Bai tap buoi 12

ll ciei bni to6n gdc bang phuong ph6p tlon hinh: - Dua bii to6n tIf, cho vi dang chinh tdc vu vd phii cfra c6c p.trinh kh0ng Am... Vay t4p PATU cria crp BTDN In 0.

Bli t$p tru6i tZ Bii 1: Cho biri toin

f(x) = 8x1 + 3x2+ 6x3 + 4x4 +3x5 + 3x6+ min

5*1 + 6x3 -2*4+ 3x5 = 90 (1)

-312x1 - x2- x3 + x4 -ll2x5- 3*6 = -24 (2)

5/3x1 +2x3- 2x4+ 12x6 > 0 (3)

2*l *x3 * 213x4- x5 <23 (4)

*i '0CI:1'2'"''6)

vi vgcto xo = (0, O, lS, 0,0,3)

u) Phao tich c6c tinh ch6t cfia vecttr x0 dtii vri,i biri to6n df, cho.

b) nn t$p PATTI cria c{p bii to6n tliii ngiu

Biri 2: Cho bii toin

f(x): 2x1* xa-124 * cr& +3xs- x6* min

3xr - Xs * xa * 2x5 - 3x6 :45 (1)

-2x1 *xz+Z*- Xl -Xs *Lxe :$ Q) X1 -3x; - Zxa *x5 = 20 (3)

x;20CI:112r ,6)

a) Ding thu$t to6n tlon hinh tim PACB cria bii tofn

b) Tim tlii)u kiQn cria ca AO fai to6n tliii ng6u kh6ng c6 PA

c) Bi6t c4= -9, hf,y tim I PATTI cria bii toin giic khi c6 th6m tlidu kiQn f(x) > 52.

d) Bi6t ct: -9 vir f(x) + max, tim tflp PATII cria c[p bii to6n tliii ng6u

Biri 3: Cho bii toin:

f(x):3ax1 + 9ax2+ 3x3 + min

5x1 *3x2* 6x3 > 8

X3 >ll4

-X1 -Zxz <-2

3x1* x2 + X3 >4 -X1 -X3 <-1

x;)0(i:1,2,3)

a)Tim tli6u kiQn

"riu u.16 x0: (2,,0,114) ln PATU

b)Iley x6c tlinh t$p phuon g 6n titiuu cria c{p biri to6n tliii ng6u khi a : ll3.

a) Biri toin tliii ngiu: f(y) = 8y, + ;Vr-2yr+ 4yo- ys -) max

5y, - Ys * 3yo - Y, ( 3a (1')

3y, - 2yt+ yt < 9a (2') 6yr+yz *yn-ys <3 (3')

YrrY1oYt ) 0; y:, Ys ( 0

G/s x0 : (2,00 ltq IdaPATtI cria BT giic Do *' t/- l6ng cic rb (1), (4), (5) vi x1, x3 ) 0 n6n mgi PATTI cria BTDN phii t/m:

I yr= lq= lr=o t_

l5Yr- lr*3!q- ls='3a

I

[ 6y, + lz*lq*ls=3

HQ phutrng trinh tr6n c6 nghiQm duy nh6t y0 : (0, 3,'3a,0, 0) Vfy d6 x0 Dr

PATTI thi y0 phii thoi m6n c6c ring buQc cdn l4i cria BTDN n6n a 2 0

b) a: 1/3 thi x0 = (2,0,1t4) Ii PATU cria bii toin giic n6n Tfp PATU cria BTDN Iir

y:{ yo = (0,3,,_1, 0: 0)}

Tt PATU y0 : (0, 3, -1,0, 0) cria bii toin tl6i ngAu, taxic ilinh iluqc t4p PATU cria btoan g6c llr

X={(2,0,114)}

-2x1+3x2 -x3 >-4 x1- 413x2 + 2* >2

X3 <2

-3x1 + x2 + 24 >l

Zxr - 5x2 - 3x3 >-3

x1(0rxz)0 a)Tim tflp phuon g 6n t6.i uu cia c{p bni toin tliii ng6u z 4 _

b)Tim cic phuong 6n t6i uu kh6ng cgc bi6n cria bhi to6n d6i ng6u vir mQt

phuong 6n tiii uu cria biri toSn tliii ngSu c6y2= 6.

