Genetic Algo 1

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Nontraditional Optimization

Algorithms

This chapter describes two nontraditional search

and

optimization

methods which are becoming popular in engineering optimization

problems in the recent past . These algorithms are included in

this book not because they are new but because they are found

to be potential search and optimization algorithms for complex

engineering optimization problems. enetic algorithms GAs) mimic

the principles of natural genetics and natural selection to constitute

search and optimization procedures.

Simulated annealing

mimics

the

cooling phenomenon of molten metals

to

constitute a search

procedure. Since both these algorithms are abstractions from a


2/11

on

Nontraditional Optimization lgorithms

2

91

1975; Michalewicz, 1992)

and through

a number

of internation

al

conference proceedings (Belew

and

Booker, 1991; Forrest , 1993·

Grefenstette, 1985, 1987; Rawlins, 1991; Schaffer, 1989, Whitley

1993).

An

exten.;ive list of GA-related papers is referenced elsewhere

(Goldberg, et. al, 1992). GAs are fundamentally different than

classical optimization algorithms we have discussed in

Chapters

2

through

5. We begin

the

discussion of GAs by first outlining the

working principles of GAs

and then

highlighting

the

differences GAs

have with the

traditional

search methods. Thereafter,

we

show a

computer

simulation

to

illustrate

the

working of GAs.

6.1.1

Working

principles

To illustrate

the

working principles of GAs,

we

first consider

unconstrained optimization problem. Later, we shall discuss how

GAs can

be

used

to

solve a constrained optimization problem . Let

us consider the following maximization problem:

Maximize

f x

),

(L) . (U) . _

..


3/11

292

Optimization for Engineering Duign: Algorithm ll and Ea amples

to a fixed mapping rule. Usually, the following linear mapping rule

is used: .

(L) x ~ U ) -

X ~ L )

. :

Xi= xi

2

l;

_

1

decoded value (si)· (6.1)

n the above equation, the variable Xi is coded in a substring

·Si

of

length li. The decoded value of a binary substring-si is calculated

as E f ; ; ; ; ~ 2i

Si,

where si E (0, 1) and the .string s is represented as

(st-ISt-2

. . .

s2s1so).

For example, a

f o u r ~ b i t

string

(0111)

has a

decoded value equal

to

((1)2° + (1)2

1

+ (1)2

2

+ (0)2

3

)

or

7.

It is

worthwhile to mention here that with four bits to code each variable,

there are only 2

4

or 16 distinct substrings possible, because each

b i t ~ p o s i t i o n can take a value either 0 or 1. The accuracy that can

be obtained with a

f o u r ~ b i t

coding is only approximately 1/16th of

the search space. But as the string length is increased by one,_the

obtainable accuracy increases exponentially to 1 /32th of the search

space.

It is

not necessary' to code all variables · n equal substring

length. The length of a substring representing a variable depends

on

the

desired accuracy in

that

variable. Generalizing tills concept,

we may say that with an l i ~ b i t coding for a variable, the obtainable

accuracy in

that

variable is approximately

( x ~ U ) -

x L))j2l;.

Once

the coding of the variables has been done, the corresponding point

X= ( x ~ ,

x2, XN f can be found using .Equation (6.1). Thereafter,

the

function value at the point x can also be calculated by

SUbstituting X in fhe given objective function f( X .

Fitness

function

As pointed

out

earlier, GAs mimic

the

u r v i v a l ~ o f ~ t h e ~ f i t t e s t principle

of

nature

to make a search process. -Therefore, GAs are naturally

suitable for solving mci.ximization problems. Minimization problems

are usually transformed into maximization _ problems by some

suitable transformation.

n

general, a

fi ness

f u n c t i o l

F ( ~ )

is

derived from the objective function _and used in successive

genetic operations . Certain genetic operato rs require that the

i t n ~ s s

fu ;,tjon be,nonnegative, although certain.operators do no.t ~ a v e t ~ s

~ . 9 u i r e m e n t . For maximization problems, the fitness function can be

considered to be the same as the objective function or F( x) = f( x).

