Prof. Cengiz Kahraman İ stanbul Technical University Department of Industrial Engineering Fuzzy Logic and Modeling

Example

Type-Two Fuzzy Sets In type-two fuzzy sets, one assumes that the concept of membership is captured by some fuzzy set defined in the unit interval, instead of single membership values or intervals. In other words, for finite X we can look at the type-two fuzzy set as a collection of individual fuzzy sets. This type of generalization is convenient in organizing information about the concept under consideration. A brief example may help clarify this point.

Type two fuzzy sets Consider traffic on a highway, which is usually a mixture of several categories of vehicles: trucks, buses, automobiles, motocycles, and so forth. To characterize the traffic, we specify an intensity for each of the categories of vehicles, the intensities being characterized by fuzzy sets. For instance we may have traffic = {heavy/trucks, light/motorcycles, moderate/automobiles, light/buses}, where heavy, light, moderate,are relevant fuzzy sets in the space of traffic intensity attached to the corresponding category of the vehicles.

Type II Fuzzy Sets The concept of a type-II fuzzy set was introduced first by Zadeh (1975) as an extension of the concept of an ordinary fuzzy set, i.e. a type-I fuzzy set. Type-II fuzzy sets have grades of membership that are themselves fuzzy. At each value of the primary variable, the membership is a function (and not just a point value) – the secondary membership function (MF)-, whose domain -the primary membership- is in the interval [0,1], and whose range -secondary grades- may also be in [0,1]. Hence, the MF of a type-II fuzzy set is three-dimensional, and it is the new third dimension that provides new design degrees of freedom for handling uncertainties. Such sets are useful in circumstances where it is difficult to determine the exact MF for a fuzzy set (FS), as in modeling a word by a FS. Interval type-II fuzzy sets, each of which is characterized by the footprint of uncertainty, are a very useful means to depict the decision information in the process of decision making. Type-II fuzzy sets are handled in two ways: interval type-II fuzzy sets and generalized type-II fuzzy sets. A triangular interval type-II fuzzy set is illustrated in the following figure. 5

Type II fuzzy sets Triangular Interval Type-II Fuzzy Set 6

Type II fuzzy sets A trapezoidal type-II fuzzy number is illustrated in the following figure. The nine points to determine a footprint of uncertainty. (a, b, c, d) determines a normal trapezoidal upper membership function and (e, f, g, i, h) determines a trapezoidal lower membership function with height h. Fig. 2. Trapezoidal Interval Type-II Fuzzy Set 7

Type II fuzzy sets 8

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Type II fuzzy sets 11

Fuzzy Arithmetics 12

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Fuzzy Ranking Methods 75

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