Question

How do you know when to use Linear Programming to solve a word problem?

Answer 1

Please see below.

Explanation:

Linear programming is a simple technique where we depict complex relationships through linear relations. These relations are constraints which put restrictions on values of the output, which are non-negative i.e. zero or positive. The results in such cases are generally located at points and then we find the most optimal point based on , at which the desired objective, which is again a linear relation among decision variables and is either maximised or minimised.

Hence, we use Linear Programming to solve a word problem when (i) we have linear relations; and (ii) certain function has to be maximized or minimized.

For example, let us have available `x` hours of labour and `y` cubic feet of wood, which we can use to make either tables or chair. A chair requires `a_1` hours of labour and `a_2` cubic feet of wood and a table requires `b_1` hours of labour and `b_2` cubic feet of wood. We have a profit of `p_a` on chair and `p_b` on table. How can we maximise profits.

Let the result be `n_a` chairs and `n_b` tables. So our constraints are

`n_axxa_1+n_bxxb_1 <= x`
`n_axxa_2+n_bxxb_2 <= y`

We are to maximise `n_axxp_a+n_bxxp_b`

and non-negative outputs are `n_a` and `n_b`.