LINEAR PROGRAMMING. Vitamin A Vitamin B Vitamin C Food X Food Y 2 2 1

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1 LINEAR PROGRAMMING 6 Marks Questions 1. 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 kg of food is given below: Vitamin A Vitamin B Vitamin C Food X Food Y kg of food X costs Rs6 and 1 kg of food Y costs Rs10.Formulate the above problem as a linear programming problem and find the least cost of the mixture which will produce the diet graphically. Ans.2 kg of food X and 4 kg of food Y. The minimum cost of the mixture is Rs If a young man rides his motor-cycle at 25 km per hour, he had to spend of 2 per km on petrol with very little pollution in the air.if he rides it at a faster speed of 40 km per hour, the petrol costs increases to Rs5 per km and rate of pollution also increases. He has Rs100 to spend on petrol and wishes to find what is the maximum distance he can travel within one hour. Express this problem as an LPP. Solve it graphically to find the distance to be covered with different speeds. Ans.50/3 km at the rate of 25 km/h and 40/3 km at the rate of 40 km/h., max. distance30 km.. 3. One kind of cake requires 200 g of flour and 25 kg of fat, 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 5kg of flour and 1kg 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 grammatically. Ans. Max. no. of cakes is 30, 20 cakes of Ist kind and 10 cakes of IInd kind. 4. A dealer in rural area wishes to purchase a number of sewing machines. He has only Rs5760 to invest and has space for almost 20 items for storage. An electronic sewing machine cost his Rs360 and manually operated sewing machine Rs240. He can sell an electric sewing machine at a profit of Rs22 and a manually operated sewing machine at a rate of Rs18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Make it as an LPP and solve it graphically. Ans. 8 electronic and 12 manually operated sewing machines, max. profit isrs A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour of finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30, respectively. The company makes a profit of Rs 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 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? Ans. Max. profit isrs 1680 per week, 12 pieces of type A and 6 pieces of type B 6. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2h on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1h on the grinding/cutting machine and 2 h on the

2 sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 h and the grinding/cutting machine for at the most 20 h and the grinding/cutting machine for at most 12 h. The profit from the sale of a lamp is Rs25 and that from a shade is Rs15. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit? Formulate an LPP and solve it grammatically. Ans. 4 pedestal lamps and 4 wooden shades daily, max. profit isrs A cooperative society of farmers has 50hec of land to grow two crops A and B. the profits from crops A and B per hectare are estimated as Rs and Rs 9000, respectively. To control weeds, a liquids herbicide has to be used for crops A and B at the rate of 20L per hectare and 10Lper hectare,respectively. Further, not more than 800L of herbicide should be used in order to protect fish and wildlife using a pond which collects drainage from this land. Keeping in mind that the protection of fish and other wildlife is more important than earning profit. How much land should be allocated to each crop so as to maximize the total profit?formulated the above as an LPP and solve it graphically. Ans. Max. profit is Rs , land allocated 30 hect. for crop A and 20 hect. for crop B. 8. A manufacture considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 13 units of workers ( male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A,2 workers and 1 unit of capital are required, while 3 workers and 1 unit of capital are is required to produce one unit of B. if A and B are priced at Rs 100 and Rs 120 per units respectively, then how should he use his resources to maximize the total revenue? Formulate the above as an LPP and solve it graphically. Ans. 3 units of goods A and 8 units of goods B, max. revenue isrs A manufacture produces nuts and bolts. It takes 1h of work on machine A and 3h on machines B to produces package of nuts. It takes 3h on machine A and 1h on machine B to produce of a package of bolts. He earns a profit of per package of bolts. How many packages of each should be produced each day so as to maximize his profits, if he operates his machines for at most 12h a day? Formulate above as a Linear Programming Problem and solve it graphically. Ans. Max.profit is Rs 73.50, 3 packages of nuts and 3 packages of bolts. 10. A dietician wishes to mix two types of foods in such a way that the vitamin contents of mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food 1 contins 2 units per kg of vitamin A and 1 unit per kg of vitamin C, while food II contains 1 units per kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 5 per kg to purchase food 1 and Rs 7 per kg to purchase food II. find the minimum cost of such a mixture. Formulate above as an LPP and solve it graphically.. Ans. The minimum cost is Rs 38, food 1 is 2units and food II is 4units 11. A factory make tennis rackets and cricket bats. Tennis rackets takes 1.5 h of machine time and 3 h of craft man s time and 1h of craft man s times.in a day,the factory has the availability of not more than 42h of machine time and 24h of craft man s time. if the profits on a rackets and a bat are Rs 20 and Rs 10 respectively, then find the number of the tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an LPP and solve it. Ans. Maximum profit is Rs 200, 4 tennis rackets and 12 cricket bats.

