A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man - hours of labour required in each shop for the production of each unit of A and B, and the number of man - hours the firm has available per week are as follows:

The profit on the sale of A is Rs 30 per unit as compared with ₹ 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.

A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit of product A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LPP.

A firm manufactures 3 products A, B and C. The profits are Rs. 3, Rs. 2 and Rs. 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product.

Machines M₁ and M₂ have 2000 machine minutes respectively. The firm must manufacture 100 A’s, 200 B’s and 50 C’s but not more than 150 A’s. Set up a LPP to maximize the profit.

A firm manufactures two types of products A and B and sells them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines M₁ and M₂. Type A requires one minute of processing time on M₁ and two minutes of M₂; type B requires one minute on M₁ and one minute on M₂. The machine M₁ is available for not more than 6 hours 40 minutes while machine M₂ is available for 10 hours during any working day. Formulate the problem as a LPP.

A rubber company is engaged in producing three types of tyres A, B and C. Each type requires processing in two plants, Plant I and Plant II. The capacities of the two plants, in the number of tyres per day, are as follows:

The monthly demand for tyre A, B and C is 2500, 3000 and 7000 respectively. If plant I costs Rs 2500 per day, and plant II costs Rs 3500 per day to operate, how many days should each be run per month to minimize cost while meeting the demand? Formulate the problem as LPP.

A company sells two different products A and B. The two products are produced in a common production process and are sold in two different markets. The production process has a total capacity of 45000 man - hours. It takes 5 hours to produce a unit of A and B hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of units of A that can be sold is 7000 and that of B is 10, 000. If the profit is Rs 60 per unit for the product A and Rs 40 per unit for the product B, how many units of each product should be sold to maximize profit? Formulate the problem as LPP.

To maintain his health a person must fulfil certain minimum daily requirements for several kinds of nutrients. Assuming that there are only three kinds of nutrients – calcium, protein and calories and the person’s diet consists of only two food items, I and II, whose price and nutrient contents are shown in the table below:

What combination of two food items will satisfy the daily requirement and entail the least cost? Formulate this as a LPP.

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

Vitamins A and B are found in two different foods F₁ and F₂. One unit of food F1 contains 2 units of vitamin A and 3 units of vitamin B. One unit of food F₂ contains 4 units of vitamin A and 2 units of vitamin B. One unit of food F₁ and F₂ cost ₹ 50 and 25 respectively. The minimum daily requirements for a person of vitamin A and B is 40 and 50 units respectively. Assuming that anything in excess of daily minimum requirement of vitamin A and B is not harmful, find out the optimum mixture of food F₁ and F₂ at the minimum cost which meets the daily minimum requirement of vitamin A and B. Formulate this as a LPP.

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man - days on each truck but only 2 man - days on each automobile. Shop B which performs finishing operations, must work 3 man - days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man - days per week available while shop B has 135 man - days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

Two tailors A and B earn Rs 150 and Rs 200 per day respectively. A can stitch 6 shirts and 4 pants per day while B can stitch 10 shirts and 4 pants per day. Form a linear programming problem to minimize the labour cost to produce at least 60 shirts and 32 pants.

An airline agrees to charter planes for a group. The group needs at least 160 first class seats and at least 300 tourist class seats. The airlines must use at least two of its model 314 planes which have 20 first class and 30 tourist class seats. The airline will also use some of its model 535 planes which have 20 first class seats and 60 tourist class seats. Each flight of a model 314 plane costs the company Rs 100, 000 and each flight of a model 535 plane costs Rs 150, 000. How many of each type of plane should be used to minimize the flight cost? Formulate this as a LPP.

Amit’s mathematics teacher has given him three long lists of problems with the instruction to submit not more than 100 of them (correctly solved) for credit. The problem in the first set are worth 5 points each, those in the second set are worth 4 points each, and those in the third set are worth 6 points each. Amit knows from experience that he requires on the average 3 minutes to solve a 5 point problem, 2 minutes to solve a 4 point problem, and 4 minutes to solve a 6 point problem. Because he has other subjects to worry about, he cannot afford to devote more than hours altogether to his mathematics assignment. Moreover, the first two sets of problems involve numerical calculations and he knows that he cannot stand more than hours work on this type of problem. Under these circumstances, how many problems in each of these categories shall he do in order to get maximum possible credit for his efforts? Formulate this as a LPP.

