For a linear programming problem,the objective function is $Z = 3x + 9y$. The corner points of the feasible region are $(0, 10), (5, 5), (15, 15),$ and $(0, 20)$. The maximum value of $Z$ is . . . . . . .

The corner points of the feasible region determined by the system of linear constraints are $(2, 72)$,$(15, 20)$,and $(40, 15)$. Let $Z = 6x + 3y$ be the objective function. The minimum value of $Z$ occurs at:

The corner points of the feasible region determined by the following system of linear inequalities: $2x + y \leq 10$,$x + 3y \leq 15$,$x, y \geq 0$ are $(0,0)$,$(5,0)$,$(3,4)$,and $(0,5)$. Let $Z = qx + py$,where $p, q > 0$. The condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is . . . . . . .

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

If an $LPP$ admits an optimal solution at two consecutive vertices of a feasible region,then:

The feasible region for a $LPP$ is shown in the figure. Find the minimum value of $Z=11x+7y$.