For minimization problems, the fitness function is an equivalent

maXimization problem chosen such

that

the optimum point remains

unchanged. A number of such transformations are possible. The

following fitness function is often used: ·

F(x) = 1/(1 (x)).

6

.2

)

Nontraditional p t i m i a ~ i o n Algorithm

298

This transformation does not alter the location of the minimum,

but converts a minimization problem to an equivalent maximization

problem. The fitness function value of a string is known as the

st ring's fitness.

The operation of GAs begins with a population of random strings

representing design or decision variables. Thereaft er, each string is

e v ~ u a . t e d to find the fitness value. The population is then operated

by three main

operators-reproduction,

c r Q ~ § l .

e r , ana

mutation

+ to

create a new population of points. The new population is fur ther

evaluated and tested for termination. f the termination criterion is

not met, the population is iteratively operated by the above three

operators and evaluated. This ptocedure · is continued until the

termination criterl.on

is

m,et . One cycle of these operatio:riS and the

subsequent evaluation procedure

is

known as a generation in GA's

terminology. The operators are described next.

G A operators

.l

Reproduction is usually the first operator applied on a population.

Reproduction sele.cts good strings in a population

and

forms a mating

pool.

That

is why'

the

reproduction operator is sometimes known

as. the selection operator. There exist a number of reproduction

operators in GA literature, but the essential idea in all of them is

that

the

b o v e ~ a v e r a g e

strings are picked from

the

current popvJation

and their multiple copies are inserted

i1 1

the mating poof in a

probabilistic fi .anner.

The c o m m o n l y ~ u s e d r e p r o d u c ~ i 9 n

operator is

the proportionate reproduction operator where a string is selected

fqr_the

~ l t

pool with a J : > a b i l i t y proportional to its. ~ t n e

Thus,

the

i ~ t h string in

the

population is selected with a probabili'ty

proportional to Fi. Since the population size is usually kept fixed in

a simple GA,

the

sum of

the

probability of each string being selected

for the mating pool must be one. Therefore, the probability for

selecting

the

th

string is

Fi

Pi= -n--,

F i

i=l

where

n

is the popula tion size. One way to implement ,this selection

scheme is

to

imagine a r o u l e i t e ~ w h e e l with

it s

circumference marked

for each string proportionate to the string's fitness , The r o u l e t t ~

wheel is spun

n

times, each time selecting ·

a.n

instance of the string

chosen by the roulet te-wheel pointer. Since the circumference of the

wheel is marked

a,c;cor

dlna to

a

string's fi.tness, this roulette-wheel


4/11

294 OptlmiJalfotl / rEngin ee

ring

Du

n

:

l

gorlthm

1

and

mechanism is expected

to

make Fi/

F

copies of

the

i-th string in

the

mating pool. The average fitness of the population is calculated as

n

F= EFifn .

i= l

Figure 6.1 shows

a.

roulette-wheel for

five

individuals having different

fitness values. Since

the third

individual

ha.S

a. higher fitness value

Point Fitness

1 25 .0

2

5.0

3

40.0

4 10.0

5

20.0

5

Figure 6.1 A rouiette-wheel marked for five individuals according

to their fitness values. The third individual has a higher probability

of selection

than

any other

than

any other it

is

expected

that

the roulette-wheel selection will

choose

the

thi rd individual more

than

any oth er individual. This

roulette-wheel selection scheme can be simulated easily.

0

Using the

fitness value

Fi of

all strings,

the

probability

of selecting a.

string

Pi

can be calculated.

T h e r e ~ f t e r t h ClJ

J

.nula.t

.iY:

e probabilitj p

i

of

each string being copied can be calculated by adding the individual

pr

obabilities from the top of

the

list. Thu_, the bottom-most string

in

the

population should have

a.

cumulative probability

Pn)

equal

to

l · The

roulette-wheel concept can be simulated by

realizi

11g

that the

i th

string in

the

population represents

the

cumulative probability

- a l u e s f r o m Pi-1 to Pi. The first string represents the cumulative

values from zero to P

1

• Thus,

the

cumulative probabilityof any string

lies .between 0

to

1. lri order

to

choose n strings, n random numbers

between zero to one are ·created

at

random.

