Q.Solve each of the following L.P.P. : Maximize $z = 4x + 2y$ subject to $3x + y \ge 27,\ x + y \ge 21,\ x + 2y \ge 30;\ x > 0,\ y > 0$

This repeats part 5)ii) with one extra constraint, $x+2y\ge30$, added. The feasible region of $3x+y\ge27,\ x+y\ge21,\ x+2y\ge30,\ x>0,\ y>0$ has corner points $(30,0)$, $(12,9)$, $(3,18)$, $(0,27)$, and remains unbounded upward for the same reason as before (three lower-bound-only constraints, no upper cap). Taking $(x,y)=(100,0)$ still satisfies all three constraints and gives $z=400$, growing without bound — so as a literal maximization this has no finite answer. Reading it as Minimize $z=4x+2y$ instead: $z(30,0)=120$, $z(12,9)=48+18=66$, $z(3,18)=12+36=48$, $z(0,27)=54$ — the minimum is still $z=48$ at $(3,18)$ (the extra constraint $x+2y\ge30$ is satisfied the …

Same corner-point method as always, but first verify the region is bounded in the direction relevant to the stated optimization; adding one more $\ge$ constraint to an alread …

Assuming an extra constraint must change the answer is a common but false intuition — a NON-binding added constraint (one the existing optimal po …