Fuzzy logic theory and systems
Introduction to fuzzy logic
Background
To deal with vagueness in human thought, Lotfi A. Zadeh (1965) first introduced the fuzzy set theory, which has the capability to represent and manipulate data and information possessing based on non-statistical uncertainties. Moreover fuzzy set theory has been designed to mathematically represent uncertainty and vagueness and to provide formalized tools for dealing with the imprecision inherent to decision making problems. Some basic definitions of fuzzy sets, fuzzy numbers and linguistic variables are reviewed from Zadeh (1965; 1975); Buckley (1985), Negi (1989), Kaufmann and Gupta (1991).
In opposite to the true or false world of Boolean logic, fuzzy logic concept allows the use of degrees of truth to compute results. The expression “fuzziness” means a sense of vagueness in defining the measures rather than the lack of knowledge about the value of the variables and the causal relationships.
The key idea of fuzzy logic is that it uses a simple and easy way in order to get the output(s) from the input(s), actually the outputs are related to the inputs using if-statements and this is the secret behind the easiness of this technique. The most fascinating thing about Fuzzy logic is that it accepts the uncertainties that are inherited in the realistic inputs and it deals with these uncertainties in such away their affect is negligible and thus resulting in a precise outputs.
Although fuzzy control was not the first engineering application of fuzzy logic, it was the first application that drew huge attention to the practical potential of fuzzy set theory. It uses many elements of fuzzy logic to define a rule-base for the controller.
Fuzzy set theory
Fuzzy set theory is a generalization of the ordinary Boolean theory. A fuzzy set is a set whose elements belong to the set with a degree of membership μ. Let X a set of objects. It is called universe of discourse. A fuzzy set A X is characterized by membership function μA(x) represents the degree of membership whose elements are between 0 and 1.
A = {(x, μA(x)); xX}
There are various types of membership function in fuzzy logic. Some standard membership functions are given here. Membership functions contain the membership values of elements in fuzzy set. Membership values can lie between 0 and 1. A membership function associated with a given fuzzy set maps an input value to its appropriate membership value.
Figure 2.2: Fuzzy membership functions
The simplest membership functions are formed using straight lines. Of these, the simplest is the triangular membership function. This function is nothing more than a collection of three points forming a triangle. The trapezoidal membership function has a flat top and really is just a truncated triangle curve. These straight line membership functions have the advantage of simplicity.
Some membership functions are built on the Gaussian distribution curve. The generalized bell membership function is specified by three parameters and has the function name gbellmf. The bell membership function has one more parameter than the Gaussian membership function, so it can approach a non-fuzzy set if the free parameter is tuned (Matlab fuzzy logic toolbox, 2015).
As an example, if we consider a universe of discourse from 40 inches to 90 inches, then, to describe height, we can use three term values such as short, average, and tall. In practice, the terms short, medium, and tall are not used in the strict sense. Instead, they imply a smooth transition. Fuzzy membership functions representing these sets are shown in Figure below. The Figure shows that a person with height 65 inches will have membership value 1 for set medium, whereas a person with height 60 inches may be a member of the set short and also a member of the set medium; only the degree of membership varies with these sets; unlike classical logic, membership function of fuzzy logic not only gives two states true or false (0 or 1), but it can also give values between 0 and 1. Therefore, it is obvious that fuzzy logic greatly differs from classical logic in basic principles and potential use.
Figure 2.3: Demonstrating membership levels
The Fuzzy Logic System deals with fuzzy parameters, which address imprecision and uncertainties, by mapping out the path of a given input to an output using the computing framework is called the Fuzzy Inference System (Jang J., et al, 1997).
This framework consists of three main processes: the Fuzzification Process, the Inferences Process from Fuzzy Rules, and the Defuzzification Process (Shin Y. C. and Xu C. 2009). Figure 2.4 is a diagram of the fuzzy inference system with the three main processes. The mapping then provides a basis from which decisions can be made, or patterns discerned.
The process of fuzzy inference involves all of the pieces that are described in Membership Functions, Logical Operations, and If-Then Rules.
