Fuzzy intersection

Under incompatible objective circumstances, a DM must make a compromise decision that provides a maximal degree-of-satisfaction for all these conflict objectives. The new optimization problem, Eq. (14), can be interpreted as the synthetic notation of a conjunction statement (maximize jointly all objectives). The result of this aggregation can be viewed as a fuzzy intersection of all fuzzy goals, Js,s e S, and is still a fuzzy set, V. 

The final degree-of-satisfaction resulting from certain variable set, nv(xy) can be determined by aggregating the degree-of-satisfaction for all objectives, /ijs (xv), s e S. via specific fuzzy intersection operator, T. 

Using the fuzzy intersection operator, the original multiobjective optimization problem, Eq. (11), is converted into a single objective problem, Eq. (17). Several operators for implementing fuzzy intersection can be selected for T, therein two most popular ones are shown below. 

Any fuzzy power set with the subsethood relation is a lattice, in which the standard fuzzy intersection and union play the roles of the meet and the join, respectively. The lattice is distributive and complemented under the standard fuzzy complement. Contrary to the Boolean lattice, which is associated with classical power sets, it does not satisfy the law of the excluded middle and the law of contradiction. Such a lattice is usually called a DeMorgan lattice. 

For fuzzy sets used in this study various set operations are required. If A and B are fuzzy subsets of U, then the fuzzy intersection, that is, the result of the operation A and B is denoted by A i B, and is interpreted as a fuzzy subset C of set U, where the corresponding membership... 

In fuzzy logic, operators such as AND, OR, and NOT are implemented by fuzzy intersection or conjunction (AND), fuzzy union or disjunction (OR), and fuzzy complement (NOT). There are various ways to define these operators, but commonly, AND, OR, and NOT logic operators are implemented by the min, max, and complement operators. The fuzzy truth, T, of a complex sentence is evaluated in this way ... 

Fuzzy set operations are a generalization of crisp set operations, each of which is a fuzzy set operation. Infuzzy logic, three operations, including fuzzy complement, fuzzy intersection and fuzzy union, are the most commonly used. Let fuzzy sets ... 

Fuzzy intersection Fuzzy intersection is the fuzzy operation for creating the intersection of fuzzy sets A and B on the universe of discourse X, which can be obtained as ... 

Figure.7 Comparison of the measured set (D) with the fuzzed combination (C) of reference spectra (A, B) by a fuzzy intersection (E) (from Ref. 12)...

In more general terms, fuzzy intersection is defined by fuzzy AND operator, fuzzy union is defined by fuzzy OR operator, and complement by fuzzy NOT operator. All properties of crisp set are also applicable for fuzzy sets except for the exeluded-middle laws. In fuzzy set theory, the union of fuzzy set with its complement does not yield the universe and the intersection of fuzzy set and its complement is not null. This difference is shown below ... 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

Fig. 2.19. Various sets of analytical data (A) Hard reference data set, mR(x). (B) Hard test data set, mT(x), which is slightly shifted compared with (A). (C) Fuzzy set of test data, mT(x) = exp (—(x — a)2/b2). (D) Intersection mT R(x) of test data and reference data which is empty in this case. (E) Intersection of fuzzed test data and reference data with a membership value of about 0.8 in this case...

Zadeh [1975] extended the classical set theory to the so-called fuzzy set theory, introducing membership functions that can take on any value between 0 and 1. As illustrated by the intersection of the (hard) reference data set (A) and the fuzzed test data set (C), the intersection (E) shows an agreement of about 80%. Details on application of fuzzy set theory in analytical chemistry can be found in Blaffert [1984], Otto and Bandemer [ 1986a,b] and Otto et al. [1992],... 

Figure 8. Diagram showing that complexes with partial mechanical bonding (P) character, i.e., pseudorotaxanes and hemicarceplexes, are represented by the intersection set [AA n I] of the set of (wholly) mechanically-bound molecules (AA) and the set of isolated molecules (I) - in other words, the fuzzy region in between these two sets. Thus, the complexes in set P are endowed simultaneously with characteristics associated with species belonging to both AA and I. The numbers 1 and 0 have been assigned arbitrarily to the species that belong either entirely or not at all to the sets AA and I.

Fuzzy complements, intersections, and unions have been characterized and studied on axiomatic grounds. Efficient procedures are now available by which various classes of functions can be generated, each of which covers the whole recognized semantic range of the respective operation. In addition, averaging operations for fuzzy sets, which have no counterparts for crisp sets, have also been investigated in this way. This rather theoretical subject, which is beyond the scope of this overview, is thoroughly covered in ref. 18. 

In this work a fuzzy matching procedure is suggested which takes the foregoing uncertainties into account at least in principle. The approach is based on a soft definition of a surface that is defined in terms of membership functions (see Fig. 8) These functions are and, .(r) and they measure to what extent a given space point belongs to the surface and the bulk of a molecule, respectively. The matching of two molecules A and B can then be calculated in many different ways. In a first attempt we used the intersection of two fuzzy sets... 

If the clusters are close it is possible that the hyperplane of equal membership will intersect the greater cluster. Some points of the greater cluster will be captured by its neighbor, as shown in Fig. 4. This represents a pathological situation that may be avoided by using a data-dependent (or adaptive) distance.With an adaptive metric the apparent sizes of clusters become equal. An adaptive distance may be induced by the radius or by the diameter of each fuzzy class A,. The diameter 6 of the fuzzy class Ai is defined as... 

The intersection of two sets A and B corresponds, according to classical theory, to all elements that are simultaneously contained in both sets. For two fuzzy sets, the intersection, AcB, is derived from the minimum of both of the membership functions m (x) and mg x) ... 

Figure 8.22 Intersection (a) and union (b) of two fuzzy sets and the cardinality of a fuzzy set (c).

As depicted in Fig. 5, sometimes the GPS logs may be not exactly intersecting but should be considered for having physical reachability. In such a case, fuzzy approaches provide flexibility for dealing with uncertainty as well as with constraints that accept some degree of error. 

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. 

The method for calculating tb is exactly that described in the introduction to this chapter for binary and multi-valued logic. The process is one of calculating fuzzy truth restrictions for the first and second lines of the deduction on the space Ux X Uy, intersecting them to produce an equivalent restriction and then projecting the result on to Uy Thus... 

AND is used to evaluate the conjunction of rule antecedents. Typically, fuzzy logic systems utilize the classical fuzzy operation intersection to implement this operation. Consider fuzzy rule 1 ... 

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