3 12. A merchant plans to sell two types of personal computers, a desktop model and a portable model that will cost Rs and Rs40000, respectively. He estimates that the 12. A merchant plans to sell two types of personal computers, a desktop model and a portable model that will cost Rs and Rs40000,respectively.He estimates that the total monthly demand of the computers will not exceed 250 units. Determine the number of the units of ʆeach type of computers which the merchant should stock to get maximum profits, if he does not to invest more than Rs 70 lakhs and his profit on the desktop model is Rs 4500 and on the portable model is Rs Make an LPP and solve it graphically. Ans. Max. profit is Rs , 200 desktop computers and 50 portable computers. 13. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/ cutting machine and a sprayer. It takes 2 h on the grinding/ cutting machine and 3 h on the sprayer to manufacture a pedestal lamp. It takes 1 h on the grinding/ cutting machine and 2 h on the sprayer to manufacture a shade. On any day, the sprayer is available for atmost 20 hand the grinding/ cutting machine for atmost 12 h. The profit from the sale of a lamp isrs and that from a shade is Rs3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit?make an LPP and solve it graphically. Ans. Maximum profit is Rs 32, when 4 lamps and 4 shades are sold. 14. A dietician wishes to mix two type of food in such a way that the vitamin contents of the mixture contain atleast 8 unit of vitamin A and 10 unit of vitamin C. food I contains 2 unit per kg of vitamin A and 1 kg per unit of vitamin C. food II contains 1 unit per kg of vitamin A and 2 unit per kg of vitamin C. it cost RS 50 per kg to purchase food I and RS 70 per kg to purchase food II formulate the problem as a linear programming problem to mimimise the cost of such mixture and find the minimize cost graphically. Ans. Min. cost of the mixture is Rs 380, food I contains 2 units and food II contains 4 units. 15. A library has to accommodate two different type of books on a shelf.the book are 6cm and 4cm thick and weight 1 kg and 1 ½ kg each respecyively the shell is 96 cm long and almost and can support a weight of 1kg. how should the shelf would be filled with a book of two type in order to include the greatest number of books?make it as an LPP and solve it graphically. Ans. Max. no. of books is 18, books of I type is 12 and books of II type is A dealer deals in two items A and B he has RS to invest and a space to store almost 80 pieces.item A cost him RS 300 and item B cost him RS 150. he can sell item A and B at profit of RS 40 and RS 25, respectively. assuming that he can sell all that he buys, formulate the above as a linear programming problem for maximum profit and solve it graphically. Ans. Max. profit is Rs 2300, 20 items of type A and 60 items of type B. 17. One kind of cake require 300g of flour and 15 g of fat, another kind of cake required 150g of flour and 30 g of fat.find the maximum number of cakes which can be made from 7.5kg of flour and 600g of fat, assuming that there is no shortage of other ingredients used in making the cakes. make it as anlpp and solve it graphically. Ans. Max. number is 30, one kind of cakes is 20 and second kind is A small firm manufactures gold rings and chains. The total number of rings and chains manufactures per day is almost 24 days to take 1h to make a ring and 30 min to make a chain.the maximum number of hours available per dayis 16. if the profit of a ring is RS300 and that on a chain is RS 190, then find the number of rings and chains that should be manufactured per day so as to earn the maximum profit. make it as an LPPand solve it graphically.

4 Ans. Max. profit of Rs 5440, 8 gold rings and 16 chains. 19. A man has RS1500 for purchasing wheat and rice and. A bag of rice and a bag of wheat cost RS180 and RS 120, respectively. he has a storage capacity of only 10 bags. he earns a profit of RS 11 and RS 9 per bag of rice and wheat, respectively. formulate the problem as an LPP to find the number of bags of each type he should buy for getting maximum profit and solve it graphically. Ans. Max. profit is Rs 100, 5 rice bags and 5 wheat bags. 20. Two tailor 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 pants 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 maximum labour cost? Make it as an LPP and solve the problem graphically. Ans. Minimum labour cost is Rs 1350, tailor A works for 5 days and tailor B works for 3 days. 21. A diet is to contain at least 80 units of vitamin A and 100 unit of minerals.two food F 1 and F 2 are available. food F1 cost RS 4 per unit and food F2 cost RS 6 per unit. one unit of food F 1 contain 3 units of vitamin A and 4 unit of minerals. one unit of F 2 contain 6 unit of vitamin A and 3 unit of minerals. formulate this as linear programming problem and find graphically the minimum cost of diet that consist of mixture of these two foods and also meet the minerals nutritional requirements. Ans. Minimum cost is Rs 104, 24 units of food F 1 and 4/3 units of food F A diet for a sick person must contain at least4000 units of vitamins,50 units of minerals and 1400 units of calories. Two foods A and Bare available at a cost of Rs5 and Rs4 per unit, respectively. One unit of food A contains200 units of vitamins, 1 unit of minerals and 40 units of calories, while one unit of food Bcontains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the food A and B should be used to have least cost, but it must satisfythe requirements of the sick person? Formulate the question as an LLP and solve it graphically. Ans. Min. cost is Rs 145, 5 units of food A and 30 units of Food B. 23. A factory owner purchases two types of machines A and B for his factory. Therequirements and the limitations for the machines are as follows: Machines Area occupied Labour force Daily inputs (in units) A B 1000 m m 2 12 men 8 men He has maximum area of 9000 m 2 available and 72 skilled persons who can operate both the machines. How many machines of each type be bought to maximise the daily output?formulate the problem as an LPP and solve it graphically. Ans. The maximum output is at B and C, machines of type A is 6 and machines of type B is An aeroplane can carry a maximum of 200 passengers.a profit of Rs.500 is made on each executive class ticket out of which 20% will go to the welfare fund of the employees. Similarly a profit of Rs.400 is made on each economy ticket out of which 25% will go for The improvement of facilities provided to economy class passengers. In both cases, the

