ieee robio 2009 conference paper first submission_1 7 2009

8/21/2019 IEEE ROBIO 2009 Conference Paper First Submission_1 7 2009

1/9

Generalization of Arithmetic and Visual Fuzzy Logic-based

Representations for Nonlinear Modeling and Optimization

in Fully Fuzzy n!ironment

Hassen Taher Dorrah, Member, IEEE and Walaa Ibrahim Mahmoud Gabr

Abstract-"his paper is directed to#ards thede!elopment of the Arithmetic and Visual Logic-based

representations for classical nonlinear systems modeling

and optimization$ "he concept #as originally proposed by

Gabr and %orrah &'-() for linear system as an e*tension of

the notion of the normalized fuzzy matrices$ +n this

concept, the arithmetic fuzzy logic-based representation type

is suggested based on dual cell representation, e*pressed

by replacing each parameter #ith a pair of parentheses,

the first is the actual !alue and the second is the

corresponding fuzzy le!el, or eui!alently .Value Fuzzy

Le!el/$ "he visual fuzzy logic-based type is proposed based

on colored cells representation e*pressed by replacing eachparameter by its !alue and a coded corresponding to its

fuzzy le!el$ For both cases, the theoretical foundations of

the fuzzy logic algebra, different properties and

implementation rules are further elaborated in this paper

for !arious cases of operations$

"he suggested approach is generalized to classical

nonlinear modeling and optimization problems that are

normally sol!ed by either the Lagrangean Function

Method or the 0acobian "echniue$ "he t#o methods #ere

then modified by incorporating the suggested fuzzy logic-

based representations assuming the fuzziness of all the

optimization formulation parameters$ 1sing a

representati!e nonlinear optimization numerical e*ample,

the proposed fuzzy logic-based formulation is applied$2oth the Lagrangean Function Method and the 0acobian

"echniue fuzzy logic-based formulations ga!e identical

results for all the solution parameters and their

corresponding fuzzy le!els$ "hese results demonstrate the

consistency and robustness of the de!eloped approach for

incorporation #ith classical nonlinear optimization

problems$ Finally, it is sho#n that the presented concept

pro!ides a unified theory for !arious linear and nonlinear

systems in fully fuzzy en!ironments$

I. I NTRODUCTION

FUZZY mari!es ha"e been a##lied b$ man$ resear!herso sol"e some real li%e a##li!aions. Ne"erheless, hea##li!aion o% %u&&$ mari!es o real li%e #roblem is sill"er$ limied. In %a!, he noion o% %u&&$ mari!es hasunlimied !a#abiliies ha ha"e no $e e'#lored in real

a##li!aions. Mos o% he s$sem daa are normall$e'#ressed in he %orm o% mari!es o% "arious $#es hama(e his noion o% %u&&$ mari!es mos #erinen oheir mani#ulaion. There%ore, here has been a real needo s!ruini&e he o#eraion o% hese %u&&$ mari!es andsear!h %or oher #h$si!all$ #ro%ound o#eraional

e!hni)ues.

Manus!ri# re!ei"ed *ul$ +, --. Dr. H. T. Dorrah is /ih he De#armen o% 0le!ri!al 0n1ineerin1,

Cairo Uni"ersi$, 01$# 2e3mail4 dorrahh5aol.!om6.

Dr. Walaa I. M. Gabr is /ih 7D8 0n1ineerin1 2Consulans6 In!.,

01$#, and he 01$#ian 0le!ri!i$ Holdin1 Com#an$, Minisr$ o% 0le!ri!i$ and 0ner1$, 01$# 2e3mail4 Walaa91abr5$ahoo.!om6.

Re!enl$, Gabr and Dorrah #resened he ne/ !on!e#o% boh 8rihmei! and :isual %u&&$ lo1i!3based

re#resenaions ;+3e"el6. The a##roa!h /as hen a##lied o man$ !lasseso% linear s$sem modelin1 and o#imi&aion o% o#eraional en1ineerin1 s$sems as a 1enerali&aion o%

he noion o% %u&&$ mari!es. This has in!luded sol"in1linear, muli3ob?e!i"e and )uadrai! #ro1rammin1o#imi&aion #roblems.

In order o !ir!um"en he shor!omin1s, he !on!e#o% %u&&$ mari!es ;@3= /as normali&ed o he realsiuaion usin1 he e'ended ran1e o%

jiaij ,=,+,+; ∀−∈ , and hen se#araes is o#eraiona%er normali&aion 2s!alin16 %rom he ori1inal #roblem"alues. This means ha ea!h #roblem #arameer ise'#ressed ino /o !om#onens4 he ori1inal

deerminisi! !om#onen and he relai"e %u&&iness!om#onen normali&ed or s!aled o sais%$ he abo"e boundaries. This /ill re)uire ha he e%%e! "alues o%he %u&&$ !om#onen are less han he main ori1inaldeerminisi! #roblem. The ori1inal #roblem soluion%ollo/s in is !om#uaions he normal #ro!edure, and

he oher %u&&iness !om#onen /ill be sub?e! o as#e!ial %u&&$ al1ebra o#eraion #ro!edure.