A farmer has a 100 - acre farm. He can sell the tomatoes, lettuce, or radishes he can raise. The price he can obtain is Rs 1 per kilogram for tomatoes, Rs 0.75 a head for lettuce and Rs 2 per kilogram for radishes. The average yield per acre is 2000 kgs for radishes, 3000 heads of lettuce and 1000 kilograms of radishes. Fertilizer is available at Rs 0.50 per kg and the amount required per acre is 100 kgs each for tomatoes and lettuce and 50 kilograms for radishes. Labour required for sowing, cultivating and harvesting per acre is 5 man - days for tomatoes and radishes and 6 man – days for lettuce. A total of 400 man - days of labour are available at ₹ 20 per man - day. Formulate this problem as a LPP to maximize the farmer’s total profit.

A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.

A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows :

The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.

Solve each of the following linear programming problems by graphical method.

Maximize Z = 5x + 3y

Subject to :

3x + 5y ≤ 15

5x + 2y ≤ 10

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 9x + 3y

Subject to :

2x + 3y ≤ 13

3x + y ≤ 5

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Minimize Z = 18x + 10y

Subject to :

4x + y ≥ 20

2x + 3y ≥ 30

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 50x + 30y

Subject to :

2x + y ≤ 18

3x + 2y ≤ 34

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 4x + 3y

Subject to :

8x + 6y ≤ 48

3x + 4y ≤ 24

x ≤ 5 , y ≤ 6

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 15x + 10y

Subject to :

2x + 3y ≤ 70

3x + 2y ≤ 80

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 10x + 6y

Subject to :

2x + 5y ≤ 34

3x + y ≤ 12

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 3x + 4y

Subject to :

2x + 2y ≤ 80

2x + 4y ≤ 120

Solve each of the following linear programming problems by graphical method.

Maximize Z = 7x + 10y

Subject to :

x + y ≤ 30000

y ≤ 12000

x ≥ 6000

x, y ≥ 0

x ≥ y

Solve each of the following linear programming problems by graphical method.

Maximize Z = 2x + 4y

Subject to :

x + y ≥ 8

x + 4y ≥ 12

x ≥ 3

y ≥ 2

Solve each of the following linear programming problems by graphical method.

Minimize Z = 5x + 3y

Subject to :

2x + y ≥ 10

x + 3y ≥ 15

x ≤ 10

y ≤ 8

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Minimize Z = 30x + 20y

Subject to :

x + y ≤ 8

x + 4y ≥ 12

5x + 8y = 20

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 4x + 3y

Subject to :

8x + 6y ≤ 48

3x + 4y ≤ 24

x ≤ 5 , y ≤ 6

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Minimize Z = x - 5y + 20

Subject to :

x - y ≥ 0

- x + 2y ≥ 2

x ≥ 3

y ≤ 4

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 3x + 5y

Subject to :

x + 2y ≤ 20

x + y ≤ 15

y ≤ 5

X, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Minimize Z = 3x₁ + 5x₂

Subject to :

x₁ + 3x₂ ≥ 3

x₁ + x₂ ≥ 2

x₁, x₂ ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 2x + 3y

Subject to :

x + y ≥ 1

10x + y ≥ 5

x + 10y ≥ 1

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = - x₁ + 2x₂

Subject to :

- x₁ + 3x₂ ≤ 10

x₁ + x₂ ≤ 6

x₁ - x₂ ≤ 2

x₁, x₂ ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = x + y

Subject to :

- 2x + y ≤ 1

x ≤ 2

x + y ≤ 3

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = - x₁ + 2x₂

Subject to :

- x₁ + 3x₂ ≤ 10

x₁ + x₂ ≤ 6

x₁ - x₂ ≤ 2

x₁, x₂ ≥ 0

Solve each of the following linear programming problems by graphical method.

