Su' dung thuat toan tien hoa vi phan toi uu cac kich thuoc dam dang hop chju uon

235 Tuyen tap cdng trinh Hdi nghi Cff hoc todn qudc Ky niem 30 nam Vien Cff hge vd 30 ndm Tgp chi Cff hoc Hd Ndi, ngdy 8-9/4/2009

Su' dung thuat toan tien hoa vi phan toi uu cac kich thuoc dam dang hop chju uon

Nguyen Quan Thang Bg tie linh Cdng binh

Nguyen The Minh, Biii Due Nang

Hge viin ky thuat Qudn sie; nguyenniinhl97I@vnn.vn

Tom tat: Bdi bdo trinh bdy kit qud nghien cieu tinh todn tdi icii doi vdi mot sd dgng ddm thep dgng hop chiu udn. Cdc tham sd hinh hoc ciia ddm thep dgng hop dugc lua chon theo chi tieu khdi lugng cue tieu.

Phuang phdp tinh dd sic dung Id thudl todn tien hod vi phdn.

1. Dat van de

Trong nganh Co' khi ndi chung, nganh Co' khi xay dung ndi rieng riit nhieu cae ehi tiet co' khi cd dang diim vdi tiet dien hinh hop. Day la mdt dang ket cau dirge diinh gia kha hgp ly ve cac mat nhu: sir dung vat lieu, kha nang chju lue... Ket eau dam thep dang hop dien hinh trong ket cau may xiiy dirng thudng la: ket eau can eiia can true thuy hie (hinh.l), cau true dang dam hay ket eau can eiia may xiie mgt gaii diin ddng thuy liic (hinh.2).

Hinh 1. Cau true dan ddng thuy lue Hinh 2. Miiy xiic dan ddng thuy lire Khi tinh toan cac kit cau tren thudng su dung so do tinh nhu hinh 3 vdi eae to hgp tai khac nhau tiiy thude vao dac tinh lam viec ciia mdi thiet bj. Mac dii la nhung chi tiet het sire quen thuge vdi so dd tinh don gian, nhung hien nay tren the gidi cac nha khoa hge van chii trgng nghien ciru vdi mue dich toi uu duge hinh dang, trgng lugng eiia kit cau, tii' do tang duge dac tinh kinh t l - ky thuat cua thiit bj cdng tac cung nhu eiia toan bg may.

Nguyen Qudn Thdng; Nguyin The Minh; Biii Diec Ndng

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Ll ->!

A

L2

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Hinh 3. So dd tinh toan eiia ket eau can trong trudng hgp chju tiii don gian nhat

Trong do cd the mdt sd cdng trinh khoa hge cdng bd gan dii\' ciia cac tac gia O-Kaung Lim

\a Eun-Ho Choi, niim 2005 [6] ha)' eiia tiie giii Menmet Yener, niim 2005 [5]. Ddi vdi cac cong trinh nghien ciru trong nude, cung cd riit nhieu eae nha khoa bgc dii quan tiim ciing nhu cdng bo cae till lieu \e tdi uu ket eiiu nhu GS. Vo Nhu Cau [1], GS. Le .Xuiin Huynh [2]... Diiy la nhung tiii lieu het sue qui gia, riit de sir dung. Tuy nhien trong qua trinh nghien ciru ap dung cac thuat loan toi uu cho ket ciiu may xii\' dung, chiing tdi cimg manh dan irng dung mdt sd thuiit toiin toi uu mdi \'di h_\ \ gng cd the phat trien va irng dung duge trong nganh eiia minh.

2. Lua chon bai toan va thuat toan,ung dung de tdi uu ket can 2.1 Lira chgn bdi todn tdi leu ket cdu

Theo till lieu [2] cac bai toim tdi uu ket ciiu thudng cd cae dang: Tdi iru hinh dang, tdi uu thiet dien ngang, tdi uu ciiu triic \^a tdi uu tdng chi phi. Trong do cae bai toan tdi uu ket ciiu \d'i ham muc tieu la ket ciiu cd trgng lugng circ tieu ma viin bao dam ciic dieu kien lam \iee thudng chiem t\ trgng riit Idn, Dii) cung la dang bai toan tdi uu ket cau dugc lua chgn trong bai biio nay.

