Question

 

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 

                     xy ≥ 0

Solution

Correct option is

 and values of Z are 63/13 and 19

 

Converting the given inequation into equations, we get

 2x + 3y = 6, x – 2y = 2, 3x + 2y = 12, –3x + 2y = 3, x = 0 and y = 0

Region represented by –2x – 3y ≤ –6: The line –2x – 3y = –6 or 2x + 3y = 6 cuts OX and OY at A1 (3, 0) and B1 (0, 3) respectively. Join these points to obtain the line 2x + 3y = 6.  

Since O (0, 0) does not satisfy the inequation –2x – 3y ≤ –6. So, the represented –2x – 3y ≤ –6 is that part of XOY-plane which does not contain the origin.  

Region represented by x – 2y ≤ 2: The line x – 2y = 2 meets the coordinate axes at A2 (2, 0) and B2 (0, –1). Join these points to obtain x – 2y = 2. Since (0, 0) satisfies the inequation x – 2y ≤ 2, so the region containing the origin represents the solution set of this inequation.

Region represented by 3x + 2y ≤ 12. The line 3x + 2y ≤ 12 intersects OX andOY at A3 (4, 0) and B3 (0, 6). Join these points to obtain the line 3x + 2y = 12. Clearly, (0, 0) satisfies the inequation  3x + 2y ≤ 12. So, the region containing the origin is the solution set of the given inequations.

Region represented by –3x + 2y ≤ 3: The line –3x + 2y = 3 intersects OX andOY at A4 (–1, 0) and B4 (0, 3/2). Join these points to obtain the line –3x + 2y = 3. Clearly, (0, 0) satisfies this inequation. So, the region containing the origin represents the solution set of the given inequation.

Region represented by x ≥ 0, y ≥ 0: Clearly, XOY quadrant represents the solution set of these two inequations. 

The shaded region shown in fig. represents the common solution set of the above inequations. This region is the feasible region of the given LPP.

                                                                

The coordinates of the corner-points (vertices) of the shaded feasible regionP1P2P3P4 are 

. These points have been obtained by solving the equations of the corresponding intersecting lines, simultaneously.

The value of the objective function at these points are given in the following table:  

Point (xy)

Value of the objective function

Z = 5x + 2y

Clearly, Z is min at  and maximum at . The minimum and maximum values of Z are 63/13 and 19 respectively.

SIMILAR QUESTIONS

Q1

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.   

Q2

A diet is to contain at least 400 units of carbohydrate, 500 units of fat, and 300 units of protein. Two foods are available: F1 which costs Rs 2 per unit, and F2 which costs Rs 4 per unit. A unit of food F1 contains 10 units of carbohydrate, 20 units of fat, and 15 units of protein; a unit of food F2 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.

Q3

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 ABC. 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?

Q4

 

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 AB 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.              

Q5

 

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 PQand 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.

Q6

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.

Q7

 

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.

Q8

 

Solve the following LPP graphically:  

Maximize    Z  = 5x + 3y  

Subject to  

           3x + 5y ≤ 15  

           5x + 2y ≤ 10 

And,    xy ≥ 0.

Q9

 

Solve the following LPP by graphical method: 

Minimize    Z = 20x + 10y  

Subject to   x + 2y ≤ 40

                   3x + y ≥ 30  

                   4x + 3y ≥ 60 

And,           xy ≥ 0

Q10

 

Solve the following LPP graphically:

Maximize and Minimize   Z = 3x + 5y  

Subject to   3x – 4y + 12 ≥ 0

                       2x – y + 2 ≥ 0 

                   2x + 3y – 12 ≥ 0 

                               0 ≤ x ≤ 4 

                                      y ≥ 2.