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, $x,y\ge 0$

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

$$\begin{gathered} 2x+4y\le 16 \\ 6x+2y\le 30 \end{gathered}$$

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

$$\begin{gathered} 2x+4y\le 16 \\ 6x+2y\le 30 \end{gathered}$$
$x,y\ge 0$

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.