Ndi chuns, bai toan tdi uu ket eiiu theo tronsj luonsi diioe \ let dudi dang toan hoc nhu sau:

Cue tieu hoa ham: F{x) - /,y,AJ^

1=]

Vdi cac dieu kien raim budc: g, =C —C > 0;

(1)

C-)

trong dd: Fixj- Trgng lugng ciia toan bd ket can; n- Tdng sd phiin lit cua ket ciiu; ;>,- Khdi lugng rieng ciia phan tir thir i; A,- Dien tich mat ciit ngang eiia phan tir thir i; /,- Chieu dai eiia phiin tir thir i; g- Cae dieu kien ban che ve co' bgc va hinh bgc; C',- Gidi ban eho phep eiia cac dieu kien ban che; C^ Trang thai thuc eiia ket eau dang xem xet.

2.2 Lua chgn thuiit todn iing dung gidi bdi todn toi uu kit cau

Theo tai lieu [2], de giai bai toan (1) ed cac nhdm phuong phap nhu sau:

Cae phuong phap qui hoaeh toiin hge;

Cae phiro'ng phiip nhiiii tir Lagrang;

Ciic phuang phap dua tren co sd md phdng Monte-Carlo.

Nhdni hai phuo'ng phap diiu lii nhdm cac phuong phap trii)'cii thdng dii dugc irng dung nhieu trong ciic bai toan tdi uu ket cau va dii dat dugc nhieu thanh cdng trong ciic nganh nhu che tao may ba). ket ciiu XLT)' dung... Tuy \a)' van cdn nhirng ban che nhu: thuat toan cdng kenh, tinh cd dao hiiin, tinh loi, khdng cd kha niing tim kiem tdi uu toan cue... nen hien nav it duge sir dung. Cac phuong phap dira tren co sd md phdng Monte-Carlo dien hinh nhu: Thuat toan di

Sie dung tlnidt todn tien hod vi phdn tdi mi cdc kic/i thude ddm dgng hop chiu udn 237

truyen (Genetic Algolrithm -GA) [8], thuiit toan md phdng luyen kim (Simulated Annealing SA) [7], thuiit toan tien hoii vi phiin (Differential Evolution - DE) [4] la Idp thuat toiin lira chgn ngau nhien dang duge irng dung trong nhieu bai toan ky thuat. Qua qua trinh lap trinh, tinh toiin thir nghiem vdi eae bai toiin tdi uu dien hinh eiia toan hge, ehiing tdi nhiin thiiy: thuat toiin DE cd tieu ehuiin dirng bai toiin mach lac, ro rang lam thdi gian tinh toiin ngiin (ngiin nhiit so vdi thuat toiin SA va GA). Ben canh do thuiit toan DE ludn eho ket qua dimg, va on dinh. Vi vay de giai bai toan (1) chiing tdi lira chgn thuiit toiin DE.

3. Tiiuat toan tien hoa vi phiin (DE) va iing dung trong tdi uu ket cau

3.1 Thuiit todn tien hod vi phdn (DE)

Tren eo' sd y tudng ciia thuiit toiin di truyen GA, vao niim 1995, Rainer Storn vii Kenneth Price da hoan thien co' che dot bien vii lai ghep de tao ra mgt thuiit toiin mdi tin ciiy, hieu quii ban. Diem khiic biet Idn nhiit ciia DE so vdi GA ludn duy tri va bd sung mdt cap quiin the, mdi quiin the bao gdm (njsiopsize) ea the tao nen tir /77 chieu cae tham sd thirc, thuat toiin tren da irng dung thanh cdng eho nhieu bai toiin tdi uu d eiie ITnh vuc khac nhau.

So dd thuat toan DE duge trinh bay tren hinh 4. Cimg nhu cac thuat toiin thude phuong phiip Monte-Carlo khiic, thuat toan tien boa vi phiin cung khdi lao quan the ban diiu [X] theo qui luat ngiiu nhien phiin bd deu trong mien xac djnh ciia tirng bien (khdi 2). Mdi phiin tir trong quiin the ban diiu nay eung duge DE thuc hien tren mien tham sd thuc vdi cdng thirc sau:

x„=ran6/(0,l)(Z?[/,j-M,^) + i?Lij (3)

trong dd: .v,,- Gia tri eiia phiin tir ij vdi: i- Sd eii the xem xet eiia biii toan; j - Sd bien ciia bai toan toi uu; BU,,, BL,,- Gidi ban tren va gidi ban dudi eiia bien .Y,,; rand(0,l)- Sd ngiiu nhien phan bd deu trong khoang [0,1].

