Representation, aggregation and classification models for decision

Jornadas de Seguimiento de Proyectos, 2009
Programa Nacional de Tecnologı́as Informáticas
Representation, aggregation and classification
models for decision making aiding under
imprecise information
TIN2006-06190
Javier Montero ∗
Faculty of Mathematics
Complutense University of Madrid
Abstract
This project represents an evolution within a consolidated research group, which has
been granted from the Spanish Programme for the Promotion of General Knowledge since
1988. The project is based on past results obtained in the framework of previous projects.
The main core of this project is to study different components of decision aid tools under
different kinds of uncertainty, being these tools mainly understood as representation techniques. In particular, a key paper published by this research team allows measurements
of the quality of each classification system, taking into account particular families of operational and consistent aggregation rules, being therefore the basis for subsequent learning
and optimization procedures. This approach is specially significative when classes have
no clear borders and large transition zones appear, since in this context a pure statistical
analysis may be inappropriate. Hence, particular emphasis is devoted to develop algorithm representation techniques with non probabilistic uncertainty. The project pursues
the design of classification aid models by putting these pillars together, to be checked in
several scientific fields where we have already some experience, in collaboration with other
research groups (stars or cells classification, land cover use, systems reliability, preference
representation, etc.)
Keywords: Decision making, preference representation, knowledge representation, aggregation information, classification, fuzzy sets, optimization.
1
Objectives
Being this project a project on basic research, it must be made clear that the first objective we
pursue is the development of models and tools potentially useful for some specific problems,
not the practical resolution of these problems. Nevertheless, the policy of this research team
has been always to strength scientific links with partners in direct contact to those problems.
Without them we can not produce useful results. We think this procedure is in the best
∗ Email:
monty@mat.ucm.es
TIN2006-06190
scientific tradition and indeed they provide us the inspiration framework for most theoretical
objectives. Whenever we get so close to a real implementation, a contract is signed in advance
in order to clarify scientific achievements from their industrial implications. In this sense, this
team has always stressed the operationality of each approach, avoiding the development of tools
that can not be taken by our partners, because its computational or conceptual requirements.
There were five main theoretical objectives being declared in the application of this still
ongoing project:
1. The bottom aggregative subproject has deserved a special attention, being a main component of the project. In particular, we have stressed the recursive approach for aggregation
operations as a constructive approach that contains a relevant proportion of aggregation
rules that can be found in practice. Although we have shown also some limits, perhaps
the most relevant consequence of our study is the paper that appeared in Fuzzy Sets and
Systems [18] pointing out that structure should be made always explicit in any context
(objects, criteria, classes, alternatives, etc.) The general model proposed in this paper
will become one of the key foundations for next next project we are actually applying to. As a first consequence, this paper offers an interesting explanation of the great
dispute that has been created around Atanassov’s ”Intuitionistic” fuzzy sets: it is the
absence of such a structure the missing argument that makes a difference between two
models with different intuitive support. It is a extremely original ongoing research. In
parallel, several studies have been developed on interval fuzzy sets, in collaboration to
H. Bustince’s research group at the Public University of Navarra [33, 35]. In addition,
we have obtained a functional characterization of migrativity [34] in collaboration to R.
Mesiar from Slovakia.
2. The consistency issue has also devoted an attention that has been always present in past
projects. In particular, we have developed several algorithms to approximate inconsistent preferences to the closet consistent preference, according to several transitive-based
consistencies [7]. Related to consistency we have also presented original approaches to Arrow’s paradox in group decision making [58] together with some positional paper pointing
out the role of fuzziness in decision making and Science [19, 60].
3. In order to deal with the computational problems related to dimensional representation,
we have developed an alternative representation approach [51]. Although this approach
has been shown useful to estimate criteria weights, we are still working on the uniqueness
issue (i.e., which one is the good representation among the family of possible representations. It is again an extremely original research that has lead in the past to the first
algorithm to evaluate dimension within the crisp context (published in Journal of Algorithms).
4. Great results have been obtained within the segmentation and coloration subproject,
developing and applying our algorithms to remotely sensed images, with the collaboration
of G. Biging at the University of California, Berkeley (USA). Improved algorithms have
been applied to real images [47, 49, 50]. Classical crisp models have been translated
to the fuzzy context, allowing a fuzzy clustering that is conceptually simpler than the
equivalent crisp clustering, and more informative due to the structure of regions being
created. It is again quite original and it is increasingly cited by other researchers.
TIN2006-06190
5. First steps towards a fuzzy verification process have been developed [56, 57], an issue
that we think deserve more effort since quite a number of practical situations do not
allow an appropriate statistical analysis to check results (quite often the consistency
of the supporting argument or algorithm is the only support for a decision). There is
still a lot of work ahead, but we are missing a more sophisticated partner to fix the
mathematical foundation for a general model, and we are actually trying to introduce
some new elements [26, 27].
