Category: fuzzy Logic

Fuzzy Logic – Useful Resources The following resources contain additional information on Fuzzy Logic. Please use them to get more in-depth knowledge on this. Useful Links on Fuzzy Logic − Wikipedia Reference for Fuzzy Logic Useful Books on Fuzzy Logic To enlist your site on this page, please drop an email to [email protected] Learning working make money

Adaptive Fuzzy Controller In this chapter, we will discuss what is an Adaptive Fuzzy Controller and how it works. Adaptive Fuzzy Controller is designed with some adjustable parameters along with an embedded mechanism for adjusting them. Adaptive controller has been used for improving the performance of controller. Basic Steps for Implementing Adaptive Algorithm Let us now discuss the basic steps for implementing adaptive algorithm. Collection of observable data − The observable data is collected to calculate the performance of controller. Adjustment of controller parameters − Now with the help of controller performance, calculation of adjustment of controller parameters would be done. Improvement in performance of controller − In this step, the controller parameters are adjusted to improve the performance of controller. Operational Concepts Design of a controller is based on an assumed mathematical model that resembles a real system. The error between actual system and its mathematical representation is calculated and if it is relatively insignificant than the model is assumed to work effectively. A threshold constant that sets a boundary for the effectiveness of a controller, also exists. The control input is fed into both the real system and mathematical model. Here, assume $xleft ( t right )$ is the output of the real system and $yleft ( t right )$ is the output of the mathematical model. Then the error $epsilon left ( t right )$ can be calculated as follows − $$epsilon left ( t right ) = xleft ( t right ) – yleft ( t right )$$ Here, $x$ desired is the output we want from the system and $mu left ( t right )$ is the output coming from controller and going to both real as well as mathematical model. The following diagram shows how the error function is tracked between output of a real system and Mathematical model − Parameterization of System A fuzzy controller the design of which is based on the fuzzy mathematical model will have the following form of fuzzy rules − Rule 1 − IF $x_1left ( t_n right )in X_{11} : AND…AND: x_ileft ( t_n right )in X_{1i}$ THEN $mu _1left ( t_n right ) = K_{11}x_1left ( t_n right ) + K_{12}x_2left ( t_n right ) : +…+ : K_{1i}x_ileft ( t_n right )$ Rule 2 − IF $x_1left ( t_n right )in X_{21} : AND…AND : x_ileft ( t_n right )in X_{2i}$ THEN $mu _2left ( t_n right ) = K_{21}x_1left ( t_n right ) + K_{22}x_2left ( t_n right ) : +…+ : K_{2i}x_ileft ( t_n right ) $ . . . Rule j − IF $x_1left ( t_n right )in X_{k1} : AND…AND : x_ileft ( t_n right )in X_{ki}$ THEN $mu _jleft ( t_n right ) = K_{j1}x_1left ( t_n right ) + K_{j2}x_2left ( t_n right ) : +…+ : K_{ji}x_ileft ( t_n right ) $ The above set of parameters characterizes the controller. Mechanism Adjustment The controller parameters are adjusted to improve the performance of controller. The process of calculating the adjustment to the parameters is the adjusting mechanism. Mathematically, let $theta ^left ( n right )$ be a set of parameters to be adjusted at time $t = t_n$. The adjustment can be the recalculation of the parameters, $$theta ^left ( n right ) = Theta left ( D_0,: D_1, : …, :D_n right )$$ Here $D_n$ is the data collected at time $t = t_n$. Now this formulation is reformulated by the update of the parameter set based on its previous value as, $$theta ^left ( n right ) = phi ( theta ^{n-1}, : D_n)$$ Parameters for selecting an Adaptive Fuzzy Controller The following parameters need to be considered for selecting an adaptive fuzzy controller − Can the system be approximated entirely by a fuzzy model? If a system can be approximated entirely by a fuzzy model, are the parameters of this fuzzy model readily available or must they be determined online? If a system cannot be approximated entirely by a fuzzy model, can it be approximated piecewise by a set of fuzzy model? If a system can be approximated by a set of fuzzy models, are these models having the same format with different parameters or are they having different formats? If a system can be approximated by a set of fuzzy models having the same format, each with a different set of parameters, are these parameter sets readily available or must they be determined online? Learning working make money