Xi\.,. , X r ^ivin non \ V bid todn J

n\ '^ Khdi tao q/thd ban d&u

Ci\

Dot bien

Ci\

Lui ghep

i?)

Chgn Igc

(Rh

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Tni sinh

jyyK/tni^^^^^sai

dung 1 IsfK/trn"^^~4uii£

\^^s<fepsj^^,..S>~^

r^

^•1.= ^+'

• In ket qun

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•

''

Hinh 4. So dd thuiit toan DE

238 Nguyin Qudn Thdng; Nguyin Thi Minh; Bid Diec Ndng

Ngay sau qua trinh tao quan the ban dau, khac vdi GA, thuat toan DE ngay tir dau thuc hien tiln trinh dot biin (khdi 3). Trong tien trinh nay, DE tiep tuc tao ra mgt quan the dugc dot biin [V] dua tren quSn thi ban dau. Ky thuat dot bien trong thuat toan DE la sir ket hgp giua he so ty le eho trirde va cac qua trinh lira chgn ngau nhien.

Phuong trinh (4) bilu diin gia trj phan tii' dot bien Vy tir viec td hgp ba phan tir khac nhau dugc chgn ngau nhien trong quan the ban dau [X].

•^r„.j+Fi^,yj-^r2,j) (4)

trong dd: rn, ri, r?- khac nhau va la eae gia trj chgn ngau nhien phan bd deu trong khoang [1, Np]\ F- Hang sd ty le, F G (0,1) la mdt sd thuc duong dieu khien mirc do tien hda eiia quan the.

Trong tiln trinh lai ghep (khdi 4), DE eung tien hanh lai ghep theo kieu cap ddi (dual crossover) tao ra mdt quan the lai ghep [U] ed gia trj cac tham sd dugc lira chgn ngau nhien tir cae quan the [X] va [V] ban dau. Ky thuat lai ghep sir dung trong lap trinh eiia DE duge bieu dien nhu sau:

fv ; if rand(0,l)<C or j = jrand

" . = (5)

in otherwise

trong do: Cr Xac suat lai ghep. C,. e (0,1) duge xac dinh bdi ngudi sir dung nham dieu khien mdt phan cac tham sd dugc sao chep tir quan the dot bien. Them vao dd gia trj eiia phiin tir lai ghep u,i vdi ehi sd chgn ngau uh'xcn jrand dugc lay tir quan the dot bien [V] se dam bao chae chan phan tir lai ghep khdng triing vdi phan tir ban dau x,,.

Trong tien trinh chgn Igc va tai sinh (khdi 4,5), cac ca the trong quan the lai ghep [U] dirge so sanh vdi eae ca the trong quan the ban dau [X] theo hudng ca thi nao cd gia trj ham mue tieu thap hon se duge lira chgn viio quiin the mdi [Y]. Ky thuat lira chgn eiia DE cd thi bilu dien nhu sau:

y,

\u,-,iff{u,,)<f{x,,)

otherwise (6)

Cudi ciing, qua trinh tai sinh ket thiic b5ng phep gan [X]=[Y] eho thi he sau.

Cac khdi 6,7,8 bieu dien dieu kien kiem tra dirng va xuat kit qua ciia thuat toan. Cac gia trj ve sd the he tien hoa (S,/,) hoac mgt gia trj vd cimg be (EPS) dugc dua ra so sanh vdi sai lech giua F(x)„„„ va gia trj trung binh ciia ham muc tieu d thi he dang xet. Bilu thiic dieu kien dimg eua thuat toan DE theo sai lech gia trj hiim cd thi vilt nhu sau:

^(^).,„ -

JlE(x),

/=!

N„ <s- (7)

trong dd: F(x),„,„- Gia trj nhd nhat eiia bam mue tieu tai thi he xet; F(x)r Gia trj ham mue tieu eiia ea the thir i; A'',,- Tdng sd ca the trong quan thi dang xet; e- Gia trj vd eiing be eho trudc (thudng chgn c = 10''^10"'' tiiy theo loai bai toan).

