Linear Programming

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

Let the company make $x$ no of $1^{\text {st }}$ type of teaching aid and y no of $2^{\text {nd }}$ type of teaching aid. $\therefore$ According to the question, $9 x+12 y \leq 180, x+3 y \leq 30, x...

read more

One kind of cake requires 200 \mathrm{~g} of flour and 25 \mathrm{~g} of fat, another kind of cake requires 100 \mathrm{~g} of flour and 50 \mathrm{~g}, of fat. Find the maximum number of cakes which can be made from 5 \mathrm{~kg} of flour and 1 \mathrm{~kg} of fat, assuming. that there is no shortage of the other ingredients used in making the cakes. Make it an LPP and solve it graphically.

Let the company make $x$ no of $1^{\text {st }}$ kind and $y$ no of $2^{\text {nd }}$ cakes. $\therefore$ According to the question, $200 x+100 y \leq 5000,25 x+50 y \leq 1000, x \geq 0, y \geq 0$...

read more

A company manufacture two types of toys A and B. type A requires 5 minutes each for cutting and 10 minutes for each assembling. Type B requires 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. He earns a profit of \pm 50 each on type A and \pm 60 each on type B. How many toys of each type should the company manufacture in a day to maximize the profit?

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

read more

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

Let the invested money in bond $A$ be $x$ and in bond $B$ be $y$. $\therefore$ According to the question, $\mathrm{X}+\mathrm{y} \leq 12000, \mathrm{x} \geq 2000, \mathrm{y} \geq 4000$ Maximize...

read more

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs.5760 to invest and has space for at most 20 items. A fan costs him Rs.360 and a sewing machine \pm240. He expects to sell a fan at a profit of \approx 22 and a sewing machine at a profit of \pm 18 . Assuming that he can sell all the items that he buys, how should he invest his money to maximize the profit? Solve the graphically and find the maximum profit.

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

read more

Kellogg is a new cereal formed of a mixture of bran and rice, that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilograms, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost producing this new cereal if bran costs \pm 5 per kilogram and rice costs \pm4 per kilogram.

Let $x$ and $y$ be number of kilograms of bran and rice. $\therefore$ According to the question, $80 x+100 y \geq 88,40 x+30 y \geq 36, x \geq 0, y \geq 0$ Minimize $Z=5 x+4 y$ The feasible region...

read more

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 \pm 50 each. on type A and \pm 60 each on type B. how many toys of each types should the company manufactures in a day to maximize the profit?

Let the company manufacture $x$ and y numbers of toys $A$ and $B$. $\therefore$ According to the question, $5 x+8 y \leq 180,10 x+8 y \leq 240, x \geq 0, y \geq 0$ Maximize $Z=50 x+60 y$ The...

read more

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

Let the manufacturer manufacture $\mathrm{x}$ and y numbers of type 1 and type 2 trunks. $\therefore$ According to the question, $3 x+3 y \leq 18,3 x+2 y \leq 15, x \geq 0, y \geq 0$ Maximize $Z=30...

read more

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 \pm 300 and that on one unit of item B be \pm 160, how many of each type of item should be produced to maximize the profit? Solve the problem graphically.

Let the firm manufacture $x$ number of A and y number of B products. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+160 y$ The feasible...

read more

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 A, 12 units of vitamin B and 8 units of vitamin C . The vitamin contents of 1 \mathrm{~kg} of each food are given below.

If $1 \mathrm{~kg}$ of food $\mathrm{X}$ cost $\pm 6$ and $1 \mathrm{~kg}$ of food $\mathrm{Y}$ costs $\pm 10$, find the minimum cost of the mixture which will produce the diet. Solution: Let $x$...

read more

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 \pm 4 and \pm 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.

Let $x$ and $y$ be number of units of food $A$ and $B$. $\therefore$ According to the question, $200 x+100 y \geq 4000, x+2 y \geq 50,40 x+40 y \geq 1400, x \geq 0, y \geq 0$ Minimize $Z=4 x+3 y$...

read more

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 A and 10 units of vitamin C. Food X contains 2 units/kg of vitamin A and 1 unit / \mathrm{kg} of vitamin C, while food Y contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs \pm 5 per \mathrm{kg} to purchase the food \mathrm{X} and \pm7 per kg to purchase the food Y. Determine the minimum cost of such a mixture.