Dfp 6n

Bii todn tldi ng6u: 7O) = -4y, + 2y, + 2y, + y+ - 3ys + min

-2yr+ y, -3yo+Zyt 33

3yr-413y, +yr- Syt>44

-yr+Zyz+ ys + Zyo-3yt =18 YitYzrYqrYs< 0;Yr>0

Giei bei to6n d6i ng6u bang phuong phdp'tlon hinh: Dua bli toi{n vd dang chfnh tdc DAt tr=-yr i tz=-Yzl tq=-Y ai ts=-Ys

7 O) = 4tr - 2t2 + 2Yt - to +3trJ min

2t, - t2 +3to - 2tt +Yo - 3 -3t, + 413t, - ta + 5t -h - 44 t1 -2tr+yr-Zto+3t, = 18 trrt ry rrtortsryrryz) 0

t, t,

)

Yr

-1 t4

30

ts Ye

01

Yt lez 0

1

0

Yo

Ysz

Ys

3

44

18

-3 413 0

3 -1

-)

-)

5

t3l

1

0

0

0 -1 0

0

1

0

s/3 -713 213 -1413 lt4l3l -sl3

Ll3 -213 tl3

sl3

713' -213

01 00 10

0 -1 0

0 1

0r

Yo

YEz

ts

15 14 6

-1413 1.413 -513 713

rl3

-1

-tl3

0

1 0

-1.16 1716

-s114 tlz

212L -tl3

01 00 10

-Ll2 -3114 -117

a)Biri to6n c6 PATU Y* = (0, 3, 0, 0, 8, 22,0),

z7 = (0, 31t4, 0, 0, ll7, ll2, t).

y(0) = (0,3 +3l1.4e, 0, 0, 8+1.170, 22+L120,0), e20 li c6c

PATU cira bii to6n chinh t6c tuong tludng cria bhi to6n ddi ngiu Do tr='yrl tr='

yzi tq=-yd ts=-Ys n6n ta c6:

b)Tap phuong 6n tdi uu kh6ng crlc bion cira bhi to6n ddi ngSu:

{y= (0, -3-31t49,0,0,-8-ll7e) /0 > 0}.

Yz= 6 e -3-31140=6 <+ 0 = -54 (loai) V4y khOng c6 PATIIcAn tim

Bii 5:

Cho biri to6n:

f(x) = 40x1+ 6x2* 30x3 + 4xa + max

-Zxr-x2-3x3+2xa2l (1)

-2y- 4x3 * xa S-2 (2) -X1 '2rg *Zxa < 2 (3)

3x1 -x2 -2*+3l2xt < 5 (4)

a)Tim t$p phuong 6n ttii uu cfia c{p bni toin

'I6i ngiu'

-TQp PATLIcta bii to6n tldi ng6u:

Y = { ,= (.0, -3-31149,0,0, -8-1 l7e) /0 > 0}'

-Xr{c tlinh tap PATTI cira biri to6n gdc: ta c6 hQ p'trinh

I x=o f*t -o

I | ^ l,xz=o

1:';i I o*t .'i;,**'i;,=:', 1;; =;

Vay tip PATU cira bii to6n gdc: X = {x* = (0, 0, 1)}'

b)Tim cic phuong 6n tiii uu kh6ng cgc bi6n cria bii to6n tliii ng4, (n6u c6) c) Khi th6m rirng buQc: 5x1 * 4 - ll2xa2 7, tim t$p phucrn g 5n tiii uu cria biri

toin tl6i ngiu

Giii:

a) Bii to6n ddi ng6u:

|fy> = yr - 2 y, +2yr+ 5yu-> min

- 2y, - ys + 3ya < 40 (1')

-yr-2!z - yn = 6 (2') -3y, - 4yr- 2y, -2yo = 30 (3') 2y1 +y2+ 2yr+3l2yo <4 (4')

yr(0 iyztyt,yr)0

- {}i&i trii to6n ildi nsf,u beng phu:uns ph6p d{tn hinh:

+Dua bii to6n vd dqng chinh ticvon v€ phii cira cdc phuong trinh khdng Am:

Dat tr=-y,

7rv> - -t,-2vz

2tr tL 3tr -2tr

Lop bni todn phu : P = ysz * yss -> min

t, -Zy, - yt t lsz =

- 2y,

+2\ + 5yo -> min

y:+3yr +ys =

*yz* Zyr+3l2yo+ le =

' trrYzrYstY+Ys,

Ye )0

40

6

30

4

3\ - 4yr - 2yt -2yo -2t1 +y2 + 2y, +3l2yo +

*Yss = Yo=

40

6

30

4

tr ,Yz tYttY$ Ys, Yo ) 0 ; Ygr,Ys: ) 0

tr Yz

J Yr

5

Ya

0

Ys

011

Ye Ysz Ysr 0

1

1

0

Ys Yez Yss

Yr

40

6

30

4

20-1

t1l -2 0

3-4-2 -2 l2

31

-20

312 0

000 010 001 100

0

0

L

0

Ys

tr

Ysr

Ya

28

6 12

t6

-1

-2

0

Ys

tL

Yz Ya

4 18 6

34

dc

10-2000

00-L101

)

-1

-)

0

Ys

tr

Yz Ye

413 6213

2213

10613

dc

0011u30 10022130 010312tl30

- Bii to6n ddi ng6u c6 Patu duy nhdt y* =(-6213,2213,413,0), v6i l(y)*i, = - 98/3.