3.2. Ung dung thuc'it todn tiin hod vi phdn (DE) cho hid todn tdi uu ket cdu:

De giai bai toan tdi uu ket cau (1) theo thuat toan DE, chiing tdi de nghj so do thuat toan nhu tren hinh 5 tao ra su ket ndi giua bai toan tinh ket cau vdi bai toan tdi uu. Theo do viec xay

Sie dung thuat todn tiin hod vi phdn tdi ini cdc kich thude ddm dgng hop chiu uon 239

dung so do tinh eiia bai toiin ket eau bao gdm: cae biin thiit kl, eae dieu kien lien kit, dieu kien tai trgng, cac dieu kien gidi ban...duge xac djnh trong khoi 1. Trong khoi 2, eiie thanh phfin eiia bai toan ket cau duge ma hoii biing eae ky hieu toan hge va xac djnh tirng biin trong thuat toan tdi uu. Tuong tir nhu trinh bay d mue 3.1, tai khoi 3, mdt quan the {npop_size) ban dau eae phuang an ket eSu duge xiiy dung ngau nhien. Mdi ca thi trong qukn thi nay tuong irng vdi mdt tap bien thiet ke Xj khac nhau, ddng thdi eung xac djnh duge khoi lugng FfxJ ciia mdi ea the (phuong an).

V bill loiin J

r < ^

^{Khai tao}

\q/ thi bun diiu

^Kiem tni cac d/kien cahgc:

US^iUS]

CV< [CV]

US.,K[USJ

Hinh 5. So do thuiit toan ghep ndi thuat toan DE vdi bai toan tinh ket eau.

Qua trinh dot bien, lai ghep, chgn Igc, tiii sinh va kiem tra tieu chuan dirng tuong tir nhu da trinh bay. Dieu khac biet duy nhat khi giai bai toan tdi uu ket cau la mdi khi ed su thay ddi eiia eae bien thiet ke (khdi 2, 3, 4), tirng phuong an phai duge kiem tra ciic dieu kien co' bgc. Chi cd cae ca the (phuong an thiet ke) nao thda man cac dieu kien ban che ve mat co hge mdi dugc xap xep vao quan the dang xem xet. Hay ndi mdt each khac ea the dd dii dieu kien tdn tai.

De giai quyet van de nay, chiing ta phai tien hanh phan tich ket cau. Do dac thii eiia bai toan tdi uu la khdi lugng tinh toan danh cho qua trinh phiin tich ket eau se rat Idn, dac bjet la doi vdi bai toan ed nhieu bien thiet ke, dan den thai gian tinh se tang len ddng thdi tai nguyen eiia may tinh cung anh hudng.

Vi vay vdi-bai toan dang dam phiing, phuong phap phan tich ket cau dugc lira chgn d day la phuong phap giai tich vdi viec tinh irng suat theo cdng thirc sau [3,9]:

- Ung suat phap:

A^. PL F J..

PF

^+-^y + - A-\

⁽⁸⁾

trong do: a- Ung suat "phap |a,i diem tinh toan; Nr Lire dgc true tren mat eat ngang ciin tinh irng suat; A/v, A/,,- Md men udn tren mat ciit ngang tinh toan; J^, J,- Md men quan tinh eiia mat eat

240 Nguyen Qudn Thdng; A^guven The Afinh; Biti Diec Ndng

ngang doi vdi true quan tinh chinh trung tiim .v va v. F- Dien tich mat eat ngang: x.y- Toa do diem tinh irng suiit.

U'ng suat tiep:

a.^

J,.h

T^M-

⁽⁹⁾

trong dd: Or Lire ciit tai tiet dien dang khiio sat; S'r Md men tTnh ciia phan mat eat tir mirc ngang \'di diem tiiih irng suiit tiep tdi mep ngoai ciia mat eat ddi vdi dudng trung boa; .7,- Mo men quan tinh ciia miit ciit ngang ddi vdi duiTiig trung hoa; F- Be rdng ciia mat ciit lai diem tinh irnsi suat.

Kiem tra ben tai \'\ tri co md men udn va lire eat tuong ddi Idn:

f7,./=V^i'+3r,' < 1,15[a];

⁽¹⁰⁾

trong dd: oj- Ung suat phiip tai vj tri kiem tra; T/- Ung suiit tiep tai vj tri kiem tra. {oi; TI dirge xac djnh trong tirng trudng hgp cu the)

Viec tinh ehu)'en \ i dugc xac dinh being phirong phap nang lugng theo cdng thire:

A.„<[Cr];

Vdi: A,

-m

,/=! 0 FjF_i ^{' • ~ : - \ '}

ill ^'•cMiMi

,/ = ! 0 '^ l"^ I

^t^Zj- ^'QiQi

, / = I O '^.1^1

dr.

