xlj + x2j + + xij + + xmj =L xij
1.2Dl}t hili tmin dttlti d~ng ban g:
Bai toan v~n tai la bai toan Q.H.T.T nen c6 th~ giai nhu a
ch11<1ng 2. Nhung khi d6 s6 ~n kha nhi'eu (m x n va m + n ~n
gia) va s6 rang bu9c cung IOn (m + n rang bu9c. Trong d6 m rang bu9c dam bao cho m di~m phat phat h,~t hang va n rang buqc dam bao cho n di~m thu nMn du hang) nhung do tinh chB:t d~c bi$t cua bai toan v~n tai (xem l~ ph~n 1.3 chUdng 1), n~n nguoi ta nghr ra m9t thu~t toan hi$u qua hdn. Trude h~t ta trinh bay bai toan duoi df}.ng bang :
Thu Bl B2 Bj B CttCic n bl b2 ... bj ... b Phat n ell c12 clj cln Al : al ... ... xll xl2 xlj xln A2 c21 c22 c2j c2n : a2 ... ... x21 x22 x2j x2n ... ... ... ... .. . ... .. .
cil ci2 c .. cin
Ai : ai ... lJ ...
xil xi2 xij xin
... ... ... ... ... ... .. .
c ml c m2 cmj c
A mn
m am ... ...
X ml xm2 xmj xmn
Trong bang m6i himg d~c tnrng cho m(jt di~m phat va m6i c(jt d~c tnrng cho m(jt di~m thu. M6i 6 d~c tnrng cho m(jt tuyen duC1ng tU n1(jt di€lm thu den m(jt di€m phat. 6 nilm tren hang i, c(jt j tuc la d~c tnrng cho tuyen duC1ng tu Ai den Bj gQi la 6 (i, j)
D\nh nghia : M(jt day cac 6 cua bang ma 2 (va kh6ng qua
2) 6 lien tiep cua day lu6n nilm tren cling m(jt hang ho~c ding m(jt c(jt gQi la m(jt day chuy'en. M(it day chuy'en khep kin gQi la
m(it vong. x- -x I X- - X I I -X
Thi di! : Trang bang ben, cac 6 c6 danh dau "x" l~p thanh m(it day chuy'en. Day 6 o cua tala :
(1) (2) (3) (4) (5) (6) '
(1, 2) (1, 3) (2, 3) (2, 4) (4, 4) (4,1) 88
Ta thay hai o HEm tiep (tinh theo cac chi s6 danh a tren) bao gio cung nilm tren cung hang ho~C cung m~t CQt.
X- - - X
I
X-- X
Con trong hang ben, ta c6 6 o t~o
thanh m9t vong. x- - --1---x
D\nh nghia : Nhung o ung vei xij > 0 trong m9t phuong an niw d6 duqc
gQi la o chqn. Nhung o con l~i duqc gQi la o lo~i.
6 chQn d~c trung cho tuyen duong ta co v~n tai hang qua. D!nh nghia : M9t phttong an ma cac o chQn khOng t?o thanh vong gQi la phuong an co biin. M9t PACB co du m + n - 1 o chQn gQi la khOng suy bien, neu c6 it hon m + n - 1 o chQn la suy bien (*)
§ 2. TINH CHAT CUA BAI TOAN V~N TAl :
Tinh chat 1 : Bai toan v~n tai can b~ng thu phat luon c6 phuong an toi uu. (**) ;
(*) Khai ni~m phuung an ca ban & chuung 1 chi xet cho b~ti toan d~ng chuiin. (adsbygoogle = window.adsbygoogle || []).push({});
(**)
Tht,rc ra ngu&i ta con d!nh nghia cho bai toan d?ng chinh t!lc : trong ma tr~n
A ch<;>n ra m c9t d9c l~p tuyen tinh (kh6ng can Ia v~cta dan vi). m iin (mg v6i m c{)t cl6 Ia w ban. Cac iin khac Ia kMng w ban. Ngu&i ta ch(mg minh v6i m c{)t cl6 Ia w ban. Cac iin khac Ia kMng w ban. Ngu&i ta ch(mg minh dugc so rang bu{)c d{)c l~p tuyen tinh trong bai toan v~n tai Ia m + n -1 va m + n - 1 c9t (mg v6i m + n - 1 iin x;i trong cac 0 kh6ng t~o thanh vong Ia d9c l?p tuyen tinh. a; bj D~t Xij = ~; Xij ~ 0 va .LJ Si = L a; bj = bj L an = bj ; La; La;
V?y bai toan c6 phuung an.
