If the vertices of a feasible region are $O(0,0), A(10,0), B(0,20), C(15,15)$,then the minimum value of the objective function $Z = 10x - 20y + 30$ is . . . . . . .

For the $LP$ problem,minimize $z = 2x + 3y$,the coordinates of the corner points of the bounded feasible region are $A(3, 3), B(20, 3), C(20, 10), D(18, 12),$ and $E(12, 12)$. The minimum value of $z$ is:

The objective function of an $LP$ problem is . . . . . . .

The maximum value of $Z = 3x + 4y$ subject to the constraints $x + y \leq 4, x \geq 0, y \geq 0$ is . . . . . . .

The linear programming problem $(LPP)$ to maximize $z = 2x + 5y$ subject to the constraints $x + 3y \leq 6$, $2x + 6y \leq 18$, $x \geq 0$, and $y \geq 0$ has:

Consider a $LPP$ given by minimize $Z = 6x + 10y$. Subject to $x \geq 6, y \geq 2, 2x + y \geq 10, x \geq 0, y \geq 0$. Redundant constraints in this $LPP$ are $....$