- X6c dinh tip PATLIcfra bii to6n g6c: ta c6 hQ p.trinh

(

I

| -2x, - x, - 3x, + Zxo =1

i - 2*r-4xr+xq=-2ex* =(-213;7 l3;-213;O) l-x, -2xr-rZxo-2

Nghi6m x* th6a man tdt ci c6c ring buOc cdn lai cira btoan gdc Vay bii to6n g6c c6

PATU duy.nhdt: x* = (-213, 7 13, -213, 0).

Tap PATU cfia btoan gdc: X= {x* = (-213,713, -213,0)}

b) Bni to6n ddi ngiu khOng c6 PATLI khOng cuc biOn.

- Tr) bing don hinh thrl3 ta tim du-o c PATU duy nhdt cfia bii to6n chinh tdc tuong tluong cfia bii todn ddi ngiu li : y* -(6213,2213,413,0, 0, 106/3), vfi T(y)_,, = - 98/3.

Eai 5;

Cho bii toin: f(x):3x1 * 44x2+ 18x3 + max

2x1-3x2 * X3 <4 (1)

x1- 413x2+ 2x3 > 2 (2)

X3 <2 (3) 3xr - x2 - 2x3 < -1 (4)

-2x1+5x2*3x3 <3 (5) x1(0rxz)0

a)Tim tflp phuong 6n tiii uu cfia cflp b]ri to6n tltii ngiu.

b) Tim cic phuong 6n tiii uu kh6ng cqc bi6n cria bii to6n tliii ngiu vi mQt

.phuong 5n ttii uu cfia bii to6n tltii ng6u c6 y = 16.

B&i 7: Tim x = (Xr, x2s , xr) sao cho

f(x) = -x1- 14x2 - 2xr - 3xo + 8x5 -+ min

912x, +2xo-3l2xt > -7 (1)

-712x, - X+ * 3l2xt >'40 (3)

x;20(i=1,5) a)Tim t4p phudng 6n tdi uu cfia c4p bii todn ddi ngiu

b) Tim c6c phuong 6n tdi uu kh0ng cqc bi6n cira bii to6n d6 cho.

Grar

ll Giei bei to6n gdc bang phuong ph6p tlon hinh

- Dua bhi tor{n vd dang chinh tdc v6i vd phii cfra c6c phuong trinh kh6ng Am

f(x) = -x1- ltx, - 2xr - 3xa + 8x5 -+ min

-912x, '2xo +3l2xs * Xo - J (1)

x, + 4x, * Xl '2x, = 50 (2) 712x, * xr - 3l2xt + x, -40 (3)

xr)0(l=1,7)

0

X7

-1 -14-2-380

0

-1

0

-2

X6 X1

x7

X3

7

50 40 5

10 00 01.

00

t40r-2 071201-312

0

-1

0

-14

x6

X1 X7 X2

52 10 5 10

009-2-310 10-8r200 00-7tt2701

0 L 2 0 -1 0 Odc

r(x) -150 00-182 4 0 0

0

-1

8

-14

x6

X1 X5

x2

t1912 5

sl2

2512

0 0 -312 -rl2 0

0 0 -712 tlz L

0 | -312 tlz 0

L 312

0-1

0 ll2 dc

0 ll2

Tt bing don hinh thfr 3 ta suy ra bii toi{n chinh tdc tuong duong vdi bei to6n gdc

c6 PATU i *=(5, 2512, 0,0, 512,11912,0) nen bdi to6n gdc c6 Patu

x* = (5, 25l2r0,0r Sl2)

-X6c dinh mp PATUcira bii toSn gdc:

Tt bing don hinh thf 3 ta c6: i *=(5, 2512,0,0,512,t1912,0)

za =(0, -L12,0,l, -112,ll2,0') X6t c6c PA :

x(0)= x *+0.2a

= (5, 2512-ll2g, 0, 0, 512 -L120,11912+1120,0) vdi 0 < 0 < 5

Ta c6 f(x(O)=f(i *)-0.& = -160 = f(x)-i,

suy ra : x(g)=;s*+0.2a = (5,2512-1,129,0,9,512'1.l2g,Ll9l2+1120,0) vdi 0 < 0 <5 Ii

c6c PATU cira bii to6n chinh tic tuong tludng cira bii to6n gdc.

