Linear Programming

There are two types of fertilizers 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?

Determine graphically the minimum value of the objective function Z = – 50x + 20y ... (1)

subject to the constraints:

2x – y ≥ –5....(2)

3x + y ≥ 3…..(3)

2x – 3y ≤ 12...(4)

x ≥ 0, y ≥ 0.....(5) 

Factory makes tennis rackets and cricket bats. A tennis basket 1.5 hours of a machine time and 3 hours of maching time and 1 hour of craftman’s tome. In a day, the factory has the availability of not more than 42 hour sof machine time and 24 hours of craftman’s time.

(i) What number of rackets and bats must be made if the factory is to work at full capacity.

(ii) If profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit to the factory when 

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

A factory manufactures two types of screws, A and B, each type 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 automatic and 3 minutes on the hand operated machines to manufacture a package of screws B. Each machine is available for at the most 4 hours on any day. The manufacturer can sell a package of screws A at a profit of Rs 7 and screws B at a profit of Rs. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produced in a day in order to maximize his profit? Determine the maximum profit.

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

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?

Dealer wishes to purchase a number of fans and sewing machines. He has only Rs. 5,760 to invest and has a space for at most 20 items. A fan costs him Rs. 360 and a sewing machine Rs. 240. His expectation is that he can sell a fan at a profit of Rs. 22 and a sewing machine at a profit of Rs. 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to minimize the profit? Formulate This as a linear programming problem and solve it graphically.