Thus

a string

that

.

represents

the

chosen random number in

the cumulative

probability

range (calculated from

the

fitness values) for the

string is

copied

to the

mating pool. This way,

the

string with a higher fitness

value will represent

a.

rla.rger range in

the

cumula

ti

ve probability

values and there

fo

re has a. higher probabj]lty of bolng copied into

th

e

matin

g pool. On

the

other

han

d,

a.

string with

a.

smaller

fi

tness

value represents a. smaller range in eumula.tive probability values and

has

a.

smaller probability of being copied into

the

mating pool. We

illustrate

the

working

ofthis

roulette-wheel simulation

later

through

a computer simulation of GAs.

lri

reproduction, good strings in a. populat ion are proba.bilistically

assigned a. larger number of copies and a. mating pool is formed.

It

is

important to

note

that

no new strings are formed in

the

reproduction phase.

In

the crossover operator, new strings are

(

created }>y exchanging information among stnngs oof the mating

pool. Many crossover operators exist hi

the

GA literature. In

·

most

crossover operators, two strings are picked

fro:qt the mating pool at

random

and

s·ome portions of

the strh

igs ·are exchanged betwee n

the

strings. A single-point crossover operator is performed by ra.ndonily

choosing

a.·

crossing site along

the

string

and

by exchanging all bits

on

the

right side

of

the crossing site as shown:

f

.

0 010 0 0 0 011 1 1

:::}

111111

111 d

The

t

wo

strings participating in

the

crossover operation are known •

as parent strings and the resulting strings are known a s children

strings.

It is

intuitive from

this

construction

that

good substrings

from parent strings can be combined to form a.

better

child string, if

an

appro priate site is chosen. Since

the

knowledge

of an

appropriate

site is usually

not

known beforehand, a. random site is often chosen.

With a.

random site,

the

children strings produced may

or

may

not

have a. combination of good substrings

f;rom

parent strings, depending

on whether or nQt the crossing site fall1 in the appropriate place. But

we

do

not

worry

about

this too much, because

if

good s.t rings are

created by crossover, there will be more copies

of them

in

the

next

mating pool generated by

the

reproduction operator.

But if

good

strings are

not

created by crossover, they will

not

survive too long,

o

eca.u

se-reproduction will select against those strings in subsequent

r a t i o n s · ·

It

is

cle

ar

from this discussion

that the

effect of crossover may be

g e

trime ntal or beneficial. Thus,, in order to preserve some of

th

e good

strings

that

are already present in

the

mating pool, not all s

tr

ings in

the m

at

ing pool are used in crossover. When a. crossover probability

of

Pc is

used, only

100pc

per cent strings in

the

popul

at

ion are used

in

th

e cross

ov

er operation

and

100(1 -

Pc

per

ce

nt

of the populatjon


5/11

296

Optimization for En gin eering Design: Algorithms and Examples

remains

a.s they

a.re

in the

current population

1

.

A crossover

operator

is mainly responsible for

the

search of new

strings, even though a.

mutation operator

is also used for this purpose

sparingly.

The mutation operator

changes 1

to

0. a.nd vice versa.

with

a. smail

mutation

proba.bilit.YL..Em. The b it-wise

mutation

is

..- .

~ ~

performed

bit

by bit by flipping

a.

coin

2

with

a.

p r o b a . b i l i t ~

at'a:'ny

oit the

outcome is

t r u e o t h e r w i s e

~

·

The

need for

mutation

is

to create a.

point in the neighEourhood

of the

current po nt, thereby achieving

a.

local search

around the

current solution. Tile

mutation

is

also U'Sed

9

E . J - ~ j t a i n . J i v e r _ s i t y

_

~ ~ _ 2 i u o ~ .

For

e x

a.mple, 7 onsider

the

following population having four eight-bit strings:

0110 1011

0011 1101

0001 0110

0111 1100

Notice

that

a.ll

four strings have

a.

