Question

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of 400 square feet of 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?

Answer 1

Let required production of chairs and tables be x and y respectively.

Since, profits of each chair and table is Rs. 45 and Rs. 80 respectively. So, profits on x number of type A and y number of type B are 45x and 80y respectively.

Let Z denotes total output daily, so,

Z = 45x + 80y

Since, each chair and table requires 5 sq. ft and 80 sq. ft of wood respectively. So, x number of chair and y number of table require 5x and 80y sq. ft of wood respectively. But,

But 400 sq. ft of wood is available. So,

5x + 80y ≤ 400

x + 4y ≤ 80 {First Constraint}

Since, each chair and table requires 10 and 25 men - hours respectively. So, x number of chair and y number of table require 10x and 25y men - hours respectively. But, only 450 hours are available . So,

10x + 25y ≤ 450

2x + 5y ≤ 90 {Second Constraint}

Hence mathematical formulation of the given LPP is,

Max Z = 45x + 80y

Subject to constraints,

x + 4y ≤ 80

2x + 5y ≤ 90

x, y ≥ 0 [Since production of chairs and tables can not be less than zero]

Region x + 4y ≤ 80: line x + 4y = 80 meets the axes at A(80, 0), B(0, 20) respectively.

Region containing the origin represents x + 4y ≤ 80 as origin satisfies x + 4y ≤ 80

Region 2x + 5y ≤ 90: line 2x + 5y = 90 meets the axes at C(45, 0), D(0, 20) respectively.

Region containing the origin represents 2x + 5y ≤ 90

as origin satisfies 2x + 5y ≤ 90

Region x, y ≥ 0: it represents the first quadrant.

The corner points are O(0, 0), D(0, 18), C(45, 0).

The values of Z at these corner points are as follows:

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

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