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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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