By Christer Carlsson, Robert Fuller
This ebook begins with the elemental ideas of fuzzy arithmetics and progresses throughout the research of sup-t-norm-extended mathematics operations, possibilistic linear structures and fuzzy reasoning ways to fuzzy optimization. 4 functions of (interdependent) fuzzy optimization and fuzzy reasoning to strategic making plans, venture administration with actual suggestions, strategic administration and provide chain administration are awarded and thoroughly mentioned. The ebook ends with an in depth description of a few clever software program brokers, the place fuzzy reasoning schemes are used to augment their performance. it may be necessary for researchers and scholars operating in smooth computing, utilized arithmetic, operations learn, administration technology, info platforms, clever brokers and synthetic intelligence.
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12) is x = (al - a2 vanishes). Which ends the proof. 2 [123} Let 2 < '"'I < 00 and iii = (ai, a), i = 1,2. Then their H,-sum A2 := iiI + ii2 has the following membership function 1 hl(Z) ifl- - '"'1-1 h2(Z) if o IA2 - zl a ~ IA2 - zl < 2, a 1 < 1 - --, '"'1-1 otherwise, where [1 - (A2 - z)/(2a)]2 hl(Z) = 1 + b -1)[(A2 - z)/(2a)j2' The following theorems can be interpreted as central limit theorems for mutually H,-related fuzzy variables of symmetric triangular form (see ). 1 [123} Let '"'I = 0 and iii = (ai,a), i E N.
Let iii = (ai,a,f3)LR, 1 SiS n be fuzzy numbers of LR-type. 11) iflog R and log L are concave functions. However, Hong  pointed out that the condition given by Triesch is not only sufficient but necessary, too. 3) with parameter 'Y. In the next two lemmas we shall calculate the exact membership function of H'"'(-sum of two symmetric triangular fuzzy numbers having common width a > 0 for each permissible value of parameter 'Y. 6 Hamacher-sum of triangular fuzzy numbers 25 = (ai, a), i = 1,2.
1 has been improved and generalized later by Kawaguchi and Da-Te [204, 205], Hong [171, 178], Hong and Kim , Hong and Hwang [170,172,177,179,180]' Markova , Mesiar [249, 250, 251]' De Baets and Markova [4, 5]. 1 remains valid for convex additive genrator f, and concave shape functions L and R. In 1994 Hong and Hwang () provided an upper bound for the membership function of T-sum of LR-fuzzy numbers with different spreads. 1 remains valid if both L 0 I and Ro I are convex functions.