Let $x$ and $y$ be number of units of $X$ and $Y$. $\therefore$ According to the question, $2 x+y \geq 8, x+2 y \geq 10, x \geq 0, y \geq 0$ Minimize $Z=5 x+7 y$ The feasible region determined $2...

read more

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 \pm 20 per unit and product B costs \pm40 per unit. How many units of each product must be taken to minimize the cost? Also, find the minimum cost.

Let $x$ and $y$ be number of units of products of $A$ and $B$. $\therefore$ According to the question, $36 x+6 y \geq 108,3 x+12 y \geq 36,20 x+10 y \geq 100, x \geq 0, y \geq 0$ Minimize $Z=20 x+40...

read more

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

    \[\begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Distance $($ in $\mathrm{km})$} \\ \hline To & $A$ & $B$ \\ \hline From & $A$ & $B$ \\ \hline$D$ & 7 & 3 \\ \hline$E$ & 6 & 4 \\ \hline$F$ & 3 & 2 \\ \hline \end{tabular}\]

Assuming that the transportation cost per \mathrm{km} is re 1 per litre, how should the delivery be scheduled in order that the transportation cost is minimum?

Let $x$ liters of petrol be transported from $A$ to $D$ and y liters of petrol be transported from $A$ to $E$. Therefore, $7000-(x+y)$ will be transported to $F$. Also, ( $4500-x$ ) liters of...

read more

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 \mathrm{P}, \mathrm{Q} and \mathrm{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. Solution: Let $x$ packets of medicines be...

read more

A brick manufacture 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 fulfill the orders so as to keep the cost of transportation minimum?

Let $x$ bricks be transported from $P$ to $A$ and y bricks be transported from $P$ to $B$. Therefore, $30000-(x+y)$ will be transported to $C$. Also, (15000-x) bricks, ( $20000-y)$ bricks and...

read more

Two godowns, A and B, have a grain storage capacity of 100 quintals and 50 quintals respectively. Their supply goes to three ration shops, D, E and F, whose requirements are 60,50 and 40 quintals respectively. The costs of transportation per quintal from the godowns to the shops are given in the following table.

$$ \begin{tabular}{|c|c|c|} \hline & \multicolumn{2}{|c|}{ Cost of transportation (in 2 perquintal) } \\ \hline To & From & B & B \\ \hline$D$ & $6.00$ & 400 \\ \hline$E$ & $3.00$ & $2.00$ \\...

read more

A gardener has a supply of fertilizers of the type 1 which consist of 10 \% nitrogen and 6 \% phosphoric acid, and of the type II which consist of 5 \% nitrogen and 10 \% phosphoric acid. After testing the soil condition, he finds that he needs at least 14 \mathrm{~kg} of nitrogen and 14 \mathrm{~kg} of phosphoric acid for his crop. If the type – I fertilizer costs 60 paise per kg and the type – II fertilizer costs 40 paise per kg, determine how many kilograms of each type of fertilizer should be used so that the nutrient requirement are met at a minimum cost. What is the minimum cost?

Let $x$ and $y$ be number of kilograms of fertilizer I and II, $\therefore$ According to the question, $0.10 x+0.05 y \geq 14,0.06 x+0.10 y \geq 14, x \geq 0, y \geq 0$ Minimize $Z=0.60 x+0.40 y$...

read more

A publisher sells a hardcover edition of a book for \pm 72 and a paperback edition of the same for \pm 40 . Costs to minutes of printing time although the hardcover edition requires 10 minutes of binding time and the paperback edition requires only 2 minutes. Both the printing and binding operations have 4800 minutes available each week. How many of each type of books should be produced in order to maximize the profit? Also, find the maximum profit per week.