Tinh cha:'t 2 : Gia sit ta c6 bang m hang, n c¢t va E la m¢t t~p hqp g<>m m + n - 1 6 cua bang kMng chU'a vong. Gia su (i, j) la o cua biing kMng thu¢c E. Ne'u ta b6 sung (i, j) vao E
d~ duqc E1 thi E1 se chua m¢t vong duy nha't la V. Cu6i cung, ne'u
lo~i khOi E1 m¢t o tuy y thu¢c vong V M duqc E2, thi E2 l~i g<>m m + n - 1 0 cua biing khong chua vong.
Thi df!- : Trong bang 4 hang 4 c¢t Mn c6 t~p E g()m m + n - 1 = 4 + 4 -
1 = 7 o khOng chua vong c6 danh da'u "x". (4, 4) Ia o cua biing kMng thu¢c E.
Khi b6 sung (4, 4) vao E se-c6 vong, duy nhl!t duqc danh dau trong biing. Vi
X X X X X-- -X ' I , __ I X• _ ; I I I ' ' I I I I X---(4,4) vong Ia duy nhl!t, nen ta't nhien la ma't di m¢t o cua V thi se ma't vong.
Chu y : Trong dinh nghia phue1ng an cd biin khOng suy bie'n, ta doi hOi s6 o chQn la m + n - 1. Trong truong h<Jp suy bie'n, ta c6
th~ b6 sung mc)t s6 o lo~ sao cho phudng an cd biin c6 m + n - 1 o chQn. Cac o lo~i duqc b6 sung nay gQi la cac "0 chc;m 0".
A , 9 'w\
§ 3. ~p PHUONG AN CO BAN BAN DAU :·
Ta dung phudng phap uu tien phdn ph6i nhi'eu nha't vao o c6 cudc phi nhO nhl!t.
Gia su trong rna tr~n c = (cij)mXn' crs la nho nha't trong cac cif Khi d6, ta phan ph6i t6i da vao o (r,s), clJ. th~ :
ar neu ar - 8
f
A' < b
X =
rs b8 ne'u ar > b8
Trong truong hqp thtt nha't, di~m Ar da phat he't hang nen c6 th~ x6a di hang r ~ua bang; d di~m thu B8 chi con din b8 - ar ta'n hang.
Trong truong hqp thu hai, di~m thu B8 da nMn du hang,
nen co th~ xoa di c<}t s cua hang va a di~m phat Ar chi con l~i
ar - b8 ta"n hang.
Trong hang, con l~i voi so hang va c(lt it hdn, ta lf;l.i tie'p tqc phftn phoi nhu tren cho de'n khi he't hang.
Cac o chQn tim du<Jc se khOng chtta vong va la phudng an
cd biin. Ne'u chua du m + n - 1 o thi ta b6' sung them ml)t s6 o chQn o cho du m + n - 1 o khOng t~o thanh vong.
