The maximum value of $z=6x+8y$ subject to the constraints $x-y \geq 0$,$x+3y \leq 12$,$x \geq 0$,$y \geq 0$ is:

If $Z = 7x + y$ subject to $5x + y \geq 5$,$x + y \geq 3$,$x \geq 0$,$y \geq 0$,then the minimum value of $Z$ is

$A$ manufacturing company makes two models $A$ and $B$ of a product. Each piece of Model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of Model $B$ requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing,the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of Rs $8000$ on each piece of model $A$ and Rs $12000$ on each piece of Model $B$. How many pieces of Model $A$ and Model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?

$A$ company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ require $5 \text{ minutes}$ each for cutting and $10 \text{ minutes}$ each for assembling. Souvenirs of type $B$ require $8 \text{ minutes}$ each for cutting and $8 \text{ minutes}$ each for assembling. There are $3 \text{ hours } 20 \text{ minutes}$ available for cutting and $4 \text{ hours}$ for assembling. The profit is $Rs. 5$ each for type $A$ and $Rs. 6$ each for type $B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

Two godowns $A$ and $B$ have grain capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops,$D$,$E$ and $F$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in $Rs$)
From/To $A$ $B$
$D$ $6$ $4$
$E$ $3$ $2$
$F$ $2.50$ $3$

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?

$A$ manufacturing company produces two items,$A$ and $B$. Each item must be processed by two machines,$I$ and $II$. Machine $I$ can be operated for a maximum of $10$ hours $40$ minutes ($640$ minutes). It takes $20$ minutes for an item $A$ and $15$ minutes for an item $B$. Machine $II$ can be operated for a maximum of $8$ hours $20$ minutes ($500$ minutes). It takes $5$ minutes for an item $A$ and $8$ minutes for an item $B$. The profit per item of $A$ is ₹ $25$ and per item of $B$ is ₹ $18$. The formulation of an $L.P.P.$ to maximize the profit (where $x$ is the number of items $A$ and $y$ is the number of items $B$) is . . . . . . .