Fuzzy logic and its application in environmental engineering

Introduction to fuzzy logic and its
application in Environmental
Engineering
Presented by
Drashti V. Kapadia

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 2

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

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 membership
Anything between them called graded membership

• 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.

• 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.

Operations of Crisp Set and Fuzzy
Set
Complement
0
x
1
(x)
0
x
1
Containment
0
x
1
0
x
1
AB
Not A
A
Intersection
0
x
1
0
x
AB
Union
0
1
AB
AB
0
x
1
0
x
1
B
A
B
A
(x)
(x) (x)
Intersection Union
Complement
Not A
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

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

Fuzzification
Types 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.

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.

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

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

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.

Types of Fuzzy Inference System
• Mamdani

Types of Fuzzy Inference System
• Sugeno

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

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

Defuzzification
• Methods
Max membership principle
Centroid method
Weighted average method
Mean max membership
Center of sums
Center of largest area
First (or last) of maxima

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

Defuzzification

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

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.

Three Level Hierarchy
Criteria
Alternatives
Goal
1 2 3 4
A B C

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

Application in Environmental
Engineering
• Water Engineering
To check the ground water vulnerability
To decide the type of water treatment giving to the water body
To optimise coast in Water Distribution Network
• Wastewater Engineering
To design control strategies to keep the process in good
working condition
Comparison of input and output datas for each unit
To evaluate wastewater Index

Application in Environmental
Engineering
• Solid Waste Management
To allocate best landfill site
To give preference of treatments
• Hazardous Waste Management
To give ranking to the treatment
• Air Pollution
To calculate Air Quality Index
• Noise Pollution
Eﬀects of noise pollution on speech interference

Overview
Sr no. Title of paper Application Method used
1
2001, Fuzzy logic
observers for a biological
wastewater treatment
process
Monitoring and
control of biological
processes in WWTP
Fuzzy Estimator
2
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
3
2007, Rule-Based Fuzzy
System for Assessing
Groundwater
Vulnerability
Recognition of
groundwater
vulnerability to
pollution
FIS with rule
based by using
DRASTIC
parameters

Overview
Sr no. Paper Application Method used
4
2007, Optimal Allocation
Of Landfill Disposal Site:
A Fuzzy Multi-Criteria
Approach
Allocation of landfill
site
MCDM
5
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
6
2009, Fuzzy logic Water
Quality index and
importance of Water
Quality Parameters,
Determination of WQI Fuzzy logic with
UNIQ 2007
model

Overview
Sr no. Paper Application Method used
7
2011, Fuzzy logic based
model for monitoring air
quality index
Calculation of AQI Fuzzy knowledge
based system
8
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
9
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

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

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

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