Thi d71- 1 : ~ Bl B2 B3 20 40 30 10 1 3 5 A1 : 30 X '·· X 20 -' -1~, 5 4 2 A2 : 25 X -. l:x:. 25 8 5 4 A3 : 35 ~--· x· ·• 30 .5 Thtt t1,t phftn nhu sau : (adsbygoogle = window.adsbygoogle || []).push({});
1. PMn vao 0 (1, 1) 20 tA'n, cqt 1 hi x6a va (J A1 con 10 tA'n. 2. Phdn vao o (2, 3) 25 tA'n, hang 2 ht x6a, B3 con dln 5 tan. 3. PMn vao o (1, 2) 10 tA'n, hang 1 b\ x6a, B2 con clln 30 tl:i'n.
4. Ph_+ vao o (3, 3) 5 tA'n, cqt 3 bt x6a, A3 con 30 tA'n. 5. Phftn 30 tl:i'n ciia A3 vao (3, 2). Ki~m l~iti ta c6 5 0 chQn, dung bAng m + n - 1 nen kMng phai b6' sung 0 chQn 0.
Thi d~J, 2 : ~ 25 25 10 5 3 1 10 " 10 7 6 8 30 ~v 25 ~ 5 • 3 2 2 20 Cx 20
1. Phan vao (1,3) 10 tan, hang 1 va c(lt 3 b\ x6a.
2. Phan vao (3, 2) 20 tan, hang 3 b\ x6a, di~m thu 2 con thieu 5 tan.
3. Phan vao (2, 2) 5 tan, c(lt 2 b\ x6a, di~m phat 2 con 25 tan.
4. Phan 25 tan cua di~m phat 2 cho di~m thu 1 t~i 6 (2, 1), Dem l~i, thay c6 4 6 chQn trong khi m + n 1 = 3 + 3 - 1 = 5. Vf?_y con thieu m(lt 6, ta b6 sung them 6 chQn 0, chhg h~n
6 (2, 3).
§ 4. THU~ T TOAN "QUY 0 CuOC PHI cAc 6 CHQN"
4.1 l>lnh ly : Neu ta c(lng vao hang i cua rna trf}.n cudc phi C = ~<J.i_}mxn s6 ri tuy y (i = !,";;;) va c(lng vao c(lt j s6 sj tuy y C = ~<J.i_}mxn s6 ri tuy y (i = !,";;;) va c(lng vao c(lt j s6 sj tuy y (j = 1,n), ta se c6 bai toan vey.n tai mdi vdi rna trey.n cudc phi
C' = (c~)mxn (c~ = cij + ri + sj) tttong dttong voi bai toan ban d'au (nghia la phttong an t6i u'U C1Ja bai toan nay Cung la phttong
an tO'i u'U CUa bai toan kia va ngu'yC l~i) ( *)
4.2 Thu~t toan g'Om 3 bttoe :
Bttac 1: Qui o cl.(dc rthr'cac ochon ·
' .
Gia sti' ta da c6 mqt phttong an co ban ban d'au voi m + n
- 1 o chQn (c6 thli c6 mqt sO' o chQn O)o Ta cqng v~w hang i cua rna tr?n cttoc phi C s5 ri (j = i,m) va cqng vao cqt j so sj (j =
1, n) 0 Ta chQn cac r i va sj the' nao cho (J rna tr?l:t cttoc p~i moi C ',
' ' h d'' , , 0
cac o c Qn eu co c ij' = 0
Bttac 2 : Ki~m tra tinh t6i U'u
1. Ne'u sau khi quy 0 cttoc phi cac o chQn, rna cac o lo~i d~u CO Cu'OC phi ;:::: 0 thi phttong an dang xet la t6i u'Uo
20 Ne'u sau khi quy 0 cttoc phi cac o chQn rna c6 it nba't mqt 0 lo~i c6 Cu'OC phi < 0, thi phttdng an dang xet khOng pbai toi u'U va ta chuyEin sang bttoc 30
Bttac 3 : Xay <h.;fng phttdng an mai tot hdn
1. Tim o dzta vao: Gia sti' o (i*, j*) c6 cttoc phi am nho nhato
' * *
0 (i ' j ) la 6 dtta vaoo
