622Về biến thiên của vết mùi trong phương pháp ACO và thuật toán mới

T9.p chi Tin hQC vk Dieu khien hpc, T.27, S.3 (2011), 263 - 274 VE BIEN THIEN CUA VET MUI TRONG P H I / O N G PHAP AGO VA CAC THUAT TOAN Mdfl* DO DIJfC D N G \ HOANG XUAN HUAN2 ^ Vi$n Cong nghe thdng tin, Dg.i hoc Quoc gia Hd Noi '^Dg.i hoc Cdng nghe Dg,i hoc Quoc gia Hd Ndi Tdm tat Trong cac thudt todn tdi Uu hda ddn kien (Ant Colony Optimization-ACQ), vgt miii bieu thi th6ng tin hgc tang cudng dl tim kiem ldi gidi cho cdc bdi todn toi Uu to hgp NP kho Bdi bao trinh bay mgt so phan tfch todn hgc ve tfnh bien thien ciia v^t miii hai thup todn ACO thdng dung nhat: ACS (Ant Colony System) vd MMAS (Max-Min Ant System), cdc phan tich la cd sd cho de xuat cdc quy tdc cdp nhdt miii mdi Abstract In Ant colony optimization (ACO) algorithms, pheromone trails express reinforcement leaming information to find solutions for NP-hard problems in combinatorial optimization This paper presents mathematical analysis on behavior of pheromone trail in two most popular ACO algorithms: ACS (Ant Colony System) and MMAS (Max-Min Ant System), and suggests new ideas for improving pheromone update rules G l l THIEU Toi um hda dan kien (Ant colony optimization-ACO) Dorigo de xudt [6] da thu hiit nhieu ngudi quan tdm nghien ciiu vd iing dung de giai cdc bai todn toi Uu td hop khd [1, 2, 7, 11, 15, 16] Tren dudng di, mdi kidn thuc di lai mdt vgt hod chat ggi Id vet miii (pheromone trail) va theo vdt miii cua cdc kien khdc de tim dudng di Dudng cd nong vet miii cdng cao thi cang cd nhidu kha nang dugc cdc kign chgn Nhd cdch giao tigp gidn tiip ndy [7] dan kiln tim dugc dudng di ngdn nhat Theo y tudng ndy, cdc thudt todn ACO sut dung kgt hgp thdng tin kinh nghiem (heuristic) vd hgc tang cudng qua cdc vet miii cua cdc kign nhdn tao de giai cdc bai todn toi Uu td hgp bdng each dUa vg bai todn tim dUdng di toi Uu tren thi cau triic tUdng iing cua bdi todn Thuat todn ACO dau tign [6] la he kign (AS) gidi bdi todn ngudi chao hang (TSP), den da cd nhieu bien the dugc dg xudt, thdng dung nhat Id he dan kiln (ACS) [7, 8] va he kiln Max-Min (MMAS) [18] Chiing dugc dp dung rdng rdi [1, 7, 11, 16, 18] dl giai nhieu bdi toan toi uu to hgp vd hieu qua noi trgi cua chiing da dugc chiing td bang thuc nghiem Ngodi cac ling dung, cd nhiing nghien ciiu ve dac tfnh ciia thudt todn [5, 9, 11, 12, 13, 15, 17] nhu anh hudng cua vlt miii, thdi gian chay vd tfnh hgi tu, ••• Trong [17,18] Stiitzle vd Dorigo dd chiing minh tinh hgi tu tdi ldi gidi tdi Uu cua cac thudt todn MMAS nhung chua tinh tdi anh hudng cua tham so heuristic, khuynh hudng hgi tu cua *Nghien cfiu dtfdc hoan dvcdi sU h6 trd tit Quy phat triln KHCNQG NAFOSTED 264 D6 ui'rc DON(;, HOANG XUAN HUAN vet miii dugc xet gia thiet da tim dugc ldi giai toi Uu Cdn Gutjahr [10,11] thi dua ve xet bien ihien eiia mut qua trinh Markov de nit tfnh hOi tu tdi ldi giai toi Uu vd sU hgi tu cua cudng dg vet miii eho eae bien the eiia, thuat todn