Question

A painter makes two paintings A and B. He spends 1 hour for drawing and 3 hours for colouring the painting A and he spends 3 hours for drawing and 1 hour for colouring the painting B. He can spend at most 8 hours for drawing and at most 9 hours colouring. The profit per painting of type A is Rs. 4000 and that of type B is Rs. 5000. Formulate as Linear Programming Problem to maximize the profit. ​

Answer 1

Answer:

Please be clear about the last sentence.

Answer 2

Given:

Two paintings A and B.

No. of hours spent for drawing in A = 1

No. of hours spent for coloring in A = 3

No. of hours spent for drawing in B = 3

No. of hours spent for coloring in A = 1

Total hours spent in drawing = 8

Total hours spent in coloring = 9

Profit per painting of type A = Rs 4000

Profit per painting of type B = Rs 5000

To find:

formulate as linear programming problem and maximize profit

Solution:

Let the hours spent on type A painting be x and hours spent on type B painting be y to maximize profit. Here, no. of hours spent in drawing and coloring each for both A and B are decision variables. The total time spent on drawing and coloring the paintings are constraints.

$Maximize, P = 4000x+5000y$

subject to

$\\ x+3y\leq 8\\ 3x+y\leq 9\\ x\leq 0, y\leq 0$

Using a graph sheet, the equations of 2 lines $x+3y\leq 8$ , $3x+y\leq 9$ are plotted as shown in figure.

$x+3y\leq 8$ is shown by red region and $3x+y\leq 9$ is shown by blue region in the graph.

The common region determined by these constraints is called feasible region. Points in the feasible region that is on the boundary lines is called corner points. The corner points obtained (approximately) are (0,0), (3,0), (2,2), (0,2.5).

Substitute these values into $P = 4000x+5000y$ and test for each corner points to check which provides maximum profit.

(0,0)⇒ $P=4000(0)+5000(0)=0$

(3,0)⇒ $P=4000(3)+5000(0)=12000$

(2,2)⇒ $P=4000(2)+5000(2)=8000+10000=18000$

(0,2.5)⇒ $P=4000(0)+5000(2.5)=12500$

As we can see, the maximum profit is obtained when $P=18000$ for (2,2).

Final Answer

The maximum profit obtained is 18,000 for (2,2)