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6 Handling uncertainty with fuzzy systems
6.1 Introduction
Ours is a vague world. We humans, talk in terms of `maybe', `perhaps', things
which cannot be defined with cent percent authority. But on the other hand,
conventional computer programs cannot understand natural language as
computers cannot work with vague concepts. Statements such as: "Umar is tall",
are difficult for computers to translate into definite rules. On the other hand,
"Umar's height is 162 cm", doesn't explicitly state whether Umar is tall or short.
We're driving in a car, and we see an old house. We can easily classify it as an
old house. But what exactly is an old house? Is a 15 years old house, an old
house? Is 40 years old house an old house? Where is the dividing line between
the old and the new houses? If we agree that a 40 years old house is an old
house, then how is it possible that a house is considered new when it is 39 years,
11 months and 30 days old only. And one day later it has become old all of a
sudden? That would be a bizarre world, had it been like that for us in all
scenarios of life.
Similarly human beings form vague groups of things such as `short men', `warm
days', `high pressure'. These are all groups which don't appear to have a well
defined boundary but yet humans communicate with each other using these
6.2 Classical sets
A classical set is a container, which wholly includes or wholly excludes any given
element. It's called classical merely because it has been around for quite some
time. It was Aristotle who came up with the `Law of the Excluded Middle', which
states that any element X, must be either in set A or in set not-A. It cannot be in
both. And these two sets, set A and set not-A should contain the entire universe
between them.
Days of the week
Figure : Classical Set
Let's take the example of the set `Days of the week'. This is a classical set in
which all the 7 days from Monday up until Sunday belong to the set, and
everything possible other than that that you can think of, monkeys, computers,
fish, telephone, etc, are definitely not a part of this set. This is a binary
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classification system, in which everything must be asserted or denied. In the case
of Monday, it will be asserted to be an element of the set of `days of the week',
but tuna fish will not be an element of this set.
6.3 Fuzzy sets
Fuzzy sets, unlike classical sets, do not restrict themselves to something lying
wholly in either set A or in set not-A. They let things sit on the fence, and are thus
closer to the human world. Let us, for example, take into consideration `days of
the weekend'. The classical set would say strictly that only Saturday and Sunday
are a part of weekend, whereas most of us would agree that we do feel like it's a
weekend somewhat on Friday as well. Actually we're more excited about the
weekend on a Friday than on Sunday, because on Sunday we know that the next
day is a working day. This concept is more vividly shown in the following figure.
Days of the weekend
Figure : Fuzzy Sets
Another diagram that would help distinguish between crisp and fuzzy
representation of days of the weekend is shown below.
Figure : Crisp v/s Fuzzy
The left side of the above figure shows the crisp set `days of the weekend', which
is a Boolean two-valued function, so it gives a value of 0 for all week days except
Saturday and Sunday where it gives an abrupt 1 and then back to 0 as soon as
Sunday ends. On the other hand, Fuzzy set is a multi-valued function, which in
this case is shown by a smoothly rising curve for the weekend, and even Friday
has a good membership in the set `days of the weekend'.
Same is the case with seasons. There are four seasons in Pakistan: Spring,
Summer, Fall and Winter. The classical/crisp set would mark a hard boundary
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between the two adjacent seasons, whereas we know that this is not the case in
reality. Seasons gradually change from one into the next. This is more clearly
explained in the figure below.
Figure: Seasons [Left: Crisp] [Right: Fuzzy]
This entire discussion brings us to a question: What is fuzzy logic?
6.4 Fuzzy Logic
Fuzzy logic is a superset of conventional (Boolean) logic that has been extended
to handle the concept of partial truth -- truth values between "completely true" and
"completely false".
