By Rafik A Aliev, Oleg H Huseynov, Rashad R Aliyev, Akif A Alizadeh
Real-world details is imperfect and is mostly defined in normal language (NL). in addition, this knowledge is usually partly trustworthy and a level of reliability is usually expressed in NL. In view of this, the idea that of a Z-number is a extra sufficient thought for the outline of real-world info. the most serious challenge that obviously arises in processing Z-numbers-based info is the computation with Z-numbers. these days, there is not any mathematics of Z-numbers steered in current literature. This ebook is the 1st to provide a complete and self-contained idea of Z-arithmetic and its functions. a number of the ideas and methods defined within the publication, with rigorously worked-out examples, are unique and seem within the literature for the 1st time. The ebook should be precious for execs, teachers, managers and graduate scholars in fuzzy good judgment, determination sciences, man made intelligence, mathematical economics, and computational economics.
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Additional info for The Arithmetic of Z-Numbers: Theory and Applications
For convenience, Ais referred to as a value of X, with the understanding that, strictly speaking, A is not a value of X but a restriction on the values which X can take. The second component, B , is referred to as certainty. Closely related to certainty are the concepts of sureness, confidence, reliability, strength of belief, probability, possibility, etc. When X is a random variable, certainty may be equated to probability. Informally, B may be interpreted as a response to the question: How sure are you that X is A?
Example: Assume that ∗ is sum. In this case, AX + AY is defined by: µ( A X + AY ) (v) = sup( µ AX (u) ∧ µ AY (v − u )), ∧ = min. 1) u Similarly, assuming that RX and RY are independent, the probability density function of R X ∗ RY is the convolution, , of the probability density functions of RX and RY . Denoting these probability density functions as pRX and pRY , respectively, we have: pRX + RY (v) = ∫ pRX (u ) pRY (v − u )du R Thus, 9575-The Arithmetic of Z-numbers 34 The Arithmetic of Z-numbers.
Assume that the factual information is that the average height of Swedes is around 170 cm. 170cm . In terms of the height density function, h, the average height of Swedes may be expressed as h ave = ∫ h max h min uh(u)du. The explanatory database consists of µ tall , µ most and h. 170cm (∫ h max h min uh(u)du . ) An important observation is in order. An internal truth value modifies the meaning of p. An external truth value does not modify the meaning of p; it places in evidence the factual information, with the understanding that factual information is a possibilistic restriction on the explanatory database.