Exercise 12.1
Question. Solve the following Linear Programming Problems graphically:
Show that the minimum of z occurs at more than two points.
Exercise 12.2
Question 1. 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 rs.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.
Question 2. One kind of cake requires 200 g of flour and 25g 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.
Question 3. A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. (i) What number of rackets and bats must be made if the factory is to work at full capacity ? (ii) If the profit on a racket and on a bat is rs. 20 and rs.10 respectively, find the maximum profit of the factory when it works at full capacity.
Question 4. A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of rs. 17.50 per package on nuts and` 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?
Question 5. A factory manufactures two types of screws, A and B, Each type of screw requires the use of two machines, an automatic and a hand-operated. It takes 4 minutes on the automatic and 6 minutes on 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 machines 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 rs. 7 and screws Bat a profit of ` 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximise his profit? Determine the maximum profit.
Question 6. A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/ cutting machine and a sprayer. It takes 2 hours on grinding/ cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp, while it takes 1 hour on the grinding/ cutting machine and 2 hours on the sprayer to manufacture a shade. On any day,the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is rs. 5and that from a shade is rs.3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?
Question 7. A company manufactures two types 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 and8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours for assembling. The profit is rs. 5 each for type A and rs. 6 each for type B souvenirs. How many souvenirs of each type should be company manufacture in order to maximise the profit?
Question 8. A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost rs. 25000 and rs. 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than rs. 70 lakhs and if his profit on the desktop model is rs. 4500 and on portable model is rs. 5000.
Question 9. 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 F1 costs rs.4 per unit food and F2 costs rs. 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. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.
Question 10. There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs rs. 6/kg and F2 costs rs. 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?
Question 11. The corner points of the feasible region are determined by the following system of linear inequalities:
Miscellaneous Exercise
Question 1. 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 maximise the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?
Question 2. A farmer mixes two brands P and Q of cattle feed. Brand P, costing rs.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 rs. 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?
Question 3. 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 rs. 16 and one kg of food Y costs rs.20. Find the least cost of the mixture which will produce the required diet?
Question 4. A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is rs. 7.50 and that on each toy of type B is rs. 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.
Question 5. An aeroplane can carry a maximum of 200 passengers. A profit of rs. 1000 is made on each executive class ticket and a profit of ` 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However, at least 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?
Sol. Let the executive class air tickets and economy class tickets sold be x and y. Now, as the seating capacity of the aeroplane is 200, so x + y <= 200. As 20 tickets for an executive class are to be reserved so we have x >= 20. And as the number of tickets for economy class should be at least 4 times that of executive class y >= 4x. Profit on the sale of x tickets of executive class and y tickets of economy class is
Question 6. 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. 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? What is the minimum cost?
Question 7. 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 petrol pumps, D, E and F whose requirements are 4500L, 3000L and 3500L respectively. The distances(in km) between the depots and the petrol pumps is given in the following table:
Assuming that the transportation cost of 10 litres of oil is ` 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?
500 litres, 3000 litres and 3500 litres of oil should be transported from depot A to petrol pumps D, E and F respectively and 4000 litres, 0 litres and 0 litres of oil be transported from depot B to petrol pumps D, E and F with minimum cost of transportation of rs. 4400.
Question 8. A fruit grower can use two types of fertilizer in his garden, brand P and brand 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 phsophoric acid, at least 270 kg of potash and at most 310 kg of chlorine.
If the grower wants to minimise 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?
Question 9. Refer to Question 8. If the grower wants to maximise the amount of nitrogen added to the garden, how many bags of each brand should be added? What is the maximum amount of nitrogen added?
Sol. From sol. 8., we have Z = 3x + 3×5y. Z is maximum at R (140, 50). Hence, to maximize the amount of nitrogen, 140 bags of brand P and 50 bags of brand Q are required. Maximum amount of nitrogen required = 595kg
Question 10. A toy company manufactures two types of dolls, A and B. Market tests and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demand for dolls of type B is at most half of that for dolls of type A. Further, the production level of dolls of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of rs.12 and rs.16 per doll respectively on dolls A and B, how many of each should be produced weekly in order to maximise the profit?
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