# Methodology Chapter: Mamdani Rule-Based System Defuzzification Genetic

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Mamdani Rule-Based System

Defuzzification

Genetic Algorithms

Genetic Encoding and Chromosomes

Fitness Calculation

Selection

Mutation

Crossover

Stopping Criteria

Artificial Neural Network

Perceptron Neurons

Sigmoid Neurons

Multi-Layers Network

Back Propagation Learning

This chapter discusses in detail the various approaches used to collect analyze and interpret data. It elaborates various important variables in relation to research methodology and approach. The various variables discussed under this section of the research include, fuzzy inference system, fuzzy set, fuzzy logic, fuzzification, Mamdani rule-based system, defuzzification, genetic algorithm, genetic encoding and chromosomes, fitness calculation, selection, mutation, crossover, stopping criteria, artificial neural network, perceptron neurons, sigmoid neurons, multi-layers network and back-propagation learning.

Fuzzy Inference System

Fuzzy inference or reasoning system is a process through which an individual maps out an output from a given input through the deployment of fuzzy logic. The process of fuzzy inference system consists of a number of components including, membership functions, if-then rules and fuzzy logic operators etcetera. (Faculty of International Burch University, 2012)

The process of fuzzy inference system has been applied to a number of fields in a successful manner. These fields include, computer vision, analysis of data, expert systems, classification of data, decision analysis and automatic control etcetera. (Faculty of International Burch University, 2012)

Because the fuzzy inference is multi-disciplinary in nature it is known by a number of names including fuzzy model, fuzzy rule-based system, fuzzy associative memory, fuzzy expert system, fuzzy logic controller, and the simple fuzzy system. (Faculty of International Burch University, 2012)

The figure below demonstrates the architecture of fuzzy inference systems. The main components of this architecture are fuzzifier, inference engine, defuzzifier and fuzzy knowledge base.

(Faculty of International Burch University, 2012)

The basic steps included in fuzzy inference system are listed below;

In the originating step, the input variables are compared with the membership rules for the determination of membership values for each of the linguistic labels. This step is also called fuzzification. (Faculty of International Burch University, 2012)

Combination of membership values on the part of premise for the evaluation of firing strength, also known as the degree of fulfilment, of various rules. (Faculty of International Burch University, 2012)

Generation of qualified consequents, which can be either fuzzy or crisp in nature, for each of the rules on the basis of their firing strength. (Faculty of International Burch University, 2012)

Aggregation of qualified consequents in order to produce an output that is crisp in nature. This step is often called defuzzification. (Faculty of International Burch University, 2012)

Components of Fuzzy Inference System

The four major components of fuzzy inference system are discussed in detail in the following section;

Fuzzy Knowledge Base

The rule base and the data base included in the fuzzy inference system are jointly known as the fuzzy knowledge base. A rule base in a fuzzy inference system consists of a number of if-then rules. A data base, on the other hand, defines and elaborates the membership functions associated with the fuzzy sets which deployed in the fuzzy rules. (Faculty of International Burch University, 2012)

Fuzzifier

The function of the fuzzifier is to convert he crisp input into a linguistic variable. This is done through the utilization of membership functions that are stored in the knowledge base of the fuzzy inference system. (Faculty of International Burch University, 2012)

Fuzzy Inference Engine

The fuzzy inference engine, with the help of a number of if-then rules, converts the fuzzy input into the fuzzy output. (Faculty of International Burch University, 2012)

Defuzzifier

The function of the defuzzifier is to convert the fuzzy input, which is generated by the inference engine, into a crisp output. This is done through the deployment of membership functions that are parallel to the membership functions used by the fuzzifier. The most commonly used defuzzifying methods include Bisector of area (BOA), Largest of maximum (LOM), Centroid of area (COA), Mean of maximum (MOM) and Smallest of maximum (SOM). (Faculty of International Burch University, 2012)

The figure below demonstrates how the defuzzifier and fuzzy inference engine work in a combination to convert a fuzzy input into a crisp output;

