Solve The Following Linear Programming Problem Graphically: Maximize  Z = 50x + 15y   Subject To             5x + y ≤ 100              x + y ≤ 60              x, y ≥ 0.

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Question

 

Solve the following linear programming problem graphically:

Maximize  Z = 50x + 15y  

Subject to

            5x + y ≤ 100

             x + y ≤ 60

             xy ≥ 0.

Solution

Correct option is

x = 10 and y = 50 will give the maximum value ofZ = 1250

 

We first convert the inequations into equations to obtain the lines 5x + y=100, x + y = 60, x = 0 and y = 0.  

The line 5x + y = 100 meets the coordinate axes at A1 (20, 0) and B1 (0, 100). Join these points to obtain the line 5x + y = 100. 

The line x + y = 60 meets the coordinates axes at A2 (60, 0) and B2 (0, 60). Join these points to obtain the line x + y = 60.  

Also,        x = 0 is the y-axis and y = 0 is the x-axis.

The feasible region of the LPP is shaded in fig. The coordinates of the corner points of the feasible region OA1 PB2 are O (0, 0), A1 (20, 0),P(10, 50) and B2 (0, 60).

                                                                        

 

Now, we take a constant value, say 300 (i.e. 2 times the 1.c.m. of 50 and 15) for Z. Then, 

                    300 = 50x + 15y

This line meets the coordinate axes at P(6, 0) and Q1 (0, 20). Join these points by a dotted line. Now move this line parallel to itself in the increasing direction i.e. away from the origin. P2Q2 and P3Q3 are such lines. Out of these lines locate a line which is farthest from the origin and has a least one point common to the feasible region. Clearly, P3Q3 is such line and it passes thorough the vertex P (10, 15) the convex polygon OA1PB2. Hence, x = 10 and y = 50 will give the maximum value of Z. The maximum value of Z is given by  

                

Testing

SIMILAR QUESTIONS

Q1

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?

Q2

 

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.              

Q3

 

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.

Q4

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.

Q5

 

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.

Q6

 

Solve the following LPP graphically:  

Maximize    Z  = 5x + 3y  

Subject to  

           3x + 5y ≤ 15  

           5x + 2y ≤ 10 

And,    xy ≥ 0.

Q7

 

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

Q8

 

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

Q9

 

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.

Q10

 

Solve the following LPP graphically:  

Maximize   Z = 5x + 7y  

Subject to

              x + y ≤ 4

             3x + 8y ≤ 24  

            10x + 7y ≤ 35

            xy ≥ 0