Dr. Lotfi Zadeh of UC/Berkeley introduced it in the 1960's as a means to model
the uncertainty of natural languages. He was faced with a lot of criticism but
today the vast number of fuzzy logic applications speak for themselves:
 Self-focusing cameras
 Washing machines that adjust themselves according to the dirtiness of the
 Automobile engine controls
 Anti-lock braking systems
 Color film developing systems
 Subway control systems
 Computer programs trading successfully in financial markets
6.4.1 Fuzzy logic represents partial truth
Any statement can be fuzzy. The tool that fuzzy reasoning gives is the ability to
reply to a yes-no question with a not-quite-yes-or-no answer. This is the kind of
thing that humans do all the time (think how rarely you get a straight answer to a
seemingly simple question; what time are you coming home? Ans: soon. Q: are
you coming? Ans: I might) but it's a rather new trick for computers.
How does it work? Reasoning in fuzzy logic is just a matter of generalizing the
familiar yes-no (Boolean) logic. If we give "true" the numerical value of 1 and
"false" the numerical value of 0, we're saying that fuzzy logic also permits in-
between values like 0.2 and 0.7453.
"In fuzzy logic, the truth of any statement becomes matter of degree"
We will understand the concept of degree or partial truth by the same example of
days of the weekend. Following are some questions and their respective
­ Q: Is Saturday a weekend day?
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A: 1 (yes, or true)
Q: Is Tuesday a weekend day?
A: 0 (no, or false)
Q: Is Friday a weekend day?
A: 0.7 (for the most part yes, but not completely)
Q: Is Sunday a weekend day?
A: 0.9 (yes, but not quite as much as Saturday)
6.4.2 Boolean versus fuzzy
Let's look at another comparison between boolean and fuzzy logic with the help
of the following figures. There are two persons. Person A is standing on the left of
person B. Person A is definitely shorter than person B. But if boolean gauge has
only two readings, 1 and 0, then a person can be either tall or short. Let's say if
the cut off point is at 5 feet 10 inches then all the people having a height greater
than this limit are taller and the rest are short.
Not Tall
Figure: Boolean Logic
On the other hand, in fuzzy logic, you can define any function represented by any
mathematical shape. The output of the function can be discreet or continuous.
The output of the function defines the membership of the input or the degree of
truth. As in this case, the same person A is termed as `Not very tall'. This isn't
absolute `Not tall' as in the case of boolean. Similarly, person B is termed as
`Quite Tall' as apposed to the absolute `Tall' classification by the boolean
parameters. In short, fuzzy logic lets us define more realistically the true functions
that define real world scenarios.
Quite Tall
Not Very Tall
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Figure: Fuzzy Logic
6.4.3 Membership Function (
The degree of truth that we have been talking about, is specifically driven out by
a function called the membership function. It can be any function ranging from a
simple linear straight line to a complicated spline function or a polynomial of a
higher degree.
Some characteristics of the membership functions are:
 It is represented by the Greek symbol
 Truth values range between 0.0 and 1.0
o Where 0.0 normally represents absolute falseness
o And 1.0 represent absolute truth
Consider the following sentence:
"Amma ji is old"
In (crisp) set terminology, Amma ji belongs to the set of old people. We define
OLD, the membership function operating on the fuzzy set of old people. OLD
takes as input one variable, which is age, and returns a value between 0.0 and
If Amma ji's age is 75 years
 We might say OLD(Amma ji's age) = 0.75
­  Meaning Amma ji is quite old
For Amber, a 20 year old:
 We might say OLD(Amber's age) = 0.2
­  Meaning that Amber is not very old
For this particular age, the membership function is defined by a linear line with
positive slope.
6.4.4 Fuzzy vs. probability
It's important to distinguish at this point the difference between probability and
fuzzy, as both operate over the same range [0.0 to 1.0]. To understand their
differences lets take into account the following case, where Amber is a 20 years
old girl.
OLD(Amber) = 0.2
In probability theory:
There is a 20% chance that Amber belongs to the set of old people, there's an
80% chance that she doesn't belong to the set of old people.
In fuzzy terminology:
Amber is definitely not old or some other term corresponding to the value 0.2. But
there are certainly no chances involved, no guess work left for the system to
classify Amber as young or old.
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6.4.5 Logical and fuzzy operators
Before we move on, let's take a look at the logical operators. What these
operators help us see is that fuzzy logic is actually a superset of conventional
boolean logic. This might appear to be a startling remark at first, but look at Table
1 below.