20 Tim vong di'eu chinh : B6' sung o*' j*) V~lO m + n - 1 0 chQn ban d'au se xuat hi$n vong duy nhat la v gQi la vong di'eu
(*) f'(x) L L c;i x1i = L L (cij + ri + si)xij
j j 2: 2: C .. X .. + 2: r. 2: X .. + 2: S. 2: X .. II II I II · I 11 j . j j j f(x) + L ro a. + L s. b. = f(x) + C I I J I . j (adsbygoogle = window.adsbygoogle || []).push({});
Hai ham ml,lc ti~u chi khac nhau mqt hang so n~n di~m ct,rc tri cua chung tr(mg nhauo
chink. (T/C 2)
3. Phan o ch&n le cua vong V : Ta danh s6 thu tv cac 6 cua vong V Mt d'au tu 6 (i*, j'\ Khi do, V phan thanh hai Mp :
vc : Cac 6 co s6 thu t\f chiin
yL : Cac 6 c6 s6 thu tq' le
4. Tim o dua ra va lu(/ng di'eu chink :
Gia sti : min x.. = x.o .o
( 1,1 .. ) yC E 1J 1 J
Khi do (i0, t) la o dua ra va xio jo la luc;ng di'eu chinh. N6i CtJ. th6 : Tim xem trong cac 6 c6 s6 thu tv chiin, 6 nao c6 phan it hang nha't thi o do la 6 dU'a ra, con lU'gng hang (J o nay la, lU'gng di'eu chinh.
5. L¢-p phucmg an m(Ji : X' = (x~)mxn dU'gc tinh nhU' sau :
l
x .. - x.o.o ne'u (i, j) E yC
1J l J
xij = xij + xio jo ne'u (i, j) E yL
xij ne'u (i, j) (/; V
Tuc la : - 6 c6 s6 thu tv chAn cua vong dU'<;Jc bOt hang di - o c6 s6 thu tq' le du<;Jc c9ng them hang - 6 ngoai vong di'eu chinh v:in giu nguyen lU'<;Jng hang cu.
Nh~n xet: 6 (i0, j0
) trU'dc c6 xiojo tan. Vi la 6 chiin nen bi tru di xio jo tan va thanh 6 lo~i.
- 6 (i*, j*) trudc la 6 lo~i, vi la 6 }e (o SO 1) nen CQng vao xio jo tan trd thanh 6 chQn.
- (xij)mxn la ph11c:Jng an Vi X~ ~ 0 (trong tr11Cfng h<;Jp bt tru x1.oJ.o, vi x.o.o la nhO nhat trong cac x .. nen khOng Am).
1 J lJ
- M6i hang ho~c c9t vong V di qua d'eu c6 1 6 chAn 1 6 le
n€m t6ng 2: x .. va 2: x.. v~n kh6ng d6i.
lJ lJ
i j
- PhttCing an (x~)mxn la cCI biin vi cac o chqn khong tl',lo thanh vong (T/C 2).
- PhttCing an nay tot hCin vi da lol;l.i ra m()t o c6 cttoc phi 0 va thay vao 6 c6 cu'oc phi < 0.
Sau khi c6 phttCing an CCI biin moi ta ll;l.i quay v'e bu'oc 1 r'oi bttoc
2 ... Cu tie'p tvc nhtt v~y, vi bill toan v~n tru luon c6 phttelng an t6i
uu (T/C 1) va s6 phttelng an cCI biin la huu h:;m (t/e 2 §1 ehttelng 2) n€m sau huu hl;l.n fan di'eu ehinh phttdng an, ta e6 phttelng an t6i ttu. (adsbygoogle = window.adsbygoogle || []).push({});
4.3 Thi d\1 : Giai bai toan v~n tai sau :
&o 0 ~ Bl B2 B3 80 20 60 5 4 1 Al : 50 X 50 3 2 6 Az'-~·40 X X 20 '2-2o 7 9 11 >:;: ____ , ' A3 : 70 X .X 60 . '10
Trude het ta tim phttdng an eel bim ban d'iiu :