From a practical point of view, consequences of theoretical results in the subsequent modeling have been tested in different practical frameworks, together with a number of decision
making fields that allow a more complete view of the general project that guides this particular
three-years research. Additional and relevant results have been obtained in related fields, that
not being included in the list of main objectives, will perform the next three-years research
(there is for example an underlying absolutely needed research related to graphs and networks
as key structures to be taken into account in the next future). The long-term objective of the
research group, perhaps never formally stated in a three-years project, is to create a interdisciplinary group with theoretical, technological and human capacity to address every stage of
a decision making process, from experimentation, data analysis and representation tools, decision analysis (amalgamating and decomposition), choice and final verification of the process
itself and its results. At this point, important results relative to decision making modeling have
been obtained in scheduling and social networks. These collaborations have been developed
with the Department of Natural Resources at UC Berkeley, the US technological company
Smiths Aerospace (Grand Rapids, Michigan), the Department of Astronomy at Complutense
University of Madrid, some Non-Governmental Organizations (like Red Cross and Médicos sin
Fronteras) and some national energy and transport companies, among others, taking advantage of the interdisciplinary composition of our research group and in some particular cases
bringing some complementary economic (and human) support from other institutions.
2
Status of the project
Below we include a quite complete list of articles produced during the first two years of the
project. It is important to realize that the project has almost 1/3 of its time ahead, so we
expect to complete results during 2009 (we plan to ask on time for an extension of the project
till December 31 because of the initial delay in the real access to funding. Nevertheless, we
think results already obtained are extremely good not only because of the number and quality
of publications produced, but because of the maturity reached by the research group, which has
extended contacts with other research groups in Madrid, Spain and the whole word. The research group has been able to get involved into this project to new young and senior researchers
(some of them barely needed in order to cover the whole decision making process), and has
attained major impact in the scientific community. The organization of the 8th international
FLINS conference (200 participants, see http://www.mat.ucm.es/congresos/flins2008) including the active participation of leading researchers in the field like L.A. Zadeh, P. Bonissone,
W. Pedrycz, J. Kacprzyk, E. Trillas and F. Herrera, among others, and the organization of the
1st FuzzyMAD meeting for the whole fuzzy community in Madrid (40 attendants), had helped
the group to get ready to more important scientific objectives.
TIN2006-06190
In the present research team we are 12 researchers: V. Cutello, L. Garmendia, D. Gómez,
E. Kerre, V. López, J. Montero (I.P.), S. Muñoz, T. Ortuño, J.T. Rodrı́guez (Ph.D. student),
B. Vitoriano, J. Yáñez plus E. Roano, recently incorporated. It is indeed relevant the number
of collaborations (in terms of joint published papers) with other research groups that the new
team (that will hopefully include profs. T. Calvo, L. de Ledesma and A. Pradera plus 2 Ph.D.
students) will present as a guarantee for the next three-years project application within the
national research framework:
• At Complutense University, the research groups of A.I. Gómez de Castro (Dept. of
Astronomy), M. Santos (Faculty of Informatics) and J. Tejada (Statistics and Operational
Research).
• In Spain, the groups of H. Bustince (Public University of Navarra) and L. Escudero (Rey
Juan Carlos University) among the rest, but also the groups of E. Carreño (National
Geographical Institute), E. Castiñeira and S. Cubillo (Technical University of Madrid), F.
Herrera (Granada University), J.L. Garcı́a Lapresta (Valladolid University), L. Martı́nez
(University of Jaén), G. Mayor (University of the Balearic Islands), P. Sobrevilla and E.
Montseny (Technical University of Catalunya) and E. Trillas (European Centre for Soft
Computing, Mieres).
• And around the world, A. Amo (GE Aviation Systems, Grand Rapids, Michigan, USA),
G. Beliakov (Australia), G. Biging (University of California at Berkeley, USA), K. Cios
(moving from the University of Ohio, Toledo, USA), V. Cutello (University of Catania,
Italy), B. De Baets (University of Gent, Belgium), J. Fodor (Hungary), E.E. Kerre (University of Gent, Belgium), J. Lu (Australia), A.D. Pearman (University of Leeds, UK)
and R.R. Yager (University of New York, USA)
A weakness of the present team is the low number of Ph.D. students, but the fact is that
during the last years quite a number of students asked us to join our research group, but finally
we had to redirect them to join other research teams with Ph.D. grants. Getting Ph.D. grants
associated to the new research project we are actually applying to is a major priority.
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TIN2006-06190
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