(II) (12) trong do: [CV]- Chu)'en vi cho phep ciia ket cau [3,9]; J^„,- Chuyen \j tai diem tinh toan;

Nk.Mk.Qi. Cac thanh phiin luc dgc, md mne udn va lire cat do lire don \'i dat tai diem va theo phuong ciin tinh chii) en \i gii)' nen tren doan thiry; A''J,, A/„',. O,', - Cac tbiiiib phiin lire dgc, mo men udn \ii lire ciit do tai trgng giiy nen tren doan thanh tbiry; £,, G,- Md dun dan hdi \a dim hdi trugt eiia

\iit lieu trong doan thir /; F- Dien tich mat ciit ngang tai vj tri xiie djnh ndi lue thude doan thir/; .7,- Md men quan tinh chinh trung tiim ciia tiet dien tinh toim; f- Chieu diii doan thir /.

Viec tinh toiin dn djnh eiia dam duoc xiie dinh theo cac cona. thirc sau:

Kiem tra dn djnh tdng the:

9',7^^•.

<

[-];

⁽¹³⁾

trong dd: A/„- Md men udn tai vi tri tinh toan; fl',- Md men khang udn ciia tiet dien ngu)'en eho thd' bien eiia canh chiu nen; (p,r He sd giiim kha nang chju udn eiia diim; faj- Ung suiit eho phep - Kiem tra dn dinh cue bg thanh dirna va biin bimc diim:

a a

— + — -

a

+

/ ^ r

,'' J

y.. 14)

\'.r )

trong dd: rr- Ung suiit phap; r- Ung suat tiep; or Ung suat phap cue bd; o,r Ling suat phap tdi han;Tj,-- Ung suat tiep tdi ban; cr,„,- U'ng suiit phiip cue bd tdi ban; }',- He sd dieu kien lam viec eiia ket eiiu.

Sie dung tlnidt todn tiin hod vi phdn tdi ini cdc kich thude ddm dgng hop chiu uon 241

Tren eo sd ly thuyet da trinh bay chiing tdi da lap trinh tinh toan cho mdt so so do dam hop chju udn. Dudi diiy se trinh bay mgt trong sd eae ket quii tinh do.

4. Vi du tinh toan 4.1 Phdt bieu bdi todn

Tim phuong an tdi uu theo trgng lugng cho dam hop thep gom 6 doan thSng ed kich thude hinh hge va dieu kien chju tai nhu hinh 6. Dieu kien ban chi vl irng suat [a]=0,18KN/inm' Chuyen vj cho phep d niit dat lire P la [CV]= 6mm. Be rdng eiia toan dam dugc coi la khdng doi (B_= 80inm). Cac gia trj hinh hge cdn lai ciia tiet dien ngang mdi doan thing dirge coi nhu mdt bien thiet ke.

P= lOKN 1500

A .3L Q-'l ®

1500

@ .SL M.

i

Hinh 6. So dd tinh eho dam cho dam chju udn vdi tai trgng tac dung tai dau miit thira.

4.2 Gidi quyet bdi todn:

Biii toan dat ra gdm 18 bien thiet ke, dieu kien rang budc ddi vdi tirng bien:

4 < a, < 12; 4 < b, < 12; 50 < H, < 250;

Dieu kien riing budc ve CO hge: o < [o]; Cv < [CV]; Khdng mat dn djnh eu bd theo cdng thirc (14).

Sir dung phan mem tdi uu ket eau vdi sd lan tien hda la 8953 the be khi dieu kien dirng eiia bai toan la e=10' , mdi the he xem .xet 30 ca the ta cd ket qua duge dua ra trong bang 2 nhu sau.

Biing 2. Ket qua tinh toan sau tdi uu (mm - KNj

^ ^ ^ ^

Doan I Doan 2 Doan 3 Doan 4 Doan 5 Doan 6

a, 4 4 4 4 4 4

b, 4 4 4.5 4.5 4 4

H, 135 235 250 250 235 135

M„„, 5.0e+03

l.Oe+04 1.5e+04 1.5e+04 1 .Oe+04 5.0e+03

W 75145 143642 153054 153054 175382 64193

o 0,067' 0,052 0,082 0,082 0,043 0,077 Trgng lugng toan bd ciia dSm: 0,544 KN; Chuyen vi tai vj tri dat luc P: 5,9inm

Chiing tdi dua kit qua tren vao chuong trinh tinh ket cau SAP2000 de kiem tra. Ket qua, chuyen vj tai vi tri dat lire P la: 6,21 mm.