FUZZY
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FUZZY
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INFERENCE PROCESS
DEFUZZIFICATION PROCESS
Fuzzifier
Fuzzy Rule Base
Fuzzy inference Engine
Defuzzifier
Crisp
Input
Crisp
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FUZZY
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FUZZY
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INFERENCE PROCESS
DEFUZZIFICATION PROCESS
Fuzzifier
Fuzzy Rule Base
Fuzzy inference Engine
Defuzzifier
Crisp
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Crisp
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Figure 2.4: Fuzzy inference system: source (Shin Y. C. and Xu C. 2009).
Fuzzification
The Fuzzification Process consists of a fuzzifier that transforms crisp input into a fuzzy set of values based on its membership function (MF). A fuzzy set is a mathematical model comprised of vague qualitative or quantitative data, which is frequently generated by means of the natural language. The membership function is a curve that maps the inputs to a membership value that ranges between 0 and 1.The Fuzzification process allows the input to the system to be expressed in linguistic terms, using the membership functions.
Inference process
The inference process involves the fuzzy inference engine that is used to perform the mapping between the input from the Fuzzification process and the output based on expert knowledge or rules. The role of fuzzy rules in the inference process is to capture the imprecise modes of reasoning and to act as the means to produce the fuzzy output from the fuzzy input.
A fuzzy rule is also known as the Fuzzy IF-THEN rule and is generally expressed as follows (Sumathi S. and Surekha P, 2010):
IF (x is A) AND (y is B) THEN (z is Z)
Where x, y, z represent the variables, and A, B, Z are the linguistic values in the universe of discourse.
This rule can be divided into two parts, the IF part, which is referred to as the antecedent or premise that contains the fuzzy description of the measured input values, and the THEN part, which is referred to as the consequent or conclusion that defines a possible fuzzy output for every corresponding input. The inference process creates the fuzzy output as the aggregation from several fuzzy rules.
Defuzzification
The defuzzification process produces and translates an aggregate fuzzy output from the inference process into a quantifiable result or crisp output. The most popular defuzzification method is the centroid calculation, which returns the center of an area under the curve according to the following Formula.
Where n is the number of discrete elements, xi is the value of the discrete element, and μA (xi) represent the corresponding MF value at the point xi.
The centroid defuzzification method finds the “balance” point of the solution fuzzy region by calculating the weighted mean of the output fuzzy region. It is the most widely used technique because, when it is used, the defuzzified values tend to move smoothly around the output fuzzy region. The technique is unique, however, and not easy to implement computationally. The method of centroid defuzzification is depicted in Figure 2.5.
Center Of gravity
X
μA(x)
Center Of gravity
X
μA(x)
Figure 2.5: Defuzzification by centroid method
The following figure shows the actual full-size fuzzy inference diagram. There is a lot to see in a fuzzy inference diagram. For instance, from this diagram with these particular inputs, we can easily see that the implication method is truncation with the min function. The max function is being used for the fuzzy OR operation.
Figure 2.6: Overall fuzzy inference process source: (simulink fuzzy logic toolbox)
With the ability to mimic the human mind and to deal with uncertainty, the fuzzy logic system provides the right tool for risk management that involves imprecision in the form of the likely outcome from an uncertain event.
Fuzzy logic advantages and limits in PRA
There are some important benefits of fuzzy approach for risk management process that are not possible to achieve with fully qualitative methods.
Some advantages of fuzzy logic model are:
1) It provides a way to combine expert opinions with quantitative data. Fuzzy logic provides a way to translate human language into mathematical equivalents, which is necessary for a decision model in which many of the input variables are not precise quantities.
2) With Fuzzy logic, we can aggregate the input values into ranges to reduce the number of logic rules that need to be specified. This is particularly important in this case given the likely need to continue to update and refine the parameters of the model. With several survey questions requesting sourcing managers to give answers based on a scale of 1 to 10, the number of logic rules that would have to be evaluated would likely be too large to maintain and update on a regular basis.
3) Fuzzy concept eliminates the need to specify arbitrary, precise thresholds for the ranges of the input values. Instead, with fuzzy logic, one can define ranges of sets in which the threshold values are extreme cases, beyond which it is extremely unlikely that a value would be considered part of the set.
