The objective function is maximum or minimum, which lies on the boundary of the feasible region.

Solve the following linear programming problem graphically: Minimise Z = 200 x + 500 y subject to the constraints
\(x+2y\ge 10,\)
\(3x+4y\le 24,\)
\(x\ge 0,y\ge 0.\)

Maximise Z = 3x + 4y subject to the constraints:
\(x+y\le 4,x\ge 0,y\ge 0.\)

Solve the following Linear Programming Problems graphically:
Minimise Z = 3x + 5y
such that x + 3y ≥ 3, x + y ≥ 2, x, y ≥ 0.

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 while 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. 7per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Formulate this mathematically and then solve it.

There are two types of fertilizers F₁ and F_2. 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 fertilizer should be used that nutrient requirement are met at a minimum cost. What is the minimum cost?

An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made 0n each executive class ticket and a profit of Rs. 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 to in order to maximise the profit for the airline. What is the maximum profit?

Solve the following LLP graphically:
Maximise Z = 2x + 3y, subject to \(x+y\le 4,x\ge 0,y\ge 0.\) 

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 Rs. 10,500 and Rs. 9,000 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, no 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 maximise the total profit of the society?

Solve the following Linear Programming Problem graphically:
Maximize Z = 3x + 4y subject to the constraints:  
\(x+y\le 4,\)
\(x\ge 0 \ \text {and} \)
\(y\ge 0.\)

A manufacturer produces nuts and bolts. It takes 2 hours work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 2 hours on machine B to produce a package of bolts. He earns a profit of Rs. 24 per package on nuts and Rs. 18 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 both for at the most 10 hours a days. Make an LPP from above and solve it graphically?

A company manufactures two types of sweaters, type A and B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make at most 300 sweaters and can spend Rs. 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs 200 on each unit of type A. The company charging a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve for max. profit.

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 class ticket out of which 25% will go for the improvement of facilities provided to economy class passengers. In both cases, the remaining profit goes to the airline's fund. 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 maximize the profit for the airline. Make the above as an LPP and solve it graphically. Do you think, more passengers would prefer to travel by such an airline than by others?