0

in the

left-most

bit

p o s i ~ i o n

f

the true

optimum

solution requires 1 in that position,

then

neither

fe"proauction

nor

crossover

operator

described above will

be

able

to

create

1

in

that position.-

The

inclusion

of mutation

introduces some

piOba.bilitYT NPm) ofturning

o into 1.

- -

These

three

operators a.re simple a.nd straightforward. The

reproduction operator

selects good s trings a.nd

the

crossover

operator

recomljiiies -good substrings from good st_ J:ggs

together

to honefully

crea.te_

a. better

substrin$

;_ l;:_he m u t a . t i ~ n operator

alters

a.

.

string

1

loca.lly to hopefully create a. better string. Even

though

none o

these c l a . i ~ s ~ guaranteed a.nd}or t ~ s t ~ d while crea.tin_g a.

is expected

that

if ba.d strings a.re created thE Y will _ e eliminated by

the

r e ~ d u c t i o n

operator

in .

the

next genera.tio

"ii"'an

d.

f

good

a.re ~ t ~ g ,

they

will

be

in.crea.singly e m J t h ~ ~ ~ t e r e s t e d readers

ma.y refer to Goldberg (1989)

a.nd other

GA

literature

given

in

the

references for

further

insight a.nd some

mathematical

foundations

of

genetic algorithms.

Here, we outline some differences

a.nd

similarities

of

GAs

with

traditional

optimization methods.

_

1

Even though

the

best 1 -

Pc)100%

of the current population can be copied

deterministically to the new population, ·this is usually performed at random.

2

Flipping

of

a coin with a probability

p

is simulated as follows. A number

between 0 to 1 is chosen at random.

f the random

number is smaller than p, th

ontcomc of coin-flipping is true, othcrwiAe the

out

come is fals

e.

Nontraditional Optimization A lgorithms

297

6.1.2

Differences between GAs a,nd traditional

methods

As seen from

the

above description of

the

working principles

of

GAs,

they

a.re ra.dica.lly different from

most of

the

traditional

optimization

methods

described

in Chapters

2

to

4. However,

the

fundamental

differences a.re described

in the

following

paragraphs.

·

GAs

~ o r k

w_ h a. string-coding

of

~ e s instead

thQ

variables.

The

a.dva.nta.ge

of

working

with a.

coiling

of

variables is

that

the

d i n g

discretizes the search space, even

though

the funct

op.

ma.y be

continuous. On

the other hand,

_ince GAs require only

ronchon

v a t u various discrete a. discrete

or

discontinuous

function ca.n

be

lia.naled with no extra. cost. T his a.llows GAs to

be

applied to

a.

wide .variety

of r o h l

An

other

a.dva.nta.ge is that

the

GA

opera.

ors

exploit

the r i t i e s

in st ring-st

:t;.l &.tJ

res

to

_

ma._ke

a.n

effective sea.rcli.

Let

us discuss this

important

aspect

of

GAs

in somewhat more details. A schema (pl. schemata.) repre sents 'a.

number of

strings

~ i t h

similarities

a.t

certain

string

positions. For

example,

in

a.

·five-bit problem,

the

·schema.

10h*) (a. ' '

denotes

either a. 0 or a.1) represents four strings

10100), 10101), 10110),

a.nd (

10111).

In

the

decoded

parameter

space,

a.

schema. represents

a. continuous

or

discontinuous region in

the

search space. Figure 6.2

shows that the above schema. represents one-eighth

of

the search

space. Since in

a.n

£-bit schema., every position ca.n

take either

0, 1,

- -1

I I I I I

I OOO••I 001••1 010••1 011••1 100••1 101••

I I

110••1 111••1

I I I I I

I

I

I

I I I I

I I

xmin

X

max

Figure 6.2

A schema with three fixed positions divides the search

space into eight regions.

