C291-78 Tópicos Avanzados: Inteligencia Computacional...

C291-78

Tópicos Avanzados:

Inteligencia Computacional I

Dra. Ma. del Pilar Gómez Gil

Primavera 2014 [email protected]

1

Ver: 08-Mar-2016

Introducción a la Lógica Difusa

mailto:[email protected]

Este material ha sido tomado de varias fuentes,

y forma parte del “Tutorial en Lógica Difusa”

impartido por el Dr. Juan Manuel Ramírez C.

Coordinación de Electrónica, 2012

2

FUZZY LOGIC

In 1965 Lotfi Zadeh, published his famous paper “Fuzzy sets”.

Zadeh extended the work on possibility theory into a formal system of mathematical logic, and introduced a new concept for applying natural language terms.

This new logic for representing and manipulating fuzzy terms was called fuzzy logic.

3

Introduction

Many decision-making and problem-solving tasks are too complex to be understood quantitatively, however, people succeed by using knowledge that is imprecise rather than precise.

Fuzzy set theory resembles human reasoning in its use of approximate information and uncertainty to generate decisions.

4

Example: driving decisions

5

Introduction (2)

Since knowledge can be expressed in a more natural way by using fuzzy sets, many engineering and decision problems can be greatly simplified.

It was specifically designed to mathematically represent uncertainty and vagueness and provide formalized tools for dealing with the imprecision intrinsic to many problems.

Fuzzy theory may be seen as an extension to classical set theory, where a membership function is added to a set. This function is defined in

]1,0[

6

Membership function

(a) Boolean Logic. (b) Multi-valued Logic.

0 1 10 0.2 0.4 0.6 0.8 100 1 10

Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership.

Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It deals with degrees of membership and degrees of truth.

Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1(completely true). Instead of just black and white it accepts that things can be partly true and partly false at the same time.

7

Membership function (2)

A membership function

represents the “degree” in that variable t is

included in the concept represented by

the tag A.

)(tA

8

Example:

fuzzy variable “temperatura”

Linguistic value/variable:

temperatura

Fuzzy subsets:

{frío, fresco, normal, tibio, caliente}

Membership functions:

)}(),(),(),(),({ ttttt calientetibionormalfrescofrío

9

Example:

fuzzy variable “temperatura” (2)

)(tA

t

The t-axis represents the universe of discourse

Each colour line is a membership function

10

The membership function of a

“Crisp set”

11

The membership function of a

“Fuzzy set”

12

Crisp sets and characteristic

function

Ax

Axxf A

if0,

if 1,)(

Let X be the universe of discourse and its elements be

denoted as x. In the classical set theory, crisp set A of X

is defined as function fA(x) called the characteristic

function of A:

fA(x) : X {0, 1}, where

This set maps universe X to a set of two elements. For

any element x of universe X, characteristic function fA(x) is

equal to 1 if x is an element of set A, and is equal to 0 if x

is not an element of A.

13

Fuzzy sets and membership functions

In the fuzzy theory, fuzzy set A of universe X is defined by function µA(x) called the membership function of set A

µA(x) : X [0, 1], where µA(x) = 1 if x is totally in A;

µA(x) = 0 if x is not in A;

0 < µA(x) < 1 if x is partly in A.

This set allows a continuum of possible choices. For any element x of universe X, membership function µA(x) equals the degree to which x is an element of set A. This degree, a value between 0 and 1, represents the degree of membership, also called membership value, of element x in set A.

14

Fuzzy Set Representation

First, we determine the membership functions. In our

“height” example, we can obtain fuzzy sets of tall, short and

average men.

The universe of discourse – the person’s heights – is

represented by three sets: short, average and tall men. As you

will see, a man who is 184 cm tall is a member of the average

men set with a degree of membership of 0.1, and at the same

time, he is also a member of the tall men set with a degree of

0.4. (see graph on next page)

15


150 210170 180 190 200160

Height, cmDegree ofMembership

Tall Men

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160

Degree ofMembership

Short Average ShortTall

170

1.0

0.0

0.2

0.4

0.6

0.8

Fuzzy Sets

Crisp Sets

Short Average

Tall

Tall

16


Typical functions that can be used to represent a fuzzy set

are sigmoid, gaussian and pi. However, these functions

increase the time of computation. Therefore, in practice,

most applications use linear fit functions.