Fuzzy Logic – Control System Fuzzy logic is applied with great success in various control application. Almost all the consumer products have fuzzy control. Some of the examples include controlling your room temperature with the help of air-conditioner, anti-braking system used in vehicles, control on traffic lights, washing machines, large economic systems, etc. Why Use Fuzzy Logic in Control Systems A control system is an arrangement of physical components designed to alter another physical system so that this system exhibits certain desired characteristics. Following are some reasons of using Fuzzy Logic in Control Systems − While applying traditional control, one needs to know about the model and the objective function formulated in precise terms. This makes it very difficult to apply in many cases. By applying fuzzy logic for control we can utilize the human expertise and experience for designing a controller. The fuzzy control rules, basically the IF-THEN rules, can be best utilized in designing a controller. Assumptions in Fuzzy Logic Control (FLC) Design While designing fuzzy control system, the following six basic assumptions should be made − The plant is observable and controllable − It must be assumed that the input, output as well as state variables are available for observation and controlling purpose. Existence of a knowledge body − It must be assumed that there exist a knowledge body having linguistic rules and a set of input-output data set from which rules can be extracted. Existence of solution − It must be assumed that there exists a solution. ‘Good enough’ solution is enough − The control engineering must look for ‘good enough’ solution rather than an optimum one. Range of precision − Fuzzy logic controller must be designed within an acceptable range of precision. Issues regarding stability and optimality − The issues of stability and optimality must be open in designing Fuzzy logic controller rather than addressed explicitly. Architecture of Fuzzy Logic Control The following diagram shows the architecture of Fuzzy Logic Control (FLC). Major Components of FLC Followings are the major components of the FLC as shown in the above figure − Fuzzifier − The role of fuzzifier is to convert the crisp input values into fuzzy values. Fuzzy Knowledge Base − It stores the knowledge about all the input-output fuzzy relationships. It also has the membership function which defines the input variables to the fuzzy rule base and the output variables to the plant under control. Fuzzy Rule Base − It stores the knowledge about the operation of the process of domain. Inference Engine − It acts as a kernel of any FLC. Basically it simulates human decisions by performing approximate reasoning. Defuzzifier − The role of defuzzifier is to convert the fuzzy values into crisp values getting from fuzzy inference engine. Steps in Designing FLC Following are the steps involved in designing FLC − Identification of variables − Here, the input, output and state variables must be identified of the plant which is under consideration. Fuzzy subset configuration − The universe of information is divided into number of fuzzy subsets and each subset is assigned a linguistic label. Always make sure that these fuzzy subsets include all the elements of universe. Obtaining membership function − Now obtain the membership function for each fuzzy subset that we get in the above step. Fuzzy rule base configuration − Now formulate the fuzzy rule base by assigning relationship between fuzzy input and output. Fuzzification − The fuzzification process is initiated in this step. Combining fuzzy outputs − By applying fuzzy approximate reasoning, locate the fuzzy output and merge them. Defuzzification − Finally, initiate defuzzification process to form a crisp output. Advantages of Fuzzy Logic Control Let us now discuss the advantages of Fuzzy Logic Control. Cheaper − Developing a FLC is comparatively cheaper than developing model based or other controller in terms of performance. Robust − FLCs are more robust than PID controllers because of their capability to cover a huge range of operating conditions. Customizable − FLCs are customizable. Emulate human deductive thinking − Basically FLC is designed to emulate human deductive thinking, the process people use to infer conclusion