A firm can produce three types of cloth, say C_{1}, C_{2}, C_{3}. Three kinds of wool are required for it, say red wool, green wool and blue wool. One unit of length C_{1} needs 2 metres of red wool, 3 metres of blue wool; one unit of cloth C_{2} needs 3 metres of red wool, 2 metres of green wool and 2 metres of blue wool; and one unit of cloth C_{3} needs 5 metres of green wool and 4 metres of blue wool. The firm has only a stock of 16 metres of red wool, 20 metres of green wool and 30 metres of blue wool. It is assumed that the income obtained from one unit of length of cloth C_{1} is Rs. 6, of cloth C_{2} is Rs. 10 and of cloth C_{3} is Rs. 8. Formulate the problem as a linear programming problem to maximize the income.
Solution
Correct option is
Subject to the constraints
And,
The given information can be put in the following tabular form:

Cloth C_{1} Cloth C_{2 } Cloth C_{3}

Total quantity of wood available

Red Wool
Green Wool
Blue Wool

2 3 0
0 2 5
3 2 4

16
20
30

Income(Rs.)

6 10 8


Let x_{1}, x_{2}, and x_{3} be the quantity produced in metres of the cloth of type C_{1}, C_{2} and C_{3} respectively.
Since 2 metres of red wool are required for one metres of cloth C_{1} and x_{1}metres of cloth C_{1} are produced, therefore 2x_{1} metres of red wool will be required for cloth C_{1}. Similarly, cloth C_{2} requires 3x_{2} metres of red wool and cloth C_{3} does not require red wool. Thus, the total quantity of red wool required is
But the maximum available quantity of red wool is 16 metres.
Similarly, the total quantities of green and blue wool required are
.
But the total quantities of green and blue wool available are 20 metres and 30 metres respectively.
Also, we cannot produce negative quantities, therefore
The total income is
Hence, the linear programming problem for the given problem is
Subject to the constraints
And,