Fuzzy Sets and Systems 20 (1986) 45-53 North-Holland 45 SOME PROBLEMS ON THE DEFINITION OF FUZZY PREFERENCE RELATIONS F.J. MONTERO and J. TEJADA Deparramento de Estadistica e I. 0.) Faculrad de Ciencias Matemdticas, Universidad Complutense, Madrid 28040, Spain Received August 1985 Revised October 1985 In this paper we deal with decision-making problems over an unfirzxy set of alternatives. On one hand, we propose the problem of finding a max-min transitive relation as near as possible to a given initial preference relation, under the least-squares criterion and such that it does not introduce deep qualitative changes. On the other hand, we define a linear extension of the initial preference relation between alternatives to a preference relation between lotteries. Keywords: Decision making, Fuzzy relations, Nondominated alternatives. 1. Introduction Since much of our information is of a fuzzy nature, decision making needs Fuzzy Set Theory in order to get good enough decisions. In this way, Fuzzy Set Theory allows us to develop some aspects of the concept of ‘satisficing’ introduced by Simon [8] to explain a more realistic behaviour of human being. Given a set X of n unfuzzy alternatives, we will suppose that the basic information is a binary preference relation defined over X, reflecting the fuzziness of individual estimates of preferences. The rationality of this approach is pointed out by Basu [l]: in many situations an appropriate tool to capture the inherent imprecision of human value judgements may be fuzzy preferences. Decision making will look for maximal or nondominated alternatives. If this set is nonempty, rational choice will be inside it. A fuzzy preference relation in X is a fuzzy subset of the Cartesian product X X X, with membership function /.A :x x x+ [O, l] in such a way that ,u(x, y) should represent the degree of preference of the alternative x to the alternative y. For a given fuzzy preference relation p, its fuzzy strict preference relation CL%, Y) = m=M4 Y) - PCY, -GO> is introduced by Orlovsky [5] in order to define the fuzzy set of nondominated 01650114/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland)