## Formulating the problem - Linear Programming (LP)

In order to formulate your LP problem, you have to do three things.

This is incredibly important, and if not done correctly could lose you all your marks. Make sure you define them carefully – you

Utilising the information provided, you must accurately formulate a set of inequalities which you can draw onto a graph to find the ‘feasible region’. Most of the time our variables cannot take negative values – we call these ‘non-negativity constraints’; i.e.

Although there are rarely marks attached to including these, it is usually a good idea to.

This is an equation for a quantity which needs to be either maximised (e.g. profit) or minimised (e.g. cost).

Omkar wants to start selling Economics guides and branded stationery on his website, A Rational Econ. The cost of making a guide is £2, it requires 2 hours to make each one and accrues £3 profit when one is sold. The cost of making a set of stationery is £3, it requires 4 hours to make each one and accrues £1 in profit when one is sold. Omkar has £30 to spend on this project in the first month, and has to spend at least 20 hours on the project to stop him from getting bored. 5 of his friends have already pre-ordered guides. Assuming everything is sold, what is the maximum profit Omkar can make, and how?

Let

Let

Take the costs and multiply them by the quantities, i.e. the variables, to get the inequality for total cost, which must be less than or equal to £30 which Omkar has available.

Take the hours and multiply them by the quantities, i.e. the variables, to get the inequality for total hours spent, which must be greater than or equal to (at least) 20 hours.

Since 5 of his friends have already pre-ordered guides, he must make at least 5 guides.

Now we include the non-negativity constraints, because we cannot have negative quantities.

We need to maximise profit, so we make an equation in terms of x and y for profit. This means taking the profit accrued from each unit and multiplying by the variable, and setting the equation to equal another variable representing profit. Here, we take profit as

Now we have to solve this problem. To see how to do this, click here.

*1. Define the variables*This is incredibly important, and if not done correctly could lose you all your marks. Make sure you define them carefully – you

*must*define them to be a quantity; i.e.- Let x be
**the number of**[litres of oil used] - Let y be
**the number of**[litres of water used]

*2. Find the constraints*

Utilising the information provided, you must accurately formulate a set of inequalities which you can draw onto a graph to find the ‘feasible region’. Most of the time our variables cannot take negative values – we call these ‘non-negativity constraints’; i.e.

*x*≥ 0*y*≥ 0

Although there are rarely marks attached to including these, it is usually a good idea to.

*3. Find the objective function*This is an equation for a quantity which needs to be either maximised (e.g. profit) or minimised (e.g. cost).

**Example:**Omkar wants to start selling Economics guides and branded stationery on his website, A Rational Econ. The cost of making a guide is £2, it requires 2 hours to make each one and accrues £3 profit when one is sold. The cost of making a set of stationery is £3, it requires 4 hours to make each one and accrues £1 in profit when one is sold. Omkar has £30 to spend on this project in the first month, and has to spend at least 20 hours on the project to stop him from getting bored. 5 of his friends have already pre-ordered guides. Assuming everything is sold, what is the maximum profit Omkar can make, and how?

*1.**Define the variables*Let

*x*be the number of guides madeLet

*y*be the number of sets of stationery made*2.**Find the constraints*Take the costs and multiply them by the quantities, i.e. the variables, to get the inequality for total cost, which must be less than or equal to £30 which Omkar has available.

**2***x***+ 3***y*≤ 30Take the hours and multiply them by the quantities, i.e. the variables, to get the inequality for total hours spent, which must be greater than or equal to (at least) 20 hours.

**2***x*+ 4*y*≥ 20Since 5 of his friends have already pre-ordered guides, he must make at least 5 guides.

*x***≥ 5**Now we include the non-negativity constraints, because we cannot have negative quantities.

*x***≥ 0***y***≥ 0***3. Find the objective function*We need to maximise profit, so we make an equation in terms of x and y for profit. This means taking the profit accrued from each unit and multiplying by the variable, and setting the equation to equal another variable representing profit. Here, we take profit as

*P*.*P***= 3***x*+*y*Now we have to solve this problem. To see how to do this, click here.