I don't know if I've posted this in the right place but I thought I'd give it a go anyway.
For question see attachment below.
So for part a) max z:150X_{11}+350X_{12}+300X_{13}+100X_{21}+500X_{22}+400X_{23}
s.t. X_{11}+X_{12}+X_{13}≤40
X_{21}+X_{22}+X_{23}≤90
X_{11}+X_{21}≥30
X_{12}+X_{22}≥40
X_{13}+X_{23}≥60
For (b) eliminate first constraint and label constraints: u_{2},v_{1},v_{2},v_{3}
Then solve for us and vs and i get u_{1}=0 u_{2}=100 v_{1}=0 v_{2}=400 and v_{3}=300
Then c*_{12}=50 so not optimal
(c) Enter X_{12} into basis create loop etc. but that give new solution as X_{12}=40, X_{21}=30 and X_{23}=60. But then when you go through us and vs again to test for optimality, you've got 5 variables to solve for and only four equations so I'm stuck. Help please!