Gabr and Dorrah ;+3


2/9

The ne/l$ su11esed Fu&&$ >o1i!3based 8rihmei!Re#resenaion a##roa!h de"elo#ed b$ Gabr and Dorrah

;+3


3/9

{ }62 Z Y X ++=

---

---

Z Y X

Z z

Y y

X x

++

++=

.

2e +−= X % hen

{ } { } X % =−+ .26

The same abo"e rules a##l$ %or di"ision andsubra!ion.

e) &t'er Implementation ("les

When a##l$in1 he 8rihmei! Fu&&$ >o1i!3basedre#resenaion la/s, i mus be obser"ed durin1im#lemenaions he %ollo/in1 se)uen!e4

i6 Firs a##l$ he arihmei! re#resenaion al1ebra ohe muli#li!aiondi"ision o#eraions.

ii6 7e!ond a##l$ he al1ebra o he

addiionssubra!ion o#eraions.iii6 Inner bra!(es are a##lied %irs hen %ollo/ed b$

he su!!eedin1 bra!(es, endin1 b$ he mos ouer bra!(e.

i"6 Rules 2i6 and 2ii6 are se)ueniall$ a##lied /henmo"in1 %rom one inner #ro!eedin1 o ouer

bra!(es.

I !an be seen ha he o#eraion se)uen!e o% heabo"e im#lemenaion rules are similar o ha o%radiional arihmei! o#eraion. This /ill #ermi eas$

o#eraion o% #arallel o#eraion o% he !orres#ondin1%u&&$ le"el b$ he !on"enional arihmei! !al!ulaions.

I:. BRI0F D07CRITION OF :I7U8> FUZZY >OGIC3B870DR 0R070NT8TION

In he arihmei! re#resenaion o% %u&&$ lo1i! le"els,

ea!h #arameer /as re#la!ed b$ /o !ells %orm 2:alue,Fu&&$ >e"el6, hus doublin1 he si&e o% he soluion.8lernai"el$, he same ori1inal soluion !ells are used,

and he !olor o% he !ell is sele!ed as is e)ui"alen%u&&$ lo1i! le"el ;+30 08M>0 OF 70>0CT0D O7ITI:0 8ND N0G8TI:0

COD0D CO>OR7 FUZZY >OGIC3B870D 7C8>0.

7 e r .

C o l o r Color Code

RBG Color

Inde'

0'!el

Color

Inde'

Corres#ondin1

Fu&&$

>e"el T $ # e

+:iolen

2>a"ender62i1h6

2-,+


4/9

2++6

/here 6,...,,2 + n x x x X = and-

m , , , , 6,...,2 += . The %un!ions 62 X f and62 X , i , iJ+, , ,m are /i!e !oninuousl$

di%%ereniable. De%ine

62626,2 X , X f X L λ λ −= . 2+6su!h ha he %un!ion L desi1naes he >a1ran1ean%un!ion o% he #roblem and he #arameers λ are he

>a1ran1e muli#liers.

The e)uaions

-,- =∂∂

=∂∂

X

L L

λ 2+A6

#ro"ide he ne!essar$ !ondiions %or deerminin1

saionar$ #oins o% 62 X f sub?e! o -62 = X , .The su%%i!ien!$ !ondiions %or he >a1ran1ean mehod/ill be saed as %ollo/s ;+-=. De%ine

6262

-

nmnm

-

.

/ 0

0 1

+×+

= 2+6

/here

nm ,

,

X

X

0

m ×

∇

∇

=

62

62+

.

and

nn ji x x

X L/

×∂∂

∂= 6,2 λ

%or all i and j.

2+0 A

8R8M0T0R7 :8>U07 OF NUM0RIC8> 08M>0 8NDTH0IR CORR07ONDING FUZZY >0:0>7.

arameer 2:alue, Fu&&$

>e"el6arameer

2:alue, Fu&&$>e"el6

+c 2+,+ 6 +a 2


5/9

-A +AA

=−−=∂∂

λ λ x x

L2


6/9

A Met'o$ *orm"lation an$ &ptimization +ol"tion

In his se!ion, /e #resen %irs a brie% summar$ o%he *a!obian Mehod ;+-= as an alernae mehod o be

e'ended %or he !ase /here are #arameers aree'#ressed in a %ull$ %u&&$ en"ironmen.

