L-fuzzy sets

Abstract

This paper explores the foundations of, generalizes, and continues the work of Zadeh in [I] and [2]. Perhaps the most important generalization is the consideration of order structures beyond the unit interval. Because of this we have been able to develop a new point of view toward optimization problems. The significance of this work may lie more in its point of view than in any particular results. The theory is still young, and no doubt many concepts have yet to be formulated, while others have yet to take their final form. However it should now be possible to visualize the outlines of the theory. Throughout the development of the theory of fuzzy sets, pattern classification has been a seminal influence. One reason for this is the natural feeling that probability theory is not appropriate for treating the kind of uncertainty that appears in pattern classification; this uncertainty seems to be more of an ambiguity than a statistical variation. Similar difficulties arise in a wide variety of problems. It is characteristic of attempts to apply probability theory to them that it is difficult or impossible to estimate the distributions assumed to be involved, that there is uncertainty about the nature of the statistical assumptions (independence, etc.), or that certain parameters are ignored, taken as given, or found difficult to estimate. Under these circumstances, the chief use of probability theory has been to partially justify intuitively appealing procedures, to suggest procedures already found useful in statistics, or to provide some sort of insight into the nature of things. We believe that fuzzy sets should be able to do at least this much. Let us consider some specific examples. A housewife faces a fairly typical optimization problem in her grocery shopping: she must select among all possible grocery bundles one that meets as well as possible several conflicting criteria of optimality, such as cost, nutritional value, quality, and variety. The partial ordering of the bundles is an intrinsic quality of this problem.