(Faculty of International Burch University, 2012)

3.2.1. Fuzzy Sets

According to the classical theory a set can be represented through the enumeration of all of its constituting elements by using (Saini & Singh, 2012);

A = {a1, a2, a3, a4a an}

(Saini & Singh, 2012)

According to the classical theory, the sets that have only two values 0 and 1 or only two values of truth are called crisp sets. The non-crisp sets are known as fuzzy sets. A characteristic function can be evaluated for fuzzy sets. This function can be developed through the generalization of the function of the classical set and is generally termed as the 'membership function.' The membership function of the fuzzy set, in contrast to the classical set theory, can have an arbitrary truth value in the closed and normalized interval. (Saini & Singh, 2012)

Each of the fuzzy set is particularly and wholly defined by a single and specific membership function. Apart from that, the symbols of the membership functions are used to label the fuzzy sets that are associated with these membership functions. (Saini & Singh, 2012)

In other words, each of the fuzzy sets and their accompanying membership functions are denoted by the same capital letter. (Saini & Singh, 2012) The figure below demonstrates the membership functions of a fuzzy set ? And a crisp set C.

(Saini & Singh, 2012)

Fuzzy sets were introduced in the year 1965 by Zadeh. The basic purpose behind the development and formulation of these sets was the mathematical representation of the vagueness and uncertainty that was associated with various variables. These sets also provide the researchers with formalized and sophisticated tools that can be used to deal the imprecision that is an inherent to a number of problems. (Fuzzy Sets and Fuzzy Logic, 2010)

The fuzzy sets and their membership functions can give a better representation of variables and elements that are vague in nature. (Faculty of Calvin College Engineering Department, 2011) For example, if we consider that all the people whose height is equal to or greater than six feet are tall then the ordinary set of these people can be demonstrated in the following manner;

(Faculty of Calvin College Engineering Department, 2011)

This type of a set consists of only two values of truth. It demonstrates that either a person is tall or he is not tall. A sharp edge fuzzy set, on the other hand, gives a more appropriate representation of the tallness of a person. (Faculty of Calvin College Engineering Department, 2011) The fuzzy set of tall people can be represented by a continuous inclining function and can be demonstrated as follows;

(Faculty of Calvin College Engineering Department, 2011)

The binary set, or the ordinary set, tells us that whether a person is tall or not. If a person is 6 feet 1 inch tall and the other is 7 feet 1 inch tall, this type of set characterizes both of these people as tall. The difference that exists between the heights of these people is ignored. In case of fuzzy sets, the degree of tallness of each person is appropriately represented. (Faculty of Calvin College Engineering Department, 2011)

3.2.2. Fuzzy Logic

Fuzzy logic can be defined as a computational model that works in a manner in which the human beings think. This logic comes into consideration when the conventional logic fails. Fuzzy logic understands world in an imprecise manner, similarly as our brain thinks. For instance, temperature is high or hot, water is deep etcetera. After looking at the world in an imprecise manner, this type of logic takes precise actions. (Massey University, 2012)

The interest in fuzzy logic was initiated by Seiji Yasunobu and Soji Miyamoto, who belonged to Hitachi, in the year 1985. They employed fussy systems to control accelerating and breaking in the Sendai Railway in the year 1987 after the inauguration of the line. (Massey University, 2012)

In the present era fuzzy logic is one of the most talked about concept in the technological and engineering industry. This is because it enables the inexpensive and cost efficient micro controllers to perform the functions that were traditionally being performed by heavy duty expensive machines. This, in return, enables the organizations to produce and deliver high quality goods and services in a cost efficient manner. As a result there is a significant increase in the productivity and efficiency of the organizations. (Massey University, 2012)

As discussed above fuzzy logic is a computer based approach that aims at mimicking the human way of thinking and solving problems. Formally, fuzzy logic can be defined as a knowledge base that aims at addressing the problems that are vague in nature and the propositions that are addressed in simple language. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

For instance, a proposition such as it is very unlikely that there would be a significant increase in the prices of gold…
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