The

schema

10h*)

is highlighted.

or *,

there

a.re a. total of 3l schemata. possible. A finite population of

size n contains only n strings,

but

contains many schemata.. Goldberg

(1989) ha.s shown

that

due to

the

action of GA operators, the number

of strings

m H,

t) representing a. schema. H

a.t a.ny

generation t grows

to a.

number

m H,

t

+1) in the next generation a.s follows:

:F H) [ IJ H) ]

m H, t +1 m H, t) --=y- 1 - Pet= Pmo H)J

(6.3)

growth factor , P

wht•ro

:T

II)

Is th

o

fitn

CRil

of

tho rhoma ca lculated

by a.vcra,;in,;


6/11

the fi

t ness

of

all strings representing

the

schema,

8( /)

is the defining

length of the schema

H

calculated as the difference in the outermost

defined positions,

and o(H)

is

the

order of

the

schema

H

calculated

as

the number of

fixed positions

in the

schema. For example,

the

schema

H =10h

has a defining length equal

to o(H) 3

1 2

and

has

an order o(H)

= 3.

The growth

factor

P

defined ;

in the

above

equation can be

g r ~ t r than,

less

than,

or equal to 1 depending

on the

schema

H

and the

chosen

GA

parameters.

I f

for a schema

the growth factor

P

: 1, the number

of

strings representing that

schema grows with generation, otherwise

the

representative strings

of the

schema reduce with generation.

The

above inequality suggests

that

the

schema having a small defirung length (small

o(H)),

a few

fixed positions (small o(H)),

and

above-average fitness

F H)

>F),

the

growth factor

P

s likely

to

be

greater

than 1. Schemata for

which

the

growth factor is

greater

than 1 grows exponentially with

generation. These

schemata

usually represent a large, good region

(a region with many high fitness points)

in the

search space. These

schemata

are known as

building blocks

in GA parlance. These

building blocks representing different good regions

in the

search space

get exponentially more copies

and

get combined with each

other

by

the

action of GA operators

and

finally form

the optimum o:r

a

near-optimum

solution. Even

though

this is

the

basic

understanding

of how GAs work,

there

exists some

mathematical

rigour

to

this

hypothesis (Davis and Principe, 1991; Vose

and

Liepins, 1991).

Holland (1975) has shown that even though

n

population members

are modified in a generation, about

n

schemata

get processed

in

a generation.

This

leverage comes

without any extra

book-keeping

(Goldberg, 1989)

and

provides

an implicit parallelism

in the working

of genetic algorithms. Even though there are a number of advantages

of using a coding of variables , there are also some disadvantages. One

of the drawbacks of using a coding representa tion is that a meaningful

and an

appropriate

coding of the problem needs to be used, otherwise

GAs

may not

converge

to

the right solution (Goldberg, et. al, 1989;

Kargupta, et, al, '1992). However, a general guideline would be to

use a coding that does

not

make

the

problem

harder

than the original

problem.

The

most striking difference between GAs

and many traditional

optimization methods is

that

GAs work with a eopulation of

points instead

of

a single point. Because

there

are more th

an

One string 6eing processed. simultaneously, it is very likely that

the

expected GA solution may

be

a global solution. Even though some

traditional

algorithms are population-based, like Box's evolutionary

optimization and complex search'methods, those methods do

not

use

previous ly obtu.ncd

lnforma.tlon

offldouUy.

In

GA8 previou sly found

good

lnform.a

tlon ls

o

mpl1asizc

d

using reproduction OJHirator and

propagat

ed adaptively through crossove·r attd mu ta.tiou

o:pe

.

rator

s.

Another

advanta

ge

wi

th a population-based search algorithm is

that multiple optimal solutions can

be capt

ured in the P,Opulation

easily, thereby reducing

th

e effort to use the same a1gorit hm many

times. Some extensions of GAs along these directions- multimo dal

function optimization (Deb, 1989; Goldberg

and

Richardson, 1987)

and multiobjective function optimization (Horn

and

Nafpliotis, 1993;

Schaffer, 1984; Srinivas

and

Deb, 1995)- have been researched

and

are

outlined

in

Section 6 1.

7.

In discussing GA operators or their working principles

in

the

previous section, nothing has been mentioned

about

the gradient or

any

other

auxiliary problem information.

In

fact , GAs do not require

any auxiliary information except the objective function values.

