworks in which gradual preferences can be modeled: fuzzy preferences,
which are a generalization of Boolean (2-valued) preference structures, and recip-
rocal preferences, also known as probabilistic relations, which are generalization of
the three-valued representation of complete Boolean preference relations. The au-
thors consider both frameworks. Since the whole exposition makes extensive use of
(logical) connectives, such as conjunctors, quasi-copulas and copulas, the authors
provide an appropriate introduction on the topic.
RadkoMesiarandLuciaVavr´ ıkov´ apresentfuzzyset andfuzzylogic-basedmeth-
ods for MCDA. Alternatives are evaluated with respect to each criterion on a scale
between 0 and 1, which can be seen as membership function of fuzzy sets. There-
fore, alternatives can be seen as multidimensional fuzzy evaluations that have to be
ordered according to the decision maker’s preferences. This chapter considers sev-
eralmethodologiesdevelopedwithinfuzzysettheorytoobtainthispreferenceorder.
After discussion of integral-based utility functions, a transformation of vectors of
fuzzy scores x into fuzzy quantity U.x/ is presented. Orderings on fuzzy quantities
induce orderings on alternatives. Special attention is paid to defuzzification-based
orderings,in particular,the meanof maxima method.Moreover,a fuzzylogic-based
construction method to build complete preference structures over the set of alterna-
tives is given.
Wassila Ouerdane, Nicolas Maudet, and Alexis Tsouki` as discuss argumentation
theory in MCDA. The main idea is that decision support can be seen as an activity
aiming to construct arguments through which a decision maker will convince first
herself and then other actors involved in a problem situation that “that action” is the
best one. In this context the authors introduce argumentationtheory (in an Artificial
Intelligence oriented perspective) and review a number of approaches that indeed
use argumentative techniques to support decision making, with a specific emphasis
on their application to MCDA.
Valerie Belton and Theodor Stewart introduce problem structuring methods
(PSM) in MCDA providing an overview of current thinking and practice with re-
gard to PSM for MCDA. Much of the literature on MCDA focuses on methods of
analysis that take a well-structured problem as a starting point with a well-defined
set ofalternativesfromwhicha decisionhas tobe madeanda coherentset of criteria
against which the alternatives are to be evaluated. It is an erroneous impression that
arriving at this point is a relatively trivial task, while in reality this is not so simple
evenwhenthe decisionmakersbelieveto havea clearunderstandingof theproblem.
Thus, PSM provides a rich representation of a problematic situation in order to en-
able effective multicriteria analysis or to conceptualize a decision, which is initially
simplistically presented, in order for the multicriteria problem to be appropriately

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xivM. Ehrgott et al.
framed. The chapter outlines the key literature, which explores and offers sugges-
tions on how this task might be approachedin practice, reviewing several suggested
approaches and presenting a selection of case studies.
Salvatore Greco, Roman Słowi´ nski, Jos´ e Rui Figueira, and Vincent Mousseau
present robust ordinal regression. Within the disaggregation–aggregationapproach,
ordinal regression aims at inducing parameters of a preference model, for example,
parameters of a value function, which represent some holistic preference compar-
isons of alternatives given by the decision maker. Usually, from among manysets of
parameters of a preference model representing the preference information given by
the DM, only one specific set is selected and used to work out a recommendation.
For example, while there exist many value functions representing the holistic pref-
erence information given by the DM, only one value function is typically used to
recommend the best choice, sorting, or ranking of alternatives. Since the selection
of one from among many sets of parameters of the preference model compatible
with the preference information given by the DM is rather arbitrary, robust ordinal
regression proposes taking into account all the sets of parameters of the preference
model compatible with the preference information, in order to give a recommenda-
tion in terms of necessary and possible consequences of applying all the compatible
preference models on the considered set of alternatives. For example, the necessary
weak preference relation holds for any two alternatives a and b if and only if all
compatible value functions give to a a value greater than or equal to the value pro-
vided to b, and the possible weak preference relation holds for this pair if and only
if at least one compatible value function gives to a a value greater than or equal to
the valuegivento b. This approachcan be appliedto manymultiplecriteria decision
models such as multiple attribute utility theory, fuzzy integral modeling interaction
between criteria, and outranking models. Moreover, it can be applied to interactive
multiple objective optimization and can be used within an evolutionarymultiple ob-
jective optimization methodology to take into account preferences of the decision
maker. Finally, robust ordinal regression is very useful in group decisions where it
permits to detect zones of consensus for decision makers.
RistoLahdelmaandPekkaSalminenpresentStochasticMulticriteriaAcceptabil-
ity Analysis (SMAA). SMAA is a family of methods for aiding multicriteria group
decision making in problems with uncertain, imprecise, or partially missing infor-
mation. SMAA is based on simulating different value combinations for uncertain
parameters, and computing statistics about how the alternatives are evaluated. De-
pendingon the problemsetting, this can meancomputinghow ofteneach alternative
becomes most preferred, how often it receives a particular rank, or obtains a partic-
ular classification. Moreover, SMAA proposes inverse weight space analysis, using
simulation with randomized weights in order to reveal what kind of weights make
each alternative solution most preferred.After discussing several variants of SMAA
the authors describe several real-life applications.
D. Marc Kilgour, Ye Chen, and Keith W. Hipel discuss multiple criteria ap-
proaches to Group Decision and Negotiation (GDN). After explaining group deci-
sion and negotiation, and the differences between them, the applicability of MCDA
techniques to problems of group decision and negotiation is discussed. Application