A manufacturing unit produces two products Pl and P2. For each piece of P1 and P2, the table below provides quantities of materials M1, M2, and M3 required, and also the profit earned. The maximum quantity available per day for M1, M2 and M3 is also provided. The maximum possible profit per day is ₹ ______

	M1	M2	M3	Profit per piece ( ₹)
P1	2	2	0	150
P2	3	1	2	100
Maximum quantity available per day	70	50	40

Two models, $$P$$ and $$Q,$$ of a product earn profits of Rs. $$100$$ and Rs. $$80$$ per piece, respectively. Production times for $$P$$ and $$Q$$ are $$5$$ hours and $$3$$ hours, respectively, while the total production time available is $$150$$ hours. For a total batch size of $$40,$$ to maximize profit, the number of units of $$P$$ to be produced is ____________.

Maximize $$\,\,\,\,\,\,\,\,\,Z = 5{x_1} + 3{x_2}$$
Subject to $$\,\,\,\,\,\,\,\,\,\,{x_1} + 2{x_2} \le 10,$$
$$\eqalign{ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1} - {x_2} \le 8, \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_1},{x_2} \ge 0 \cr} $$

In the starting simplex tableau, $${x_1}$$ and $${x_2}$$ are non-basic variables and the value of $$Z$$ is zero. The value of $$Z$$ in the next simplex tableau is __________________.

Maximize $$\,\,\,\,Z = 15{x_1} + 20{x_2}$$
Subject to
$$\eqalign{ & 12{x_1} + 4{x_2} \ge 36 \cr & 12{x_1} - 6{x_2} \le 24 \cr & \,\,\,\,\,\,\,\,\,{x_1},\,\,{x_2} \ge 0 \cr} $$

The above linear programming problem has