$z = 30x - 30y + 1800$ is an objective function. The corner points of the feasible region are $(15, 0), (15, 15), (10, 20), (0, 20),$ and $(0, 15)$. $z$ has the minimum value at $\ldots$ point.

For a Linear Programming Problem $(LPP)$,if the objective function is $Z = 4x + 3y$ and the corner points of the bounded feasible region are $(0,0), (25,5), (16,16),$ and $(5,24)$,then the maximum value of $Z$ occurs at the point . . . . . . .

Show that the minimum of $Z$ occurs at more than two points.
Maximize $Z = -x + 2y$,subject to the constraints:
$x \geq 3, x + y \geq 5, x + 2y \geq 6, y \geq 0$

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:

An aeroplane can carry a maximum of $200$ passengers. $A$ profit of $Rs. 1000$ is made on each executive class ticket and a profit of $Rs. 600$ is made on each economy class ticket. The airline reserves at least $20$ seats for executive class. However,at least $4$ times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

An oil company has two depots $A$ and $B$ with capacities of $7000 \, L$ and $4000 \, L$ respectively. The company is to supply oil to three petrol pumps,$D, E$ and $F$ whose requirements are $4500 \, L, 3000 \, L$ and $3500 \, L$ respectively. The distances (in $km$) between the depots and the petrol pumps are given in the following table:

From/To	$A$	$B$
$D$	$7$	$3$
$E$	$6$	$4$
$F$	$3$	$2$

Assuming that the transportation cost of $10 \, L$ of oil is $Rs. \, 1$ per $km$,how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?