5 Remaining profit goes to the airline s fund. The airline reserves at least 20 seats for executiveclass. However at least four times as many passengers prefer to travel by economy class thanby the executive class. Determine how many tickets of each type must be sold in order tomaximize the net profit of the airline. Formulate the L.P.P and solve graphically. Do you think,more passengers would prefer to travel by such an airline than by others? Ans. Max. profit is Rs 84000, Executive class tickets 40 and Economy class tickets A retired person wants to invest an amount of upto Rs His agent recommends investing into two types of bonds A and B, bond A yielding 8% return on the amount invested and bond B yielding 10% return on the amount invested. After some consideration, he decides to invest atleastrs in bond A and atleastrs in bond B. What should his agent suggest if he wants to maximize his return on investment? Find also the maximum income. Formulate the above as a L.P.P and solve the problem graphically Ans.Max. income is Rs 1160, Rs 2000 in bond A and Rs in bond B 26) A dealer wishes to purchase a number of fans and sewing machines. He has only Rs to invest and has space for at most 20 items. A fan costs Rs. 360 and a sewing machine cost Rs He can sell a fan at a profit Rs. 22 and a sewing machine at Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximise the profit? Translate the problem Mathematically and solve it Ans- Maximum profit is Rs. 392 when he sells 8 fans and 12 sewing machines. 27-) 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. 28. 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 How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost? Ans-Minimum transportation cost is Rs. 510 and this is so when 10 quintals, 50 quintals, 40 quintals are transported from the godown at A and 50 quintals, 0 quintals, 0 quintal are transported from the godown at B to the ration shops at D, E, F respectively.

6 30-)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 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? Ans- Cost is minimum when tailor A and tailor B work for 5 days and 3 days respectively. 31-)A company sells two different products A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 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 70 and that of B is 125. If the profit is Rs.20 per unit for the product A and Rs.15 per unit for the product B, how many units for each product should be sold to maximize profit. Ans- For maximum profit, 25 units of product A and 125 units of product B are produced and sold. 32-) An aeroplane can carry a maximum of 250 passengers. A profit of Rs 500 is made on each first class ticket and a profit of Rs350 is made on each economy class ticket. The airline reserves at least 25 seats for first class. However, at least 3 times as many passengers prefer to travel by economy class than by the first class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. Ans- For maximum profit, 62 first class tickets and 188 economy class tickets should be sold. 33-)A small firm manufactures gold ring and chains. the total number of ring and chains manufactured per day is at most 24.it takes 1 hour and 30 min to make a chain.max number of hrs available per day is 16.if the profit on the ring is Rs 300 and that on the chain is Rs190.find the number of rings and chain that should be manufactured per day so as to earn max profit.make it as an Lpp and solve it graphically. Ans- 8 rings and 8 chains 34-)A cooperative society of farmers has 50 hectare of land to grow two crops X and Y. The profit from crops X and Y per hectare are estimated as Rs10,500 and Rs9,000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare. Further, not more than 800 litres of herbicide should be used in order to protect fish and wild life using a pond which collects drainage from this land. How much land should be allocated to each crop so as to maximize the total profit of the society? Ans- Maximum profit is Rs by allocating 30 hectares for crop X and 20 hectares for crop Y.

7 35-)A library has to accommodate two types of books on a shelf. the books are 6cm and 4cm thick and weigh 1kg and 3/2kg. the shelf is 96cm long and almost can support a weight of 21kg. how should the self be filled with the books of two types in order to include the greatest number of books? make it as an L.P.P and solve it graphically. Ans- 12 books of first type and 6 books of second type 36-) A dealer in rural area wishes to purchase a number of sewing machines. He has only rupees to invest and has space for at most 20 items. Electronic sewing machines cost him rupees 360 and manually operated sewing machine Rs.240. He can sell an electronic sewing machine at a profit of rupees 22 and a manually operated sewing machine at a profit of Rs.18 Assuming that he can sell all the items he can buy, how should he invest his money in order to maximize his profit. Make it as a Linear Programming Problem and solve it graphically. Ans- Maximum profit is Rs. 392 when 8 electric and 12 manually operated sewing machines are sold. 37-)A company manufactures two types of sweaters :type A sweaters type B.It costs Rs 360 to make a type A sweater and Rs 120 to make a type B sweater. The company can make at most 300 sweaters and spend at most Rs72,000 a day. The number of sweaters of type A cannot exceed the number of sweaters of type B by more than 100.The company makes a profit of Rs 200 for each sweater of type A but considering the difficulties of a common man the company charges a nominal profit of Rs. 20 on a unit of type B. What will be the maximum profit. Ans- Maximum profit- Rs and occurs when 175 sweaters of type A and 75 of type B are manufactured.