from what they know. Reliability − FLC is more reliable than conventional control system. Efficiency − Fuzzy logic provides more efficiency when applied in control system. Disadvantages of Fuzzy Logic Control We will now discuss what are the disadvantages of Fuzzy Logic Control. Requires lots of data − FLC needs lots of data to be applied. Useful in case of moderate historical data − FLC is not useful for programs much smaller or larger than historical data. Needs high human expertise − This is one drawback as the accuracy of the system depends on the knowledge and expertise of human beings. Needs regular updating of rules − The rules must be updated with time. Learning working make money

Fuzzy Logic – Set Theory Fuzzy sets can be considered as an extension and gross oversimplification of classical sets. It can be best understood in the context of set membership. Basically it allows partial membership which means that it contain elements that have varying degrees of membership in the set. From this, we can understand the difference between classical set and fuzzy set. Classical set contains elements that satisfy precise properties of membership while fuzzy set contains elements that satisfy imprecise properties of membership. Mathematical Concept A fuzzy set $widetilde{A}$ in the universe of information $U$ can be defined as a set of ordered pairs and it can be represented mathematically as − $$widetilde{A} = left { left ( y,mu _{widetilde{A}} left ( y right ) right ) | yin Uright }$$ Here $mu _{widetilde{A}}left ( y right )$ = degree of membership of $y$ in widetilde{A}, assumes values in the range from 0 to 1, i.e., $mu _{widetilde{A}}(y)in left [ 0,1 right ]$. Representation of fuzzy set Let us now consider two cases of universe of information and understand how a fuzzy set can be represented. Case 1 When universe of information $U$ is discrete and finite − $$widetilde{A} = left { frac{mu _{widetilde{A}}left ( y_1 right )}{y_1} +frac{mu _{widetilde{A}}left ( y_2 right )}{y_2} +frac{mu _{widetilde{A}}left ( y_3 right )}{y_3} +…right }$$ $= left { sum_{i=1}^{n}frac{mu _{widetilde{A}}left ( y_i right )}{y_i} right }$ Case 2 When universe of information $U$ is continuous and infinite − $$widetilde{A} = left { int frac{mu _{widetilde{A}}left ( y right )}{y} right }$$ In the above representation, the summation symbol represents the collection of each element. Operations on Fuzzy Sets Having two fuzzy sets $widetilde{A}$ and $widetilde{B}$, the universe of information $U$ and an element 𝑦 of the universe, the following relations express the union, intersection and complement operation on fuzzy sets. Union/Fuzzy ‘OR’ Let us consider the following representation to understand how the Union/Fuzzy ‘OR’ relation works − $$mu _{{widetilde{A}cup widetilde{B} }}left ( y right ) = mu _{widetilde{A}}vee mu _widetilde{B} quad forall y in U$$ Here ∨ represents the ‘max’ operation. Intersection/Fuzzy ‘AND’ Let us consider the following representation to understand how the Intersection/Fuzzy ‘AND’ relation works − $$mu _{{widetilde{A}cap widetilde{B} }}left ( y right ) = mu _{widetilde{A}}wedge mu _widetilde{B} quad forall y in U$$ Here ∧ represents the ‘min’ operation. Complement/Fuzzy ‘NOT’ Let us consider the following representation to understand how the Complement/Fuzzy ‘NOT’ relation works − $$mu _{widetilde{A}} = 1-mu _{widetilde{A}}left ( y right )quad y in U$$ Properties of Fuzzy Sets Let us discuss the different properties of fuzzy sets. Commutative Property Having two fuzzy sets $widetilde{A}$ and $widetilde{B}$, this property states − $$widetilde{A}cup widetilde{B} = widetilde{B}cup widetilde{A}$$ $$widetilde{A}cap widetilde{B} = widetilde{B}cap widetilde{A}$$ Associative Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$(widetilde{A}cup left widetilde{B}) cup widetilde{C} right = left widetilde{A} cup (widetilde{B}right )cup widetilde{C})$$ $$(widetilde{A}cap left widetilde{B}) cap widetilde{C} right = left widetilde{A} cup (widetilde{B}right cap widetilde{C})$$ Distributive Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$widetilde{A}cup left ( widetilde{B} cap widetilde{C}right ) = left ( widetilde{A} cup widetilde{B}right )cap left ( widetilde{A}cup widetilde{C} right )$$ $$widetilde{A}cap left ( widetilde{B}cup widetilde{C} right ) = left ( widetilde{A} cap widetilde{B} right )cup left ( widetilde{A}cap widetilde{C} right )$$ Idempotency Property For any fuzzy set $widetilde{A}$, this property states − $$widetilde{A}cup widetilde{A} = widetilde{A}$$ $$widetilde{A}cap widetilde{A} = widetilde{A}$$ Identity Property For fuzzy set $widetilde{A}$ and universal set $U$, this property states − $$widetilde{A}cup varphi = widetilde{A}$$ $$widetilde{A}cap U = widetilde{A}$$ $$widetilde{A}cap varphi = varphi$$ $$widetilde{A}cup U = U$$ Transitive Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$If : widetilde{A}subseteq widetilde{B}subseteq widetilde{C},:then:widetilde{A}subseteq widetilde{C}$$ Involution Property For any fuzzy set $widetilde{A}$, this property states − $$overline{overline{widetilde{A}}} = widetilde{A}$$ De Morgan’s Law This law plays a crucial role in proving tautologies and contradiction. This law states − $$overline{{widetilde{A}cap widetilde{B}}} = overline{widetilde{A}}cup overline{widetilde{B}}$$ $$overline{{widetilde{A}cup widetilde{B}}} = overline{widetilde{A}}cap overline{widetilde{B}}$$ Learning working make money

Fuzzy Logic – Traditional Fuzzy Refresher Logic, which was originally just the study of what distinguishes sound argument from unsound argument, has now developed into a powerful and rigorous system whereby true statements can be discovered, given other statements that are already known to be true. Predicate Logic This logic deals with predicates, which are propositions containing variables. A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. Following are a few examples of predicates − Let E(x, y) denote “x = y” Let X(a, b, c) denote “a + b + c = 0” Let M(x, y) denote “x is married to y” Propositional Logic A proposition is a collection of declarative statements that have either a truth value “true” or a truth value “false”. A propositional consists of propositional variables and connectives. The propositional variables are dented by capital letters (A, B, etc). The connectives connect the propositional variables. A few examples of Propositions are given below − “Man is Mortal”, it returns truth value “TRUE” “12 + 9 = 3 – 2”, it returns truth value “FALSE” The following is not a Proposition − “A is less than 2” − It is because unless we give a specific value of A, we cannot say whether the statement is true or false. Connectives In propositional logic, we use the following five connectives − OR (∨∨) AND (∧∧) Negation/ NOT (¬¬) Implication / if-then (→→) If and only if (⇔⇔) OR (∨∨) The OR operation of two propositions A and B (written as A∨BA∨B) is true if at least any of the propositional variable A or B is true. The truth table is as follows − A B A ∨ B True True True True False True False True True False False False AND (∧∧) The AND operation of two propositions A and B (written as A∧BA∧B) is true if both the propositional variable A and B is true. The truth table is as follows − A B A ∧ B True True True True False False False True False False False False Negation (¬¬) The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false. The truth table is as follows − A ¬A True False False True Implication / if-then (→→) An implication A→BA→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true. The truth table is as follows − A B A→B True True True True False False False True True False False True If and only if (⇔⇔) A⇔BA⇔B is a bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true. The truth table is as follows − A B A⇔B True True True True False False False True False False False True Well Formed Formula Well Formed Formula (wff) is a predicate holding one of the following − All propositional constants and propositional variables are wffs. If x is a variable and Y is a wff, ∀xY and ∃xY are also wff. Truth value and false values are wffs. Each atomic formula is a wff. All connectives connecting wffs are wffs. Quantifiers The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic − Universal Quantifier Existential Quantifier Universal Quantifier Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀. ∀xP(x) is read as for every value of x, P(x) is true. Example − “Man is mortal” can be transformed into the propositional form ∀xP(x). Here, P(x) is the predicate which denotes that x is mortal and the universe of discourse is all men. Existential Quantifier Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃. ∃xP(x) for some values of x is read as, P(x) is true. Example − “Some people are dishonest” can be transformed into the propositional form ∃x P(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people. Nested Quantifiers If we use a quantifier that appears within the scope of another quantifier, it is called a nested quantifier. Example ∀ a∃bP(x,y) where P(a,b) denotes a+b = 0 ∀ a∀b∀cP(a,b,c) where P(a,b) denotes a+(b+c) = (a+b)+c Note − ∀a∃bP(x,y) ≠ ∃a∀bP(x,y) Learning working make money

