O.R. IN THE CLASSROOM SwedeBuild

SwedeBuild SwedeBuild 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? 15/11/2018 www.LearnAboutOR.co.uk

Modelling You have some resources: 6 rectangular LEGO® bricks 8 square LEGO® bricks How many tables and chairs can you make? 15/11/2018 www.LearnAboutOR.co.uk

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. 15/11/2018 www.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 15/11/2018 www.LearnAboutOR.co.uk

Modelling How 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? 15/11/2018 www.LearnAboutOR.co.uk

Constraints The 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 produced c = the number of chairs produced 15/11/2018 www.LearnAboutOR.co.uk

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. 15/11/2018 www.LearnAboutOR.co.uk

The Car Wash A car wash can take both cars and caravans 15/11/2018 www.LearnAboutOR.co.uk

Using the Car Wash This car and caravan want to use the car wash, will they fit? 15/11/2018 www.LearnAboutOR.co.uk

Using the Car Wash Another car and caravan would also like to use the car wash, will they fit? 15/11/2018 www.LearnAboutOR.co.uk

Using the Car Wash A third car and caravan would like to use the car wash, will they fit? 15/11/2018 www.LearnAboutOR.co.uk

L C K Finding an equation If we label the car wash L, the car C and the caravan K, ... L K C 15/11/2018 www.LearnAboutOR.co.uk

L C K Finding an equation ... can you write a formula that links the car and caravan to the length of the carwash? L K C 15/11/2018 www.LearnAboutOR.co.uk

So the equation is C + K ≤ L Graphing the equation So the equation is C + K ≤ L If the length of the car wash is 8 metres, can you plot the line of the equation on a graph? 8 metres 15/11/2018 www.LearnAboutOR.co.uk

Graphing the equation In 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? 15/11/2018 www.LearnAboutOR.co.uk

Graphing the equation 15/11/2018 www.LearnAboutOR.co.uk

Graphing the equation Since the original equation was ‘less than or equal to’ where will the possible solutions be on the graph? 15/11/2018 www.LearnAboutOR.co.uk 17

Graphing the equation The possible solutions will be on or below the line of the graph. Let us try a point in the yellow region to check. 15/11/2018 www.LearnAboutOR.co.uk

The Car Wash There is another type of car wash, this one can take 2 cars and a caravan 15/11/2018 www.LearnAboutOR.co.uk

L K C C Finding an equation Using the same letters C, K and L to represent the length of a car, caravan and car wash... L K C C 15/11/2018 www.LearnAboutOR.co.uk

L K C C Finding an equation ... How would you write the equation now to represent having 2 cars in the car wash? L K C C 15/11/2018 www.LearnAboutOR.co.uk

So the equation is 2C + K ≤ L Graphing the equation So the equation is 2C + K ≤ L If the length of this car wash is 12 metres, can you plot the line of the equation on a graph? 12 metres 15/11/2018 www.LearnAboutOR.co.uk

Graphing the equation 15/11/2018 www.LearnAboutOR.co.uk

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 available Tables (t) Chairs (c) 2 1 6 15/11/2018 www.LearnAboutOR.co.uk

Tables & Chairs Constraints 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 available Tables (t) Chairs (c) 2 8 15/11/2018 www.LearnAboutOR.co.uk

Tables & Chairs Constraints So the equations are: 2T + C ≤ 6 and 2T + 2C ≤ 8 What co-ordinates would you plot in order to graph these equations? 15/11/2018 www.LearnAboutOR.co.uk

Tables & Chairs Constraints 2t + c ≤ 6 2t + 2c ≤ 8 15/11/2018 www.LearnAboutOR.co.uk

Plotted solutions 15/11/2018 www.LearnAboutOR.co.uk

Feasible region 15/11/2018 www.LearnAboutOR.co.uk

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. 15/11/2018 www.LearnAboutOR.co.uk

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 Can you think of any other examples? 15/11/2018 www.LearnAboutOR.co.uk