Fuzzy arithmetics

Kaufmann, A., and Gupta, M.M. (1985), Introduction to Fuzzy Arithmetic Theory and Applications, Van Nostrand Reinhold, New York, NY. 

Fuzzy arithmetic Fuzzy arithmetic is the arithmetic embodied in operations snch as addition, subtraction, multiplication, and division of fnzzy nnmbers. Fnzzy nnmbers are unimodal distribution functions of the real line that grade all real numbers according to the possibility that each might be a valne the fnzzy number could take on. The minimum of the function is 0, which represents impossible values, and the maximum is 1, which represents those... 

The application of fuzzy logic to the risk assessment of the use of solvents in order to evaluate the uncertainties affecting both individual and societal risk estimates is an area with relevance to the present considerations (Bonvicini et al., 1998). In evaluating uncertainty by fuzzy logic, fuzzy numbers describe the uncertain input parameters and calculations are performed using fuzzy arithmetic the outputs will also be fuzzy numbers. The results of these considerations work are an attempt to justify some of the questions the use of fuzzy in the field of risk analysis stimnlates. 

Fuzzy arithmetic enables us to evaluate algebraic expressions in which values of variables are fuzzy numbers. It also enables us to deal with algebraic equations in which coefficients and unknowns are fuzzy numbers. Furthermore, fuzzy arithmetic is a basis for developing fuzzy calculus and, eventually, fuzzy mathematical analysis Although a lot of work has already been done along these lines, enormous research effort is still needed to fully develop these mathematical areas. 

Uncertainties inherent to the risk assessment process can be quantitatively described using, for example, statistical distributions, fuzzy numbers, or intervals. Corresponding methods are available for propagating these kinds of uncertainties through the process of risk estimation, including Monte Carlo simulation, fuzzy arithmetic, and interval analysis. Computationally intensive methods (e.g., the bootstrap) that work directly from the data to characterize and propagate uncertainties can also be applied in ERA. Implementation of these methods for incorporating uncertainty can lead to risk estimates that are consistent with a probabilistic definition of risk. 

For fuzzy modeling or for comparison of fuzzy functions, the fundamentals of fuzzy arithmetics are needed (cf. Figure 8.25). These fundamentals are given, for example, in references [19] or [20]. Applications are known for calibration of analytical methods and for qualitative and quantitative comparison of chromatograms, spectra, or depth profiles. 

ABSTRACT By analyzing the hierarchy of safety factors in purification plant of natural gas, they are divided into personnel, equipment, environment and management. After the study of safety index, evaluation methods and fuzzy arithmetic method for each hierarchy, its fuzzy evaluation flow is given. During the evaluation, the weight of the factors and each hierarchy is decided by analytical hierarchical process, and the operational criterion adapts maximum membership degree. And this model is exemplified in purification plant of natural gas. The results show that the second fuzzy evaluation is effective to assess purification plant of natural gas. 

Notice that (f2, e, P/) refers to a fuzzy probability space, denotes one of the fuzzy arithmetical operations, i.e., addition, subtraction, multiplication and division. It should also be noted that lx and Ox refer to the fuzzy subset of the real numbers 1 and 0, respectively. 

You should be aware that ANNs and ESs can be combined into expert networks. This often requires the use of fuzzy arithmetic and logic (see Chapter 9 of Ref. 62 for an introduction). So when should you use an ANN as opposed to an ES or statistical method Only you can provide the answer. We have presented a variety of issues to consider when making a choice. Howevei you are most famihar with your data and therefore best equipped to consider the issues involved in deciding. In the final analysis, if you are looking for the best answers to your problems, you may need to try several methods to see which one in fact performs best for you. 

Avoid the traditional fuzzy arithmetic, the form of the fuzzy membership function will not widely spread due to the arithmetic, especially for the complex systems (2) the artificial bee colony algorithm has a higher efficiency in the optimization process, so it could reduce the time cost (3) the ABC algorithm has improved compared to the traditional ABC, it increase the ability of the global sought, so the results will more precise. 

After performing the aggregation step, the classical fuzzy arithmetic is applied to the FuBI bounds of BEs belonging to the Top Event (TE) minimal cut-sets in order to obtain the TE s FuBI. 

Fuzzy arithmetic operations - In standard fuzzy arithmetic, basic operations on real numbers are extended to those on fuzzy intervals. A fiizzy interval A is a normal fuzzy set on R (set of real numbers) whose a-cuts for all a e (0,1] are closed intervals of real numbers and whose support is bounded by A. 

The membership function used (i.e. trapezoidal function) allows a membership value of 1 for a range of probabilities unlike the triangular function. This function is thought to model the probability of occurrence close to what it is in reality (Pillay (2001)). Figure 6.19 shows the membership function along with its ordinal scale. The limits and the centre point values of the ordinal scale are given by the dotted line and will be used to perform the fuzzy arithmetic. 

Figure 6.21 Graphical representation of fuzzy arithmetic operation on two basic events...

Kaufinann, A. Gupta, M. (1986) Introduction to fuzzy arithmetic Theory and applications. New York 1986. 

---