Fuzzy Logic Tutorial Job Search Fuzzy Logic resembles the human decision-making methodology and deals with vague and imprecise information. This is a very small tutorial that touches upon the very basic concepts of Fuzzy Logic. Audience This tutorial will be useful for graduates, post-graduates, and research students who either have an interest in this subject or have this subject as a part of their curriculum. The reader can be a beginner or an advanced learner. Prerequisites Fuzzy Logic is an advanced topic, so we assume that the readers of this tutorial have preliminary knowledge of Set Theory, Logic, and Engineering Mathematics. Learning working make money

Fuzziness in Neural Networks Artificial neural network (ANN) is a network of efficient computing systems the central theme of which is borrowed from the analogy of biological neural networks. ANNs are also named as “artificial neural systems,” parallel distributed processing systems,” “connectionist systems.” ANN acquires large collection of units that are interconnected in some pattern to allow communications between units. These units, also referred to as nodes or neurons, are simple processors which operate in parallel. Every neuron is connected with other neuron through a connection link. Each connection link is associated with a weight having the information about the input signal. This is the most useful information for neurons to solve a particular problem because the weight usually inhibits the signal that is being communicated. Each neuron is having its internal state which is called the activation signal. Output signals, which are produced after combining the input signals and the activation rule, may be sent to other units. It also consists of a bias ‘b’ whose weight is always 1. Why to use Fuzzy Logic in Neural Network As we have discussed above that every neuron in ANN is connected with other neuron through a connection link and that link is associated with a weight having the information about the input signal. Hence we can say that weights have the useful information about input to solve the problems. Following are some reasons to use fuzzy logic in neural networks − Fuzzy logic is largely used to define the weights, from fuzzy sets, in neural networks. When crisp values are not possible to apply, then fuzzy values are used. We have already studied that training and learning help neural networks perform better in unexpected situations. At that time fuzzy values would be more applicable than crisp values. When we use fuzzy logic in neural networks then the values must not be crisp and the processing can be done in parallel. Fuzzy Cognitive Map It is a form of fuzziness in neural networks. Basically FCM is like a dynamic state machine with fuzzy states (not just 1 or 0). Difficulty in using Fuzzy Logic in Neural Networks Despite having numerous advantages, there is also some difficulty while using fuzzy logic in neural networks. The difficulty is related with membership rules, the need to build fuzzy system, because it is sometimes complicated to deduce it with the given set of complex data. Neural-Trained Fuzzy Logic The reverse relationship between neural network and fuzzy logic, i.e., neural network used to train fuzzy logic is also a good area of study. Following are two major reasons to build neuraltrained fuzzy logic − New patterns of data can be learned easily with the help of neural networks hence, it can be used to preprocess data in fuzzy systems. Neural network, because of its capability to learn new relationship with new input data, can be used to refine fuzzy rules to create fuzzy adaptive system. Examples of Neural-Trained Fuzzy system Neural-Trained Fuzzy systems are being used in many commercial applications. Let us now see a few examples where Neural-Trained Fuzzy system is applied − The Laboratory for International Fuzzy Engineering Research (LIFE) in Yokohama, Japan has a back-propagation neural network that derives fuzzy rules. This system has been successfully applied to foreign-exchange trade system with approximately 5000 fuzzy rules. Ford Motor Company has developed trainable fuzzy systems for automobile idle-speed control. NeuFuz, software product of National Semiconductor Corporation, supports the generation of fuzzy rules with a neural network for control applications. AEG Corporation of Germany uses neural-trained fuzzy control system for its water – and energy conserving machine. It is having total of 157 fuzzy rules. Learning working make money

