Methodology Chapter on "Mamdani Rule-Based System Defuzzification Genetic"

Methodology Chapter 22 pages (6138 words) Sources: 30

[EXCERPT] . . . .

Fuzzy logic, however, can address this proposition in an appropriate manner with the help of test score semantics. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

One of the important components in fuzzy logic depends on linguistic variables. These are the variables whose value is defined in terms of words and numbers that in the form of natural language. Any relation that exists between two linguistic variables is expressed in terms of if-then rules. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

The if-then rules, when developed and formulated appropriately by the experts, form the basis of the knowledge base for the fuzzy logic controllers and the fuzzy expert systems. Once the meaning of and the relationship between the linguistic variables is determined, fuzzy logic provides the researchers with appropriate reasoning that consists of both deductive inferences as well as the interpolative inferences, (Faculty of Department of Computer Science and Engineering University of Nevada, 2006) as shown in the following example;

'Old coins are usually rare collectibles

Rare collectibles are generally expensive

Old coins are generally expensive'

(Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

In order to mimic and approximate the human way of thinking fuzzy logic uses neutral language terms rather than the quantitative language terms. A number of similarities exist between fuzzy language and neutral networks because behavioral systems can be created through the deployment of both of these strategies. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

Fuzzy logic through the exploration of uncertainty and unpredictability, continuously shapes the world in which we live. Fuzzy logic has also been used tremendously in the field of business and technology. The businesses and commercial institutions deploy fuzzy logic for the purpose of appropriate forecast and analysis. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

As discussed above, fussy logic enables various organizations to enhance their efficiency by using micro controllers, therefore, a number of organizations have made fuzzy logic an integral part of their production processes which makes their production cost efficient. (Faculty of Department of Computer Science and Engineering University of Nevada, 2006)

The figure below demonstrates the role played by fuzzy logic in the production of various controls;

3.2.3. Fuzzification

Fuzzification can be defined as a process in which the crisp quantities are converted into fuzzy quantities. The process of converting imprecise data and variables into fuzzy variables is useful but this process is not necessary when this fuzzy data is to be used in the fuzzy systems. (Faculty of Graduate School of Dalian Maritime University, 2011)

The above mentioned idea is demonstrated in the following figure in which the data has been considered as both a fuzzy reading as well as a crisp reading;

(Faculty of Graduate School of Dalian Maritime University, 2011)

In the above mentioned diagram, figure (a) represents fuzzy sets and crisp readings, whereas, figure (b) represents fuzzy sets along with fuzzy readings. (Faculty of Graduate School of Dalian Maritime University, 2011)

3.2.4. Mamdani Rule-Based System

The Mamdani Rule-based system consists of the following four major steps;

Step 1: Fuzzification

The first step involved with identifying the fuzzy inputs and then determining the degree to which these inputs belong to the appropriate fuzzy sets. (Kusiak & Negnevitsky, 2011)

Step 2: Rule Evaluation

The second step is of rule evaluation. In this steps the fuzzy inputs are taken and then applied to the antecedents of the fuzzy rules. If a fuzzy rule consists of a number of antecedents then either the fuzzy operator 'AND' or the fuzzy operator 'OR' is used to determine a single number that represents result of the evaluation of antecedents of fuzzy rules. (Kusiak & Negnevitsky, 2011)

The value determined is known as the truth value and it is then applied to the membership functions. In order to evaluate the disjunction of the antecedents of the fuzzy rule the fuzzy operator 'OR' is used. The fuzzy operator 'AND', on the other hand, is used to determine the conjunction of fuzzy rule antecedents. (Kusiak & Negnevitsky, 2011)

The typically used classical operation union of fuzzy rule-based system is;

A-B (x) = max [?A (x), ?B (x)]

(Kusiak & Negnevitsky, 2011)

The AND fuzzy operation intersection is;

A-B (x) = min [?A (x), ?B (x)]

(Kusiak & Negnevitsky, 2011)

The common method used to correlate the rule consequent with the truth value of the antecedent of the fuzzy rule is cutting the consequent membership function at a level that aligns with the truth of the antecedent of the fuzzy rules. This method is also known as clipping. (Kusiak & Negnevitsky, 2011)

The major drawback of clipping method is that the top of the membership function is sliced and hence it loses a considerable amount of information. Clipping, however, is preferred over other methods because it involves less complex and fast mathematical operations. In addition to that, the outputs produced by clipping are easy to defuzzify. (Kusiak & Negnevitsky, 2011)

Even though, clipping is a preferred approach but the scaling approach preserves the shape of the fuzzy states in an appropriate manner. In this method the rule consequents are multiplied by the truth of the rule antecedents. (Kusiak & Negnevitsky, 2011)

Step 3: Aggregation of the Rule Outputs

Aggregation is the process in which the outputs of the rules are unified. Under this step the membership functions of all the rule consequents, which were previously clipped or scaled, are and then combined into a single fuzzy set. (Kusiak & Negnevitsky, 2011)

The input in the step of rule evaluation is a list of clipped or scale consequents of membership functions and the output is unique fuzzy set for each of the variables of output. (Kusiak & Negnevitsky, 2011) The figure below is the pictorial representation of this step;

Step 4: Defuzzification

The last step of the Mamdani Rule-based system is defuzzification. Fuzziness enables the researchers to determine appropriate rules for the fuzzy variables but the final output of the Mamdani Rule-based system has to crisp in nature. The input in the process of defuzzification is the aggregate fuzzy sets and the output of this process is a single number. (Kusiak & Negnevitsky, 2011)

3.2.5. Defuzzification

It can be defined as a process of converting fuzzy quantities to crisp quantities. It is contradictory to fuzzification, which is the process of converting precise and crisp values into fuzzy quantities. (Faculty of Graduate School of Dalian Maritime University, 2011)

There various methods for defuzzifying fuzzy quantities some of these methods are mentioned below;

Max membership principle: It is also known as the height method. (Faculty of Graduate School of Dalian Maritime University, 2011) It can be mathematically and graphically represented as follows: (in the given formula z* is the defuzzified value)

(z*) ? For all

(Faculty of Graduate School of Dalian Maritime University, 2011)

Centroid Method: This method is also called center of area or the center of gravity method. (Faculty of Graduate School of Dalian Maritime University, 2011) The centroid method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Weighted Average Method: This method is generally restricted to the membership functions that produce symmetrical outputs. (Faculty of Graduate School of Dalian Maritime University, 2011) This method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Mean max membership: This method is also regarded as middle of maxima method. The maximum representation of a variable can be plateau rather than a single point. (Faculty of Graduate School of Dalian Maritime University, 2011) This method can be represented graphically and mathematically in the following manner;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

Center Of Sums: This method is faster than many of the methods of defuzzification. In addition to that, this method is not restricted to the membership functions that have symmetric outputs. Apart from that, this method involves the algebraic sums of individual fuzzy outputs rather than the union of these outputs. (Faculty of Graduate School of Dalian Maritime University, 2011) Graphically and mathematically, this method can be represented as follows;

Z* =

(Faculty of Graduate School of Dalian Maritime University, 2011)

In the above diagram, figure (a) represents the first membership function. The second membership function is represented by figure (b). Figure (c), on the other hand, represents the defuzzification step of the center of sum method. (Faculty of Graduate School of Dalian Maritime University, 2011)

3.3. Genetic Algorithms

The genetic algorithms were first introduced and deployed by John Holland. John Holland's book 'Adaptation in Natural and Artificial Systems' was the basis for the creation of a flourishing field in research and application that spreads beyond the field of genetic algorithms. (Reeves, 2008)

A biology analogy acted as the basic approach… READ MORE

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