A producer of furniture manufactures two products - tables and chairs. Processing of these products is done on two machines A and B. A chair needs 2 hours on machine A and 6 hours on machine B. A table needs 5 hours on machine A and no time on machine B. There are 16 hours of time per day accessible on machine A and 30 hours on machine B. Profit earned by the manufacturer from a chair and a table is Rs 2 and Rs 10 correspondingly. What must be the everyday production of each of two products?

Answer

Assume x₁ indicates the number of chairs

Assume x₂ indicates the number of tables

LPP

Max Z = 2x₁ + 10x₂

Subject to

2x₁+ 5x₂ ≤ 16

            6x₁ + 0x₂ ≤ 30

 x₁ ≥ 0 , x₂ ≥ 0 

Solve graphically

The first constraint 2x₁+ 5x₂ ≤ 16, can be written in the form of equation

2x₁+ 5x₂ = 16

Place x₁ = 0, then x₂ = 16/5 = 3.2

Place x₂ = 0, then x₁ = 8

The coordinates are (0, 3.2) and (8, 0)

The second constraint 6x₁ + 0x₂ ≤ 30, can be written in the form of equation

6x₁ = 30 → x₁ =5

The corner positions of feasible region are A, B and C. So the coordinates for the corner positions are

A (0, 3.2)

B (5, 1.2) (Solve the two equations 2x₁+ 5x₂ = 16 and x₁ =5 to obtain the coordinates)

C (5, 0)

We are given that Max Z = 2x₁ + 10x₂

At A (0, 3.2)

Z = 2(0) + 10(3.2) = 32

At B (5, 1.2)

Z = 2(5) + 10(1.2) = 22

At C (5, 0)

Z = 2(5) + 10(0) = 10

Max Z = 32 and x₁ = 0, x₂ = 3.2

The manufacturer must manufacture about 3 tables and no chairs to obtain the max profit.