17

Funciones de membresía (1)

Función de membresía triangular

18

Función de membresía trapezoidal


19

Función de membresía gaussiana


20

Funciones de membresía sigmoide


21

Linguistic Variables and Hedges At the root of fuzzy set theory lies the idea of linguistic variables.

A linguistic variable is a fuzzy variable. For example, the statement “John is tall” implies that the linguistic variable John takes the linguistic value tall.

In fuzzy expert systems, linguistic variables are used in fuzzy rules. For example:

IF wind is strong

THEN sailing is good

IF project_duration is long

THEN completion_risk is high

IF speed is slow

THEN stopping_distance is short

22

Linguistic Variables and Hedges

The range of possible values of a linguistic variable represents the universe of discourse of that variable. For example, the universe of discourse of the linguistic variable speed might have the range between 0 and 220 km/h and may include such fuzzy subsets as very slow, slow, medium, fast, and very fast.

A linguistic variable carries with it the concept of fuzzy set qualifiers, called hedges.

Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly.

23


Short

Very Tall

Short Tall

Degree ofMembership

150 210180 190 200

1.0

0.0

0.2

0.4

0.6

0.8

160 170

Height, cm

Average

TallVery Short Very Tall

24


25


26

Membership Functions

For the sake of convenience, usually a fuzzy set is denoted as:

A = A(xi)/xi + …………. + A(xn)/xn where A(xi)/xi (a singleton) is a pair “grade of

membership” element, that belongs to a finite universe of discourse:

A = {x1, x2, .., xn}

27

Logic operators (1/2)

28

(Matlab, 2015)

Operations of Crisp Sets

Intersection Union

Complement

Not A

A

Containment

AA

B

BA BAA B

29

Operations of Fuzzy Sets

Complement

0 x

1

( x )

0 x

1

Containment

0 x

1

0 x

1

A B

Not A

A

Intersection

0 x

1

0 x

A B

Union

0

1

A B

A B

0 x

1

0 x

1

B

A

B

A

( x )

( x ) ( x )

30

Blue line is the

resulting

membership function

Logic operators (2/2)

31

(Matlab, 2015)

Complement Crisp Sets: Who does not belong to the set?

Fuzzy Sets: How much do elements not belong to the set?

If A is the fuzzy set, its complement A can be found as follows:

A(x) = 1 A(x)

The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement.

32

Containment

Crisp Sets: Which sets belong to which other sets?

Fuzzy Sets: How much sets belong to other sets?

Similar to a Chinese box, a set can contain other sets.

The smaller set is called the subset. For example, the

set of tall men contains all tall men; very tall men is a

subset of tall men. However, the tall men set is just a

subset of the set of men. In crisp sets, all elements of a

subset entirely belong to a larger set. In fuzzy sets,

however, each element can belong less to the subset

than to the larger set. Elements of the fuzzy subset have

smaller memberships in it than in the larger set.

33

Intersection Crisp Sets: Which element belongs to both sets?

Fuzzy Sets: How much of the element is in both sets?

In classical set theory, an intersection between two sets contains the elements shared by these sets. For example, the intersection of the set of tall men and the set of fat men is the area where these sets overlap. In fuzzy sets, an element may partly belong to both sets with different memberships.

A fuzzy intersection is the lower membership in both sets of each element. The fuzzy intersection of two fuzzy sets A and B on universe of discourse X:

AB(x) = min [A(x), B(x)] = A(x) B(x),

where xX

34

Union

Crisp Sets: Which element belongs to either set?

Fuzzy Sets: How much of the element is in either set?

The union of two crisp sets consists of every element that falls into

either set. For example, the union of tall men and fat men contains

all men who are tall OR fat.

