This book constitutes the major results of the EU COST (European Cooperation in the field of Scientific and Technical Research) Action 274: TARSKI - Theory and Applications of Relational Structures as Knowledge Instruments - running from July 2002 to June 2005.

The 17 revised full papers were carefully reviewed and selected for presentation. The papers are devoted to further understanding of interdisciplinary issues involving relational reasoning by addressing relational structures and the use of relational methods in applicable object domains such as non-classical logics, multimodal logics and relational logics, binary relation logic, algebraic logic, fuzzy preference relations, lattices, dominance relationship, extending aggregation operators, and various applications.

Relational structures abound in our daily environment: relational databases, data mining, scaling procedures, preference relations, etc. As the documentation of scientific results achieved within the European COST Action 274, TARSKI, this book advances the understanding of relational structures and the use of relational methods in various application fields.

The 12 revised full papers were carefully reviewed and selected for presentations. The papers are devoted to mechanization of relational reasoning, relational scaling and preferences, and algebraic and logical foundations of real world relations.

The title of this book seems to indicate that the volume is dedicated to a very specialized and narrow area, i. e. , to the relationship between a very special type of optimization and mathematical programming. The contrary is however true. Optimization is certainly a very old and classical area which is of high concern to many disciplines. Engineering as well as management, politics as well as medicine, artificial intelligence as well as operations research, and many other fields are in one way or another concerned with optimization of designs, decisions, structures, procedures, or information processes. It is therefore not surprising that optimization has not grown in a homogeneous way in one discipline either. Traditionally, there was a distinct difference between optimization in engineering, optimization in management, and optimization as it was treated in mathematical sciences. However, for the last decades all these fields have to an increasing degree interacted and contributed to the area of optimization or decision making. In some respects, new disciplines such as artificial intelligence, descriptive decision theory, or modern operations research have facilitated, or even made possible the interaction between the different classical disciplines because they provided bridges and links between areas which had been developing and applied quite independently before. The development of optimiiation over the last decades can best be appreciated when looking at the traditional model of optimization. For a well-structured, Le.

Bernard ROY Professor, University of Paris-Dauphine Director of LAMSADE 11 is not unusual for a dozen or so loosely related working papers to be published in book form as the natural outgrowth of a scientific gathering. Although many a volu- me of collected papers has come into point in this way, the homogeneity of the arti- cles included will often be more apparent than real. As the reader will quickly ob- serve, such is not the case with the present volume. As one can judge from its ti- tle, 1t is in fact an outcome of an ed~torial project by J. Kacprzyk and M. Roubens. T~ey asked contributing authors to submit recent works which would examine. within a non-traditional theoretical framework, preference analysis and preference modeliing 1n a fuzzy context oriented towards decision aid. The articles by J.P. Ooignon, B. Monjardet, T. Tanino and Ph. Vincke empnasize the analysis of oreference structures, mainly in the presence of incomparability. In- transitivlty, thresholds and, more generally, inaccurate determination. Considera- ble attention is devoted to the analysis of efficient and non-dominated (in Pareto's sense of the term) decisions in the four papers presented by S. Ovchinnikov and M.

The following scheme summarizes the different families introduced in this chapter and the connections between them. Family of interval orders f Row-homogeneous Column-homogeneous Family of family of interval semi orders family of interval orders orders Homogeneous family of i nterva 1 orders Homogeneous family of semi orders Family of weak orders 85 5.13. EXAMPLES We let to the reader the verification of the following assertions. Example 1 is a family of interval orders which is neither row-homogeneous nor column-homogeneous. Example 2 is a column-homogeneous family of interval orders which is not row-homogeneous but where each interval order is a semiorder. Example 3 is an homogeneous family of interval orders which are not semiorders. Example 4 is an homogeneous family of semi orders . . 8 ~ __ --,b ~---i>---_ C a .2 d c Example Example 2 .8 .6 c .5 a 0 a d Example 3 Example 4 5.14. REFERENCES DOIGNON. J.-P ** Generalizations of interval orders. in E. Degreef and J. Van Buggenhaut (eds). T~ndS in MathematiaaZ PsyahoZogy. Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1984. FISHBURN. P.C., Intransitive indifference with unequal indifference intervals. J. Math. Psyaho.~ 7 (1970) 144-149. FISHBURN. P.C., Binary choice probabilities: on the varieties of stochastic transitivity. J. Math. Psyaho.~ 10 (1973) 327-352.