O.R. IN THE CLASSROOM
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Transcript of O.R. IN THE CLASSROOM
O.R. IN THE CLASSROOM
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
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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
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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.
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The Car Wash
A car wash can take both cars and caravans
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Using the Car Wash
This car and caravan want to use the car wash, will they fit?
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Using the Car WashAnother car and caravan would also like to use the car wash, will they fit?
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Using the Car WashA third car and caravan would like to use the car wash, will they fit?
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Finding an equationIf we label the car wash L, the car C and the caravan K, ...
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L
KC
Finding an equation... can you write a formula that links the car and caravan to the length of the carwash?
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L
KC
Graphing the equation
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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
Graphing the equation
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Car Wash
C + K = 8
Length (m) of Car (C)
Leng
th (m
) of C
arav
an (K
)
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Car Wash
C + K = 8
Length (m) of Car (C)
Leng
th (m
) of C
arav
an (K
)Since the original equation was ‘less than or equal to’ where will the possible solutions be on the graph?
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Graphing the equation
Graphing the equation
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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.
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Car Wash
c + v = 8
Length of car (m)
Leng
th o
f car
avan
(m)
There is another type of car wash, this one can take 2 cars and a caravan
The Car Wash
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Using the same letters C, K and L to represent the length of a car, caravan and car wash...
Finding an equation
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L
KC C
... How would you write the equation now to represent having 2 cars in the car wash?
Finding an equation
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L
KC C
Graphing the equation
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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?
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Car Wash
2C + K = 12
Length (m) of Car (C)
Leng
th (m
) of C
arav
an (K
)
Graphing the equation
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Tables & Chairs Constraints
Can you write an equation linking the number of tables and chairs to the number of rectangle bricks?
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Number of rectangle bricks
needed Total rectangle
bricks availableTables
(t)Chairs
(c)
2 1 6
Tables & Chairs Constraints
Can you write an equation linking the number of tables and chairs to the number of square bricks?
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Number of square bricks needed Total square
bricks availableTables
(t)Chairs
(c)2 2 8
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?
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Tables & Chairs Constraints
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Constraints for the rectangle and square bricks
Rectangle BricksSquare Bricks
Number of Tables
Num
ber o
f Cha
irs
2t + c ≤ 6
2t + 2c ≤ 8
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Constraints for the rectangle and square bricks
Rectangle BricksSquare BricksFurniture combinations
Number of Tables
Num
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f Cha
irs
Plotted solutions
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-3 -2 -1 0 1 2 3 4
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Constraints for the rectangle and square bricks
Rectangle BricksSquare BricksFurniture combinations
Number of Tables
Num
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f Cha
irs
Feasible region
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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.
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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?