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0) e atfons if) \ pare'—'—) p vy = 9. Menie “Duare: Mast Important Finding mm the deve topme wt of Linear Programming Probleme fo the exfetence of duality Fn LPP. Lineay Programming Probleme extat tm pols. That fc fn LPP every mondtimtzakon problem ic 7 Wh associated wfth a Minimization problem. Conversely , assocfated yorth enemy minim zaston Probie” fe a wartmi zation protien. One wr Wie problem worth Pe objedive f? OL Baxi zation, woe Can Lome oy dualtty YuaHonch! p of LPP , th minimization _Yvexion — ‘The orginal LEP te broom at primal prbem, and the derived problem te cnobr af ’ dual prbem. Procedure for converting a primal Mto a dual. Case 41: When the given _prololem Le _navimization onet The Objeckive” £" of primal te of waxlmized” bype 2 the chuchural donctramnt: are of < type. Now FE: tha batic vanables are 2C),%a,- te . give Atif ferent name for vorbbles of dual. let Wem be = | Y,,Ya,-- Um. Consider ta example below: ne aS PasTTB) Lane: LE QA BX, Set Min: Z= loyti2y, St a X, 43% £10 Yy+24, % 2 L 2%, 44% £12 & aytoy, 22 & -p—___ 1, % ZO Yi i42 7 © --———__ ae p>—___| Cv *| Gace dO! When the problem is of minimizod” type C Bimal Dual» Min: Z= 10%), H1D%) ob May: X= OY +BY, St BQ t+2% yD 4,434, £10 3% +4% 73 V oy trey, £12. % XX RO Y¢4 7-0 ¥| Case 3. When the problem lal got both » << Consheiyk, then depend ng Ubon tye pbieoHwe . 4 convert all the ondheinte to eftrey yor < type. thad fc, If the objeakve fic of minimizad”? type, then see that all conghaink are of & Luce & if the Objeckve {7 ic of marximize¥” type, they see that al the congbalnts ane oft < kupe.. To jnvert > to < OR SF to 2, Simpy multiply Tre Conshaintk by -1, Once you convert the Oxndheint , ten tonte the dual at explained In cate 1% 2. Gusidey the example. TTT TT SS WO CBiimal> Dual Mae: Z= 2%, 423%, St Mm: Ze loy,-I2y5 St AVA, Z10 (i Y-2yy 2 2x, 43%, 712 & | 4y-34,% 2 2 11% 70, Wie %O- { 4 May: Z=O2%,43%q Gt / — 4, tU% £10 _, a TT = 2%,- 37, £~12 ¥ > 1 V2 7D ~ — a io Write the dual. anare —=— NY Examples Mae Zs OY, ~¥, 4UY, ot HX, +2%o~ %o 25° QOXN- Vo: + Xn <6 med + %p 4 3% 210. UM+ %,_ £12 Xi %0,%,% O > Dual Mtn Ze Su, +64, +104 A!2-2. t ‘ UY t2Y,44o +44y > 2 2y,- Yo + Ys tou, 7-1 ~Yy +49 +3Yst Yq % 4 2 S42 Ya Yu % 0 Crampe. Min. Ze %,4+3%. St ° Q2H,+% 42 Wy tQ%ytO%27rS * — HAM t2%e= Il %1 1%, %2O The Savne pra Gn tk wHtten Yo Mint 2 %yt3B%q b= —297,-%2>%-3- -— %,t2%o +6Y2%S5 ° — FH t%t2%e_ > 2 UH % y= 2% % =~ 2 F %s72,%7O Bolic. Mar: Ke = Ba sSoat 24, -24.) ot -2Y,- Yo - yh ry! ETRE: Z TES ee £3, YM, ya 2 May: = 39,754, +24, at = DY) 7 Yo 7H £Q 4 “UTAH AYS SX! 64, +24, $3 A YY 20 LY, unresnctedl Bromple (@)! Man: Za Br d1F%o tIW%y. WH oA >%B — BY, % O%e 4) QMNAVoS%,=15 MsVa5V%y7O | The awe Con be wmtion al , Mav: Ze BvitlT MHtIVn 2 t- — HM t%y~-%q 4B, — 3y,4 2%, 41 QYt%e-S% oq 4) — 2%, -%otS%_ 4-1 2% V% MRO 1 C Dual) —_, me Ze ~ S34 +Yot3aYea Qt Mae: @ —¥,-39, 4+24)-297 > 2 Y, +045 +4,-Ul y IF ~¥,+24,-Sul teu! > 4 Yr Yo 4a 4s >% O Vv Mt Ze 34,4 YotYg GE ~ Y.~ 34,4 2Y4a% 8 Y,th7 IF 4, $24, -SR7 9 U, oD & Ya Unrermncel 