Fuzzy Logic – Applications In this chapter, we will discuss the fields where the concepts of Fuzzy Logic are extensively applied. Aerospace In aerospace, fuzzy logic is used in the following areas − Altitude control of spacecraft Satellite altitude control Flow and mixture regulation in aircraft deicing vehicles Automotive In automotive, fuzzy logic is used in the following areas − Trainable fuzzy systems for idle speed control Shift scheduling method for automatic transmission Intelligent highway systems Traffic control Improving efficiency of automatic transmissions Business In business, fuzzy logic is used in the following areas − Decision-making support systems Personnel evaluation in a large company Defense In defense, fuzzy logic is used in the following areas − Underwater target recognition Automatic target recognition of thermal infrared images Naval decision support aids Control of a hypervelocity interceptor Fuzzy set modeling of NATO decision making Electronics In electronics, fuzzy logic is used in the following areas − Control of automatic exposure in video cameras Humidity in a clean room Air conditioning systems Washing machine timing Microwave ovens Vacuum cleaners Finance In the finance field, fuzzy logic is used in the following areas − Banknote transfer control Fund management Stock market predictions Industrial Sector In industrial, fuzzy logic is used in following areas − Cement kiln controls heat exchanger control Activated sludge wastewater treatment process control Water purification plant control Quantitative pattern analysis for industrial quality assurance Control of constraint satisfaction problems in structural design Control of water purification plants Manufacturing In the manufacturing industry, fuzzy logic is used in following areas − Optimization of cheese production Optimization of milk production Marine In the marine field, fuzzy logic is used in the following areas − Autopilot for ships Optimal route selection Control of autonomous underwater vehicles Ship steering Medical In the medical field, fuzzy logic is used in the following areas − Medical diagnostic support system Control of arterial pressure during anesthesia Multivariable control of anesthesia Modeling of neuropathological findings in Alzheimer”s patients Radiology diagnoses Fuzzy inference diagnosis of diabetes and prostate cancer Securities In securities, fuzzy logic is used in following areas − Decision systems for securities trading Various security appliances Transportation In transportation, fuzzy logic is used in the following areas − Automatic underground train operation Train schedule control Railway acceleration Braking and stopping Pattern Recognition and Classification In Pattern Recognition and Classification, fuzzy logic is used in the following areas − Fuzzy logic based speech recognition Fuzzy logic based Handwriting recognition Fuzzy logic based facial characteristic analysis Command analysis Fuzzy image search Psychology In Psychology, fuzzy logic is used in following areas − Fuzzy logic based analysis of human behavior Criminal investigation and prevention based on fuzzy logic reasoning Learning working make money

Discuss Fuzzy Logic Fuzzy Logic resembles the human decision-making methodology and deals with vague and imprecise information. This is a very small tutorial that touches upon the very basic concepts of Fuzzy Logic. Learning working make money

