Linear Programming

State the limitations of L.P.P.

Define L.P.P and its advantages.

A candle plant manufactures two types of products $X$ and $Y$ and sells them at a profit of $₹5$ on type $X$ and $₹3$ on type $Y$. Each product is processed on two machines $G$ and $H$. Type $X$ requires one minute of processing time on $G$ and two minutes on $H$; type $Y$ requires one minute on $G$ and one minute on $H$. The machine $G$ is available for not more than $6 \mathrm{hours} 40 \mathrm{minutes}$, while machine $H$ is available for $8 \mathrm{hours} 20 \mathrm{minutes}$ during any working day; formulate the problem as linear programming problem.

Define unbounded feasible region in linear programming.

The bounded feasible region for a linear programming problem (LPP) is given as below:

Which one of the following can be optimal solution for the given LPP.

Which one of the following is infeasible solution to the linear programming problem (LPP), whose feasible region is given by:

The linear inequalities or equations or restrictions on the variables of a linear programming problem (L.P.P.) are called:

A manufacturing company makes two types of teaching aids $M$ and $N$ of mathematics for class XII. Each type of $M$ requires $9$ labour hours of fabricating and $1$ labour hour for finishing. Each type of $N$ requires $12$ labour hour for fabricating and $3$ labour hour for finishing. For fabricating and finishing, the maximum labour hours available per week are $180$ and $30$ respectively. The company makes a profit of $₹80$ on each piece of type $M$ and $₹120$ on each piece of type $N$. How many pieces of type $M$ and type $N$ should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

A company manufactures two types of toys, $a$ and $b$. Type $a$ requires $5$ minutes each for cutting and $10$ minutes each for assembling. Type $b$ required $8$ minutes each for cutting and $8$ minutes each for assembling. There are $3$ hours available for cutting and $4$ hours available for assembling in a day. The profit is $₹50$ each on type $a$ and $₹60$ each on type $b$. how many toys of each types should the company manufactures in a day to maximise the profit?

Anil wants to invest at the most $₹12000$ in bonds $M$ and $N$. According to rules, he has to invest at least $₹2000$ in bond $M$ and at least $₹4000$ in bond $N$. if the rate of interest of bond $M$ is $8\%$ per annum and on bond $N$, it is $10\%$per annum, how should he invest his money for maximum interest? Solve the problem graphically.

A manufacturer produces two types of steel trunks. He has two machines, $M_1$ and $M_2$. The first type of trunk requires $3$ hours on machine $M_1$ and $3$hours on machine $M_2$. The second type required $3$ hours on machine $M_1$ and $2$ hours on Machine $M_2$. Machine $M_1$ and $M_2$ can work at most for $18$ hours and $15$ hours per day respectively. He earns a profit of $₹30$ and $₹25$ per trunk of the first type and second type respectively. How many trunks of each type must he make each day to make the maximum profit?

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.

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 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.

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.

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

A manufacturer makes two product, $A$ and $B$. product $A$ sells at $₹200$ each and takes $\frac{1}{2}$hour to make. Product B sells at $₹300$ each and takes $1$ hour to make. There is a permanent order for $14$ of product $A$ and $16$ of product $B$. A working week consist of $40$ hours of production and the weekly turnover must not be less than $₹10000$. If the profit on each of the product $A$ is $₹20$ and on product $B$, it is $₹30$ then how many of each should be produced so that the profit is maximum? Also, find the maximum profit.

A manufacturer makes two types, $A$ and $B$, of teapots. Three machines are needed for the manufacture and the time required for each teapot on the machines is given below.

Each machine is available for a maximum of $6$hours per day. If the profit on each teapot of type $A$ is $75$ paise and that on each teapot of type $B$ is $50$ paise, show that $15$ teapots of type $A$ and $30$ of type $B$ should be manufactured in a day to get the maximum profit.   

A manufacturer of a line of patent medicines is preparing a production plan on medicines $A$ and $B$. There are sufficient ingredients available to make $20000$ bottles of $A$ and $40000$ bottles of $B$ but there are only $45000$ bottles into which either of the medicines can be put. Furthermore, it takes $3$ hours to prepare enough material to fill $1000$ bottles of $A$ and it takes $1$ hour to prepare enough material to fill $1000$ bottles of $B$, and there are $66$ hours available for this operation. The profit is $₹8$ per bottle for $A$ and $₹7$ per bottle for $B$. How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

A dealer wishes to purchase a number of fans and sewing machines. He has only $₹5760$ to invest and space for at most $20$ items. A fan costs him $₹360$ and a sewing machine,$₹240$. He expects to gain $₹22$ on a fan and $₹18$ on a sewing machine. Assuming that he can sell all the items he can buy, how should he invest the money in order to maximise the profit?

Mr. Dass wants to invest $₹12000$ in public provident fund ($\mathrm{PPF}$) and in national bonds. He has to invest at least $₹1000$ in $\mathrm{PPF}$ and at least $₹2000$ in bonds. If the rate of interest on $\mathrm{PPF}$ is $12\%$ per annum and that on bonds is $15\%$ per annum, how should he invest the money to earn maximum annual income? Also find the maximum annual income.