1. Phan viw 6 ( 1, 3) 50 ta'n, hang 1 bi x6a, B3 con e'iin 10 tan.
2. Phan v?:w 6 (2, 2) 20 ta"n, c(>t 2 hi x6a, A2 con l:;<i 20 ta"n. 3. Phan vao 6 (2, 1) 20 ta"n, hang 2 hi x6a, B1 con 60 ta"n. 4. Phan vao 6 (3, 1) 60 tan, c(>t 1 b\ x6a, a A3 con 10 ta"n.
5. 10 tan con l:;<i cua A phlln vao 6 (3, 3).
Ki&'m tra l~i ta da c6 5 6 chQn dung b~ng m + n 1 = 3 + 3 - 1 = 5
Ta b:it cfau ap dl.}ng thu~t toan.
Bttdc 1 : Quy 0 cttoc phi cac 6 chc;m
Ta c(>ng vao hang i so ri (i = 1, 2, 3) va c(>t j so sj (j 1, 2, 3) sao cho cttoc phi cac 6 c6 danh dau "x" tra thanh 0.
Thtfc chat cua vi~c d6 la ta giai M phttong trlnh. 1 + rl + s3 = 0 (1)
3 + rz + sl 0 (2) 2 + r2 + sz = 0 (3) 7 + r3 +,sl 0 (4) -' 11 + r3 + s:-1 0 (5)
Ta c6 5 phuong trlnh (t6'ng quat la m + n - 1 phuong trinh) voi 6 lfn so (t6'ng quat la m + n lfn s6), nen c6 th&' chQn m()t lfn nMn gia tr\ tuy y. Chiing h~n cho r 2 = 0 tU (2) ta c6 s1 = - 3, tu (3) ta c6 r 2 = - 2, tU (4) cho s1 = - 3 ta c6 r 3 = - 4. Tu (5) vdi r 3 = ,_ 4 ta c6 s3 = - 7. Tu (1) vdi s3 = - 7 ta c6 r1 ~ 6.
Chu y : 0 tren, d&' thay ban chat van d'e, ta l~p M 5 phttong trinh d&' giai tim rl' r 2, r 3, 1sl' s2, s3. Trong thtfc te, ta tien hanh nhtt sau : cho m()t lin nao d6 b~ng 0. Tiep theo, can cu vao cac 6 chon, neu 6 nay a hang da biet r. thi tinh s. = - c .. - r. ; con neu
. 1 ' J 1J 1
6 nay a c6t c6 s. biet r'Oi thi tinh r. = 4 c .. - s ..
. J 1 I 1J J
Ma tr~n cuoc phi moi se la :
8 8 0 X 50 0 0 - 1 ~- --- 1 10 I '· X / - - - - - - - - X -- 20 CD : 10 I 3 0 I ~< - - - - @/ " 0 X - - - - 70 ®
Bttetc 2 : Ki~m tra tinh toi U'u : Ph11ong an chu'a toi 11u vi con o lo::,l.i (2, 3) c6 cu'oc phi - 1 < 0. Ta chuy~n sang b11oc 3.
Bttetc 3 : L~p phtt<1ng an mCii tot hdn :
1. Tim o dua vao : d6 la o (2, 3)
2. Tim vong di'eu chinh : B6 sung 6 (2, 3) vao se xuat hi~n
vong v g'om 4 6 : {(2, 3) ; (3, 3), (3, 1), (2, 1)}
3. Phan o chdn le : Danh so thu tv nh11 (J bang tren, ta c6 yC = {(2, 1) ; (3, 3)}
yL = {( 2, 3) ; (3, 1)}
4. Tim o dua ra: min {x21' x33} = min {20, 10} = 10 = x33. Tuc trong 2 6 chlin, thi o (3, 3) phan it hang hon n€m n6 la o du'a ra va lu'qng hang 10 tan (J day la I11<;1ng di'eu chinh.
