fuzzy logic and its application in environmental engineering

Introduction to fuzzy logic and its application in Environmental

Engineering

Presented byDrashti V. Kapadia

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Content• Introduction

• Fuzzy Set vs Crisp Set

• Operation on Fuzzy System

• FMCDM

• Application in Environmental Engineering

• Overview of Research Papers

• Advantages and drawbacks

Presented By : Drashti V. Kapadia

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Introduction

• Fuzzy Logic is a rigorous methodology for dealing with elements of uncertainty and vagueness.

• It is a set of mathematical principles for knowledge representation based on degrees of membership.

• Lotfi A. Zadeh in 1965, introducing the concept of fuzzy sets, that opened a totally new view of systems, logic and models of reasoning


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Crisp Set vs Fuzzy Set• Crisp set A is a mapping for the elements of S to the set {0,1} A: S {0,1}

µ A(x) = 1 If x is an element of set A µ A(x) = 0 If x not an element of set A

• Fuzzy set F is a mapping for the elements of S to the interval [0,1]

F : S [0,1] Characteristic function: 0≤ µ F(x) ≤ 1

For 1 full membership and for 0 no membershipAnything between them called graded membership


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Crisp Set vs Fuzzy Set• Working with binary decision• 39°c has not been included in strong fever

• Therefore about 39°c we can say that it is less strong fever compare with 42°c is more strong fever.


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Crisp Set vs Fuzzy Set• The x-axis represents the universe of discourse – the range of

all possible values applicable to a chosen variable. The variable is the man height. The universe of men’s heights consists of all tall men

• The y-axis represents the membership value of the fuzzy set. The fuzzy set of “tall men” maps height values into corresponding membership values.


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Operations of Crisp Set and Fuzzy Set

Complement

0 x

1 (x)

0 x

1

Containment

0 x

1

0 x

1

AB

NotA

A

Intersection

0 x

1

0 x

AB

Union0

1AB

AB

0 x

1

0 x

1

BA

BA

(x)

(x) (x)

Intersection Union

Complement

NotA

A

Containment

AA

B

BA AA B


Operation of Fuzzy Rule Based System

Crisp Input

Fuzzy Input

Fuzzy Output

Crisp Output

Fuzzification

Rule Evaluation

Defuzzification

Input Membership Functions

Rules / Inferences

Output Membership Functions

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Fuzzification• Fuzzification

It is the process where the crisp quantities are converted to fuzzy

• Membership Function (MF)

It is a curve that defines how each point in the input space is mapped to a membership value between 0 and 1


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FuzzificationTypes of membership Function• Trimf Simplest membership function with three points It has easy mathematical formula.

• Trapmf It has a flat top with straight lines of simplicity.


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Fuzzification• Gaussian MF

Because of their smoothness and concise notation, Gaussian and bell membership functions are popular methods for specifying fuzzy sets. Both of these curves have the advantage of being smooth and nonzero at all points.


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Fuzzification• Sigmoidal MF

Asymmetric and closed (i.e. not open to the left or right) membership functions can be synthesized using two sigmoidal functions


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Fuzzification• The function zmf is the asymmetrical polynomial curve open

to the left, smf is the mirror-image function that opens to the right, and pimf is zero on both extremes with a rise in the middle


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Rule Evaluation• Fuzzy rules are linguistic IF-THEN- constructions that have

the general form "IF A THEN B“

• A is called the antecedent (premise) and B is the consequence (End result) of the rule

• By applying fuzzy operator ‘AND’, ‘OR’ and finally using implication method we can get single fuzzy variable.


Presented By : Drashti V. Kapadia 15

Types of Fuzzy Inference System

• Mamdani


Types of Fuzzy Inference System• Sugeno

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Difference• Mamdani FIS uses the technique of defuzzification of a fuzzy

output, Sugeno FIS uses weighted average to compute the crisp output. Therefore in Sugeno FIS the defuzzification process is by passed.

• Mamdani FIS has output membership functions whereas Sugeno FIS has no output membership functions.

• It should be noted that the Mamdani FIS can be used directly for either MISO systems (multiple input single output) as well as for MIMO systems (multiple input multiple output), while the SUGENO FIS can only be used in MISO systems


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Difference• Mamdani method is widely accepted for capturing expert

knowledge. Sugeno method is computationally efficient and works well with optimization and adaptive techniques, which makes it very attractive in control problems, particularly for dynamic non linear systems.