Maximize Z = 3x + 4y

Subject to :

x – y ≤ 1

x + y ≥ 3

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Show the solution zone of the following inequalities on a graph paper:

5x + y ≥ 10

x + y ≥ 6

x + 4y ≥ 12

x ≥ 0, y ≥ 0

Find x and y for which 3x + 2y is minimum subject to these inequalities. Use a graphical method.

Solve each of the following linear programming problems by graphical method.

Find the maximum and minimum value of 2x + y subject to the constraints :

x + 3y ≥ 6,

x – 3y ≤ 3,

3x + 4y ≤ 24,

- 3x + 2y ≤ 6,

5x + y ≥ 5,

x, y ≥ 0.

Solve each of the following linear programming problems by graphical method.

Find the minimum value of 3x + 5y subject to the constraints

- 2x + y ≤ 4

x + y ≥ 3

x – 2y ≤ 2

x, y ≤0

Solve each of the following linear programming problems by graphical method.

Solved the following linear programming problem graphically :

Maximize Z = 60x + 15y

Subject to constraints

x + y ≤ 50

3x + y ≤ 90

x, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below :

2x + 4y ≤ 8

3x + y ≤ 6

x + y ≤ 4

x ≥ 0, y ≥ 0

Solve each of the following linear programming problems by graphical method.

Solve the following LPP graphically :

Maximize Z = 20x + 10y

Subject to the following constraints

x + 2y ≤ 28

3x + y ≤ 24

x ≥ 2

Solve each of the following linear programming problems by graphical method.

Solve the following linear programming problem graphically :

Minimize z = 6x + 3y

Subject to the constraint

4x + y ≥ 80

x + 5y ≥ 115

3x + 2y ≤ 150

x ≥ 0, y ≥ 0

A diet of two foods F1 and F2 contains nutrients thiamine, phosphorous and iron. The amount of each nutrient in each of the food (in milligrams per 25 gms) is given in the following table :

The minimum requirement of the nutrients in the diet is 1.00 mg of thiamine, 7.50 mg of phosphorous and 10.00 mg of iron. The cost of F1 is 20 paise per 25 gms while the cost of F2 is 15 paise per 25 gms. Find the minimum cost of diet.

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

To maintain one’s health, a person must fulfill certain minimum daily requirement for the following three nutrients : calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below :

A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamine and at least 600 calories. Each unit of A contains 0.12 milligram of thiamine and 100 calories, while each unit of B contains 0.10 milligram of thiamine and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1 kg of food X and 1 kg of food Y are given below :

One kg of food X costs ₹ 5, whereas one kg of food Y costs ₹ 8. Find the least cost of the mixture which will produce the desired diet.

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F2 costs ₹ 4 per unit F2 costs ₹ 6 per unit one unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements.

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 kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs ₹ 5 per kg and rice costs ₹ 4 per kg.

A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use of cashew nuts and 540 grams of hazel nuts. Mixture A costs ₹ 8 per kg. and mixture B costs ₹ 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture he should use to minimize the cost of the bag.

One kind of cake requires 300 gm of flour and 15 gm of fat, another kind of cake requires 150 gm of flour and 30 gm of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 gm of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an LPP and solve it graphically.

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ₹ 60 kg and Food Q costs ₹ 80 kg. Food P contains 3 units / kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains 4 units / kg of Vitamin A and 2 units / kg of vitamin B. Determine the minimum cost of the mixture.

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹ 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹ 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

A dietician 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 one kg food is given below :

One kg of food X costs ₹16 and one kg of food Y costs ₹ 20. Find the least cost of the mixture which will produce the required diet?

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

If a young man drives his scooter at a speed of 25 km/hr, he has to spend Rs2 per km on petrol. If he drives the scooter at a speed of 40 km/hour, it produces air pollution and increases his expenditure on petrol to Rs 5 per km. He has a maximum of Rs100 to spend on petrol and travel a maximum distance in one hour time with less pollution. Express this problem as an LPP and solve it graphically. What value do you find here?

