A brick manufacturer has two depots, $P$ and $Q$, with stocks of $30000$ and $20000$ bricks respectively. He receives order from three building $a,b,c$ for $15000,20000$ and $15000$ bricks respectively. The costs of transporting $1000$ bricks to the building from the depots are given below.

	Cost of transportation (in $₹$ per quintal)
From  To	$a$	$b$	$c$
$P$	$40$	$20$	$30$
$Q$	$20$	$60$	$40$

How should the manufacture fulfil the orders so as to keep the cost of transportation minimum?

A medicine company has factories at two places, $X$ and $Y$. From these places, supply is made to each of its three agencies situated at $P, Q$ and $R$. the monthly requirement of the agencies are respectively $40$ packets, $40$ packets and $50$ packets of medicine, while the production capacity of the factories at $X$ and $Y$ are $60$ packets and $70$packets respectively. The transportation costs per packet from the factories to the agencies are given as follows.

How many packets from each factory should be transported to each agency so that the cost of transportation is minimum? Also, find the minimum cost.

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 pumps, $d,e,f$, 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:

Assuming that the transportation cost per km is $1$ rupee per litre, how should the delivery be scheduled in order that the transportation cost is minimum?

A firm is engaged in breeding pigs. The pigs are fed on various products grown on the farm. They need certain nutrients, named as $X,Y,Z$. the pigs are fed on two products, $a$ and $b$. One unit of product $a$ contain $36$ unit of $X, 3$ units of $Y$ and $20$ units of $Z$, while one unit of product $b$ contain $6$ units of $X, 12$ units of $Y$ and $10$ units of $Z$. the minimum requirement of $X, Y, Z$ are $108$ units, $36$ units and $100$ units respectively. Product $a$ costs $₹20$ per unit and product $b$ costs $₹40$ per unit. How many units of each product must be taken to minimise the cost? Also, find the minimum cost.

A dietician wishes to mix two types of food, $X$ and $Y$, in such a way that the vitamin contents of the mixture contains at least $8$ units of vitamin $K$ and $10$ units of vitamin $E$. Food $X$ contains $2 units/kg$ of vitamin $K$ and $1 unit /kg$  of vitamin $E$, while food $Y$ contains $1 unit/kg$ of vitamin $K$ and $2 units/kg$ of vitamin $E$. It costs $₹5 per kg$ to purchase the food $X$ and $₹7 per kg$ to purchase the food $Y$. Determine the minimum cost of such a mixture.

A diet for a sick person must contain at least $4000$ units of vitamins, $50$ units of mineral and $1400$ calories. Two food, $a$ and $b$, are available at a cost of $₹4$ and $₹3$ per unit respectively. If one unit of $a$ contains $200$ units of vitamins, $1$ unit of mineral and $40$ calories, and $1$ unit of $b$ contains $100$ units of vitamins, $2$ units of mineral and $40$ calories, find what combination of foods should be used to have the least cost.

A housewife wishes to mix together two kinds of food, $X$ and $Y$, in such a way that the mixture contains at least $10$ units of vitamin $B_1$, $12$ units of vitamin $B_2$ and $8$ units of vitamin $B_3$. The vitamin contents of $1kg$ of each food are given below.

If $1kg$ of food $X$ cost $₹6$ and $1kg$ of food $Y$ costs $₹10$, find the minimum cost of the mixture which will produce the diet.

A firm manufactures two types of product, $a$ and $b$, and sells them at a profit of $₹5$ per unit of type $a$ and $₹3$ per unit of type $b$. Each product is processed on two machines, $M_1$ and $M_2$. One unit of type $a$ requires one minute of processing time on $M_1$ and two minutes of processing time on $M_2$, whereas one unit of type $b$ requires one minute of processing time on $M_1$ and one minute on $M_2$. Machines $M_1$ and $M_2$ are respectively available for at most $5$ hours and $6$ hours in a day. Find out how many units of each type of product the firm should produce a day in order to maximise the profit. Solve the problem graphically.

A small firm manufactures items $a$ and $b$. The total number of items that it can manufacture in a day is at most $24$. Item $a$ takes one hour to make while item $b$ take only half an hour. The maximum time available per day is $16$ hours. If the profit on one unit item $a$ be $₹300$ and that on one unit of item $b$ be $₹160$, how many of each type of item should be produced to maximise the profit? Solve the problem graphically.