Yay t{p PATUcira bii tor{n gdc:

X= {x = (5, 2512-Ll2g, 0,0, 512 -ll2g)l 0 < 0 < 5}

-TAp PATIIcira bii to6n ddi ng6u: Y = {y* - (0, '1,'2,3)l

b) T4p c6c phuong r{n tdi uu khdng crlc bi6n cira bii to6n gdc

T = {x = (5,2512-1129,0,9, 512 -Ll2g)10 < 0 < 5}

Bii 8:

Cho bii to6n: Tim x = (xr, xz, t x5) sao cho

f(x) = -2x1+ *, * 1x3 + 3xa - xs -+ max

Z

2x, - 3x, - x3 - 2xo + 6x, = -12 (1)

_7

-5x, + ,2 -x, +3xo - 5xs222 (2)

8xr - x, - Zxo + 4xs < 76 (3)

xr20CI= 1,5)

a)Tim tAp phuong rin t6i uu cfia cdp bii to6n ddi ngiu.

'b)Khi f(x) + min, tim t4p phuong 6n tdi uu cfra bii toSn tldi ng6u

ll ciei bni to6n gdc bang phuong ph6p tlon hinh:

- Dua bii to6n tIf, cho vi dang chinh tdc vu vd phii cfra c6c p.trinh kh0ng Am.

f(x) = -Zxr+ *r*

i.x3 + 3xa - xs + max

-2x, + 3x, + x, + 2xa- 6x, =L2

_7

-5x,+ -x, +3xa - Sxt'xu=22

/.

8x1 -x, 'Zxa+!s +x7=76

xr20(i=1,7)

- BAi to6n tr0n khdng phii dang chudn n€n ta lAp bii toSn phu :

P = xgz-> min

-2x,+3x,+x, +2xo-6x, 'l2 _7

-5x, + ,2 -x, +3xo - 5xr- xu **ur- 22

8xr - x, -Zxn+ 4xs * Xz = 76

x.,20A=L7)1xsrZ0

Xr Xr

312 3

Xr Xa

-1 X{

0 Xr

01

X7 x9t

0

1

0

x3 xEz x7

0 ,,,

76

-2t31 12 -571203 8-10-2

-6000 -5-10L 4010

0

1

0

x2

x8z

X7

4 8

80

-213 1

-8/3 0

2213 0

rl3 213

-716 213

1,13 -413

-2 r21

2

0 -1 0

01 10

1

-1

0

x2

xs

X7

L2 4

72

-t013 1

-413 0

100

-sl6 t4l3l

-71r2 tl3 1,1) -)

0 1 0

-1

-112

1

0

0*

1

3

-1

0

x4

X5 X7

9 1

90

-slz 314 -sl8

-rl2 -rl4 -318

s 312 tl4

L0 01 00

-314 0

-tl4 0

a)Tt bing don hinh ttrfi 4,ta c6 A6 = -2 < 0 mi x;6 S 0 lvjeD n6n bii to6n gdc

kh6ng giii tlugc, do d6 bii todn ddi ngSu cfrng kh6ng giii tluo c.

Vay t4p PATU cria crp BTDN In 0.

b) Khi f(x) -+ min, til bing tlon hinh thfr 3 ta duoc:

Tt bing tlon hinh thf 3 ta suy ra bii to6n chinh tic tuong duong vdi bhi to6n

gdc c6 PATU; * - (0,12, 0,0,4, 0, 72\ nen bhi to6n gdc c6 PATU

x* = (0, 12,0,0, 4) vu f(x)*,n = $,

i* - (0,r2, oro,4, o, 72)

zr = (1,10/3, 0, 0r 413,0, -10)

X6t cr{c PA :

x(o)=i *+0.21- (0, 12+10/30, 0, 0,4+4130,0,72-100) v6i 0 < g <7,2

Ta c6 f(x(0))=f(x *)-0.4r = 8 = f(x)-in

suy ra : x(0)= i *+0.21 - (0, t2+10/30, 0, 0, 4+4130,0,72-100) v6i 0 s g <7 ,2

li c6c PATU cira bii tor{n chinh tdc tuong tluong cfia biri to6n g6c.

Vay tap PATUcira bii toSn gdc:

X = {x - (0,12+10/30,0, 0,4+4130 ) /0 < g<7,21.

- Xdc dinh tAp PATI-I cfra bii toSn ddi ngiu: ta c6 hQ p.trinh

1

lr= 4

I

Vt

=-2 h=0

Vay tap PATUcira bii to6n ddi ng6u: Y = {y* = (L14,112,0)l

]-rr,'* 7/2yr-lz=l

I

I oy, - 5yr+4yr: -l €