MMAS md chUa khao sdt cho ACS Tii>' nhien eae bai t,()a,n toi Uu to hgp thi sd phiWng an Id hiiu hg-n ngn kit qua ve \'iee xae suat tim tiiay ldi gidi hgi tu ve so lan ldp dan vo hijin la tam thUdng Trong bai bdo se phan ti'eh chi tiet, hgn ve eae dac tfnh bien thien cua vet miii cac thuat toan ACO, tren ed sd dd de xuat eae quy tde cdp nhdt miii mdi Ket qua thuc nghiem cho thay Uu diem eiia eae de xuat, ndy Bdi bao gdm: Mue gidi thigu phugng phdp toi Uu dan kien va hai thudt todn thdng dung: ACS vd MMAS Miie trinh bay cdc ket qua, khao sat xu the thay doi ciia vet mui cac thudt toan ACO Cae quy tile edp nhdt miii mdi dugc gidi thieu Muc Muc gidi thi?u ket qua tln.fc nghigm doi vdi bai toan TSP de minh hga cho cae de xuat mdi Vd sau ciing la phan ket luan PHUdNG PHAP T I UU H A DAN KIEN (ACO) Trong cae thudt todn ACO, cac h? ACS vd MMAS la hai thuat todn thdng dung nhdt va se dugc nghien cilu bdi Vi vdy, trudc gidi thigu thudt todn ACO tong qudt, ta tdm tdt bai todn TSP vd hai thudt todn ACS, MMAS [7,8,18] 2.1 ACS va M M A S giai bai toan T S P 2.1.1 Bdi todn TSP Bdi todn TSP dugc phdt bieu nhu sau Cho n thdnh pho, ngUdi chdo hang can tim mgt chu trinh cd dudng di ngan nhat qua mdi diing mdt lan De dOn gian ta xet bai todn trgn dd thi vd hudng G = {V, E) dd tap dinh V (n dinh) ky hieu tap cdc vd E Id tap cdc canh, {i,j) bilu thi dUdng di noi cdc dinh i,j E V va cd dg dai dij {dij la khoang each tii thdnh i tdi thdnh j) De giai bdi todn ndy, phuong phdp ACO diing bien vlt mui TIJ kCt hgp \di mdi canh {i,j) va ban dau dugc khdi tao bdi gid tri TQ CO m kiln nhdn t^o, d budc ldp t chiing thuc hien cdc thu tuc xdy dUng ldi gidi ngdu nhiin va cdp nhdt mui theo tumg thudt todn Sau mgt s6 budc ldp cho trudc, ldi giai tdt nhat tim dugc Id ldi giai ciia bdi todn 2.1-2 Hi ACS (Ant Colony System) Xdy dung ldi gidi Ban dau mdi kien dugc ddt ngau nhien d mgt dinh xuat phdt vd thdm cdc dinh khdc de xdy dung dudng di vdi thu tuc budc ngdu nhien theo quy tdc chuyin trang thai sau Quy tdc chuyin trQ,ng thdi Gia sii kiln fc dang d dinh i, nd chgn dinh s tilp theo nhd quy tdc: ^ ^ {argmax^^j^^^i;){Ti^u{t)r]l^} : q < go, \ : g > 90, j ,^ dd q la gia tri chgn ngdu nhign thuge khoang {0,1), qo E (0,1) Id tham s6 cho trUdc, r]ij = ^ , cdn j la dinh dUdc chgn theo xac suat cho bdi cdng thutc vdi a = VE BIEN T H I E N CUA VET MIJI TRONG PHUONG P H A P ACO VA CAC T H U A T T O A N M I g{s') nlu f{s) < f{s') (trong bdi todn TSP g{s) la nghich dao dai dudng di tUdng umg), dd d moi budc ldp cudng vlt miii se thay ddi theo mgt cdc quy tdc sau ddy Quy tdc ACS: Quy tdc phdng theo ACS, bao gdm cd cap nhat dia phUdng va todn cue + Cap nhdt nidi dia phuOng Nlu kiln h thdm canh (i, j ) , tiic la {i,j) E s{h) thi canh se thay ddi miii theo cdng thiic TiJ ^ (^ - p)ri,j + pn (8) -I- Cop nhdt mdi todn cue Cap nhdt miii toan cue chi cho cdc canh thudc w{t): Ti,3 ^ (1 - P)Ti,j + pg{w{t)), y{i,j) E w{t) (9) Quy tdc MMAS Quy tac thuc hien theo MMAS Sau