By Olga Poleshchuk, Evgeniy Komarov
This ebook offers with professional overview types within the kind of semantic areas with completeness and orthogonality houses (complete orthogonal semantic spaces). Theoretical and useful stories of a few researchers have proven that those areas describe specialist reviews such a lot safely, and for this reason they have been frequently incorporated in additional refined types of highbrow platforms for selection making and knowledge research. equipment for developing professional overview types of features, comparative research of those types, reports of structural composition in their units and developing of generalized versions are defined. versions to acquire score issues for gadgets and teams of items with qualitative and quantitative features are provided. a few regression types combining components of classical and fuzzy regressions are awarded.
All tools and versions constructed through the authors and defined within the e-book are illustrated with examples from a number of fields of human activities.
This booklet intended for scientists within the box of computing device technological know-how, professional structures, synthetic intelligence and choice making; and likewise for engineers, post-graduate scholars and scholars who examine the bushy set thought and its applications.
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The point x = 1 corresponds to total presence of characteristic X appearance, and consequently it is considered as a typical point of term X m . Let us denote relative frequencies of objects’ occurrence for which characteristic X intensity is estimated by levels X l by al , accordingly m ∑a l =1 l = 1. Let us assume that fuzzy numbers corresponding to terms X l with membership functions μ l ( x ) belong to population Λ and satisfy a side condition (1*): If L(x ) , R( x ) are nonlinear functions, they have a central symmetry versus the point of inflexion.
R⎜⎜ ⎝ aR ⎠ ~ The unimodal A - (L − R ) number has membership function of tolerance (L − R ) number under the condition of a1 = a 2 . A unimodal (L − R ) number is written symbolically as A ≡ (a1 , a L , a R ) . If A ≡ (a1 , a 2 , a L , a R ) , B ≡ (b1 , b2 , bL , bR ) , then: ~ ~ ~ 1. A ⊕ B ≡ (a1 + b1 , a2 + b2 , aL + bL , aR + bR ); ~ ~ 2. β A ≡ (β a1 , β a 2 , β a L , β a R ) on β ≥ 0 ~ 3. β A ≡ (β a2 , βa1 , β a R , β a L ) on β < 0 ~ 16 The Basic Concepts of the Fuzzy Set Theory Fig. 3 Fuzzy Sets 17 By analogy with operation of algebraic product (multiplication), exponentiation ~ ~ operation is defined.
3. Membership function of a normal triangular number or normal FN of triangular type (Fig. c) L( x ) = R( x ) = 1 − x, 0 ≤ x ≤ 1; μ A~ ( x ) = (a1 , a L , a R ) 18 The Basic Concepts of the Fuzzy Set Theory ⎧ ⎪1 − ⎪ ⎪ μ A~ ( x ) = ⎨1 − ⎪ ⎪1, ⎪0, ⎩ a1 − x a −x , 0< 1 ≤ 1, a L > 0; aL aL x − a1 x − a1 , 0< ≤ 1, a R > 0; aR aR x = a1 ; x < a1 − a L or x > a1 + a R . 4 Linguistic Variables and Semantic Spaces One of the basic concepts of the fuzzy set theory is the concept of a fuzzy variable [4]. A triple ~ X ,U , A { } is referred to as a fuzzy variable, where X – the name of the variable; U — area ~ of its definition (universal set); A — the fuzzy set of universal set which describes possible values of the fuzzy variable.