By Iwona Skalna, Bogdan Rębiasz, Bartlomiej Gawel, Beata Basiura, Jerzy Duda, Janusz Opila, Tomasz Pelech-Pilichowski

This ebook indicates how universal operation administration tools and algorithms will be prolonged to accommodate imprecise or vague details in decision-making difficulties. It describes how one can mix determination timber, clustering, multi-attribute decision-making algorithms and Monte Carlo Simulation with the mathematical description of obscure or imprecise details, and the way to imagine such details. furthermore, it discusses a wide spectrum of real-life administration difficulties together with forecasting the obvious intake of metal items, making plans and scheduling of creation procedures, venture portfolio choice and economic-risk estimation. it's a concise, but complete, reference resource for researchers in decision-making and decision-makers in company firms alike.

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**Example text**

1985. Selecting uncertainty calculi and granularity: An experiment in trading off precision and complexity. In Proceedings of the first Workshop on Uncertainty in Artificial Intelligence, 57–66, Los Angeles. 13. A. 1975. In Fuzzy sets and their applications to cognitive and decision processes, ed. -S. Fu, K. Tanaka, and M. , Calculus of fuzzy restrictions. 14. A. 1973. Outline of a new approach to the analysis of complex systems and decision processes. IEEE Transactions on Systems, Man, and Cybernetics 3: 28–44.

Based on a centroid point, two fuzzy numbers A˜ and B˜ are compared using the following rules [21]: ˜ > x 0 ( B), ˜ Then A˜ > B. ˜ If x 0 ( A) ˜ < x 0 ( B), ˜ Then A˜ < B. ˜ If x 0 ( A) ˜ ˜ If x 0 ( A) = x 0 ( B), Then ˜ > y 0 ( B), ˜ Then A˜ > B. ˜ If y 0 ( A) ˜ < y 0 ( B), ˜ Then A˜ < B. ˜ Else If y 0 ( A) ˜ ˜ Else A = B. 4): This gives the ordering: A˜ 3 < A˜ 1 < A˜ 2 < A˜ 4 . The method is rather simple, but it requires a pairwise comparison of fuzzy numbers to be ordered. 4 Yager Ranking Index Approach In [22], Yager proposed the following index to ordering fuzzy numbers: 1 ˜ = Y ( A) ( f A˜ (y) + g A˜ (y))dy.

3 Modelling Dependencies Between Fuzzy Numbers 21 where X is an input matrix and yˆ is an output vector. The problem is to determine the parameters of interval regression: ⎤ a1 ⎢ ⎥ aˆ = ⎣ ... 41) an that comprise all possibilities determined by the model and data. More formally, aˆ must be such that [39]: y j ∈ x j1 a1 + x j2 a2 + · · · + x jn an , ∀ j = 1, . . , p. 42) It is natural to try to achieve several desirable properties of the interval parameters. They should be as tight as possible, they should be balanced and they should respect the central tendency.