4) Regular repeating of fuzzy risk analysis is important for achieving the optimal results. That is a reason why there are good results of using fuzzy approach for assessing the highly dynamic systems where there are regular changes in values of input data.
Two key disadvantages of a fuzzy logic model for this application are:
1) It requires the use of an obscure subject that does not necessarily accomplish something that traditional, or crisp logic, rules cannot. As discussed above, it is possible to establish crisp logic rules that will lead to the same conclusion as a fuzzy logic model. In the temperature example above, one could create logic rules for each value of the temperature and the sunlight measure in order to determine at what level the heater should be set. A traditional logic model would be easier to understand and eliminate the need for users of the model to learn about a subject with which they are most likely not familiar.
2) It still requires the enumeration of logic rules. While the number of rules will be fewer than in the case of a crisp set in which input values are not aggregated into ranges, there will still be many logic rules that need to be created and maintained, and the number of logic rules will still increase exponentially with the number of data input variables chosen. The implementation of the model should allow for easy updating of these logic rules, though the sheer number of logic rules that need to be created could represent a major obstacle in using a fuzzy logic-based model for this purpose.
Fuzzy Risk assessment and decision making
A risk assessment and decision-making platform built on a fuzzy logic system can provide consistency when analyzing risks with limited data and knowledge. It allows people to focus on the foundation of risk assessment, which involves the cause-and-effect relationship between key factors as well as the exposure for each individual risk. The decision making step is the last step of the assessment as shown in the figure below.
The techniques of risk analysis are powerful tools to help people manage uncertainty. Thorough risk analysis estimation and evaluation can provide valuable support for decision making. There are many risk analysis techniques currently in use that attempt to evaluate and estimate risk. These techniques can be either qualitative or quantitative depending on the information available and the level of detail that is required (J.C. Bennett, G.A. Bohoris, 1996).
RISK LEVEL
FUZZYINFERENCESYSTEM
Probability
Severity
DECISION MAKING (ACTION)
RISK LEVEL
FUZZYINFERENCESYSTEM
Probability
Severity
DECISION MAKING (ACTION)
Figure 2.7: Fuzzy decision making process
The usual way to assess risk using fuzzy logic is to model a Fuzzy inference system (FIS).A Fuzzy inference system is a rule-based system with concepts and operations associated with fuzzy set theory and fuzzy logic (Mendel, 2001; Ross, 2010). These systems map an input space to an output state; therefore, they allow constructing structures that can be used to generate responses (outputs) to certain stimulations (inputs), based on stored knowledge on how the responses and stimulations are related. There are 4 main steps in the FIS as described in the section 2.2.
Step 1. Independent variables are selected as the key determinants or indicators of the dependent variable.
Step 2. Fuzzy sets are created for both independent and dependent variables. Instead of using the numerical value, fuzzy sets in terms of human language are used to describe a variable. The degree of truth that each variable belongs to a certain fuzzy set is specified by the membership function.
Step 3. Inference rules are built in the system. A fuzzy hedge may be used to tweak the membership function according to the description of the inference rules.
Step 4. The output fuzzy set of the dependent variable is generated based on the independent variables and the inference rules. After defuzzification, a numerical value may be used to represent the output fuzzy set.
An easier way to use fuzzy logic in assessing risks is using a computer based method, namely fuzzy expert systems. Expert Systems (ES), also called knowledge-based systems, are computer programs that achieve the same level of accuracy as human experts when dealing with complex, structured specific domain problems so that they can be used by non-experts to obtain answers, solve problems or get decision support within such domains (Turban, E. et al., 2004). The strength of these systems lies in their ability to put expert knowledge to practical use when an expert is not available. Expert systems make knowledge more widely available and help overcome the problem of translating knowledge into practical, useful results. ES architecture contains four basic components: (a) a specialized Knowledge Base that stores the relevant knowledge about the domain of expertise; (b) an Inference Engine, which is used to reason about specific problems, for example using production rules or multiple attribute decision-making models; (c) a working memory, which records facts about the real world; and (d) an interface that allows user-system interaction, as depicted in the figure below.