Fuzzy Logic – Membership Function We already know that fuzzy logic is not logic that is fuzzy but logic that is used to describe fuzziness. This fuzziness is best characterized by its membership function. In other words, we can say that membership function represents the degree of truth in fuzzy logic. Following are a few important points relating to the membership function − Membership functions were first introduced in 1965 by Lofti A. Zadeh in his first research paper “fuzzy sets”. Membership functions characterize fuzziness (i.e., all the information in fuzzy set), whether the elements in fuzzy sets are discrete or continuous. Membership functions can be defined as a technique to solve practical problems by experience rather than knowledge. Membership functions are represented by graphical forms. Rules for defining fuzziness are fuzzy too. Mathematical Notation We have already studied that a fuzzy set à in the universe of information U can be defined as a set of ordered pairs and it can be represented mathematically as − $$widetilde{A} = left { left ( y,mu _{widetilde{A}} left ( y right ) right ) | yin Uright }$$ Here $mu widetilde{A}left (bullet right )$ = membership function of $widetilde{A}$; this assumes values in the range from 0 to 1, i.e., $mu widetilde{A}left (bullet right )in left [ 0,1 right ]$. The membership function $mu widetilde{A}left (bullet right )$ maps $U$ to the membership space$M$. The dot $left (bullet right )$ in the membership function described above, represents the element in a fuzzy set; whether it is discrete or continuous. Features of Membership Functions We will now discuss the different features of Membership Functions. Core For any fuzzy set $widetilde{A}$, the core of a membership function is that region of universe that is characterize by full membership in the set. Hence, core consists of all those elements $y$ of the universe of information such that, $$mu _{widetilde{A}}left ( y right ) = 1$$ Support For any fuzzy set $widetilde{A}$, the support of a membership function is the region of universe that is characterize by a nonzero membership in the set. Hence core consists of all those elements $y$ of the universe of information such that, $$mu _{widetilde{A}}left ( y right ) > 0$$ Boundary For any fuzzy set $widetilde{A}$, the boundary of a membership function is the region of universe that is characterized by a nonzero but incomplete membership in the set. Hence, core consists of all those elements $y$ of the universe of information such that, $$1 > mu _{widetilde{A}}left ( y right ) > 0$$ Fuzzification It may be defined as the process of transforming a crisp set to a fuzzy set or a fuzzy set to fuzzier set. Basically, this operation translates accurate crisp input values into linguistic variables. Following are the two important methods of fuzzification − Support Fuzzification(s-fuzzification) Method In this method, the fuzzified set can be expressed with the help of the following relation − $$widetilde{A} = mu _1Qleft ( x_1 right )+mu _2Qleft ( x_2 right )+…+mu _nQleft ( x_n right )$$ Here the fuzzy set $Qleft ( x_i right )$ is called as kernel of fuzzification. This method is implemented by keeping $mu _i$ constant and $x_i$ being transformed to a fuzzy set $Qleft ( x_i right )$. Grade Fuzzification (g-fuzzification) Method It is quite similar to the above method but the main difference is that it kept $x_i$ constant and $mu _i$ is expressed as a fuzzy set. Defuzzification It may be defined as the process of reducing a fuzzy set into a crisp set or to convert a fuzzy member into a crisp member. We have already studied that the fuzzification process involves conversion from crisp quantities to fuzzy quantities. In a number of engineering applications, it is necessary to defuzzify the result or rather “fuzzy result” so that it must be converted to crisp result. Mathematically, the process of Defuzzification is also called “rounding it off”. The different methods of Defuzzification are described below − Max-Membership Method This method is limited to peak output functions and also known as height method. Mathematically it can be represented as follows − $$mu _{widetilde{A}}left ( x^* right )>mu _{widetilde{A}}left ( x right ) : for :all:x in X$$ Here, $x^*$ is the defuzzified output. Centroid Method This method is also known as the center of area or the center of gravity method. Mathematically, the defuzzified output $x^*$ will be represented as − $$x^* = frac{int mu _{widetilde{A}}left ( x right ).xdx}{int mu _{widetilde{A}}left ( x right ).dx}$$ Weighted Average Method In this method, each membership function is weighted by its maximum membership value. Mathematically, the defuzzified output $x^*$ will be represented as − $$x^* = frac{sum mu _{widetilde{A}}left ( overline{x_i} right ).overline{x_i}}{sum mu _{widetilde{A}}left ( overline{x_i} right )}$$ Mean-Max Membership This method is also known as the middle of the maxima. Mathematically, the defuzzified output $x^*$ will be represented as − $$x^* = frac{displaystyle sum_{i=1}^{n}overline{x_i}}{n}$$ Learning working make money