A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet to teak wood and 450 man hours. It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs 80. How many items of each product should be produced by the company so that the profit is maximum?

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Solution

Let x units of chairs and y units of tables were produced

Therefore, $x, y \geq 0$

The given information can be tabulated as follows:

Wood(square feet) Man hours

Chairs(x) 5 10

Tables(y) 20 25

Availability 400 450

Therefore, the constraints are
$5 x + 20 y \leq 400 10 x + 25 y \leq 450$

It is known that to make a chair requires 5 square feet of wood and 10 man-hours and yields a profit of Rs 45, while each table uses 20 square feet of wood and 25 man-hours and yields a profit of Rs 80.

Therefore, profit gained to make x chairs and y tables is Rs 45x and Rs 80y respectively.
Total profit = Z = $45 x + 80 y$ which is to be maximised.

Thus, the mathematical formulation of the given linear programmimg problem is
Max Z = $45 x + 80 y$
subject to

$5 x + 20 y \leq 400 10 x + 25 y \leq 450$
$x, y \geq 0$

First we will convert inequations into equations as follows:
5x + 20y = 400, 10x + 25y = 450, x = 0 and y = 0

Region represented by 5x + 20y ≤ 400:
The line 5x + 20y = 400 meets the coordinate axes at $A (80, 0)$ and $B (0, 20)$ respectively. By joining these points we obtain the line
5x + 20y = 400 . Clearly (0,0) satisfies the 5x + 20y = 400 . So, the region which contains the origin represents the solution set of the inequation 5x + 20y ≤ 400.

Region represented by 10x + 25y ≤ 450:
The line 10x + 25y = 450 meets the coordinate axes at $C (45, 0)$ and $D (0, 18)$ respectively. By joining these points we obtain the line
10x + 25y = 450. Clearly (0,0) satisfies the inequation 10x + 25y ≤ 450. So,the region which contains the origin represents the solution set of the inequation 10x + 25y ≤ 450.

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 5x + 20y ≤ 400, 10x + 25y ≤ 450, x ≥ 0, and y ≥ 0 are as follows.

The corner points are A(0, 18), B(45, 0)

The values of Z at these corner points are as follows

Corner point Z= 45x + 80y

A 1440

B 2025

The maximum value of Z is 2025 which is attained at B $(45, 0)$ .

Thus, the maximum profit is of Rs 2025 obtained when 45 units of chairs and no units of tables are produced

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