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

Solve the following problem graphically:
Minimise and Maximise : Z = 3x + 9y 
subject to the constraints:
\(x+3y\le 60,\)
\(x+y\ge 10,\)
\(x\le y\)
\(x\ge 0,\)
\(y\ge 0.\)

Solve the following Linear Programming Problems graphically:
Maximise Z = 3x + 2y
subject to x + 2y ≤ 10, 3x + y ≤ 15, x, y ≥ 0.

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 Rs. 60/kg and Food Q costs Rs. 80/kg. Food P contains 3 units/kg of Vitamin A AND 4 units/kg of vitamin B. Food Q contains 5 units/kg Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

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 vitamin 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?

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 4500 L, 3000 L and 3500 L respectively. The distance (in km) between the depots and the petrol pumps are given in the following table:

Assuming that the transportation cost of 10 litres of oil is Rs. 1 per km, how should the delivery be scheduled in order that the transportation cost is minimum? What is the minimum cost?

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 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?

Minimize and Maximize Z = 5x + 2y, subject to the following constraints:
\(x-2y\le 2,3x+2y\le 12,-3x+2y\le 3,x\ge 0,y\ge 0.\)

Kellogg is a new cereal formed of a mixture of bran and rice that contains at least 88 grams of protein and at least 56 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producing this new cereal if bran costs Rs. 5 per kilogram and rice costs Rs. per kilogram.

Draw the graph of the following LLP:\(3x+y\le 17,x,y\ge 0\)

(Manufacturing Problem) A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is atmost 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs. 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an LPP and solve it graphically.

Suppose every gram of wheat produces 0.1 g of protein and 0.25 g of carbohydrates and corresponding values for rice are 0.05 g and 0.5 g respectively. Wheat cost Rs. 25 and rice Rs. 100 per kilogram. The minimum daily requirements of proteins and carbohydrates for an mixed in a daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost, assuming that both wheat and rice are to be taken in a diet? What is your opinion about healthy diet? Name few ingredients necessary for a healthy diet.

A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 unit/ kg of vitamin A and 1 unit/kg of vitamin C while food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs. 5 per kg to produce Food I and Rs. 7 per kg to produce food II. Determine the minimum cost of such a mixture. Formulate the above as a LLP and solve 3 it graphically.

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 manufacture 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 for 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 Rs. 7 profit and  that of B at a profit of Rs. 4. Find the production level per day for maximum profit graphically.

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?

Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below: 2x + 4y \(\le \) 8 \(\Rightarrow\) x + 2y \(\le \) 4
3x + y \(\le \) 6
x + y \(\le \) 4
x \(\\ \ge \) 0, y  \(\\ \ge \) 0

A retired person wants to invest an amount of Rs. 50,000. His broker recommends investing in two type of bonds 'A' and 'B' yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs. 20,000 in bond 'A'  and at least Rs. 10,000 in bond 'B'. He also wants invest at least as much in boud 'A' as in bond ·'B'. Solve this linear programming problem graphically to maximise his returns.