O.R. IN THE CLASSROOM

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SwedeBuild

SwedeBuildSwedeBuild would like to launch a new range of dining furniture that will include the table and chairs pictured below.

Can you help the production manager to find out how many tables and chairs should be made in order to get the greatest profit?22/04/2023 2www.LearnAboutOR.co.uk

Modelling

You have some resources:

• 6 rectangular LEGO® bricks • 8 square LEGO® bricks

How many tables and chairs can you make?

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Modelling

If a square brick costs £3.00 and a rectangular brick costs £5.00, how much will it cost to make the tables and chairs that you have just made?

You want to use the bricks to make tables and chairs to sell for a profit.

A common design that uses the least amount of bricks would allow us to keep the costs to a minimum.22/04/2023 4www.LearnAboutOR.co.uk

Modelling

To make a table use:• 2 rectangular

bricks • 2 square bricks

To make a chair use:• 1 rectangular

bricks • 2 square bricks

ModellingHow many combinations of tables and chairs can you make, are there any bricks leftover and what is the total cost of each combination?

• A table sells for £32.00• A chair sells for £21.00

Using the information about selling prices, what combinations of tables and chairs give you the greatest profit?22/04/2023 6www.LearnAboutOR.co.uk

ConstraintsThe amount of profit that can be made is limited by the available resources; this means the number of square and rectangular bricks.This is called a constraint on the resources.Constraints can be modelled using algebra.The constraints for this problem can be modelled using 2 variables:

t = the number of tables producedc = the number of chairs produced

Formulating constraints

The following example will help you to think about how to formulate the equations that will represent the constraints and then graph them.

The Car Wash

A car wash can take both cars and caravans

Using the Car Wash

This car and caravan want to use the car wash, will they fit?

Using the Car WashAnother car and caravan would also like to use the car wash, will they fit?

Using the Car WashA third car and caravan would like to use the car wash, will they fit?

Finding an equationIf we label the car wash L, the car C and the caravan K, ...

Finding an equation... can you write a formula that links the car and caravan to the length of the carwash?

Graphing the equation

So the equation is C + K ≤ L

8 metres

If the length of the car wash is 8 metres, can you plot the line of the equation on a graph?

Graphing the equationIn order to graph the equation C + K ≤ 8 you solve it as an equality, C + K = 8. How would you go about finding points to plot this as a graph?

If C = 0 what does K = ?If K = 0 what does C = ?This will give you 2 co-ordinates to plot on a graph.How do you write the co-ordinates from the above values for C and K?22/04/2023 16www.LearnAboutOR.co.uk

0 1 2 3 4 5 6 7 8 9 100

Car Wash

C + K = 8

Length (m) of Car (C)

) of C

0 1 2 3 4 5 6 7 8 9 100

Car Wash

C + K = 8

) of C

)Since the original equation was ‘less than or equal to’ where will the possible solutions be on the graph?

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The possible solutions will be on or below the line of the graph. Let us try a point in the yellow region to check.

0 1 2 3 4 5 6 7 8 9 100

Car Wash

c + v = 8

Length of car (m)

There is another type of car wash, this one can take 2 cars and a caravan

The Car Wash

Using the same letters C, K and L to represent the length of a car, caravan and car wash...

Finding an equation

... How would you write the equation now to represent having 2 cars in the car wash?

Finding an equation

So the equation is 2C + K ≤ L

12 metres

If the length of this car wash is 12 metres, can you plot the line of the equation on a graph?

0 1 2 3 4 5 6 7 8 9 10 11 12 130123456789

10111213

Car Wash

2C + K = 12

) of C

Tables & Chairs Constraints

Can you write an equation linking the number of tables and chairs to the number of rectangle bricks?

Number of rectangle bricks

needed Total rectangle

bricks availableTables

(t)Chairs

Can you write an equation linking the number of tables and chairs to the number of square bricks?

Number of square bricks needed Total square

bricks availableTables

(t)Chairs

(c)2 2 8

So the equations are:

2T + C ≤ 6 and 2T + 2C ≤ 8

What co-ordinates would you plot in order to graph these equations?

-3 -2 -1 0 1 2 3 4 5 6

Constraints for the rectangle and square bricks

Rectangle BricksSquare Bricks

Number of Tables

2t + c ≤ 6

2t + 2c ≤ 8

-3 -2 -1 0 1 2 3 4

Rectangle BricksSquare BricksFurniture combinations

Number of Tables

Plotted solutions

-3 -2 -1 0 1 2 3 4

Rectangle BricksSquare BricksFurniture combinations

Number of Tables

Feasible region

Linear Programming

This type of maths is called Linear Programming.

Linear Programming is concerned with finding the best solution (e.g. increase profit, reduce costs) from a set of constraints.

Linear Programming is used extensively in Operational Research.

Examples

Virgin Media• Sending marketing

information to customers– Who?– How often?– What type of

media?

British Airways• Buying new aircraft– Size– Price– Cost of operation

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Can you think of any other examples?

O.R. IN THE CLASSROOM

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