Table: Logical Operators
The table above lists down the AND, OR and NOT operators and their respective
values for the boolean inputs. Now for fuzzy systems we needed the exact
operators which would act exactly the same way when given the extreme values
of 0 and 1, and that would in addition also act on other real numbers between the
ranges of 0.0 to 1.0. If we choose min (minimum) operator in place for AND, we
get the same output, similarly max (maximum) operator replaces OR, and 1-A
replaces NOT of A.
Table: Fuzzy Operators
In a lot of ways these operators seem to make sense. When we are ANDing two
domains, A and B, we do want to have the intersection as a result, and
intersection gives us the minimum overlapping area, hence both are equivalent.
Same is the case with max and 1-A.
The figure below explains these logical operators in a non-tabular form. If we
allow the fuzzy system to take on only two values, 0 and 1, then it becomes
boolean logic, as can be seen in the figure, top row.
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Figure: Logical vs Fuzzy Operators
It would be interesting to mention here that the graphs for A and B are nothing
more than a distribution, for instance if A was the set of short men, then the graph
A shows the entire distribution of short men where the horizontal axis is the
increasing height and the vertical axis shows the membership of men with
different heights in the function `short men'. The men who would be taller would
have little or 0 membership in the function, whereas they would have a significant
membership in set B, considering it to be the distribution of tall men.
6.4.6 Fuzzy set representation
Usually a triangular graph is chosen to represent a fuzzy set, with the peak
around the mean, which is true in most real world scenarios, as majority of the
population lies around the average height. There are fewer men who are
exceptionally tall or short, which explains the slopes around both sides of the
triangular distribution. It's also an approximation of the Gaussian curve, which is
a more general function in some aspects.
Apart from this graphical representation, there's also another representation
which is more handy if you were to write down some individual members along
with their membership. With this representation, the set of Tall men would be
written like follows:
 Tall = (0/5, 0.25/5.5, 0.8/6, 1/6.5, 1/7)
­ Numerator: membership value
­ Denominator: actual value of the variable
For instance, the first element is 0/5 meaning, that a height of 5 feet has 0
membership in the set of tall people, likewise, men who are 6.5 feet or 7 feet tall
have a membership value of maximum 1.
6.4.7 Fuzzy rules
First of all, let us revise the concept of simple If-Then rules. The rule is of the
If x is A then y is B
Where x and y are variables and A and B are some distributions/fuzzy sets. For
If hotel service is good then tip is average
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Here hotel service is a linguistic variable, which when given to a real fuzzy
system would have a certain crisp value, maybe a rating between 0 and 10. This
rating would have a membership value in the fuzzy set of `good'. We shall
evaluate this rule in more detail in the case study that follows.
Antecedents can have multiple parts:
 If wind is mild and racquets are good then playing badminton is fun
In this case all parts of the antecedent are resolved simultaneously and resolved
to a single number using logical operators
The consequent can have multiple parts as well
 if temperature is cold then hot water valve is open and cold water valve is
How is the consequent affected by the antecedent? The consequent specifies
that a fuzzy set be assigned to the output. The implication function then
modifies that fuzzy set to the degree specified by the antecedent. The most
common ways to modify the output fuzzy set are truncation using the min function
(where the fuzzy set is "chopped off").
Consider the following figure, which demonstrates the working of fuzzy rule
system on one rule, which states:
"If service is excellent or food is delicious then tip is generous"
Figure: Fuzzy If-Then Rule
Fuzzify inputs: Resolve all fuzzy statements in the antecedent to a degree of
membership between 0 and 1. If there is only one part to the antecedent, this is
the degree of support for the rule. In the example, the user gives a rating of 3 to
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the service, so its membership in the fuzzy set `excellent' is 0. Likewise, the user
gives a rating of 8 to the food, so it has a membership of 0.7 in the fuzzy set of
Apply fuzzy operator to multiple part antecedents: If there are multiple parts to the
antecedent, apply fuzzy logic operators and resolve the antecedent to a single
number between 0 and 1. This is the degree of support for the rule. In the
example, there are two parts to the antecedent, and they have an OR operator in
between them, so they are resolved using the max operator and max(0,0,0.7) is
0.7. That becomes the output of this step.
Apply implication method: Use the degree of support for the entire rule to shape
the output fuzzy set. The consequent of a fuzzy rule assigns an entire fuzzy set to
the output. This fuzzy set is represented by a membership function that is chosen
to indicate the qualities of the consequent. If the antecedent is only partially true,
(i.e., is assigned a value less than 1), then the output fuzzy set is truncated
according to the implication method.
