Question
Solve the following LPP by graphical method:
Minimize Z = 20x + 10y
Subject to x + 2y ≤ 40
3x + y ≥ 30
4x + 3y ≥ 60
And, x, y ≥ 0
Solution
x = 6, y = 12 & Z = 240
Converting the given inequations into equations, we obtain the following equations:
x + 2y = 40, 3x + y = 30, 4x + 3y = 60, x = 0 and y = 0.
Region represented by x + 2y ≤ 40: The line x + 2y = 40 meets the coordinates axes at A_{1} (40, 0) and B_{1} (0, 20) respectively. Join these points to obtain the line x + 2y = 40. Clearly, (0, 0) satisfies the inequationx + 2y ≤ 40. So, the region in xyplane that contains the origin represents the solution set of the given inequation.
Region represented by 3x + y ≥ 30: The line 3x + y = 30 meets X and Yaxes at A_{2} (10, 0) and B_{2} (0, 30) respectively. Join these points to obtain this line. We find that O (0, 0) does not satisfy the inequation 3x + y ≥ 30. So that region in xyplane which does no contain the origin is the solution set of this inequation.
Region represented by 4x + 3y ≥ 60: The line 4x + 3y = 60 meets X and Yaxes at A_{3} (15, 0) and B_{1} (0, 20) respectively. Join these points to obtain the line 4x + 3y = 60. We observe that O (0, 0) does not satisfy the inequation 4x + 3y ≥ 60. So, the region not containing the origin in xyplane represents the solution set of the given inequation. Region represented by x≥ 0, y ≥ 0: Clearly, the region represented by x ≥ 0 and y ≥ 0 is the first quadrant in xyplane.
The shaded region A_{3}A_{1} QP in fig. represents the common region of the regions represented by the above inequations. This region expresents the feasible region of the given LPP.
The coordinates of the cornerpoints of the shaded feasible are A_{3} (15, 0)A_{1} (40, 0), Q (4, 18) and P (6, 12). These points have been obtained by solving the equations of the corresponding intersecting lines, simultaneously.
The values of the objective function at these points are given in the following table:
Point (x, y) 
Value of the objective function Z = 20x + 10y 
A_{3} (15, 0) 
Z = 20 × 15 + 10 × 0 = 300 
A_{1} (40, 0) 
Z = 20 × 40 + 10 × 0 = 800 
Q (4, 18) 
Z = 20 × 4 + 10 × 18 = 260 
P (6, 12) 
Z = 20 × 6 + 10 × 12 = 240 
Clearly, Z is minimum at P (6, 12). Hence, x = 6, y = 12 is the optimal solution of the given LPP. The optimal value of Z is 240.
SIMILAR QUESTIONS
A company makes two kinds of leather belts, A and B. Belt A is high quality belt, and B is of lower quality. The respective profits are Rs 4 and Rs 3 per belt. Each belt of type A requires twice as much time as a belt of type B, and if all belts were of type B, the company could make 1000 belts per day. The supply of leather is sufficient for only 800 belts per day (bothA and B combined). Belt A requires a fancy buckle, and only 400 buckles per day are available. There are only 700 buckles available for belt B. What should be the daily production of each type of belt? Formulate the problem as a LPP.
A dietician whishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of Vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units per kg of vitamin A and 1 unit per kg of vitamin C while food ‘II’ contains 1 unit per kg of vitamin A and 2 units per kg of vitamin C. It costs Rs 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Formulate the above linear programming problem to minimize the cost of such a mixture.
A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F_{1}_{’} which costs Rs 2 per unit, and F_{2}_{’} which costs Rs 4 per unit. A unit of food F_{1} contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F_{2} contains 25 units of carbohydrate, 10 units of fat, and 20 unit of protein. Find the minimum cost for a diet consists of a mixture of these two foods and also meets the minimum nutrition requirements. Formulate the problem as a linear programming problem.
The objective of a diet problem is to ascertain the quantities of certain foods that should be eaten to meet certain nutritional requirement at minimum cost. The consideration is limited to milk, beaf and eggs, and to vitamins A, B, C. The number of milligrams of each of these vitamins contained within a unit of each food is given below:
Vitamin 
Litre of milk 
Kg of beaf 
Dozen of eggs 
Minimum daily requirements 
A B C 
1 100 10 
1 10 100 
10 10 10 
1 mg 50 mg 10 mg 
Cost 
Rs 1.00 
Rs 1.10 
Re 0.50 

What is the linear programming formulation for this problem?
There is a factory located at each of the two places P and Q. From these locations, a certain commodity is delivered to each of the three depots situated at A, B and C. The weekly requirements of the depots are respectively 5, 5 and 4 units of the commodity while the production capacity of the factories at P and Q are 8 and 6 units respectively. The cost of transportation per unit is given below.
To From 
Cost (in Rs) 


A 
B 
C 

P Q 
16 10 
10 12 
15 10 
How many units should be transported from each factory to each in order that the transportation cost is minimum. Formulate the above as a linear programming problem.
A brick manufacturer has two depots, A and B, with stocks of 30,000 and 20,000 bricks respectively. He receives orders from three builders P, Qand R for 15,000, 20,000 and 15,000 bricks respectively. The cost in Rs of transporting 1000 bricks to the builders from the depots are given below:
From To 
P 
Q 
R 
A B 
40 20 
20 60 
30 40 
How should the manufacturer fulfil the orders so as to keep the cost of transportation minimum?
Formulate the above linear programming problem.
A company is making two products A and B. The cost of producing one unit of products A and B are Rs 60 and Rs 80 respectively. As per the agreement, the company has to supply at least 200 units of product B to its regular customers. One unit product A requires one machine hour whereas product B has machine hours available abundantly within the company. Total machine hours available for product A are 400 hours. One unit of each product A and B requires one labour hour each and total of 500 labour hours are available. The company wants to minimize the cost of production by satisfying the given requirements. Formulate the problem as a LLP.
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:

Product A 
Product B 
Weekly capacity 
Department 1 
3 
2 
130 
Department 2 
4 
6 
260 
Selling price per unit 
Rs 25 
Rs 30 

Labour cost per unit 
Rs 16 
Rs 20 

Raw material cost per unit 
Rs 4 
Rs 4 

The problem is to determine the number of units of produce each product so as to maximize total contribution to profit. Formulate this as a LLP.
Solve the following LPP graphically:
Maximize Z = 5x + 3y
Subject to
3x + 5y ≤ 15
5x + 2y ≤ 10
And, x, y ≥ 0.
Solve the following LPP graphically:
Minimize and Maximize Z = 5x + 2y
Subject to –2x – 3y ≤ – 6
x – 2y ≤ 2
3x + 2y ≤ 12
–3x + 2y ≤ 3
x, y ≥ 0