By P. C. Fishburn (auth.), Dr. Janusz Kacprzyk, Prof. Dr. Marc Roubens (eds.)
The $64000 factor of the way to beat rigidness, inadequacy and human inconsistency relating to traditional assumptions on personal tastes in selection making (for instance, concerning yes/no crispness or transitivity) is mentioned by means of recognized specialists during this quantity. within the introductory articles, analyses of these traditional assumptions are given and the necessity for reconsiderations and adjustments as to preference-related facets is encouraged. the subsequent contributions are ordinarily interested in concerns with regards to valued (including fuzzy) choice kinfolk, similar to research in their houses and their use in numerous determination making and selection difficulties and in staff determination making.
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Extra info for Non-Conventional Preference Relations in Decision Making
Preference, indifFerence, incomparabilite. 1SAOE n'9, Universite de Paris-Dauphine, 1980. , and VI"ICKE, Ph. Pseudo-orders definition, properties and numerical representation. l1atl7ematical Soc/al Sciences, 1987, 14 to appear. SCOTT, D. and SUPPES, P. Foundational aspects of theories of measurement. Journal of Symbolic Logie, 1958, 23,113-128. WIENER, N. Contribution to the theory of relative position Proceediflgs of tile Cambridge Pl7llosopl7lcal Society, i 914, 17, 441 -449 A SENERAlISATION OF PROBABILISTIC CONSISTENCY: LINEARITY CONDITIONS FOR YALUED PREFERENCE RELATIONS Bernard t10NJARDET Univsrsite Paris V and Centre d'Analyse et de MethematiQue Sociales 54, bd Respeil 75 270 Peris Cedex 06 - Frence ABSTRACT, We show how the theory of "probabll1stic consistency" developped in the framework of pair comparison methods with forced choice can be extended to orbitrory volued preference relations.
5 indicates that x. is indifferent to x. (denoted >- J Xi - lJ J l Max-min transitivity implies the usual transitivity: x j ). Vi, j ,k. 10 is a class of weak orders on X. 6) We may consider the class of linear orders on X as follows: E Jr Ji 1 C '}1 = l R Of course, ri j = a or 1 for iij}. 7) 'j 0 . The following proposition shows that every fuzzy preference relation is a convex combination of nonfuzzy preference relations. 1. '} = co where co Jr i ':! 0 = co 11 , denotes the convex hull of convex combinations of elements in :Pi) :, i (i.
A preference relation on X is modelled by a binary relation P on X. We write (X,y) e P or x Py It the two objects x and yare related by P, and write (x,Y) ,. P or #y otherwise; we write xly If and only if xJ"l1 and ypCx, and xtiy if and only if ypcx. We can Interpret XPy as a strict preference of x to y, xIy as an Indifference betWeen x and y. For binary relations P and Q on X, PQ is the usual relative product, that is x PQy if and only if th~r~ is som~ z c X such that xPz and :zQy. For each ty~ of model in the follOwing list we give a relational definition and an 37 equivalent characterization based on a real-valued representation (we omit the quantifications like "for each x, y in X").