In fuzzy sets, the union is the reverse of the intersection. That is,

the union is the largest membership value of the element in

either set. The fuzzy operation for forming the union of two fuzzy

sets A and B on universe X can be given as:

AB(x) = max [A(x), B(x)] = A(x) B(x),

where xX

35

Crisp and Fuzzy –union and

intersection

36

A B 𝑨 ∪ 𝑩 𝑨 ∩ 𝑩

¬𝑨 𝑨 ∪ ¬𝑨 = 𝑼 𝑨 ∩ ¬𝑨 = ∅

m𝑎𝑥(𝜇𝐴, 𝜇𝐵) min(𝜇𝐴, 𝜇𝐵) m𝑎𝑥(𝜇𝐴, 𝜇¬𝐴) min(𝜇𝐴, 𝜇¬𝐴)

0 0 0 = m𝑎𝑥(0,0)

0 = m𝑖𝑛(0,0) 1 1 = m𝑎𝑥(0,1) 0 = m𝑖𝑛(0,1)

0 1 1= m𝑎𝑥(0,1)

0 = m𝑖𝑛(0,1)

1 0 1= m𝑎𝑥(1,0)

0 = m𝑖𝑛(1,0) 0 1 = m𝑎𝑥(1,0) 0 = m𝑖𝑛(1,0)

1 1 1= m𝑎𝑥(1,1)

1 = m𝑖𝑛(1,1)

Crisp and Fuzzy – containment

(subset)

37

A B 𝑨 ∪ 𝑩 𝑨 ∩ 𝑩

m𝑎𝑥(𝜇𝐴, 𝜇𝐵) min(𝜇𝐴, 𝜇𝐵)

0 0 0 = m𝑎𝑥(0,0) 0 = m𝑖𝑛(0,0)

0 1 1= m𝑎𝑥(0,1) 0 = m𝑖𝑛(0,1)

1 0 1= m𝑎𝑥(1,0) 0 = m𝑖𝑛(1,0)

1 1 1= m𝑎𝑥(1,1) 1 = m𝑖𝑛(1,1)

If 𝐴 ⊆ 𝐵𝑡ℎ𝑒𝑛𝐴 ∪ 𝐵 = 𝐵 𝐴 ∩ 𝐵 = 𝐴,

Example:

A = “very tall”

B = “Tall”

You can’t be

“very tall” and

“not tall” at the

same time!

Properties of Fuzzy Sets

Equality of two fuzzy sets

Inclusion of one set into another fuzzy set

Cardinality of a fuzzy set

An empty fuzzy set

-cuts (alpha-cuts)

38

Equality

Fuzzy set A is considered equal to a fuzzy

set B, IF AND ONLY IF (iff):

A(x) = B(x), xX

39

Inclusion

Inclusion of one fuzzy set into another fuzzy set. Fuzzy

set A X is included in (is a subset of) another fuzzy

set, B X:

A(x) B(x), xX

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;

B = 0.5/1 + 0.55/2 + 1/3

then A is a subset of B, or A B

40

Cardinality Cardinality of a non-fuzzy set, Z, is the number of elements in Z.

BUT the cardinality of a fuzzy set A, the so-called SIGMA COUNT, is expressed as a SUM of the values of the membership function of A, A(x):

cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi), for i=1..n

Consider X = {1, 2, 3} and sets A and B

A = 0.3/1 + 0.5/2 + 1/3;

B = 0.5/1 + 0.55/2 + 1/3

cardA = 1.8

cardB = 2.05

41

Empty Fuzzy Set

A fuzzy set A is empty, IF AND ONLY IF:

A(x) = 0, xX

Consider X = {1, 2, 3} and set A

A = 0/1 + 0/2 + 0/3

then A is empty

42

Alpha-cut

An -cut or -level set of a fuzzy set A X is an

ORDINARY SET A X, such that:

A={A(x), xX}.

Consider X = {1, 2, 3} and set A

A = 0.3/1 + 0.5/2 + 1/3

then A0.5 = {2, 3},

A0.1 = {1, 2, 3},

A1 = {3}

43

Fuzzy Set Normality

A fuzzy subset of X is called normal if there exists at least

one element xX such that A(x) = 1.

A fuzzy subset that is not normal is called subnormal.

All crisp subsets except for the null set are normal. In fuzzy

set theory, the concept of nullness essentially generalises to

subnormality.

