Fuzzy numbers

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 sets that are defined on the set K of real numbers (i.e., A = J ) have special significance in fuzzy set theory. They can be interpreted as fuzzy numbers provided they satisfy the following requirements ... 

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. 

In general, a fuzzy system is any system whose variables (or, at least, some of them) range over states that are fuzzy numbers rather than real numbers. These fuzzy numbers may represent linguistic terms such as very small, medium, and so on, as interpreted in a particular context. If they do, the variables are called linguistic variables. 

A semantic rule, which assigns to each linguistic term its meaning—an appropriate fuzzy number defined on the range of the base variable... 

All of these linguistic terms except fuzzy modifiers are represented in each context by appropriate fuzzy sets. Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements to which the predicates apply. Fuzzy truth values and fuzzy probabilities are represented by fuzzy sets defined on the unit interval [0,1]. Fuzzy quantifiers are either absolute or relative they are represented by appropriate fuzzy numbers defined either on the set of natural numbers or on the interval [0,1]. Fuzzy modifiers are operations by which fuzzy sets representing the various other linguistic terms are appropriately modified to capture the meaning of the modified linguistic terms. 

Although the usual quantization is mathematically convenient, it completely ignores uncertainties induced by unavoidable measurement errors around the boundaries between the individual intervals. This is highly unrealistic. Quantization forced by the limited resolution of the measuring instrument involved can be made more realistic by replacing the crisp intervals with fuzzy intervals or fuzzy numbers. This is illustrated for our example in Fig. 4b. Fuzzy sets are in this example fuzzy numbers expressed by the shown triangular membership functions, which express the linguistic... 

Each new bond (i,/) created at the current level is thus associated with a fuzzy number m ij). Equation (11) defines a membership function associated with every new bond formed during the generation process. Precautions have been taken in the program so that the term l/R will not introduce an error in those cases where the. / ,yS approach zero (which hardly ever occurs). 

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. 

The QPS system itself has also been constructed in an hierarchical fashion. The lowest level being the definitions and methods of the primitive data structures used universally within the system, for example list and set. Also at this level are the objects that implement the system s datatypes, currently real numbers, intervals and triangular fuzzy numbers (13). The next level defines the physical schema, using a filing system based on B-Trees for efficiency. 

Tens of milligrams. Now there is a truly wishy-washy phrase. There is an art to the assignment of an exact number or, as is sometimes desperately needed, a fuzzy number, to a collection of things. In my youth (somewhere way back yonder in the early part of the century) I had been taught rules of grammer that were unquestionably expected of any well-educated person. If you used a Latin stem, you used a Latin prefix. And if you used a Greek stem, you used a Greek prefix. 

Fuzzy set operations are derived from classical set theory. In addition, there exist theories for calculating with fuzzy numbers, functions, relations, measures, or integrals. 

Modeling Fuzzy numbers Calibration with errors in x and y... 

Figure 6 shows a structure of applying fuzzy logic in control. First, two types of inputs must be obtained numerical inputs and human knowledge or rule extraction from data (i.e., fuzzy rules). Then the numerical inputs must be fuzzified into fuzzy numbers. The fuzzy rules consist of the fuzzy membership functions (knowledge model) or so-called fuzzy associative memories (FAMs). Then the... 

Triangular fuzzy numbers (TFNs), 1781—1782 TRI (Toxic Releases Inventory), 594 Trolley conveyors, 1517—1518 TRON, 2565... 

In general, fuzzy quantifiers are fuzzy numbers that take part in fuzzy propositions. They are of two kinds. Fuzzy quantifiers of the first kind are defined onR and characterize linguistic terms such as about 10, much more than 100, and at least about 5. Fuzzy quantifiers of the second kind are defined on [0,1] and characterize linguistic terms such as almost all, about half, and most. 

In this chapter, the due times of different production orders are represented as trapezoidal fuzzy numbers (TrFN) with the following definition ... 

As shown in Fig. 7.3, d, d , d and d are crisp real numbers such that 0< d membership value of these fuzzy numbers expresses the degree of satisfaction associated with corresponding job completion time complete satisfaction if the job is completed dming the time interval of d to d°-, the degree of satisfaction increases linearly from time to d and decreases linearly from time cf to d and complete dissatisfaction if the job is completed before t=d or beyond t=d. ... 

Table 10.2 Linguistic terms of fuzzy rating scales with fuzzy number representation...

Linguistic terms of the satisfaction degree S Linguistic terms of the level of importance W Approximated value of corresponding fuzzy number... 

Notice, however, that in this present case we are interested in human reliability modeling. As a result the values for conditional probabilities mentioned above are obtained from the opinions of experts, and it has to be kept in mind that reliable data is not easy to obtain in this kind of problem concerning the installation of optical monitoring system in onshore oil well. Therefore, in order to use the elicitation method most accessible to experts, in this paper a fuzzy approach is adopted. Using fuzzy numbers, one can consider the linguistic variables adopted by the experts. The next section presents the main concepts of fuzzy logic and the arithmetic operations related to fuzzy numbers, which will be used to calculate human error probabihty. 

As mentioned by Dubois and Padre (1980), one can represent fuzzy numbers by their core and respective left and right deviances, L and R. When this deviance is zero, there is a crisp number. 

In this way, one can establish the symmetric of a fuzzy number. Taking M as it is above mentioned, its symmetric —M is achieved as follows ... 

Considering the same development, the multiphca-tion of two positive fuzzy numbers is as foUows ... 

Considering equation (9), one can estabhsh an expression for the division of two fuzzy numbers. 

Through the definition of these operahons involving fuzzy numbers the necessary operahons can be established for the calculus involved in the quanhfi-cation of the Bayesian network. It should be nohced that the above operahons was presented by Dubois and Padre (1980). 

As above mentioned, algebraic operations with fuzzy numbers constitute a generalization of crisp number operations (Dubois and Padre, 1980). 

Note that, having defined aU the necessary operations related to the Bayesian network and the algebraic operations associated with fuzzy numbers, these concepts can be combined to establish what is called the Fuzzy Bayesian Network (HaUiweU et al, 2003 Leon-Rojas et al., 2003). 

In equation (15), K x) is the defiizzified value, c is the minor value considered in the domain, d is the highest value considered in the domain, ai and bj are, respectively, the possibility degree of I for the lower and higher extremities. For the triangular fuzzy number, aj = bj. 

In this paper we presented an approach where fuzzy logic and BBN concepts are combined to estimate human error probability. This combination leads to a fuzzy Bayesian network approach based on the concept o fuzzy number and on extension principles applied to discrete fuzzy probabilities calculation. 

---