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

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

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Answer 1 · 2017-10-19T16:46:09

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 $x$ $x$ hours of labour and $y$ $y$ cubic feet of wood, which we can use to make either tables or chair. A chair requires $a_{1}$ $a_1$ hours of labour and $a_{2}$ $a_2$ cubic feet of wood and a table requires $b_{1}$ $b_1$ hours of labour and $b_{2}$ $b_2$ cubic feet of wood. We have a profit of $p_{a}$ $p_a$ on chair and $p_{b}$ $p_b$ on table. How can we maximise profits.

Let the result be $n_{a}$ $n_a$ chairs and $n_{b}$ $n_b$ tables. So our constraints are

$n_{a} \times a_{1} + n_{b} \times b_{1} \leq x$ $n_axxa_1+n_bxxb_1 <= x$
$n_{a} \times a_{2} + n_{b} \times b_{2} \leq y$ $n_axxa_2+n_bxxb_2 <= y$

We are to maximise $n_{a} \times p_{a} + n_{b} \times p_{b}$ $n_axxp_a+n_bxxp_b$

and non-negative outputs are $n_{a}$ $n_a$ and $n_{b}$ $n_b$ .

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

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