Let $x$ and $y$ be number of hardcover and paperback edition of the book. $\therefore$ According to the question, $5 x+5 y \leq 4800,10 x+2 y \leq 4800, x \geq 0, y \geq 0$ Maximize $Z=(72 x+40...

read more

A man owns a field area 1000 \mathrm{~m}^{2}. He wants to plant fruit trees in it. He has a sum of \pm 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 \mathrm{~m}^{2} of ground per trees and costs \pm 20 per tree, and type B requires 20 \mathrm{~m}^{2} of ground per tree and costs \pm 25 per tree. When full grown, a type A tree produces an average of 20 \mathrm{~kg} of fruit which can be sold at a profit \pm 2 per \mathrm{kg} and type -\mathrm{B} tree produces an average of 40 \mathrm{~kg} of fruit which can be sold at a profit of \pm 1.50 per \mathrm{kg}. How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?

Let $x$ and $y$ be number of $A$ and B trees. $\therefore$ According to the question, $20 x+25 y \leq 1400,10 x+20 y \leq 1000, x \geq 0, y \geq 0$ Maximize $Z=40 x+60 y$ The feasible region...

read more

A manufacture makes two product, A and B. product A sells at \mp 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 \mp10000. If the profit on each of the product A is \mp 20 and on product B, it is \mp 30 then how many of each should be produced so that the profit is maximum? Also, find the maximum profit.

Let $x$ and $y$ be number of $A$ and $B$ products. $\therefore$ According to the question. $0.5 x+y \leq 40,200 x+300 y \geq 10000, x \geq 14, y \geq 16$ Maximize $Z=20 x+30 y$ The feasible region...

read more

A manufacture 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.

$$ \begin{tabular}{|l|l|l|l|} \hline Machine & \multicolumn{2}{|l|}{ Time (in minutes) } \\ \hline Type & I & II & III \\ \hline A & 12 & 18 & 6 \\ \hline B & 6 & 0 & 9 \\ \hline \end{tabular} $$...

read more

A small firm manufactures gold rings and chains. The combined number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and half an hour for a chain. The maximum number of hour to available per day is 16 . If the profit on a ring is 300 and that on a chain is 190, how many of each should be manufactured daily so as to maximize the profit?

Let $x$ and $y$ be number of gold rings and chains. $\therefore$ According to the question, $x+y \leq 24, x+0.5 y \leq 16, x \geq 0, y \geq 0$ Maximize $Z=300 x+190 y$ The feasible region determined...

read more

A company producing soft drinks has a contrast which requires a minimum of 80 units of chemical A and 60 , units of chemical B to go in each bottle of the drink. The chemical are available in a prepared mix from two different suppliers. Supplier X has a mix of 4 units of A and 2 units of B that costs Rs.10, and the supplier Y has a mix of 1 unit of A and 1 unit of B that costs \mp4. How many mixes from X and Y should the company purchase to honor the contract requirement and yet minimize the cost?

Let $x$ and $y$ be number of mixes from suppliers $X$ and $Y$. $\therefore$ According to the question, $4 x+y \geq 80,2 x+y \geq 60, x \geq 0, y \geq 0$ Minimize $Z=10 x+4 y$ The feasible region...

read more

A small manufacture has employed 5 skilled men and 10 semiskilled men and makes an article in two qualities, a deluxe model and an ordinary model. The making of a deluxe model requires 2 hours work by a skilled man and 2 hours work by a semiskilled man. The ordinary model requires 1 hour by a skilled man and 3 hours by a semiskilled man. By union rules, no man can work more than 8 hours per day. The manufacture gains \mp 15 on the deluxe model and \mp 10 on the ordinary model. How many of each type should be made in order to maximize his total daily profit? Also, find the maximum daily profit.

Let $x$ and $y$ be number of deluxe article manufactured and ordinary article manufactured. $\therefore$ According to the question, $2 x+y \leq 40,2 x+3 y \leq 80, x \geq 0, y \geq 0$ Maximize $Z=15...

read more

A toy company manufactures two types of dolls, A and B. Each doll of type B take twice as long to produce as one of type A, and the company would have time to make a maximum of 2000 per day, if it produces only type A. the supply of plastic is sufficient to produce 1500 dolls per day (both A and B combined). Type B requires a fancy dress of which there are only 600 per day available. If the company makes profit of Rs.3 and \pm5 per dolls respectively on dolls A and B, how many of each should be produced per day in order to maximize the profit? Also, find the maximum profit.

