Question

2. Consider the LP maximize subject to 7.0 2.0 0 3.0 +8.02 +22 < 5 +2.r2 < 4 +3.62 27 21,12 > 0 Explain, without solving the LP, that any optimal solution (yi, y, y37 of the dual problem must satisfy y = 0. Deduce what happens to the optimal value of the primal problem if we replace 27 in the third constraint by 29.

Answer 1

Ang. Given LPP in primal form. maximiza I -70, +622 Đ•+ 2 2 4 2 / 5 21, 1224 34 + 3H2 =27 1, 2 o Now Dual form of riven LPP minimize W = 64, +4₂ +2743 Bit 24 t +373 2-7 y +24₂ +3738 Y, I2, I3 con unrestoiated. Now in transformed Dual problem we have three basic variable y 2 and Y3 which are unrestricted and only two constraints equation - To obtath abtimal solution we have to choose ons basic sariwie is hero. m=24 No. of constraint ) n = 3 (no of voorables) than nom = (3-2) = 9 equal to variable is Zero- so 23 = 7 (11) Manimize z – – 71, +82 – a 2 +12 25 1 +242 24 3x +32 & 20 Now change LPP primal by substituting duck form into variably h equation from aha

M** ( +7, - ) 32) 37 Now if choosing y = 24 CHR is baie variable the so I c from simples 2 8 a 7 2 -8 y, 0 I 0 g table. solution 0 93 3 3 0 0 To get simplen optimality conclition non base Ugriple H2 is most Negative which is entering variable , whorey I so leoving varivare. a min = ( 541, 42 , 29/3) = 2 , so New simplen table 2 n. 12 I, I, I3 801 B 11 0 4 0 16 9, 3/2 O V o 12 1 0 O 2 93 3 0 0 -32 23 Based on optimality condition, call Робе the Z row cofficients are positive which soll is optimum. Do man 2 = 16 By changing right hand sich stipulate 27-29 will not change the optimal solution which is 16.