Although

the

direct search methods used in traditional optimization

methods do not explicitly require the gradient information, some of

those methods use search directions that are similar

in

concept to the

gradient of the function. Moreover, some direct search

methods

work

under the assumption that

the

function to be optimized is unimodal

and

continuous.

In

GAs, no such assumption is necessary.

__.:;; One

o t ~ r

difference

in the

operation of

GAs is

he

use

of

probabilities in

their operators

. None of

the

enetic o

era

rs work

-a:eFefmmishc

aJly. n e repro uction

operator,

even though a string

is expected to have

Fi/

F copies

in

the

mating

pool, a siinulatiori

of the

roulette-wheel selection scheme s used to assign

the true

number

of copies. In the crossover operator, even

though

good strings

(obtained from the

mating

pool) are crossed, strings to

be

crossed

are created at

random

and cross-sites are creat ed

at

random.

In the

mutation

operator

, a

random

bit is suddenly altered. The actidn

of these operators

may appear to be

l).aive,

but

careful studies

may

provide some interesting insights

about

this type

of

search.

The

basic

problem with most of

the

traditional methods is that

they

use fixed

transition

rules

to

move from one point

to

another. For instance, in

the steepest descent

method

; the search direction is always calculated

as

the

negative ofthe gradient

at

any point, because in that direction

the reduction in the function value is maximum.

In

trying to solve a

multimodal problem

with many

local

optimum

points (interestingly,

many

real-world engineering optimization problems are likely to

be

multimodal), search procedures may easily get

trapped in

one

of

the local

optimum

points. Consider the bimodal function shown

in Figure 6.3.

The objective function has one local minimum and

one global minimum. I f

the

initial point is chosen

to be

a point


7/11

800

--

~

Global

opt imum

X

Figure 6.3 An objective function

with

one local

optimum and

one

global

optiqmm. The

point x t) is in

the

local

basin

.

in

the l o c ~ l

basin (point

x t) in the

figure),

the

steepest descent

algorithm will eventually find the local optimu m point. Since the

transition

rules are rigid, there is no escape from these local optima.

The only way to solve the above problem to global optimality is to

have a

starting

point in

the

global basin. Since this information

is usually not known in any problem, the steepest-descent method

and

for

that matter

most

traditional

methods) fails .

o

locate

the

global optimum.

We

show simulation results showing inability of

the steepest descent

method

to find the global

optimum

point on

a multimodal problem

in

Section 6.3. However, these

traditional

methods

can

be

best applied to a special class of problems suitable

for those methods. For example, the gradient .search methods will

outperform

almost any a l g o

i t h m in

solving continuous, unimodal

problems, but

they

are

not

suitable for multimodal problem. Thus, in

general,

traditional

methods are not robust. A

robust

algorithm can

be designed

in

such a way that it uses

the

steepest descent direction

most

of the. time, but also uses

the

steepest ascent direction (or any

other

direction) with some probability. Such a mixed

strategy may

require more number of function evaluations to solve "continuous,

unimodal problems, because of

the

extra

computations involved in

trying with .non-descent directions.

But

this

strategy

may

be

able

to solve complex, multi modal problems to global optimalit

Y

.

In the

muHimodal problem, shown

in

the above figure, the mixed

strategy

may take the point

x t)

into

the

global basin (when t:r.ied with

non-descent directions)

and

finally find the global

optimum

point.

GAs use similar search strategies by using probability in all their

operators. ince an initial

random

population is used, to start with,

the

search can proceed

in

any direction

and

no

major

decisions are

made

in the beginning. · Later on, when the pQpulation begins to

copverge in some bit positions, tho search direction

narro

ws

and

a ;near-

opt

imal solution

is

achieved. T

hi

s na ture of

narrowi

ng t he

search space as

the

sear ch

pr

ogresses is

adaptiv

e

and

is a unique

characteristic o f genetic algorithms.

6.1.3 Similarities between GAs

and

traditional

methods

Even though GAs are different than most

traditional

search

algorithms, there are some similarities. In

traditional

search

methods, where a search direction is used to find a new p

.Qin1....