Sai sd giiia gia trj chuyen vi tinh toan theo SAP2000 va chuyen vi cho phep la:

6,21-[CV]

[CV]

!• 100% =^3,5%

242 Nguyin Qudn Thdng; Nguyin The Minh; Bid Diec Ndng

Nhu vay ket qua tinh toan tdi uu dam thep dang hop vdi gia trj cae bien thiet ke nhu trong bang 2 la chap nhan duge.

5. Ket luan

Cae ket qua trinh bay trong bai bao nay mdi chi la nhirng nghien cuu biidc dau vdi vi du tinh toan cdn don gian. Tuy vay, vdi nhung kinh nghiem tich hiy duge thdng qua qua trinh lap trinh tren nhieu ngdn ngir (MATLAB, FORTRAN), tinh thir nhieu biii toan tdi uu dien hinh vdi so lugng bien kha Idn (24 bien), chiing tdi budc dau eung ed mdt sd nhan xet nhu sau:

- Viec tinh toan tdi uu ket cau tren co' sd ket hgp thuat toan tien boa vi phan vdi bai toan thiet ke ket eau may xay dung giiip ich riit nhieu eho ngudi linn cdng tae ky thuat. Cac bien thiet ke cd the dugc gan la eae bien rdi rac hoac lien tiie de thoa man yeu ciiu ciia ket eau thuc.

- Thuat toan DE eung ed eo' sd la quii trinh ngau nhien gidng nhieu thuat toan tdi uu toan cue khac (EA,GA, SA...) nhirng nd thuc su da ed diem manh rat rieng. Cu the: Tieu chuan dirng thuat toan da dugc cai thien dang ke khi ket hgp duge eae tieu ehuiin dirng eiia cac phuong phap truyen thdng va hien dai. Tir do, khi ap dung thuat toan DE dl tinh toan kit eau, ket qua cua bai toan thudng rat dn djnh va ed do tin cay cao.

Thdi gian tinh toan tren may khi sii' diing thuat toiin DE vao thuc hien bai toan toi uu kit cau cung thudng nhanh nhat so vdi cac thuiit toan ng§u nhien khac (SA.GA). Mat khac, qua trinh kiem tra ket cau lai cd tinh dde lap tuong ddi vdi qua trinh tiln boa. Vi vay hoan toan ed kha nang giai duge cac bai toan ket eau phirc tap vdi dieu kien ban ehi chat che hon. Ngoai ra, ket qua tinh toan tdi uu ket eau se la eo sd de nit ngSn qua trinh thuc nghiem, tao hieu qua kinh te cao do cd the giam thieu viec tien hanh thi nghiem, thuc nghiem.

Tai lieu tham khiio

[I] GS.TSKH. Vo Nhu Cau (2003). Tinh ket cdu theo phuang phdp toi ini. Nha xuat ban xay duno.. Ha Noi.

[2] GS.TS. Le Xuan Huynh (2006). Tinh todn ket cdu theo ly thuyet tdi icii. Nha xuat ban Khoa hoc ky thuat, HaNoi.

[3] Nguyin TiSn Thu (2007). Kit cdu thep. Nha xuc4t ban xay dung, Ha Noi.

[4] Kenneth Price, Rainer Storn, Jouni Lampinen (2005). Differiential Evolution - A Practical Approach to Global Optimization. Springer. Verlag.

[5] Menmet Yener (2005). Design of a computer interface for automatic finite element analvsis of an excavator boom. Middle east technical university. Saudi Arabia.

[6] O-Kaung Lim, Eun-Ho Choi (2005)."An approximate optimization method for large scaled design problem by distributed process based on the internet".6th word congresses of structural and multidisciplinary optimization. Brazil

[7] S.Kirkpatrick, CD. Gelatt, M,P. Vecchi (1983)."Optimization by Simulated Annealing" Volum 220 SCIENCE

|8| Zbigniew Michalewicz (1994). Genetic Algorithms + Data Struclures = Evolution Programs.

Springer. Verlag.

|9| MeTajuiHMecKne KOHCTpyKunn cxponTejibHbix H flopo>icHbix Maiunn (1972). HiiiaxejibCTBO MamiiHOCTpoehHe. MocKBa.