For the following LP:

a$.$

Rewrite this LP using slack$/$surplus variables.

b$.$

Solve using the simplex method $$ show the final tableau and interim tableaus.

c$.$

What is the dual LP $($state it using slack$/$surplus variables$)?$

d$.$

State explicitly which primal variables correspond to which dual variables.

e$.$

For what coefficients of $1$ in the primal objective function will the final tableau still represent the optimal solution?

f$.$

If the three constraints on the right hand side of the primal are suddenly changed from $(5,4,4)$ to $(1,1,1),$ use post$-$optimal $($sensitivity$)$ analysis to quickly show what the new right$-$hand side values in the final tableau? Show your work.

g$.$

If the optimization function in the original LP is changed to $=41+3223,$ show the updated tableau.

h$.$

If a new constraint $1+2+3<=1$ is added to the original LP$,$ use post$-$optimal analysis to show the new final tableau obtained by adding in this constraint. Use the generalized simplex method to re$-$optimize this problem.