In general, one rule by itself doesn't do much good. What's needed are two or
more rules that can play off one another. The output of each rule is a fuzzy set.
The output fuzzy sets for each rule are then aggregated into a single output fuzzy
set. Finally the resulting set is defuzzified, or resolved to a single number. The
next section shows how the whole process works from beginning to end for a
particular type of fuzzy inference system.
6.5 Fuzzy inference system
Fuzzy inference system (FIS) is the process of formulating the mapping from a
given input to an output using fuzzy logic. This mapping then provides a basis
from which decisions can be made, or patterns discerned
Fuzzy inference systems have been successfully applied in fields such as
automatic control, data classification, decision analysis, expert systems, and
computer vision. Because of its multidisciplinary nature, fuzzy inference systems
are associated with a number of names, such as fuzzy-rule-based systems, fuzzy
expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic
controllers, and simply (and ambiguously !!) fuzzy systems. Since the terms used
to describe the various parts of the fuzzy inference process are far from standard,
we will try to be as clear as possible about the different terms introduced in this
Mamdani's fuzzy inference method is the most commonly seen fuzzy
methodology. Mamdani's method was among the first control systems built using
fuzzy set theory. It was proposed in 1975 by Ebrahim Mamdani as an attempt to
control a steam engine and boiler combination by synthesizing a set of linguistic
control rules obtained from experienced human operators. Mamdani's effort was
based on Lotfi Zadeh's 1973 paper on fuzzy algorithms for complex systems and
decision processes.
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6.5.1 Five parts of the fuzzy inference process
Fuzzification of the input variables
Application of fuzzy operator in the antecedent (premises)
Implication from antecedent to consequent
Aggregation of consequents across the rules
Defuzzification of output
To help us understand these steps, let's do a small case study.
6.5.2 Case Study: dinner for two
We present a small case study in which two people go for a dinner to a
restaurant. Our fuzzy system will help them decide the percentage of tip to be
given to the waiter (between 5 to 25 percent of the total bill), based on their rating
of service and food. The rating is between 0 and 10. The system is based on
three fuzzy rules:
If service is poor or food is rancid then tip is cheap
If service is good then tip is average
If service is excellent or food is delicious then tip is generous
Based on these rules and the input by the diners, the Fuzzy inference system
gives the final output using all the inference steps listed above. Let's take a look
at those steps one at a time.
Figure: Dinner for Two Fuzzify Inputs
The first step is to take the inputs and determine the degree to which they belong
to each of the appropriate fuzzy sets via membership functions. The input is
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always a crisp numerical value limited to the universe of discourse of the input
variable (in this case the interval between 0 and 10) and the output is a fuzzy
degree of membership in the qualifying linguistic set (always the interval between
0 and 1). Fuzzification of the input amounts to either a table lookup or a function
The example we're using in this section is built on three rules, and each of the
rules depends on resolving the inputs into a number of different fuzzy linguistic
sets: service is poor, service is good, food is rancid, food is delicious, and so on.
Before the rules can be evaluated, the inputs must be fuzzified according to each
of these linguistic sets. For example, to what extent is the food really delicious?
The figure below shows how well the food at our hypothetical restaurant (rated on
a scale of 0 to 10) qualifies, (via its membership function), as the linguistic
variable "delicious." In this case, the diners rated the food as an 8, which, given
our graphical definition of delicious, corresponds to
= 0.7 for the "delicious"
membership function.