moi kiln deu xdy dung xong ldi giai d moi budc lap, vlt mui dugc thay doi theo cdng thiic ^ij ^ (1 - P)n,j + A r i j , (10) dd Ar = /^^(^W) ''' \max{n-{l-p)ri,j,0} iiJ)ew{t) {i,j)^w{t), ^ ' d day ri > la tham so Chuy 1) So vdi MMAS muc 2.4.1, d cdng thiic (11) ta xem TI = Tmin vd gia thilt vdi mgi t deu cd Tmax > 9{'^{t)) (gia thilt ndy chi dl ddn gian cho trinh bay) 2).Cdc quy tac cap nhdt miii trgn Id quy tdc G-best, nlu cdc cdng thiic (9) va (11) thay w{t) bdi w^{t) thi ta ndi Id quy t i c i-best, muc sau ta chi xet cho quy tac G-best P H A N TICH T O A N H O C VE XU THE VET MUI ACO Phan se nghign ciiu tinh hdi tu cua cdc thudt todn ACS vd MMAS, sau udc lugng xdc suat tim thay mgt phUdng dn d bUdc lap t, ta se khao sU thay ddi cua vlt miii 3.1 Udc Ixidng x a c s u a t t i m t h a y m o t phifdng a n M e n h d e Cdc khang dinh sau diing a) Bdi todn tSng qudt ludn cd ldi gidi tdi Uu b) Vdi mdi kit qud thuc nghi$m, cdc gid tri f{w{t)) ludn hdi tu t ddn vd han 268 D6 DLfc DONG, HOANG XUAN HUAN c) Ky hi$u tdp ldi gidi cua bdi todn Id 5* vd gid tri toi Uu la g * {g* = / ( s ) : s E S*) thi vdi moi CQ,nh (/', j ) E E ta co ddnh gid sau < Tmin = min{ro,ri,.g(u;(l))} < r y < max{ro,ri,5(ti;(l))} = Tmax(12) Chitng minh Khdng djnh a) Id hien nhien vl tdp S (vd X* ) Id tdp hihi h^n Khdng dinh b) suy tir tfnh ddn digu gidm ciia day f{w{t)) vd day ndy h\ chdn bdi g* Khing dinh c) dl ddng nhan dugc nhd chiing minh quy n9,p theo t vdi lUu f ring d moi lin cdp nhdt miii, cudng vet miii cua cac cg-nh (A, j ) theo quy t i c ACS cd dg,ng: n^j Pmin > 0, (13) dd, Pmin xdc dinh bdi cdng thiic P„i„ = l - e a : p ( - ^ ^ ^ ) (14) Chiing minh Gid sii s dUdc xdc dinh bdi X{E X*) =< uo, ., Uk> {k < h) dUdc md rgng tut uo- ydi mdi bUdc ldp i va Vi < /i, tut (9) vd (14) ta cd danh gid sau doi vdi xdc suit p{xi+i/xi) di Xi+i = < tto, •••, Ui, Ui+i > dUdc md rOng tut Xj = < no, •••, Ui > bdi mdi kiln r la '"min^WiWt+i ^ '''•^''"Sin p(x,+i/x,) > ^ •'"'H-^^.H-i > 2^ueJr{ui) ^maxil-UiUi+i "''max •::^^ (15) Do dd, nlu ggi p{r, s) la xdc suit dl kiln r tim dugc s ta cd i—l 't ' mnv •• 'max '* 'mnnv '• 'max Vay ta cd Udc lugng sau doi vdi xdc suat lan ldp ca m kiln khdng tim thiy Suy ra, ps{t) > - exp (^ ri>^%^ j = Pmin, ni^nh de dUdc chiing minh C h u y Nlu diing quy tic chuyin trang thdi cho bdi (7) thi dg dang nhdn dUdc P„,„ = ( l - * ) [ l - e x p ( - ! ^ ^ ) ' Kf hieu P{t) la xdc suit tim dUdc ldi giai t budc ldp (hay w{t) E S*) Dinh ly sau Id md rOng cua Dinh ly 4.1 [7], dam bdo tfnh hgi tu cho cdc thuat toan VE BIEN THifiN CUA VET MUI TRONG PHUONG Dinh ly 3.2 Vdi moie>Q PHAP ACO VA CAC THUAT TOAN MCI bi y, ton tQ.i T cho vdi moit>T 269 ta diu co: P{t) > -e Chiing minh Theo Menh dl 3.1, ludn ton tai ldi gidi toi m s* E S* TH Dinh ly ta cd Udc lugng ' ' P W > l - n [ l - P - W ] = l-(l-Pmi„)* = l - e x p f - ^ ! ^ '=1 V (16) "• "^max Bilu thiic (16) cho ta kit ludn ciia dinh ly chgn T du ldn 3.2 • Dac tinh cua v l t mui Ta thiy ring thuc tl, d cdc budc ldp t dii ldn thi khd nang g{w{t)) > g{w{t + 1)) (vd do w{t + 1) ^ w{t)) rit be ngn cd thi tut budc ldp to cd cdc canh {i,j) khdng bao gid thudc vdo w{t).