A manufacturer has three machines installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas Machine III must operate at least for 5 hours a day. He produces only two items, each requiring the use of three machines. The number of hours required for producing one unit each of the items on the three machines is given in the following table :

He makes a profit of Rs 6.00 on item A and Rs 4.00 on item B. Assuming that he can sell all that he produces, how many of each item should he produce to maximize his profit? Determine his maximum profit. Formulate this LPP mathematically and then solve it.

Two tailors, A and B earn ₹ 15 and ₹ 20 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost?

A factory manufactures two types of screws, A and B, each type requiring the use of two machines - an automatic and a hand - operated. It takes 4 minute on the automatic and 6 minutes on the hand - operated machines to manufacture a package of screws ‘A’, while it takes 6 minutes on the automatic and 3 minutes on the hand - operated machine to manufacture a package of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a package of screws ‘A’ at a profit of 70 P and screws ‘B’ at a profit of ₹ 1. Assuming that he can sell all the screws he can manufacture, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

A company produces two types of leather belts, say type A and B. Belt A is a superior quality and belt B is of a lower quality. Profits on each type of belt are 2 and 1.50 per belt, respectively. Each belt of type A requires twice as much time as required by a belt of type B. If all belts were of type B, the company could produce 1000 belts per day. But the supply of leather is sufficient only for 800 belts per day (both A and B combined). Belt A requires a fancy buckle and only 400 fancy buckles are available for this per day. For belt of type B, only 700 buckles are available per day.

How should the company manufacture the two types of belts in order to have a maximum overall profit?

A small manufacturer has employed 5 skilled men and 10 semi - skilled men and makes an article in two qualities deluxe model and an ordinary model. The making of a deluxe model requires 2 hrs. work by a skilled man and 2 hrs. work by a semi - skilled man. The ordinary model requires 1 hr by a skilled man and 3 hrs. by a semi - skilled man By union rules no man may work more than 8 hrs per day. The manufacturers clear profit on deluxe model is Rs 15 and on an ordinary model is Rs 10. How many of each type should be made in order to maximize his total daily profit.

A manufacturer makes two types A and B of tea - cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below :

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

A factory owner purchases two types of machines, A and B, for his factory. The requirements and limitations for the machines are as follows :

He has an area of 7600 sq.m available and 72 skilled men who can operate the machines. How many machines of each type should he buy to maximize the daily output?

A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 gm of silver and 1 gm of gold while that of type B requires 1 gm of silver and 2 gm of gold. The company can produce 9 gm of silver and 8 gm of gold. If each unit of type A brings a profit of ₹ 40 and that of type B ₹ 50, find the number of units of each type that the company should produce to maximize the profit. What is the maximum profit?

A manufacturer of Furniture makes two products : chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 4 hrs on machine A and 2 hrs on machine B . There are 16 hrs of time per day available on machine A and 30 hrs on machine B. profit gained by the manufacturer from a chair and a table is ₹ 3 and ₹ 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet of teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man - hours and yields a profit of ₹ 45, while each table uses 20 square feet of wood and 25 man - hours and yields a profit of ₹ 80. How many items of each product should be produced by the company so that the profit is maximum?

A firm manufactures two products A and B. Each product is processed on two machines M₁ and M₂. Product A requires 4 minutes of processing time on M₁ and 8 min. on M₂; product B requires 4 minutes on M₁ and 4 min. on M₂. The machine M₁ is available for not more than 8 hrs 20 min. while machine M₂ is available for 10 hrs. during any working day. The products A and B are sold at a profit of ₹ 3 and ₹ 4 respectively.Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

A firm manufacturing two type of electric items, A and B, can make a profit of 20 per unit of A and ₹ 30 per unit of B. Each unit of A requires 3 motors and 4 transformers and each unit of B requires 2 motors and 4 transformers. The total supply of these per month is restricted to 210 motors and 300 transformers. Type B is an export model requiring a voltage stabilizer which has a supply restricted to 65 units per month. Formulate the linear programming problem for maximum profit and solve it graphically.