Figure: Fuzzify Input Apply fuzzy operator
Once the inputs have been fuzzified, we know the degree to which each part of
the antecedent has been satisfied for each rule. If the antecedent of a given rule
has more than one part, the fuzzy operator is applied to obtain one number that
represents the result of the antecedent for that rule. This number will then be
applied to the output function. The input to the fuzzy operator is two or more
membership values from fuzzified input variables. The output is a single truth
Shown below is an example of the OR operator max at work. We're evaluating
the antecedent of the rule 3 for the tipping calculation. The two different pieces of
the antecedent (service is excellent and food is delicious) yielded the fuzzy
membership values 0.0 and 0.7 respectively. The fuzzy OR operator simply
selects the maximum of the two values, 0.7, and the fuzzy operation for rule 3 is
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Figure: Apply Fuzzy Operator Apply implication method
Before applying the implication method, we must take care of the rule's weight.
Every rule has a weight (a number between 0 and 1), which is applied to the
number given by the antecedent. Generally this weight is 1 (as it is for this
example) and so it has no effect at all on the implication process. From time to
time you may want to weigh one rule relative to the others by changing its weight
value to something other than 1.
Once proper weightage has been assigned to each rule, the implication method
is implemented. A consequent is a fuzzy set represented by a membership
function, which weighs appropriately the linguistic characteristics that are
attributed to it. The consequent is reshaped using a function associated with the
antecedent (a single number). The input for the implication process is a single
number given by the antecedent, and the output is a fuzzy set. Implication is
implemented for each rule. We will use the min (minimum) operator to perform
the implication, which truncates the output fuzzy set, as shown in the figure
Figure: Apply Implication Method
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Since decisions are based on the testing of all of the rules in an FIS (fuzzy
inference system), the rules must be combined in some manner in order to make
a decision. Aggregation is the process by which the fuzzy sets that represent the
outputs of each rule are combined into a single fuzzy set. Aggregation only
occurs once for each output variable, just prior to the fifth and final step,
defuzzification. The input of the aggregation process is the list of truncated output
functions returned by the implication process for each rule. The output of the
aggregation process is one fuzzy set for each output variable.
Notice that as long as the aggregation method is commutative (which it always
should be), then the order in which the rules are executed is unimportant. Any
logical operator can be used to perform the aggregation function: max
(maximum), probor (probabilistic OR), and sum (simply the sum of each rule's
output set).
In the diagram below, all three rules have been placed together to show how the
output of each rule is combined, or aggregated, into a single fuzzy set whose
membership function assigns a weighting for every output (tip) value.
Figure: Aggregate all outputs Defuzzify
The input for the defuzzification process is a fuzzy set (the aggregate output
fuzzy set) and the output is a single number. As much as fuzziness helps the rule
evaluation during the intermediate steps, the final desired output for each variable
is generally a single number. However, the aggregate of a fuzzy set
encompasses a range of output values, and so must be defuzzified in order to
resolve a single output value from the set.
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Perhaps the most popular defuzzification method is the centroid calculation,
which returns the center of area under the curve. There are other methods in
practice: centroid, bisector, middle of maximum (the average of the maximum
value of the output set), largest of maximum, and smallest of maximum.
Figure: Defuzzification
Thus the FIS calculates that in case the food has a rating of 8 and the service
has a rating of 3, then the tip given to the waiter should be 16.7% of the total bill.
6.6 Summary
Fuzzy system maps more realistically, the everyday concepts, like age, height,
temperature etc. The variables are given fuzzy values. Classical sets, either
wholly include something or exclude it from the membership of a set, for instance,
in a classical set, a man can be either young or old. There are crisp and rigid
boundaries between the two age sets, but in Fuzzy sets, there can be partial
membership of a man in both the sets.
6.7 Exercise
1) Think of the membership functions for the following concepts, from the
famous quote: "Early to bed, and early to rise, makes a man healthy,
wealthy and wise."
a. Health
b. Wealth
c. Wisdom
2) What do you think would be the implication of using a different shaped
curve for a membership function? For example, a triangular, gaussian,
square etc
3) Try to come up with at least 5 more rules for the tipping system(Dinner for
two case study), such that the system would be a more realistic and
complete one.