\/t > to hodc ludn thuOc vao nd Ta se khao sat ddc diem ciia nj cdc trudng hgp ndy Dinh ly 3.3 Gid svC canh {i,j) thuQc vdo ldi gidi chdp nhdn duoc s ndo dd vd tdn tai T cho {i,i)Ew{t),\/t>T thi cdc khang dinh sau dung a) Tij{t) hdi tu theo xdc sudt tdi n niu dUng quy tdc cap nhat mdi ACS^) n,j{t) = n vdi moit>T+ j ^ ^ ^ niu ddng quy tdc cap nhdt mui MMAS Chiing minh a) Ve > ta can chiing minh lim^P{\Tij{t)-n\>e) = 0) Cdng thiic cap nhdt miii dia phUdng (8) cd t h i vilt nhu sau: nj fcO ta cd: JTij - n j = 1(1 - p)''[Tij - ri]| < e Ta se chiing minh lim P(fc < fco) = t—*oo Thuc vay, neu ddt po = ^'n* thi ta cd ddnh gid sau ddi vdi xdc sudt P{r; {i,j)) kiln ** ' m a x r cd cap nhdt dia phUdng canh {i,j) d mdi budc ldp: P{r; {i,j)) > po > Ngn xdc suit P{r; -i{i,j)) kiln r khdng cap nhat dia phUdng canh {i,j) d mdi budc lap thoa man P{r;-r{i,j)) T thi cdc khdng djnh sau dung a) Niu cdp nhdt miii theo A CS thi //^ I 9{w{T))-n ;imr,,(/,)>ri + ^ ^ _ ( ^ _ ^ ) ^ ^ , ,Q^ (19) b) Niu rdp nhdt miii theo MMAS thi lim r,:,,(i) > g{w{T)) (20) Chttng minh a) Ky higu fc« Id so lin cdp nhdt miii dja phUdng ciia c^nh {i,j) bUdc \^pu,khid6tac6:Ti^j{u+l) = {l-p)''-+'rij{u) + Y:tVi^-P)P''i + (^-P)Pi9iw('^))-n] (cd mgt lan cdp nhdt todn cue m5i lan ldp) Do 9{w{t)) Id hdm ddn di^u tdng theo t ngn ta suy Ti,j{t) > (1 - Prri,j{T) + f ; ( l - p ) > r i + Y,{1 - p)'^p[g{w{T)) - n ] , v=l v=l dd gt = E U T + I ( ^ « + 1) ^^ ^i = ^^ ^^+i = hv + fct-v+2 + 1, vdi mgi Bdi vi ku (1 - p)'^'rij{T) + f ^ ( l - p) V l + E v=l (1 - Pr^"'^'^P[9{w{T)) v>2 - nj (21) v=0 Liy gidi ban dudi t d i n vd han bilu thiic (20), luu y ring lim^—oo gt = oo ta cd bilu thiic (19) Khing dinh b) chiing minh tUdng tU nhd liy gidi h§in bilu thiic t-T-l n,{t) > (1 - pY-''Ti,j{T) + 5;] (1 - p)''P9{w{T)) (22) C h u y Cdc cdng thiic (21) vd (22) trongchiing minh djnh ly cho ta udc lUdng can dudi ciia g(w(t)) 4.1 THAO LUAN N h a n xet chung v§ cac thuat toan A C O Nhd kit hgp thdng tin heuristic vd thdng tin hgc tdng cudng nhd md phong hoat dgng cua ddn kiln, cdc thudt todn ACO cd cdc utu dilm ndi trOi sau: 1) Viec tim kilm ngiu nhien dua trgn cdc thdng tin heuristic ldm cho phep tim kiem hnh hoat va mem deo trgn khdng gian rOng hdn phUdng phdp heuristic sin cd, dd cho ta ldi giai tot hdn va cd t h i tim dUdc ldi giai toi Uu 2) Su kit hgp hgc tdng cudng thdng qua thdng tin ve cudng vlt miii cho phep ta tutng budc thu hep khdng gian tim kiem md vdn khdng loai bd cdc ldi giai tot, dd nang cao chit lugng thudt toan Bdy gid ta cd cdc nhdn xet chung vl dac tfnh khai thac va kham pha cua cdc thudt toan vfe BifeN THIfiN CXJA VET MIJI TRONG PHUONG 4.2 PHAP ACO vA CAC THUAT TOAN M I 271 T i n h khai t h a c va k h a m p h a Tinh khai thac la viec tap trung tim kiem ldi gidi quanh pham vi ciia cac canh {i.j) thudc