A factory uses three different resources for the manufacture of two different products,20 units of the resources a, 12 units of B and 16 units of C being available 1 unit of the first product requires 2,2 and 4 units of the respective resources and 1 unit of the second product requires 4,2 and 0 units of respective resources. It is known that the first product gives a profit of 2 monetary units per unit and the second 3. Formulate the linear programming problem. How many units of each product should be manufactured for maximizing the profit? Solve it graphically.

A publisher sells a hard cover edition of a text book for ₹ 72.00 and a paperback edition of the same ext for ₹ 40.00. Costs to the publisher are ₹ 56.00 and ₹ 28.00 per book respectively in addition to weekly costs of ₹ 9600.00. Both types require 5 minutes of printing time, although hardcover requires 10 minutes binding time and the paperback requires only 2 minutes. Both the printing and binding operations have 4,800 minutes available each week. How many of each

type of book should be produced in order to maximize profit?

A firm manufactures headache pills in two sizes A and b. Size A contains 2 grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine ; size B contains 1 grain of aspirin, 8 grains of bicarbonate and 66 grains of codeine. It has been found by users that it requires at least 12 grains of aspirin, 7.4 grains of bicarbonate and 24 grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief. Determine also the quantity of codeine consumed by patient.

A chemical company produces two compounds, A and B. The following table gives the units of ingredients, C and D per kg of compounds A and B as well as minimum requirements of C and D and costs per kg of A and B. Find the quantities of A and B which would give a supply of C and D at a minimum cost.

A company manufactures two type of novelty Souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours available for assembling. The profit is 50 paisa each for type A and 60 paisa each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

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

A manufacturer produces two type of steel trunks. He has two machines A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 2 hours on machine B . Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs 30 and Rs 25 per trunk of the first type and the second type respectively. How many trunks of each type must
he make each day to make maximum profit?

A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials 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. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hours 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 manufacturer schedule his production in order to maximize his profit?

An aeroplane can carry a maximum of 200 passengers. A profit of ₹ 400 is made on each first class ticket and a profit of ₹ 600 is made on each economy class ticket. The airline reserves at least 20 seats of first class. However, at least 4 times as many passengers prefer to travel by economy class to the first class. Determine how many each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit.

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

Anil wants to invest at most ₹ 12000 in Saving Certificates and National Saving Bonds. According to rules, he has to invest at least ₹ 2000 in Saving Certificates and at least 4000 in National Saving Bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on National Saving Bonds is 10% per annum, how much money should he invest to earn

maximum yearly income? Find also his maximum yearly income.

A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of 1400 to purchase young trees. He has the choice of two type of trees. Type A requires 10 sq.m of ground per tree and costs ₹ 20 per tree and type B requires 20 sq.m of ground per tree and costs ₹ 25 per tree. When fully grown, type A produces an average of 20 kg of fruit which can be sold at a profit of ₹ 2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of ₹ 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?

A diet of two foods F1 and F2 contains nutrients thiamine, phosphorous and iron. The amount of each nutrient in each of the food (in milligrams per 25 gms) is given in the following table :

The minimum requirement of the nutrients in the diet is 1.00 mg of thiamine, 7.50 mg of phosphorous and 10.00 mg of iron. The cost of F1 is 20 paise per 25 gms while the cost of F2 is 15 paise per 25 gms. Find the minimum cost of diet.

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

To maintain one’s health, a person must fulfill certain minimum daily requirement for the following three nutrients : calcium, protein and calories. The diet consists of only items I and II whose prices and nutrient contents are shown below :

A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamine and at least 600 calories. Each unit of A contains 0.12 milligram of thiamine and 100 calories, while each unit of B contains 0.10 milligram of thiamine and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1 kg of food X and 1 kg of food Y are given below :

One kg of food X costs ₹ 5, whereas one kg of food Y costs ₹ 8. Find the least cost of the mixture which will produce the desired diet.

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F2 costs ₹ 4 per unit F2 costs ₹ 6 per unit one unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these foods and also meets the mineral nutritional requirements.

