1.

Maximizez= x_ 1+x_ 2,subjecttox_ 1+x _ 2le10,3x _ 2-2x_ 1le15x_ 1le6,x_ 1 ,x _ 2ge0 . Showthatthemaximumvalueofzoccursat morethantwopoints. Whatisyourconclusion ?

Answer»

Solution :First wedrawthelinesAB, CD andEF whoseequationsare`x_ 1+ x_ 2=10 ,3x _ 2-2x _ 1=15andx _ 1=6`respectively.

ThefeasibleregionisOEPQDOwhichisshadedinthe figure.
Theverticesofthefeasible regionare`O( 0 , 0 ), E (6, 0) , P , Q and D ( 0,5 ) `.
Pisthepointofintersectionof the LINES`x _ 1+x _ 2=10andx _ 1=6`.
SUBSTITUTING` x _ 1=6` in` x _ 1+x _ 2=10`,weget
`6+x_ 2= 10 `
`thereforex_2=4`
`thereforeP=( 6,4 ) `
Qisthepointofintersection OFTHE lines
`3x_ 2- 2x_ 1=15 ""`...(1)
and`x _ 1+x_ 2=10"" `... (2 )
Multipyingequation(2)by2, weget
`2x _ 1+2 x _ 2=20 `
Addingthisequationwith(1) , we get,
`5x _ 2=35`
`thereforex _ 2=7 `
`therefore `from(2) ,`x_ 1+ 7 =10`
`thereforex _ 1=3`
`thereforeQ =( 3, 7) `
Thevaluesoftheobjectivefunction`z =x_ 1+x _ 2`at theseverticesare
`z(O)=0 +0= 0`
`z (E) =6 + 0=6 `
`z (P)=6+4= 10`
`z( Q)=3 + 7= 10 `
`z ( D)=0+5 =5`
`therefore z`has maximumvalues10attwoconsecutiveverticesPandQ ofthefeasibleregion.
Hence , z hasmaximumvalue10 at everypointofthesegmentjoiningthe POINTS `P ( 6, 4 ) and Q (3, 7 ) `, i.e.,maximumvalueofz occursat morethantwopoints. This LPP has infinitesolutions.


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