Fuzzy relations

When fuzzy sets are defined on universal sets that are Cartesian products of two or more sets, they are called fuzzy relations. For any Cartesian product of n sets, the relations are called n-dimensional. From the standpoint of fuzzy relations, ordinary fuzzy sets may be viewed as degenerate, one-dimensional relations. All concepts and operations applicable to fuzzy sets are applicable to fuzzy relations as well. However, fuzzy relations involve additional concepts and operations that emerge from their multidimensionality. 

Concepts that are important for general n-dimensional fuzzy relations include projections to lower-dimensional spaces, cylindric extensions of projections, and cylindric closures. These concepts are simple generalizations of their classical counterparts, and it is not essential to cover them here. It is more important to introduce some key concepts regarding fuzzy binary relations, which have a broad applicability. 

Membership functions of any fuzzy binary relation on X Y have the form 

For each pair x,y eXxY, the membership degree R(x,y) indicates how strongly element x is related to element y according to R. The inverse of R, denoted by K , is a relation on YxA defined by 

Given two fuzzy binary relationships R and Q defined on Z X y and YxZ, respectively, their standard composition R°Q is defined by the formula 

An important aspect of fuzzy logic is the ability to relate sets with different universes of discourse. Consider the relationship 

Then for the fuzzy sets L and M defined by equation (10.8), for U from 5 to 35 in steps of 5 

Several sueh statements would form a eontrol strategy and would be linked by their union 

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For fuzzy relations on XxX, the following cutworthy properties are recognized ... 

Employing these definitions, we can characterize important classes of fuzzy relations in the same way as their crisp counterparts. Fuzzy equivalence relations are reflexive, symmetric, and transitive fuzzy compatibility relations are reflexive and symmetric fuzzy partial orderings are reflexive, antisymmetric, transitive, etc. Each of these relations is cutworthy that is, each a-cut of a fuzzy relation of a particular type is a crisp relation of the same type. 

The problem of solving fuzzy relation equations arises whenever two of the relations are given and the third is to be determined. When P and Q are given, the problem of determining R is trivial. When R and Q (or P) are given, to determine P (or Q) is considerably more difficult, but it is very important in many applications. Efficient methods for solving this problem have been developed, but their coverage is beyond the scope of this overview. 

V, then the result. 4 n 5 is a fuzzy relation Ry g in the product set U V, where the corresponding membership function is defined as... 

We learned the fundamentals about inferences on rule-based systems in Section 8.1 on symbolic knowledge processing. One possibility to infer on the knowledge represented in the computer is based on IF-THEN rules. In fuzzy logic, the premises and consequences are described by a fuzzy relation. Consider the following rules ... 

To calculate membership function and establish a matrix of fuzzy relation... 

According to the fuzzy relation matrix analyse the degree of coal spontaneous combustion danger of Huanghua port. 

Now let us return to fuzzy composition. The operation, in essence, is an intersection of fuzzy relations, projected on to a particular space. Consider the composition of two fuzzy relations A C X x Y and BC Y xZ. Now as these are not contained in the same space, they both have to be cylindrically extended into a common space. A is therefore extended into X x Y x Z to give A, and B is extended into X x Y x Z to give B. This cylindrical extension is merely, as the name implies, the extending or repeating of the membership values into the third dimension of Z for A and X for B. 

This 7 factor of 0.86 gives no indication of the uncertainly involved in the answer. The same calculation will now be presented where the relation between M and X is treated as a fuzzy relation R, Fig. 6.5, and the effective length factor is a fuzzy linguistic variable, k. The applied load will be treated as a random variable with a given probability distribution function. For the purposes of illustration discrete values for selected points will be used for variables which are obviously continuous. This is not a serious limitation due to the inherent subjectivity and approximation in establishing the fuzzy relations. The discrete values will be assumed to be central values operating for a region of the continuous variable either side of the element value. 

Let us assume for this purpose that it is a unit matrix as shown in Fig. 10.5, it then represents the fuzzy relation equals (Section 6.1.1). If for example... 

Because they map onto [0,1], similarity relations are fuzzy relations [50], which differ from classical relations that map pairs of elements onto the set of binary values 0,1. Similarity relations satisfy two mathematical properties, namely, they are reflexive, S(i,j)=0 if m =m., and generally are symmetric, S(iJ)=S(j,i) for l ij n, but they are generally intransitive. Asymmetric similarities, which will be discussed in Section 15.5.1, have been employed in MSA, but the number of applications is relatively small to date. 

Zhou et al. [61] described production planning of multi-location plant and distributors on condition that unit production cost, production capacity and demand are fuzzy parameters. They built up a fuzzy expected value model and fuzzy related chance-constrained programming model in consideration of different decision criteria and discussed a clear equivalent form of the fuzzy programming model when... 

The consciousness resonance does not eliminate fuzziness, which is an eternal companion to any process of thinking and knowing. At the same time, when the consciousness resonance helps one to transcend the fuzziness related to a problem that dissolves, it opens space for new problems to emerge bringing with them new... 

The uncertainty and fnzzy concepts in understanding problems that emerge out of life complexity as it unfolds cannot be resolved at the same level of knowledge that one has when these problems appear. Only when one s consciousness is expanded i.e. raised to a higher level, then the tension fades and the problems, being seen in a new light, are no longer problems. When problems dissolve, one may say that the fuzziness related to them has been transcended, but this is on relativistic sense, not in absolute sense. 

The conditional sentence is equal to a certain fuzzy relation R e X x T we will use the max-min rule that was selected from the numerous definitions of such fuzzy relations found in the literature... 

If the elements of the universe are considered as variables then equation (9) reads as restriction if x = xq holds then y must be smaller than xq. Fuzzy relations are fuzzy subsets on X X Y determined by membership functions depending on two arguments of equal importance, e.g., fx x, y). In analogy to the former example we investigate the fuzzy relation R2. nearly smaller than . The membership function for Rj, I r, may be defined by... 

Fuzzy relations in different product spaces can be combined with each other. In the simplest case we have a fuzzy relation Ri on X X y and a fuzzy relation R2 on F x Z. To combine these relations various kinds of compositions have been developed which differ in their mathematical properties and results. The most frequently applied is the max-min composition. The max-min composition Ri QR2 combining X and Z may be defined by the membership function... 

A more general definition supplies the max- composition. An outstanding special case of a composition of fuzzy relations is the so-called extension principle. That is the most practiced way of extending algebraic operations from crisp to fuzzy numbers. ... 

Peeva, K. Kyosev, Y. Fuzzy Relational Calculus Theory, Applications and Software (with CD-ROM, vol. 22), Advances in Fuzzy Systems, 22 ISBN 9789812560766 World Scientific Singapore, 2005. 

The tmderlying presirmption of the SCRM Framework is that the risk and performance sources and drivers can be foreseert, identified, evaluated, prioritized and managed. The practical reality suggests considerably more fuzziness relating to these drivers, their impact on performance and the effectiveness of possible... 

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