.a1

~ a s t

two P?ints are 1 l e f i

m P f i c

· or_ ~ c l £ used to ·define

the seardi direction. In

the

Hooke-Jeeves pattern search method, a

pattern move

is

created using two points. In gradien t-based methods,

the

search direction requires derivative information which is usually

calculated using function values at two neighbouring points.

In the

crossover operator (which is mainly responsible for the GA search),

two points a re also used

to

create two new points. Th us, the crossover

?Peration is simi l- --. .


8/11

30

lml•• ion

tift:

t o r i l t nfl mp

/

along directions (

Ct and c2)

shown in the fi

gur

e (either along soHd

arrows

or

along dashed arrows). The exact locations

of

the children

points along these directions depend on

the

relative distance be

tw

een

the parents

(Deb

and

Agrawal, 1994).

The

points

Yt and

Y2

are the

two typical children points

obtained

after crossing

the parent

points

PI and p

2

•

Thus,

it

may

be envisaged

that

point PI has moved in the

direction from di up to the point YI

and

similarly

the

point P2 has

moved to the point Y2.

Since the two points used in the crossover operator

are

chosen

at random, many

such search directions are possible. Among

them

some directions

may

lead to the global basin and some directions

may

not

. The reproduction

operator

has

an

indirect effect

of

filtering

the good

search directions

and

help guide

the

search.

The

purpose

of

the mutation

operator

is .to create a point in the vicinity of the

current point. The search in the mutation operator is similar to

a local search

method

such as the exploratory search used in the

Hooke-Jeeves

method.

With the discussion

of

the differences and

similarities

of

GAs with

traditional

methods, we are now ready to

present

the

algorithm in a step-by-step format.

Algorithm

Step 1

Choose a coding

to

represent problem

parameters,

a

selection

operator,

a crossover

operator, and

a

mutation operator.

Choose population size, n, crossover probability, Pc and mutation

probability, Pm. Initialize a random population of strings of size f

Choose a

maximum

allowable generation number

tmax·

Set t = 0.

Step 2

Evaluate each string in

the

population.

Step 3

I f t

>

tmax or

other termination

criteria is satisfied,

Terminate.

Step 4

Perform reproduction on

the

population.

Step 5

Perform crossover on

random

pairs

of

strings.

Step 6

Perform

mutation

on every

string.

Step 7

Evaluate strings in the new population . Set = +1

and

go to Step 3.

The algorithm is straightforward with repeated application

of

three operators

(Steps 4 to 7) to a population

of

points. We show the

working

of

this algorithm to the unconstrained Himmelblau function

used in

Chapter

3.

p l i m ~

30

EXERCISE 6 1 1

The object ive is to minimi

ze the

function

,;

..

.

f xt,x2)

=(xi+

X

-11)

2

+ x i+ 7

2

in

th

e interval 0

; ;;

x

1

, x

2

;:;; 6.

Recall

that the true

solution

to

th is

problem is (3, 2 f having a function value equal to ~ e r o .

Step 1

In order

to

solve this problem using genetic algorithms,

we choose

binary

coding to represent variables Xt and x2. In

the calculati


9/11

104

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10/11

306

Optimization fo r b n

gin

eet·i ng Design; Algorithms and b xamples

~

0 - ·- - - - - - - - - -

c

5

' - . ._0

'

'

o.

-·-·-·-·-·-·- o

"

'

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4

· · · · · · · · · · · · · · · · · · · · · · · · - ·- ·- · . . " \ 0

c

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·.

a \ '

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.

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~

\ \c\·... \ I

~

.

----

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I

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1rumum :

'

oint

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0

i

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lc

\

\ c \ I I \ I I

I.

J.

:.

I

2

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'L·

0

.

0

1

2

3

4 5

6

Xt

. . . 1500.0

850.0

500.0

300.0

150.0

75.0

30.0

10.0

o Initial

population

0 Mating

pool

Figure 6.5

The

initial population (marked with

empty

circles)

and

the mating pool (marked with boxes) on a contour plot of the objective

function.

