1.

Maximizez=6x+4y ,subjecttoxle2,x+yle3,- 2 x+y le1,xge 0 ,yge 0. Also , findmaximumvalue ofz.

Answer»

Solution :Firstwedraw the lines AB, CD and EF whoseequationsarex = 2,`x+y=3 and- 2x+y =1 `respectively.

Theverticesofthefeasibleregionare` O(0, 0 ), A (2, 0 ) ,P , QandF (0 , 1 ) `.Pisthe pointof intersectionof the lines.
`x+y=3andx=2 `
Substituting`x= 2`in `x+y=3 `,weget,
`2 +y=3`
`thereforey=1`
`thereforeP-=( 2,1) `
Qis the pointof INTERSECTION ofthe lines
`x+y=3""`... ( 1 )
and` -2x+y= 1 `
Onsubtracting, weget,
`3x= 2""therefore x=(2 ) /(3)`
`therefore`from(1) , ` ( 2) /(3)+y=3"" thereforey=(7) /(3) `
` thereforeQ -= (( 2) /(3),( 7 ) /(3)) `
Thevaluesoftheobjectivefunction `Z =6x+4y`attheseverticesare
`z (O ) =6 ( 0)+4(0) = 0 `
` z (A )=6 ( 2 )+4( 0 )= 12 `
`z (P )=6 ( 2 )+4( 1)= 12 + 4 = 16 `
`z (Q)=6 (( 2 ) /(3))+ 4 ( (7)/(3)) =(12) /(3) +(28) /(3)=(40 ) /(3) = 13.33 `
` z (F)=6(0) +4 ( 1 )=4`
` therefore` z has maximum value16,whenx =2 andy =1.


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