5. L¢p phUCJng an :
6 (2, 1) chAn, tr11oc phan 20 gio con 10
6 (3, 3) chAn, tru'oc phan 10 gio con 0 thanh 6 lo~i (adsbygoogle = window.adsbygoogle || []).push({});
6 (2, 3) le, tru'oc phan 0 gio phan 10 thanh 6 chQn
6 (3, 1) le, tru'oc phan 60 gio thanh 70
6 (1, 3) ngoai vong V tru'oc phan 50 giO vh 50 6 (2, 2) ngoai vong v tru'oc phan 20 gio vh 20
X' = (1~ 70 70 PhucJng an mdi la : 0 20 0 50) 10 0
Ta da c6 phuelng an eel blm mdi g6m m + n - 1 6 chQn, nen
l~i quay trb v'e budc 1 :
Buac 1 : Quy 0 cttac phi cac 0 chqn : (bang trang 97) Cho r2 = 0 do 6 (2, 1) ta c6 s1 = 0 ; do 6 (2, 2) ta c6 s2 = 0 ; do 6 (2, 3) ta c6 s3 = 1. Tu o (1, 3) vdi s3 = 1 ta c6 r1 = - 1. Tit 6 (3, 1)~ vdi s1 = 0 ta c6 r3 = 0.
Ma tr~n cudc phi moi la :
7 7 0 X 0 0 0 X .X X 0 3 1 X
Buac 2 : Ki~m tra tinh t6i uu :
Ta thay cac 6 lm}.i d'eu c6 cudc phi ducJng ; v?y phucJng an
X' = .(1~· 2~ ~~ )·
70 0 0
la phuelng an t6i uu.
Ph ue1ng an nay nghia la the' nao ? Ta chuy~n 50T hang cua A
1 v'e ca B
3 ; 40 tan cua A
2 chuy~n
cho B1 10, B2 20 va B3 10 ; 70T cua A3 chuy~n ca cho B1. Voi phue1ng an t6i uu nay cuoc phi it nhat phiii tra la :
f(x') = 1.50 + 3.10 + 2.20 + 6.10 + 7.70 = 670 deln v~ ti'en
niw d6.
Ne'u dung phtt<1ng an ban d'Au thi
f(x) = 1.50 + 3.20 + 2.20 + 7.60 + 11.10 = 680 ddn Vl tfen.
Chu y : d tren, dg lam ro each thl!c hi~n thu~t toan, nen ta giai thich chi tie't tung bttoc va tUng cong vi~c trong buoc. Khi lam bai t?p, ta chi c'An giai thich ke't qua cu6i cung va y nghia kinh te' cu!l ke't qua.
§ 5. PHUONG PHAP THE VJ
5.1 Cd sd tmin hqc :
Bai toan v?n tai d?ng t6ng quat la :
(1) n n f(x) = 2.: c .. x .. --. min lJ lJ j= 1 n L xij = ai (i = l,m) j=l (2) m L xij i=l = b. (i = l,n) J (3) ij ~ 0 (i = l,m ; j = l,n) Bai toan d6i ngliu cua bai toan nay la :
m n
(1) g(u, v) = L a"U. + L b. v. _. max 1 1 J J (adsbygoogle = window.adsbygoogle || []).push({});
i=l j=l.
(2) ui + vj ~ cij (i = l,m ; j = l,n)
(3) ui' vj tuy y
Gac cl).p dmg bu¢c d6i ng§.u la :
xij ~ 0 - ui + vj =:: cij (i = l,m ; j = l,n)
(Xem rna tr~n A trong bai toan thi d\1 1.3 chttdng 1 M thay ro hem)
Gia sa ta da c6 phttdng an cd biin ban d'au vdi m + n - 1 o