By decision-making in a fuzzy environment is meant a decision process in which the goals and/or the constraints, but not necessarily the system under control, are fuzzy in nature. This means that the goals and/or the constraints constitute classes of alternatives whose boundaries are not sharply defined.

An example of a fuzzy constraint is: “The cost of A should not be substantially higher than α,” where α is a specified constant. Similarly, an example of a fuzzy goal is: “x should be in the vicinity of x0,” where x0 is a constant. The italicized words are the sources of fuzziness in these examples.

Fuzzy goals and fuzzy constraints can be defined precisely as fuzzy sets in the space of alternatives. A fuzzy decision, then, may be viewed as an intersection of the given goals and constraints. A maximizing decision is defined as a point in the space of alternatives at which the membership function of a fuzzy decision attains its maximum value.

The use of these concepts is illustrated by examples involving multistage decision processes in which the system under control is either deterministic or stochastic. By using dynamic programming, the determination of a maximizing decision is reduced to the solution of a system of functional equations. A reverse-flow technique is described for the solution of a functional equation arising in connection with a decision process in which the termination time is defined implicitly by the condition that the process stops when the system under control enters a specified set of states in its state space.

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Decision-making in a fuzzy environment

A functional defined on the class of generalized characteristic functions (fuzzy sets), called “entropy≓, is introduced using no probabilistic concepts in order to obtain a global measure of the indefiniteness connected with the situations described by fuzzy sets. This “entropy≓ may be regarded as a measure of a quantity of information which is not necessarily related to random experiments. Some mathematical properties of this functional are analyzed and some considerations on its applicability to pattern analysis are made.

A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory

The adjective "fuzzy" seems to be a very popular and very frequent one in the contemporary studies concerning the logical and set-theoretical foundations of mathematics. The main reason of this quick development is, in our opinion, easy to be understood. The surrounding us world is full of uncertainty, the information we obtain from the environment, the notions we use and the data resulting from our observation or measurement are, in general, vague and incorrect. So every formal description of the real world or some of its aspects is, in every case, only an approxima­ tion and an idealization of the actual state. The notions like fuzzy sets, fuzzy orderings, fuzzy languages etc. enable to handle and to study the degree of uncertainty mentioned above in a purely mathematic and formal way. A very brief survey of the most interest­ ing results and applications concerning the notion of fuzzy set and the related ones can be found in [l]. The aim of this paper is to apply the concept of fuzziness to the clasical notions of metric and metric spaces and to compare the obtained notions with those resulting from some other, namely probabilistic statistical, generalizations of metric spaces. Our aim is to write this paper on a quite self-explanatory level the references being necessary only for the reader wanting to study these matters in more details.

http://www.kybernetika.cz/content/1975/5/336/paper.pdf

Fuzzy metrics and statistical metric spaces

Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence

The concept of fuzzy sets is presented as a new tool for the formulation and solution of systems and decision problems which contain fuzzy components or fuzzy relationships. After a brief description of the basic theory of fuzzy sets, implications to systems theory and decision making are indicated. Fuzzy set theory is then applied to fuzzy linear programming problems and it is shown how fuzzy linear programming problems can be solved without increasing the computational effort. Some critical remarks concerning the presently existing axioms and necessary future research efforts conclude this introductionary paper.

C.L Chang

Papers

Fuzzy topological spaces