unit 4 duality+in+lpp

7/30/2019 Unit 4 Duality+in+LPP

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. '4 - !-^-,r'Fetafinn nf the clual variable from the loss ( lint ofThe nterpretationf thedualvariablerom he ossor economiccviewproves xtremely seful n makinguturedecisionsn the activitiesbeing rogrammed.SelfAssessmentQuestionsStateYes or No1. DualL.P.P lwayseducesheamount f computation.V f\t2. lt spossibleo reversehedualL.P.Poprimal .P.P.4.3 Formulationof dual ProblemConsiderhe ollowing.P.PMaximize Z = C/1 *gryz + . . .t CnXnSubjecto theconstraintsQ r tX t + a 1 2 X z . . . + Q t n X n 3 1d n Xr + a22Xz* . . . * 1 z n X r3 bzQ mt t * On2XZ . . . * QmnXnS ^X r . X z , . . . , X r 0 .To construct dualproblem, e adopt he ollowing uide ines:i . The maximizat ion robleproblemn thedualandviceversaii. (s) tvoe of constraintsn the primalb es of constraintsniii.*Td "*ffoients c., cz. .,cn in the objective unctionbecomeb.,, z,..,b' in theobjectiveunction f the dual'

of the primaliv. The constants r, b2,...,b, in the constraintsf the primalbecoms 1,

c2,. . ,cn n heconstraintsf thedualv. lf the primal has n variablesand m constraintshe dual will have mvariables nCn constraints

vi. Thevariablesn both he primalanddual are non-negativeThen hedualProblem illbeMin imizeW=brYr + bzYz+. . . +bmYn ubject to theconst ra in ts?rrYr + dztYz+ . . * Oi l Y^2 CrArzYr az2Yz+. . * Am2Y^2 zQ t n Y t 3 2 n Y z + . . . t 8 m n Y ^ 2 c n

Y r , Y z ,, Y ^2 0 '

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Solution: First3xr 10xz>15isconvertedo "s" ypeas-3x1 llxzs -lsThereforetsdual sMiniZ = -15h + 20yzSubjecto -3yt* 4yz>1oo10y ,+15Y2>200Yt, Yz,> 0

Example4: Wher,he constraintsontain="Sion.Write hedualof MaxZ = 4oxt+ 30xzsubjecto l0xr+6xz 1 0X1,X2, 2 0

Solution: FirstSxr 7xz2 -10fs rewritten s -5x1+7;z< 10SecondlY r+ Xz= 9 is writtenas

X r + x z s 9Xr* Xz> 9 this s sameas- X 1x 2 S - 9

ThereforeGivenproblem sm a x Z = 4 0 x r + 3 0 x 2subject o 10xt+ 6 xzs 15- Sxt+7 xzS 10

x t +x239' X 1 x 2 2 9X1,X2, 2 0

Thereforetsdual s ::M in iW =-1Sy t+10y2+gys lgy r "Subjecto 1 O y , - S y r + y r t - 9 t " > 4 06 y r * 7 y t * Y s t - Y s " > 3 9Yt, Yz,yst - ys" > 0Letys| - yr" = y,


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6/14

: Write hedualof he ollowing.P. P= 3xr Zxz+ 4xso 3x1+ 5x2 4xs276x1+ xz + 3xs247xr- 2xz xs3 10Xr '2Xz+ Sxs24x1+ 7xz 24> 2,X,t, Xz, X32 0

Since heproblems of minimizationll constraintshould e oftype.We multiplyhe hirdconstrainthroughouty-1 so that 7xr+ 2xz+g2 -10 )F>

et y1 yz,ys,y+and ys be the dual variablesassociatedwith the above iveThen he dualproblem s givenby:W = 7y, + 4 yz- 10ys+ 3 h + 2yso 3yt * 6yr- 7 yt* h + 4yss 35y, * Yz+ 2ys- 2yo+ 7ys= -24 y , * 3 y r * Y s + S Y q - 2 y s 3 4

Yt, Yz,Yg,Yq,Ys20AssessmentQuestions2

Fil l n the blanks. Thecoefficientsf decisionariable in hp objectiveunction ecomesquantitiesn righthandsid-e f T\VICL__. "


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1 1I 3541 1/rsa \The leastpositive atio I = lindicateso how the numberof hoursof[ 1 1)machineM1can be decreased.he eastnegativeatio -10) ndicatesohowmuchhenumber f hours f machineMrcanbe ncreased

4.5.4CalculatingherangeLowerimit= loo - 354 - 7461 1 1 1Upper l im i t=00 - -10 )= 10Henceherangeof hoursor Mr s ryto 110,Bythe sameway herange1 1of hours or machineMzandMs anbe calculatedThemanagementf a companyarely estrictsts interesto the numericalvaluesof an optimal olution. ctuallyt is interestedn knowinghe impactof changesn the inputparameteralueson the optimal olution. uchaprocesss knownassensitivitynalysis.SelfAssessmentQuestionsFill n he blanks r^ e t1. Sensitivitynalysiss carried uton [- [ f^aLsimplex table.2. lt helpsus to study he effectof chanSesn@


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2 . Minimise = 200yr+ 150y, 100ya 80yoS.t SYt 3Yr- SYt BYo 34y,t Syr- 4Yt 4Yo>Y't,Yz,Ys,Y+ 0Minimise = 12y,t 8y2+ 8y3S.t 4y't 4Yr* 4Yt>23 y r y z - y z 21Yt,Yz,Ys,> 0

3 .

unit 4 duality+in+lpp

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