cac ldi giai tot nhdt da bilt tdi thdi dilm dang xet cdn tuih khdm phd Id tim kiem cdc pham vi khac Ti-ong each cap nhat miii G-best ta da biet u'{t) uen viec tim kilm quanh nd se han che nhieu tinh khdm pha cap nhdt theo i-best se md rdng mien ndy hon Vi vdy thitc hanh cap nhat theo i-best tot hdn G-bcst Trong cac bai toan toi Uu td hgp thifdng thi xac suit dl mOt phUdng dn cho trUdc ditgc cac kiln tim dugc mdi ph6p ldp r^t be Vi vay cd thi sau mdt s6 bUdc ldp cUdng dd vet miii tren mdi phan ciia nd xdc dinh theo cac cdng thiic (8 11) se b6 va gidm kha ndiig kham pha dUdc chiiiig mdc dii chiing cd thi vdn rat hiia hen thudc ldi gidi tdt Ch^ng han vdi bai toan TSP ta cd menh dl sau Menh dd Trong bdi todn TSP khdng dinh hUdng, m.Si chu trinh Haminton (dUdng liin) qua canh {i.j) vd khong qua canh (fc h) co the ddi nhiiu nhdt canh di co dUOc chu trinh di qua canh (fc h) md khdng qua {i.j) Chiing minh Ket ludn ciia mgnh dl dl ddng nhdn dUdc tit hinh dUdi day Trong dd chu trinh gom cdc canh (dUdng liln) liln qua {i.j) vd nhd ddi di canh tUdng iing {(m.7).(n.fc).(7.j).(fc.o).(p./i)} thdnh {{m k).{n.i).{i.o).{k.h).{p.j)} se dUdc chu trinh di qua canh (fc h) md khdng qua (i-j) o\ (dUdng diit dgan) • ; p Hinh 4.1 Hai chu trinh khac canh dudng liln qua canh {i j) vd dUdng diit doan qua canh (fc h) Cdc diim han chi cua ACO 1) Menh de trgn cho thiy thuat tgan mdi bit dau, cac vlt mui khdi tao nhU thi mgt canh (fc h) "tdt hdn'' canh (?' j ) nd thudc chu trinh ddi hdn cd thi dao ngUdc mgt cdch r^t ngdu nhign Khi mgt canh ngdu nhien ma khdng ditdc cdp nhdt miii sau mgt s6 bUdc thi cudng miii cua nd nhanh chdng bi giam xuong va khd dugc cac kiln chgn sau dd mac du "chat hrdng" ciia nd chUa chdc da la "xau' 2) Nlu khdi tao miii nhu va khdng dimg thdng tin heuristic thi xdc suit ciia mdi canh dugc mdi kiln da cho sii dung lan lap dau la 2/(n - l),^xac suit rat be n ldn Nhu vay tiiy theo timg loai bai tgan ma ty le giita TQ va TI rat cd y nghia dg cdn bang gifra tinh khdm pha va khai thdc cua thudt todn 3) Cac hrgng miii cap nhat theo cdc cdng thiic (5.a), (5.b) hay (8)-(ll) phu thudc vao gia tri ham muc tieu ciia ldi giai ma cac kiln xay dUng dUdc cdc bUdc lap Viec xac dinh cac gia tri TQ , n hay Tmin, ^max ciing phu thuoc vao tUdng quan vdi cdc gid tri chua dUdc xdc dinh tnrdc cua timg bai todn thi thuat toan mdi tdt dUdc Dieu rit khd thUc hien Bay gid ta se binh luan cu t h i hdn ve tirng thuat todn da ngu 272 4.3 D DI?C DONG, HOANG XUAN HUAN C6c thu&t todn c a p nhfit miii t h e o quy t^c A C S Trong cdch cdp nhat imii todn ei.ie ciia ACS theo [8] bdi cdng thiic [4] vd (5.b) khdng dam bdo cudng dg vet miii thda man thuOc khodng [Tmin, Tmax], vet miii cua nhiing canh khdng dugc cdc kiln sii dung vd khdng thuge dudng di tdt se chdng dan ve vd cdc kiln sau se (6 bo qua cdc ctjiih ndy Tuy vdy, nhu da chi d trgn, C9.nh ndy van cd thi Id cgnh tham gia \ en p = 0,05 Ngodi ra, SMMAS ti lg ^ ddt bing N.k vdi fc = I 10° iv _ ou [1 N