• Easy formalization and interpretability of Mamdani-type fuzzy inference systems (FIS), while ensuring the computational efficiency and accuracy of Sugeno-type FIS


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Defuzzification

• MethodsMax membership principleCentroid methodWeighted average methodMean max membershipCenter of sumsCenter of largest areaFirst (or last) of maxima


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Centroid method

• This method is also known as center of gravity or center of area defuzzification. This technique was developed by Sugeno in 1985. This is the most commonly used technique. The only disadvantage of this method is that it is computationally difficult for complex membership functions. The centroid defuzzification technique can be expressed as

• where zCOG is the crisp output, μA(z) is the aggregated membership function and z is the output variable


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Defuzzification


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FDM

1. Individual decision making

2. Multi person decision making

3. Multi criteria decision making

4. Multistage decision making

5. Fuzzy ranking

6. Fuzzy linear programming


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FMCDM• It is the process to choose amongst alternatives based on

multiple criteria.

• Methods

MADM (Multi Attribute Decision Making)

MODM (Multi Objective Decision Making)

• MADM involve the design of a ‘best’ alternative by considering the tradeoffs within a set of design constraint.

• In MODM number of alternatives is effectively infinite, and tradeoff among design criteria are typically described by continuous function.


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Three Level Hierarchy

Criteria

Alternatives

Goal

1 2 3 4

A B C


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Steps in MCDM Methodology • Defining the problem and fixing the criteria

• Appropriate data collection

• Establishment of feasible/efficient alternatives

• Formulation of payoff matrix (alternative versus criteria array)

• Selection of appropriate method to solve the problem

• Incorporation of decision-makers preference structure

• Choosing one or more of the best/suitable alternatives for further analysis


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Application in Environmental Engineering

• Water EngineeringTo check the ground water vulnerabilityTo decide the type of water treatment giving to the water bodyTo optimise coast in Water Distribution Network

• Wastewater EngineeringTo design control strategies to keep the process in good working conditionComparison of input and output datas for each unitTo evaluate wastewater Index


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Application in Environmental Engineering

• Solid Waste ManagementTo allocate best landfill siteTo give preference of treatments

• Hazardous Waste ManagementTo give ranking to the treatment

• Air PollutionTo calculate Air Quality Index

• Noise PollutionE ects of noise pollution on speech interferenceff



OverviewSr no. Title of paper Application Method used

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2001, Fuzzy logic observers for a biological wastewater treatment process

Monitoring and control of biological processes in WWTP

Fuzzy Estimator

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2005, Energy Saving In A Wastewater Treatment Process: An Application Of Fuzzy Logic Control

Energy saving upto 10% in WWTP

Implementation of FLC to regulate aeration

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2007, Rule-Based Fuzzy System for Assessing Groundwater Vulnerability

Recognition of groundwater vulnerability to pollution

FIS with rule based by using DRASTIC parameters


OverviewSr no. Paper Application Method used

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2007, Optimal Allocation Of Landfill Disposal Site: A Fuzzy Multi-Criteria Approach

Allocation of landfill site

MCDM

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2008, An expert system for predicting the effects of speech interference due to noise pollution on humans using fuzzy approach

Effects of noise pollution on speech interference

Knowledge based Rule based fuzzy approach to make Mamdani and Sugeno model

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2009, Fuzzy logic Water Quality index and importance of Water Quality Parameters,

Determination of WQI Fuzzy logic with UNIQ 2007 model

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OverviewSr no. Paper Application Method used

72011, Fuzzy logic based model for monitoring air quality index

Calculation of AQI Fuzzy knowledge based system

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2014, Predicting Efficiency Of Treatment Plant By Multi Parameter Aggregated Index

Prediction of treated Wastewater quality and evaluation the performance of WWTP

FMCDM for WWQI with AHP and Simple Multi Attribute Rating Technique

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2015, Optimal Design of Level-1 Redundant Water Distribution Networks with Fuzzy Demands

Cost optimization in water distribution network system

Fuzzy optimization model and GA model


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Advantages• Easy to understand, test and maintain

• Easy to be prototyped

• They operate even when there is lack of rules or wrong rules.

• Combination of linguistic and numeric

• Reasoning process is simple so saving the computing power

• Less time require to develop a model than convetional


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Drawbacks

• Need more tests and simulation

• Do not learn easily

• Difficult to establish correct rules

• Lack of precise mathematical model



Thank you

fuzzy logic and its application in environmental engineering

Education