For he nonlinear o#imi&aion #roblem des!ribed in2+6, de%ine

6,2 Z Y X = 2A


7/9

-62 = X ,

or

=

+

A

+

A+

+A+++

-

b

b

x

x

x

aaa

aaa

γ β α

. 2a1ran1ean Fun!ion Mehod and he*a!obian Te!hni)ue 1a"e ideni!al resuls %or all he #arameers soluions and heir !orres#ondin1 %u&&$le"els. The onl$ di%%eren!e /as %or he %u&&$ le"els o% he su%%i!ien!$ !ondiions 1i"en b$ he Hesbian Mari'

. 1 and he deri"ai"eA-

- x f ∂∂ as he$

indi!ae di%%eren %ormulas. These %u&&$ le"els !an berans%erred o e)ui"alen un!erain$ b$ in!or#orain1

he "alue o% relai"e %u&&iness r f 2#re%erabl$ be EE

+6 o% he #roblem.

In order o anal$&e more he #ro#osed %u&&$ lo1i! based %ormulaion, si' di%%eren s!enarios o% he samenumeri!al e'am#le /ere desi1ned as sho/n in Table A.The resuls o% sol"in1 he s!enarios usin1 boh he>a1ran1ean Fun!ion Mehod and he *a!obianTe!hni)ue are sho/n in he same able. These resuls

demonsrae he !onsisen!$ and robusness o% hede"elo#ed a##roa!h %or in!or#oraion /ih !lassi!alnonlinear o#imi&aion #roblems.

T8B>0

R 07U>T7 OF :8RIOU7 7C0N8RIO7 OF DIFF0R0NT I NUTFUZZY >0:0> OF NUM0RIC8> 08M>0.

K


8/9

arameer :alue

Corres#ondin1 Fu&&$ >e"el o%

Di%%eren 7!enarios

I II III I: : :I

+c + 3A 3 3+ + A

c + A + 3+ 3 3A

Ac + 3A 3 3A 3

++a + A + 3+ 3 3A

+a + 3A 3 3+ + A

+Aa A 3 3+ 3+ + 3

+a < A + 3+ 3 3A

a A + 3+ 3 3A

Aa + A + 3+ 3 3A

+b 3+ 3 3+ +

b < A + 3+ 3 3A

+ x -.L-A + + - - 3+ 3+

x -.AKL 3< 3A 3 A <

A x -.L@ + 3+ 3 A @

+λ -.-LK- 3 A 3A 3A 3

λ -.A-A 3@ 3 <

62 X f -.LKL 3 3+ 3+ + +

. 1 @- 3+ + - - 3+ +

A-

- x f ∂∂ a1ran1ean Fun!ionMehod and he *a!obian Te!hni)ue %u&&$ lo1i! based

%ormulaions 1a"e ideni!al soluion resuls o% a sele!ednumeri!al e'am#le %or all he #arameers and heir !orres#ondin1 %u&&$ le"els. These resuls demonsraehe !onsisen!$ and robusness o% he de"elo#ed

a##roa!h %or in!or#oraion /ih !lassi!al nonlinear o#imi&aion #roblems.

I is re!ommended ha he de"elo#ed %u&&$ lo1i!3 based arihmei! and "isual re#resenaions be1enerali&ed as a uni%ied heor$ o be in!or#oraed /ihmodelin1 and 1lobal o#imi&aion o% boh linear andnonlinear s$sems ;+-3+=. This uni%ied heor$ /ill be

based on %ormin1 #arallel o#eraional al1orihm %or he!al!ulaion o% soluion %u&&iness o 1o /ih he!on"enional arihmei! modelin1 and o#imi&aion!om#uaions. Finall$, e%%ors should also !ommen!e ine'endin1 he de"elo#ed %u&&$ lo1i!3based %ormulaion

o addiional mahemai!al %ormulas 2su!h as ine1raionand di%%ereniaion6 and o d$nami!al s$sems.

R 0F0R0NC07

;+= Walaa Ibrahim Gabr and Hassen Taher Dorrah, Ne/ Fu&&$

>o1i!3based 8rihmei! and :isual Re#resenaions %or

7$semsQ Modelin1 and O#imi&aionS IEEE International !onference on (obotics an$ .iomimetics, Februar$ 3inear O#imi&aion Usin1 Fu&&$ >o1i!3based8rihmei! and :isual Re#resenaions /ih For/ard and

Ba!(/ard Tra!(in1 S, IEEE International !onference on (obotics an$ .iomimetics, Februar$ 3


9/9

U78, --A.

;++= Thomas Weise, 4lobal &ptimization Alorit'ms 5 -'eory

an$ Application, h#4///.i3/eise.de#ro?e!sboo(.#d%,

*anuar$ --L.;+= anos M ardalos, and H. 0d/in Romei?n 20diors6,

1an$boo6 of 4lobal &ptimization, :olume , Uni"ersi$ o%

Florida, Gaines"ille, lu/er 8!ademi! ublisher, Boson,

UR>4 h#4///.o#imi&aion3online.or1DB9FI>0 ---A

ieee robio 2009 conference paper first submission_1 7 2009

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