Let $x$ and $y$ be number of doll A manufactured and doll B manufactured. $\therefore$ According to the question, $x+y \leq 1500, x+2 y \leq 2000, y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=3 x+5...

read more

A manufacture 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 \mp 8 per bottle for A and \mp 7 per bottle for B. How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.

Let $x$ and $y$ be number of bottles of medicines $A$ and $B$ be prepared. $\therefore$ According to the question, $x+y \leq 45000,3 x+y \leq 66000, x \leq 20000, y \leq 40000 \cdot x \geq 0, y \geq...

read more

A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of 0.25 and \approx 0.50 each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

Let $x$ and $y$ be number of soaps be manufactured of $1^{\text {st }}$ and $2^{\text {nd }}$ type. $\therefore$ According to the question, $2 x+3 y \leq 480,3 x+5 y \leq 480, x \geq 0, y \geq 0$...

read more

A firm manufactures two types of products, A and B, and sells them at a profit of \approx 2 on type A and B. Each product is processed on two machines, M_{1} and M_{2}. Type A requires one minute of processing time on M_{1} and two minutes on M_{2} . Type B requires one minute on M_{1} and one minute on M_{2} is available for not more than 6 hours 40 minutes while M_{2} is available for at most 10 hours a day. Find how many products of each type the firm should produce each day in order to get maximum profit.

Let the firm manufacture $x$ number of Aand y number of $B$ products. $\therefore$ According to the question, $X+y \leq 400,2 x+y \leq 600, x \geq 0, y \geq 0$ Maximize $Z=2 x+2 y$ The feasible...

read more

A dealer wishes to purchase a number of fans and sewing machines. He has only \pm 5760 to invest and space and \pm 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 maximize the profit?

Let the number of fans bought be $x$ and sewing machines bought be $y$. $\therefore$ According to the question, $360 x+240 y \leq 5760, x+y \leq 20, x \geq 0, y \geq 0$ Maximize $Z=22 x+18 y$ The...

read more

A manufacture produces nuts and bolts for industrial machinery. It takes 1 hour of work on machine A and 3 hours on machine B to produces a packet of nuts while it takes 3 hours on machine A and 1 hours on machine B to produce a packet of bolts. He earns a profit \mp17.50 per packet on nuts and \mp7 per packet on bolts. How many packets of each should be produced each day so as to maximize his profit if he operates, each machine for at the most 12 hours a day? Also find the maximum profit.

Let the number of packets of nuts and bolts be $x$ and y respectively. $\therefore$ According to the question, $x+3 y \leq 12,3 x+y \leq 12, x \geq 0, y \geq 0$ Maximize $Z=17.50 x+7 y$ The feasible...

read more

A man has \pm1500 to purchase rice and wheat. A bag of rice and a bag of wheat cost \mp 180 and 120 respectively. He has storage capacity of 10 bags only. He earns a profit of \mp 11 and 78 per bag of rice and wheat respectively. How many bags of each must he buy to make maximum profit?

Let the number of wheat and rice bags be $x$ and $y$. $\therefore$ According to the question, $120 x+180 y \leq 1500, x+y \leq 10, x \geq 0, y \geq 0$ Maximize $Z=8 x+11 y$ The feasible region...

read more

A small firm manufactures necklace and bracelets. The total number of necklace and bracelet that it can handle per day is at most 24. It takes 1 hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16 . If the profit on a necklace is \pm 100 and that on a bracelet is \pm 300, how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Let the firm manufacture $x$ number of necklaces and y number of bracelets a day. $\therefore$ According to the question, $x+y \leq 24,0.5 x+y \leq 16 x \geq 1, y \geq 1$ Maximize $Z=100 x+300 y$...

read more

Mr.Dass wants to invest Rs 12000 in public provident fund (PPF) and in national bonds. He has to invest at \mathrm{~ l e a s t ~ Rs. 1 0 0 0 ~ i n ~ P P F ~ a n d ~ a t ~ l e a s t ~ } bonds is 15 \% per annum, how should he invest the money to eam maximum annual income? Also find the maximum annual income.

Let the invested money in PPF be $x$ and in national bonds be $y$. $\therefore$ According to the question, $\begin{array}{l} \mathrm{X}+\mathrm{y} \leq 12000 \\ \mathrm{x} \geq 1000, \mathrm{y} \geq...

read more