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

Oct 19, 2017

#### 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

${n}_{a} \times {a}_{1} + {n}_{b} \times {b}_{1} \le x$
${n}_{a} \times {a}_{2} + {n}_{b} \times {b}_{2} \le y$

We are to maximise ${n}_{a} \times {p}_{a} + {n}_{b} \times {p}_{b}$

and non-negative outputs are ${n}_{a}$ and ${n}_{b}$.