Question

Solve the following LPP graphically:  

Minimize Z = 3x + 5y     

Subject to  

         – 2x + y ≤ 4  

            x + y ≥ 3

           x – 2y ≤ 2   

           xy ≥ 0

Solution

Correct option is

Z is 29/3 at x = 8/3, y = 1/3

 

Converting the inequations into equations, we obtain the lines –2x + y = 4,x + y = 3, x – 2y = 2, x = 0 and y = 0     

These lines are drawn on a suitable scale and the feasible region of the LLP is shaded in fig.

                                            

Now, give a value, say 15 equal to (1.c.m. of 3 and 5) to Z to obtain the line 3x + 5y = 15. This line meets the coordinate axes at P1 (5, 0) and Q1­ (0, 3). Join these points by a dotted line. Move this line parallel to itself in the decreasing direction towards the origin so that it passes through only one point of the feasible region. Clearly, P3Q3 is such a line passing through the vertex P of the feasible region. The coordinates of P are obtained by solving the lines x – 2y = 2 and x + y = 3. Solving these equations, we get   

  

         

Hence, the minimum value of Z is 29/3 at x = 8/3, y = 1/3.

SIMILAR QUESTIONS

Q1

 

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.

Q2

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.

Q3

 

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.

Q4

 

Solve the following LPP graphically:  

Maximize    Z  = 5x + 3y  

Subject to  

           3x + 5y ≤ 15  

           5x + 2y ≤ 10 

And,    xy ≥ 0.

Q5

 

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

Q6

 

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

Q7

 

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.

Q8

 

Solve the following linear programming problem graphically:

Maximize  Z = 50x + 15y  

Subject to

            5x + y ≤ 100

             x + y ≤ 60

             xy ≥ 0.

Q9

 

Solve the following LPP graphically:  

Maximize   Z = 5x + 7y  

Subject to

              x + y ≤ 4

             3x + 8y ≤ 24  

            10x + 7y ≤ 35

            xy ≥ 0  

Q10

 

A house wife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A,12 units of vitamin B and 8 units of vitamin C.

The vitamin contents of one kg of food is given below:   

 

Vitamin A

Vitamin B

Vitamin C

Food X:

1

2

3

Food Y:

2

2

1

One kg of food X costs Rs 6 and one kg of food Y costs Rs 10. Find the least cost of the mixture which will produce the diet.