The

best point in

the

population has a function value 39.849

and the average function value of the initial population is 360.540.

pool contains strings

at

random, we pick pairs of strings from the top

of the

list. Thus, strings 3

and

10 participate in

the

first crossover

operation.

When

two strings are chosen for crossover, first a coin

is flipped with a probability Pc

=

0.8 to check whether a crossover

is desired or

not.

If the outcome of the coin-flipping is

true,

the

crossing over is performed, otherwise

the

strings are directly placed

in an intermediate population for subsequent genetic operation. It

turns

out

that

the outcome of the first coin-flipping is

true,

meaning

that

a crossover is required to be performed. The next step is to

find a cross-site

at

random.

We choose a site by creating a

random

number

between (0,

.e-

1) or

(0, 19).

It

turns out

that

the

obtained

random number

is 11. Thus, we cross

the

strings at the site 11

and

create two new strings. After crossover,

the

children strings

are placed in the intermediate population. Then, strings 14 and 2

(selected at random are used in

the

crossover operation. This time

the coin-flipping comes true again and we perform the crossover at

the site 8 found at random. The new children strings are put

into

the

intermediate

population. Figure

6.6

shows how points cross over

and form new points.

The

points marked with a small box are the

points in the mating pool and the points

marked

with a small circle

are children points created a f ~ e r crossover operation. Notice

that

not

N

mthrHltiiiiMI

O t J L i m i

t ~ t l c m

A ouf hm

6

r

:n ..• J

: ;

t • t :

............ ····· o 1iatina

pool

··--

If ·

2

1

·-

o After orosaover

·

· ~ _

·

.a\.

\

\

··.

··

o ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

0 1 2 3 4 5 6

x

ao

Figure 6.6 The population after the crossover operation. Two

points are crossed over to form two new points.

Of ten

pairs of strings,

seven pairs are crossed.

all 10 pairs of points in

the

mating pool cross with each other. With

th

e flipping

of

a coin

with

a probability Pc

=

0.8,

it

turns out

that

fourth, sevent

h,

and tenth crossovers come out to be false. Th us,

in these cases, the strings are copied directly

into

the

interm

ediat

population. The complete population at the end of the crossover

operation is shown in Table 6.2.

It

is interesting to

note

that

with

Pc =

0.8, the ex

pected number of

crossover in a population

of

size

20 is 0.8 X 20/2 or 8. In this exercise problem, we performed seven

e r s

an d in

three

cases we simply copied the strings to

th

e

ltttormediate population. Figure 6.6 shows that some good points

a.nd

some not -so-good points

are

created after crossover. In some

ascs,

po

i

nts

far away from

the parent

points

are created

and in

some cases points close to the

parent

points are created.

Step 6 The

next

step is to perform mutation on strings in th

intermediate

population. For bit-wise mutation, we flip a coin with

a proba biHty m = 0.05 for every bit.

If the

outcome is true, we alter

tho bit to 1 or 0 depending on

the

bit va]ue. With a probability of

0.0

,a

population

Rlzo

20, and

a

string length 20, we

ca.

n expect to ·

n. t.M IL

tota

.l of a.hou t 0.01) x 20 x

20

or 20 bitR ln tho population ,

l li hl< ,

0.2 Hhowll th11

11111 tlitod blts in

bold

ch

a.ra.ctcrtl In tho t 1 ~ b l

A


11/11

308

t i m i ~ t d i o n for

engineering

D

e1ign

Al

gorltAm1

and

Ex

amples

11.1

c

- M

o o c i c 0 . , ; ~

M

00 0 ' )

co 00 0 ' )

C 'O') O ' ) ~ . - < C ' L < ' : > O O C ' O

C O O M O . - < L < ' : > M . - < . - < O ~ < t ' t - M < t ' . - < L < ' : > O ' ) < t ' O

~ . , ; d ~ . , ; ~ . , ; . , ; d ~ ~ ~ ~ ~ ~ d ~ ~ ~ . , .

.-

t-

. C ' M 0

t-

...... C ' L< > t- M M M M O O M O " < t ' " < t ' C O C O L < ' : > . - < M t - L < ' : > C O

OO')"

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