Math Project

HoMEOn G Solvfi ons

Rove + fallouins Coollay If Hu objecth funth un of Mx p^ban is a bovnded abo t +k min

probn has no fihsibe solutiuns.

PruSurps X, is tuesb Soludion fur h MAX problem, po Y, i> fasibh Solu fiua f fr 4& n probm

Tuon 9.9., st hen thct X,< b Y

snu 4u ntx probum is uabundcd m al s fr a X Sveh hit

X, CX; n fAhd Xz suel h

Y.CX solkin no ftasi ble solktins.

i volts Thn 9M, So , cnnod b fensíbk p/obkm, aJ hs min piubn has

MAxini : X, sb oX, X

X, X ,Xo

Minimt: X, ¥i duul

M uso sh tLi - Ch vSe LP Assistat, or

jvs sp4 4 iqualitins

1X nut feasih ut feuhu

Maximic: Hx, +l0 X- Sx, 2x4 subet to 3x,-aX1 * 4x Xy < 26

x+6X -X +Sx, 3o

Xa,X,,X >O

has To shan that its ftesibk

3)-2() o O 26 s26 ()6 ().

-4 () +8() O O - -19.6 s10

J

Obedi svnution valve: 4()+ 10($)+0 *0:: 3S.2

The duel is

sbct to 3y, * Y- 4y,4

y,671 ?,19 -3

Y SYa Y3

To sho it's fuesi Ju 3 () () O = 4 >y

-1 ()*S(H)O 10 Jo ()-(H)+ 0 3-3

Oje i fvntion v-l: 26(3)+30(s)4O }5.

Curolluy 4.4.) 4allg ,Us ht Hsutin to +k nax prvb sd b (G,i,0, o). mia probum msi b (,0).

Thi is beLC

+solutin to th

cX, 7S.J 6Y

Pu Prn or disprow: (1,1,0,0) i n opkinSulutien poi Maxini:2 x,idY1

+X X

X, *X X 3

0 0 | S|

3 Nu 4ht An is nu slatk fr th fist 2t on stra ía ts

nd ositiu slatk in 4u 3" N dual:

svbjr to : Y Y

YY

Thus Y 0, s

4Solt is (,2,o)ih is fusibu.

Maimi su:9x,3x5x1*Jx4 Void hepblu: (o9,9,1)

Sx, +3x X y |6 4x,X X

' Xy <12

X,X,90

304) 21) 16 <16 v 3) 10 12

uro sltk n kisl . pusi in 3

Du: Mininiec: Yy, *16y4 sbc fo: 2y,* S Y* 4y,9

y,3y2* Y YY>5.

3 y 3

So , Y

Solvtiu is (1, 2,0 ulh is fslbu,

ON edy k«gbil4a: (3,0,1,0,s) 8(3 -1(s) b 5I9 J sluk of

+(S) 4 s4 v sluh O0 2s)]4I4 sluk of O

-10) 3

So, posiu skek n nstria 1, s 0 OaMinii:|8 y,* y a

S

Pl y^ 0 s. l ttY, 4

T A* hut o Sol-t is (0,1, 1). 3 is Viols h d (nstai»t y() 3(1) 13<16. So

(3,0,,0, $) s NoT optinel sols ki puit

Yu fsb: (I,0,1,0) slh of O

33 Slcl of{ O -

sbcto' y,y2 5

NoS«l ), *ay,

: 5

y. *

y, *4 A 3o)tita of (-1,3). 3it} c&phet

pt solttn punt. pu,nt