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 kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs ₹ 5 per kg and rice costs ₹ 4 per kg.

A wholesale dealer deals in two kinds, A and B (say) of mixture of nuts. Each kg of mixture A contains 60 grams of almonds, 30 grams of cashew nuts and 30 grams of hazel nuts. Each kg of mixture B contains 30 grams of almonds, 60 grams of cashew nuts and 180 grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use of cashew nuts and 540 grams of hazel nuts. Mixture A costs ₹ 8 per kg. and mixture B costs ₹ 12 per kg. Assuming that mixtures A and B are uniform, use graphical method to determine the number of kg. of each mixture he should use to minimize the cost of the bag.

One kind of cake requires 300 gm of flour and 15 gm of fat, another kind of cake requires 150 gm of flour and 30 gm of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 gm of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an LPP and solve it graphically.

Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs ₹ 60 kg and Food Q costs ₹ 80 kg. Food P contains 3 units / kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains 4 units / kg of Vitamin A and 2 units / kg of vitamin B. Determine the minimum cost of the mixture.

One kind of cake requires 200 g of flour and 25 g of fat, and another kind of cake requires 100 g of flour and 50 g of fat. Find the maximum number of cakes which can be made from 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

A dietician has to develop a special diet using two foods P and Q. Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6 units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least 240 units of calcium, at least 460 units of iron and at most 300 units of cholesterol. How many packets of each food should be used to minimize the amount of vitamin A in the diet? What is the minimum amount of vitamin A?

A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹ 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹ 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

A dietician 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 one kg food is given below :

One kg of food X costs ₹16 and one kg of food Y costs ₹ 20. Find the least cost of the mixture which will produce the required diet?

A fruit grower can use two types of fertilizer in his garden, brand P and Q. The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least 240 kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine.

If the grower wants to minimize the amount of nitrogen added to the garden, how many bags of each brand should be used? What is the minimum amount of nitrogen added in the garden?

Two godowns, A and B, have grain storage 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 :

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

A medical company has factories at two places, A and B. From these places, supply is made to each of its three agencies situated at P, Q and R. The monthly requirements of the agencies are respectively, 40, 40 and 50 packets of the medicines, while the production capacity of the factories, A and B are 60 and 70 packet respectively. The transportation cost per packet from the factories to the agencies are given below :

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

The solution set of the inequation 2x + y > 5 is

Objective function of a LPP is

Which of the following sets are convex?

Let X₁ and X₂ are optimal solutions of a LPP, then

The maximum value of Z = 4x + 2y subjected to the constraints 2x + 3y ≤ 18, x + y ≥ 10; x, y ≥ 0 is

The optimal value of the objective function is attained at the points

The maximum value of Z = 4x + 3y subjected to the constraints 3x + 2y ≥ 160, 5x + 2y ≥ 200, x + 2y ≥ 80; x, y ≥ 0 is

Consider a LPP given by

Minimum Z = 6x + 10y

Subjected to x ≥ 6; y ≥ 2; 2x + y ≥ 10; x, y ≥ 0

Redundant constraints in this LPP are

The objective function Z = 4x + 3y can be maximized subjected to the constraints 3x + 4y ≤ 24, 8x + 6y ≤ 48, x ≤ 5, y ≤ 6; x, y ≥ 0

If the constraints in a linear programming problem are changed

Which of the following statements is correct?

Which of the following is not a convex set?

By graphical method, the solution of linear programming problem

Maximize Z = 3x₁ + 5x₂

Subject to 3x₁ + 2x₂ ≤ 18

x₁ ≤ 4

x₂ ≤ 6

x₁ ≥ 0, x₂ ≥ 0, is

The region represented by the inequation system x, y ≥ 0, y ≤ 6, x + y ≤ 3 is

The point at which the maximum value of x + y, subject to the constraints x + 2y ≤ 70, 2x + y ≤ 95, x, y ≥ 0 is obtained, is

The value of objective function is maximum under linear constraints

The corner points of the feasible region determined by the following system of linear inequalities:

2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is