By Christer Carlsson, Robert Fuller
This e-book starts off with the elemental strategies of fuzzy arithmetics and progresses throughout the research of sup-t-norm-extended mathematics operations, possibilistic linear structures and fuzzy reasoning techniques to fuzzy optimization. 4 functions of (interdependent) fuzzy optimization and fuzzy reasoning to strategic making plans, venture administration with genuine recommendations, strategic administration and provide chain administration are awarded and punctiliously mentioned. The publication 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 invaluable for researchers and scholars operating in gentle computing, utilized arithmetic, operations learn, administration technological know-how, details structures, clever brokers and synthetic intelligence.
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Extra info for Fuzzy Reasoning in Decision Making and Optimization
If Land R are twice differentiable, concave functions, and f is twice differentiable, strictly convex function, then Proof. Let z ~ 0 be arbitrarily fixed. According to the decomposition rule of fuzzy numbers into two separate parts, we can assume without loss of generality that z < a. ,,+Mn )(nz)=f[-I] (n'f(L(nan~nz))) = f[-I] (n. f (L ( a: z) ) ) = f[-I] (n. , M(x n )) (M x ". x M)(z) = Xl· .. , M( y'z)) = f[-I] (n. 14) holds for some n (Mk) (z) = = k, i. e. , M(x k )) xl·····xk- Z ='T(M( ifZ), ...
We show now the relationship between the interval-valued expectation E(A) = [E*(A), E*(A)], introduced in  and the interval-valued possibilistic mean M(A) = [M*(A), M*(A)] for LR-fuzzy numbers with strictly decreasing shape functions. An LR-type fuzzy number A E F can be described with the following membership function  50 1. Fuzzy Sets and Fuzzy Logic A(u) = u - q+) R ( -(3- if q+ ::; u ::; q+ ° + (3 otherwise where [q_,q+] is the peak of A; q_ and q+ are the lower and upper modal values; L, R: [0, 1] ~ [0,1]' with L(O) = R(O) = 1 and L(1) = R(1) = are non-increasing, continuous mappings.
In a similar manner we introduce M*(A) , the upper possibilistic mean value of A, as 48 1. Fuzzy Sets and Fuzzy Logic M*(A) = 21 1 = -,,-,,-O~l-1 1 'Ya2b)d'Y 'Y a 2('Y)d-Y ~ 'Yd'Y o ~l Pos[A ~ a2b)]a2b)d'Y ~l Pos[A ~ a2b)]d'Y ~lpOS[A ~ a2b)] X max[Apd'Y ~l Pos[A ~ a2b)]d'Y where we have used the equality Pos[A ~ a2b)] = ll(a2b) , 00) = sup A(u) u~a2(-y) = 'Y. Let us introduce the notation M(A) = [M*(A), M*(A)]. that is, M (A) is a closed interval bounded by the lower and upper possibilistic mean values of A.
Fuzzy Reasoning in Decision Making and Optimization by Christer Carlsson, Robert Fuller