Solve the following (1 to 8) linear programming problems graphically:

Q1. Minimize Z=-3x + 4y subject to the constraints x + 2y ≤8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Q2. Minimize Z = 3x + 5y subject to the constraints x + 3y ≥ 3, x + y ≥ 2, x ≥ 0, y ≥ 0.

Q3. Maximize Z = 4x + y, subject to the constraints x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0.

Q4. Minimize Z = 200x + 500y, subject to the constraints x + 2y ≥ 10, 3x + 4y ≤ 24, x ≥ 0, y ≥ 0.

Q5. Minimize Z= x + 2y subject to the constraints 2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0.
Show that the minimum value of Z occurs at more than two points.

Q6. Minimize and maximize Z = 3x +9y subject to the constraints x + y ≥ 10, x + 3y ≤ 60, x ≤y, x ≥ 0, y ≥ 0.

Q7. Minimize Z = 3x + 2y subject to the constraints x + y ≥ 8, 3x + 5y ≤ 15, x ≥ 0, y ≥ 0

Q8. Minimize and maximize Z = x + 2y subject to the constraints x + 2y ≥ 100, 2x - y ≤0, 2x + y ≤ 200, x ≥ 0, y ≥ 0.

Q9. A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 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 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs 100 and Rs 120 per unit respectively, how srl is should he use his resources to maximise the total revenue? Form the above as an L.P.P. and solve graphically.

Q10. 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 Rs 7 per package on bolts. How many packages of each should be produced each day, so as to maximize his profit, if he operates his machines for atmost 12 hours a day.

Q11. A small firm manufactures gold rings and chains. The combined number of rings and chains manufactured per day is atmost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs 300 and on a chain is Rs 190, how many of each should be manufactured daily so as to maximize the profit?

Q12. A company produces two types of goods, A and B, that require gold and silver. Each unit of type A requires 3 g of silver and 1 g of gold while that of B requires 1 g of silver and 2 g of gold. The company can use atmost 9 g of silver and 8 g of gold. If each unit of type A brings a profit of Rs 40 and that of type B Rs 50, find the number of units of each type that the company should produce to maximize the profit. Formulate and solve graphically the LPP and find the maximum profit.

Q13. 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 minutes on the automatic and 6 minutes on the hand-operated machine 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 atmost 4 hours on any day. The manufacturer can sell a package of screws 'A' at a profit of Rs 7 and of screws 'B' at a profit of Rs 10. 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

Q14. A dealer in rural area wishes to purchase a number of sewing machines. He has only ₹57600 to invest and has space for atmost 20 items. An electronic sewing machine costs him ₹3600 and a manually operated sewing machine ₹2400. He can sell an electronic sewing machine at a profit of ₹220 and a manually operated sewing machine at a profit of ₹180. 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 L.P.P. and solve it graphically.

Q15. 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 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 each for type A and ₹60 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

Q16. A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at ₹7 profit and that of B at a profit of ₹4. Find the production level per day for maximum profit graphically.

Q17. A cottage industry manufactures pedastal lamps and wooden shades. Both products require machine time as well as craftsman time in making. The number of hour(s) for producing 1 unit of each and corresponding profit is given in the following table:

In a day, the factory has availability of not more than 42 hours of machine time and 24 hours of craftsman time. Assuming that all items manufactured are sold, how should the manufacturer schedule his daily production in order to maximise the profit? Formulate it as an L.P.P. and solve graphically.

Q18. An aeroplane can carry a maximum of 200 passengers. Baggage allowed for a first class ticket is 30 kg and for an economy class ticket is 20 kg. Maximum capacity for the baggage is 4500 kg. The profit on each first class ticket is Rs 500 and on each economy class ticket is Rs 300. Determine how many tickets of each type must be sold to maximize the profit of the airline. Also find the maximum profit.

Q19. A carpenter has 90, 80 and 50 running feet respectively of teak wood, plywood and rosewood which is used to produce product A and product B. Each unit of product A requires 2, 1 and 1 running feet and each unit of product B requires 1, 2 and 1 running feet of teak wood, plywood and rosewood respectively. If product A is sold for Rs 48 per unit and product B is sold for Rs 40 per unit, how many units of product A and product B should be produced and sold by the carpenter, in order to obtain the maximum revenue. Formulate the above as a linear programming problem and solve it, indicating clearly the feasible region in the graph.

Q20. A diet is to contain atleast 80 units of Vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available costing Rs 5 per unit and Rs 6 per unit respectively. One unit of food F₁ contains 4 units of vitamin A and 3 units of minerals whereas one unit of food F₂ contains 3 units of vitamin A and 6 units of minerals. Formulate this as a linear programming problem. Find the minimum cost of diet that consists of mixture of these two foods and also meets minimum nutritional requirement

Q21. A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain atleast 8 units of vitamin A, and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C. Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs ₹50 per kg to purchase food I and ₹70 per kg to purchase food II. Determine the minimum cost of such a mixture.

Q22. Reshma wishes to mix two types of foods P and Q in such a way that the vitamin contents of the mixture contains atleast 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.

Q23. A farmer has a supply of chemical fertiliser of type A which contains 10% nitrogen and 6% phosphoric acid and of type B which contains 5% nitrogen and 10% phosphoric acid. After soil test, it is found that atleast 7 kg of nitrogen and same quantity of phosphoric acid is required for a good crop. The fertiliser of type A costs Rs 5 per kg and the type B cost Rs 8 per kg. Using linear programming, find how many kilograms of each type of the fertiliser should be bought to meet the requirement and for the cost to be minimum. Find the feasible region in the graph.

Q24. David wants to invest atmost Rs 12000 in Bonds A and B. According to the rule, he has to invest atleast Rs 2000 in Bond A and atleast Rs 4000 in Bond B. If the rates of interest on Bonds A and B respectively are 8% and 10% per annum, formulate the problem as L.P.P. and solve it graphically for maximum interest. Also determine the maximum interest received in a year.

Q25. An oil company has two depots A and B with capacities of 7000 litre and 4000 litre respectively. The company is to supply oil to three petrol pumps D, E and F, whose requirements are 4500 litre, 3000 litre and 3500 litre respectively. The distance (in km) between the depots and the petrol pumps is given in the following table :

Assuming that the transportation cost per km is Rs 1 per litre, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?