The height of a fuzzy subset A is the large membership

grade of an element in A

height(A) = maxx(A(x))

44

Fuzzy Sets Core and Support

Assume A is a fuzzy subset of X:

the support of A is the crisp subset of X consisting of

all elements with membership grade:

supp(A) = {x A(x) 0 and xX}

the core of A is the crisp subset of X consisting of all

elements with membership grade:

core(A) = {x A(x) = 1 and xX}

45

Fuzzy Set Math Operations

aA = {aA(x), xX}

Let a =0.5, and

A = {0.5/a, 0.3/b, 0.2/c, 1/d}

then

Aa = {0.25/a, 0.15/b, 0.1/c, 0.5/d}

Aa = {A(x)a, xX}

Let a =2, and

A = {0.5/a, 0.3/b, 0.2/c, 1/d}

then

Aa = {0.25/a, 0.09/b, 0.04/c, 1/d}

…

46

Fuzzy Sets Examples

Consider two fuzzy subsets of the set X,

X = {a, b, c, d, e }

referred to as A and B

A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}

and

B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

47

Fuzzy Sets Examples

Support:

supp(A) = {a, b, c, d }

supp(B) = {a, b, c, d, e }

Core:

core(A) = {a}

core(B) = {o}

Cardinality:

card(A) = 1+0.3+0.2+0.8+0 = 2.3

card(B) = 0.6+0.9+0.1+0.3+0.2 = 2.1

48

Fuzzy Sets Examples

Complement:

A = {1/a, 0.3/b, 0.2/c 0.8/d, 0/e}

A = {0/a, 0.7/b, 0.8/c 0.2/d, 1/e}

Union:

A B = {1/a, 0.9/b, 0.2/c, 0.8/d, 0.2/e}

Intersection:

A B = {0.6/a, 0.3/b, 0.1/c, 0.3/d, 0/e}

49

Fuzzy Sets Examples

aA: for a=0.5 aA = {0.5/a, 0.15/b, 0.1/c, 0.4/d, 0/e}

Aa: for a=2 Aa = {1/a, 0.09/b, 0.04/c, 0.64/d, 0/e}

a-cut:

A0.2 = {a, b, c, d}

A0.3 = {a, b, d}

A0.8 = {a, d}

A1 = {a}

50

Exercises

For A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e}

Draw the Fuzzy Graph of A and B Then, calculate the following: - Support, Core, Cardinality, and Complement for A and

B independently - Union and Intersection of A and B - the new set C, if C = A2 - the new set D, if D = 0.5B - the new set E, for an alpha cut at A0.5

51

Solutions A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Support

Supp(A) = {a, b, c, d} Supp(B) = {b, c, d, e}

Core

Core(A) = {c} Core(B) = {}

Cardinality

Card(A) = 0.2 + 0.4 + 1 + 0.8 + 0 = 2.4 Card(B) = 0 + 0.9 + 0.3 + 0.2 + 0.1 = 1.5

Complement

Comp(A) = {0.8/a, 0.6/b, 0/c, 0.2/d, 1/e} Comp(B) = {1/a, 0.1/b, 0.7/c, 0.8/d, 0.9/e}

52

Solutions A = {0.2/a, 0.4/b, 1/c, 0.8/d, 0/e} B = {0/a, 0.9/b, 0.3/c, 0.2/d, 0.1/e} Union

AB = {0.2/a, 0.9/b, 1/c, 0.8/d, 0.1/e} Intersection

AB = {0/a, 0.4/b, 0.3/c, 0.2/d, 0/e} C=A2

C = {0.04/a, 0.16/b, 1/c, 0.64/d, 0/e}

D = 0.5B

D = {0/a, 0.45/b, 0.15/c, 0.1/d, 0.05/e}

E = A0.5 E = {c, d}

53

Inference Fuzzy Rules

In 1973, Dr. Lotfi Zadeh published his second most influential paper. This

paper outlined a new approach to analysis of complex systems, in which

Zadeh suggested capturing human knowledge in fuzzy rules.

