Maximal Ordinal Two-Factorizations

Given a formal context, an ordinal factor is a subset of its incidence relation that forms a chain in the concept lattice, i.e., a part of the dataset that corresponds to a linear order. To visualize the data in a formal context, Ganter and Glodeanu proposed a biplot based on two ordinal factors. For the biplot to be useful, it is important that these factors comprise as much data points as possible, i.e., that they cover a large part of the incidence relation. In this work, we investigate such ordinal two-factorizations. First, we investigate for formal contexts that omit ordinal two-factorizations the disjointness of the two factors. Then, we show that deciding on the existence of two-factorizations of a given size is an NP-complete problem which makes computing maximal factorizations computationally expensive. Finally, we provide the algorithm Ord2Factor that allows us to compute large ordinal two-factorizations.Comment: 15 pages, 6 figures, 2 algorithms, 28th International Conference on Conceptual Structure

Preference Modelling

This paper provides the reader with a presentation of preference modelling fundamental notions as well as some recent results in this field. Preference modelling is an inevitable step in a variety of fields: economy, sociology, psychology, mathematical programming, even medicine, archaeology, and obviously decision analysis. Our notation and some basic definitions, such as those of binary relation, properties and ordered sets, are presented at the beginning of the paper. We start by discussing different reasons for constructing a model or preference. We then go through a number of issues that influence the construction of preference models. Different formalisations besides classical logic such as fuzzy sets and non-classical logics become necessary. We then present different types of preference structures reflecting the behavior of a decision-maker: classical, extended and valued ones. It is relevant to have a numerical representation of preferences: functional representations, value functions. The concepts of thresholds and minimal representation are also introduced in this section. In section 7, we briefly explore the concept of deontic logic (logic of preference) and other formalisms associated with "compact representation of preferences" introduced for special purpoes. We end the paper with some concluding remarks

A theoretical look at ELECTRE TRI-nB and related sorting models

ELECTRE TRI is a set of methods designed to sort alternatives evaluated on several attributes into ordered categories. The original ELECTRE TRI-B method uses one limiting profile per category. A more recent method, ELECTRE TRI-nB, allows one to use several limiting profiles for each category. We investigate the properties of ELECTRE TRI-nB. When the number of limiting profiles used to define each category is not restricted, ELECTRE TRI-nB is easy to characterize axiomatically and is found to be equivalent to several other methods proposed in the literature. We extend this result in various directions.Comment: 40 page

Comparison of random variables from a game-theoretic perspective

This work consists of four related parts, divided into eight chapters. A ¯rst part introduces the framework of cycle-transitivity, developed by De Baets et al. It is shown that this framework is ideally suited for describing and compar- ing forms of transitivity of probabilistic relations. Not only does it encompass most already known concepts of transitivity, it is also ideally suited to describe new types of transitivity that are encountered in this work (such as isostochas- tic transitivity and dice-transitivity). The author made many non-trivial and sometimes vital contributions to the development of this framework. A second part consists of the development and study of a new method to compare random variables. This method, which bears the name generalized dice model, was developed by De Meyer et al. and De Schuymer et al., and can be seen as a graded alternative to the well-known concept of ¯rst degree stochastic dominance. A third part involves the determination of the optimal strategies of three game variants that are closely related to the developed comparison scheme. The de¯nitions of these variants di®er from each other solely by the copula that is used to de¯ne the payo® matrix. It turns out however that the characterization of the optimal strategies, done by De Schuymer et al., is completely di®erent for each variant. A last part includes the study of some combinatorial problems that orig- inated from the investigation of the transitivity of probabilistic relations ob- tained by utilizing the developed method to compare random variables. The study, done by De Schuymer et al., includes the introduction of some new and interesting concepts in partition theory and combinatorics. A more thorough discussion, in which each section of this work is taken into account, can be found in the overview at the beginning of this manuscript. Although this work is oriented towards a mathematical audience, the intro- duced concepts are immediately applicable in practical situations. Firstly, the framework of cycle-transitivity provides an easy means to represent and compare obtained probabilistic relations. Secondly, the generalized dice model delivers a useful alternative to the concept of stochastic dominance for comparing random variables. Thirdly, the considered dice games can be viewed in an economical context in which competitors have the same resources and alternatives, and must choose how to distribute these resources over their alternatives. Finally, it must be noted that this work still leaves opportunities for future research. As immediate candidates we see, ¯rstly the investigation of the tran- sitivity of generalized dice models in which the random variables are pairwisely coupled by a di®erent copula. Secondly, the characterization of the transitivity of higher-dimensional dice models, starting with dimension 4. Thirdly, the study of the applicability of the introduced comparison schemes in areas such as mar- ket e±ciency, portfolio selection, risk estimation, capital budgeting, discounted cash °ow analysis, etc