Question

A manufacturer of Furniture makes two products : chairs and tables. processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 4 hrs on machine A and 2 hrs on machine B. There are 16 hrs of time per day available on machine A and 30 hrs on machine B. Profit gained by the manufacturer from a chair and a table is Rs 3 and Rs 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

Answer 1

Let  x chairs and y tables were produced.
Number of chairs and tables cannot be negative.
Therefore, \[x, y \geq 0\]

The given information can be tabulated as follows:

	Time on machine A(hrs)	Time on machine B (hrs)
Chairs	2	6
Tables	4	2
Availability	16	30

Therefore, the constraints are

\[2x + 4y \leq 16\]
\[6x + 2y \leq 30\]

Profit gained by the manufacturer from a chair and a table is Rs 3 and Rs 5 respectively. Therefore, profit gained from x chairs and y tables is Rs 3x and Rs 5y.
Total profit = Z =  \[3x + 5y\]  which is to be maximised

Thus, the mathematical formulat​ion of the given linear programmimg problem is 

Max Z =  \[3x + 5y\]
subject to

\[2x + 4y \leq 16\]
\[6x + 2y \leq 30\]

\[x, y \geq 0\]

First we will convert inequations into equations as follows:
2x + 4y = 16, 6x + 2y =30, x = 0 and y = 0
Region represented by 2x + 4y ≤ 16:
The line 2x + 4y = 16 meets the coordinate axes at A₁(8, 0) and B₁(0, 4) respectively. By joining these points we obtain the line 2x + 4y = 16. Clearly (0,0) satisfies the 2x + 4y = 16. So, the region which contains the origin represents the solution set of the inequation 2x + 4y ≤ 16.
Region represented by 6x + 2y ≤ 30:
The line 6x + 2y =30 meets the coordinate axes at C₁(5, 0) and D₁(0, 15) respectively. By joining these points we obtain the line 6x + 2y =30 . Clearly (0,0) satisfies the inequation 6x + 2y ≤ 30. So,the region which contains the origin represents the solution set of the inequation 6x + 2y ≤ 30.
Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x ≥ 0, and y ≥ 0.
The feasible region determined by the system of constraints 2x + 4y ≤ 16, 6x + 2y ≤ 30, x ≥ 0, and y ≥ 0 are as follows.The corner points are O(0, 0), B₁(0, 4), E₁

\[\left( \frac{22}{5}, \frac{9}{5} \right)\] and C₁(5, 0). 

The values of Z at these corner points are as follows

Corner point	Z= 3x + 5y
O	0
B1	20
E1	22.2
C1	15

The maximum value of Z is 22.2 which is attained at B₁

\[\left( \frac{22}{5}, \frac{9}{5} \right)\] Thus, the maximum profit  is of Rs 22.20 obtained when \[\frac{22}{5}\] units of chairs  and \[\frac{9}{5}\]  units of tables are produced.