A fuzzy rule can be defined as a conditional statement in the form:

IF x is A

THEN y is B

where x and y are linguistic variables; and A and B are linguistic values

determined by fuzzy sets on the universe of discourses X and Y,

respectively.

54

Classical Vs Fuzzy Rules A classical IF-THEN rule uses binary logic, for example,

Rule: 1 Rule: 2

IF speed is > 90 km/h IF speed is < 40 km/h

THEN THEN

stopping_distance> 40 m stopping_distance < 15 m

The variable speed can have any numerical value between 0 and 150 km/h, and the variable stopping_distance can take any numerical value between 0 and 60 m. In other words, classical rules are expressed in the black-and-white language of Boolean logic.

55

Classical Vs Fuzzy Rules

A fuzzy IF-THEN rule uses fuzzy logic :

Rule: 1 Rule: 2

IF speed is fast IF speed is slow

THEN stopping_distance is long THEN stopping_distance is short

In fuzzy rules, the linguistic variable speed also has the range (the universe of discourse) between 0 and 150 km/h, but this range includes fuzzy sets, such as slow, medium and fast. The universe of discourse of the linguistic variable ‘stopping_distance’ can be between 0 and 60 m and may include such fuzzy sets as short, medium and long.

56

Classical Vs Fuzzy Rules

Fuzzy rules relate fuzzy sets.

In a fuzzy system, all rules fire to some extent, or in other words they fire partially. If the antecedent is true to some degree of membership, then the consequent is also true to the corresponding degree.

57

Firing Fuzzy Rules

Tall men Heavy men

180

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

190 200 70 80 100160

Weight, kg

120

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

These fuzzy sets provide the basis for a weight

estimation model. The model is based on a

relationship between a man’s height and his weight:

IF height is tall THEN weight is heavy

58

Firing Fuzzy Rules

Tall menHeavy men

180

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

Height, cm

190 200 70 80 100160

Weight, kg

120

Degree ofMembership

1.0

0.0

0.2

0.4

0.6

0.8

The value of the output or a truth membership grade of the

rule consequent can be estimated directly from a

corresponding truth membership grade in the antecedent.

This form of fuzzy inference uses a method called

monotonic selection.

59

Firing Fuzzy Rules

A fuzzy rule can have multiple antecedents, for example:

IF service is excellent

OR food is delicious

THEN tip is generous

The consequent of a fuzzy rule can also include multiple parts, for instance:

IF temperature is hot

THEN hot_water is reduced;

cold_water is increased

60

1a. Specify the problem

Air-conditioning involves the delivery of air, which can be

warmed or cooled and have its humidity raised or lowered.

An air-conditioner is an apparatus for controlling, especially

lowering, the temperature and humidity of an enclosed

space. An air-conditioner typically has a fan which

blows/cools/circulates fresh air and has a cooler. The cooler

is controlled by a thermostat. Generally, the amount of air

being compressed is proportional to the ambient

temperature.

1b. Define linguistic variables

• Ambient Temperature

• Air-conditioner Fan Speed

Example: Air Conditioner

61

2. Determine Fuzzy Sets


Fuzzy sets can have a variety of shapes. However, a triangle or a trapezoid can often provide an adequate representation of the expert knowledge, and at the same time, significantly simplifies the process of computation.

62

2. Determine Fuzzy Sets: Temperature Temp

(0C).

COL

D

COOL PLEASAN

T

WARM HO

T

0 Y* N N N N

5 Y Y N N N

10 N Y N N N

12.5 N Y* N N N

15 N Y N N N

17.5 N N Y* N N

20 N N N Y N

22.5 N N N Y* N

25 N N N Y N

27.5 N N N N Y

30 N N N N Y*

Temp

(0C).

