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

Solve the following linear programming problem graphically: Maximise Z = 4x + y subject to the constraints:
\(x+y\le 50,\)
\(3x+y\le 90,\)
\(x\ge 0,y\ge 0.\)

(Manufacturing Problem) A manufacturer has Three machines I,II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for at least 5 hours a day. She produces only two items M and N each on the three machines are given in the following table:

She makes a profit of RS.600 and Rs 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

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 + 4 y
subject to x + 2y ≤ 8, 3x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Solve the following Linear Programming Problems graphically:
Minimise Z = x + 2y
subject to 2x + y ≥ 3, x + 2y ≥ 6, 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 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 hours of machine time and 1 hour of craftman's time. In a day, that factory has the availability of not more than 42 hours of 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 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

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 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?

Verify that the following problem has no feasible solution:
Maximize Z = 4x₁ + 4x_2, subject to the constraints:
\(2x_{ 1 }3x_{ 2 }\le 18,x_{ 1 }+x_{ 2 }\ge 12,x_{ 1 },x_{ 2 }\ge 0.\)

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.

A furniture firm manufactures chairs and tables, each requiring the use of three machines -A, B and C. Production of one chair requires 2 hours on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is Rs. 30 while by selling one table the profit is Rs. 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximize profit? Formulate the problem as LLP and solve it graphically.

(Manufacturing Problem) A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs. 80 on each piece of type A and Rs. 120 on each piece of type B. How many pieces of type A and B should be manufactured per week to get maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week? 

Two tailors A and B earn Rs. 150 and Rs. 200 per day respectively. A can stitch 6 shirts and 4 pants while B can stitch 10 shirts and 4 pants per day. How many days shall each work if it is desired to produce (at least) 60 shirts and 32 pants at a minimum labour cost? Make it an LLP and solve the problem graphically.