COLD COOL PLEASANT WARM HOT

0

64

Temperature Fuzzy Sets

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Temperature Degrees C

Tru

th V

alu

e Cold

Cool

Pleasent

Warm

Hot

Temperature Fuzzy Sets

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30

Temperature Degrees C

Tru

th V

alu

e Cold

Cool

Pleasent

Warm

Hot

2. Determine Fuzzy Sets: Temperature (cont.)


2. Determine Fuzzy Sets: Fan Speed Rev/sec

(RPM)

MINIMAL SLOW MEDIUM FAST BLAST

0 Y* N N N N

10 Y N N N N

20 Y Y N N N

30 N Y* N N N

40 N Y N N N

50 N N Y* N N

60 N N N Y N

70 N N N Y* N

80 N N N Y Y

90 N N N N Y

100 N N N N Y*


65

2. Determine Fuzzy Sets: Fan Speed (cont.)


Speed Fuzzy Sets

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

Speed

Tru

th V

alu

e MINIMAL

SLOW

MEDIUM

FAST

BLAST

66

3. Bring out and construct fuzzy rules


RULE 1: IF temp is cold THEN speed is minimal

RULE 2: IF temp is cool THEN speed is slow

RULE 3: IF temp is pleasant THEN speed is medium

RULE 4: IF temp is warm THEN speed is fast

RULE 5: IF temp is hot THEN speed is blast

To accomplish this task, we might ask the expert to describe how the problem can be solved using the fuzzy linguistic variables defined previously.

Required knowledge also can be collected from other sources such as books, computer databases, flow diagrams and observed human behaviour.

67

4. Encode the fuzzy sets, fuzzy rules

and procedures to perform fuzzy

inference into the expert system

To accomplish this task, we may choose an option:

to build our system using a programming language such as C/C++ or Pascal, or

to apply a fuzzy logic development tool such as MATLAB Fuzzy Logic Toolbox or Fuzzy Knowledge Builder.

etc


68

5. Evaluate and tune the system


The last, and the most laborious, task is to evaluate and tune the system. We want to see whether our fuzzy system meets the requirements specified at the beginning.

Evaluation of the system output is performed for test situations on the several representative values of input variables. Fuzzy Logic development tools often can generate surface to help us evaluate and analyze the system’s performance.

Tuning of the system consists of reviewing, adding and/or changing the membership functions and rules in order to increase the performance of the system.

69

5a. Evaluate the system

Consider a temperature of 16oC, use the system

to compute the optimal fan speed.


RECALL: Operation of a fuzzy expert system:

Fuzzification: determination of the degree of membership of crisp inputs in appropriate fuzzy sets.

Inference: evaluation of fuzzy rules to produce an output for each rule.

Aggregation: combination of the outputs of all rules.

Defuzzification: computation of crisp output

70

• Fuzzification

Affected fuzzy sets: COOL and PLEASANT

COOL(T) = – T / 5 + 3.5

= – 16 / 5 + 3.5

= 0.3

PLSNT(T) = T /2.5 - 6

= 16 /2.5 - 6

= 0.4

Temp=16 COLD COOL PLEASANT WARM HOT

0 0.3 0.4 0 0


71

• Inference


RULE 1: IF temp is cold THEN speed is minimal

RULE 2: IF temp is cool THEN speed is slow

RULE 3: IF temp is pleasant THEN speed is medium

RULE 4: IF temp is warm THEN speed is fast

RULE 5: IF temp is hot THEN speed is blast

RULE 2: IF temp is cool (0.3) THEN speed is slow (0.3)

RULE 3: IF temp is pleasant (0.4) THEN speed is medium (0.4) 72

• Aggregation

speed is slow (0.3) speed is medium (0.4) +


73

• Defuzzification

COG = 0.125(12.5) + 0.25(15) + 0.3(17.5+20+…+40+42.5) + 0.4(45+47.5+…+52.5+55) + 0.25(57.5)

0.125 + 0.25 + 0.3(11) + 0.4(5) + 0.25

= 45.54rpm


74

Centroid of area zCOA

where A(z) is the aggregated output MF.

,dz)z(

zdz)z(

z

Z

A

Z

A

COA

Input – Output Plot

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.2

0.3

0.4

0.5

0.6

number_of_serversmean_delay

nu

mb

er_

of_

spa

res

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

temp

fan-s

peed


one input – one output

gives nonlinear transfer

characteristic More general example:

two inputs – one output gives

3D transfer surface 76

C291-78 Tópicos Avanzados: Inteligencia Computacional...

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