1. Linear Programming

2. Plot the equation of each one (ignoring the inequality)
Starter
Plot the following inequalities on a graph and shade the region that satisfies all 3…
x = 4 y
𝑥>4
𝑦≤8
2𝑥+𝑦≤18 18 16
Plot the equation of each one (ignoring the inequality)
A quick way to plot the 3rd one is to work out x when y = 0, and to work out y when x is 0, and plot from these!
We need to be right of the red line, below the blue line, and below/left of the green line! 14 12 10 8
y = 8
2𝑥+𝑦=18 6 4
If x = 0, y = 18
If y = 0, x = 9 2 x 2 4 6 8 10 12 14 16 18
2x + y = 18

3. Linear Programming
Linear programming is a mathematical method of optimisation
For example a business might want to make the most profits, or the lowest costs
Businesses have restrictions on them though (employees/space/money etc), so they seek to get the best they can using what they have
Linear programming helps with this!

4. Linear Programming The direction and the inclusion of the
‘or equal to’ is important when writing your constraints.
For example,
‘at most 4’ is stuff that is less than or equal to 4. So x ≤ 4 would be used.
‘Greater than 2’ is everything bigger than 2, but not including. So x > 2 would be used.

5. Machine time needed (hours) Craftsman’s time needed (hours)
Linear Programming
Chair A
Chair B
Machine time needed (hours) 2 3
Craftsman’s time needed (hours) 4
A company makes two different types of chair. The machine time and craftsman’s time needed to make each chair is shown in the table to the right.
In a week, the company has 30 hours of machine time and 32 hours of craftsman’s time available.
They want to make at least 2 of chair A and at least 6 of chair B.
What combinations of chair A and B can be made under the restrictions above?
To answer this type of question, you need to write several inequalities using the information given…

6. Machine time needed (hours) Craftsman’s time needed (hours)
Linear Programming
A company makes two different types of chair. The machine time and craftsman’s time needed to make each chair is shown in the table to the right.
In a week, the company has 30 hours of machine time and 32 hours of craftsman’s time available.
They want to make at least 2 of chair A and at least 6 of chair B.
What combinations of chair A and B can be made under the restrictions above?
Let ‘x’ be the number of chair A and ‘y’ be the number of chair B to be made…
Chair A
Chair B
Machine time needed (hours) 2 3
Craftsman’s time needed (hours) 4
𝑥≥2
The number of chair A has to be greater than 2
𝑦≥6
The number of chair B has to be greater than 6

7. Machine time needed (hours) Craftsman’s time needed (hours)
Linear Programming
A company makes two different types of chair. The machine time and craftsman’s time needed to make each chair is shown in the table to the right.
In a week, the company has 30 hours of machine time and 32 hours of craftsman’s time available.
They want to make at least 2 of chair A and at least 6 of chair B.
What combinations of chair A and B can be made under the restrictions above?
Let ‘x’ be the number of chair A and ‘y’ be the number of chair B to be made…
Chair A
Chair B
Machine time needed (hours) 2 3
Craftsman’s time needed (hours) 4
The total machine time taken will be:
2×𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐ℎ𝑎𝑖𝑟 𝐴 +
3×𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐ℎ𝑎𝑖𝑟 𝐵
=2𝑥+3𝑦
The total machine time available is 30 hours though. Therefore:
2𝑥+3𝑦≤30
𝑥≥2
𝑦≥6
2𝑥+3𝑦≤30

8. Machine time needed (hours) Craftsman’s time needed (hours)
Linear Programming
A company makes two different types of chair. The machine time and craftsman’s time needed to make each chair is shown in the table to the right.
In a week, the company has 30 hours of machine time and 32 hours of craftsman’s time available.
They want to make at least 2 of chair A and at least 6 of chair B.
What combinations of chair A and B can be made under the restrictions above?
Let ‘x’ be the number of chair A and ‘y’ be the number of chair B to be made…
Chair A
Chair B
Machine time needed (hours) 2 3
Craftsman’s time needed (hours) 4
The total craftsman’s time taken will be:
4×𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐ℎ𝑎𝑖𝑟 𝐴 +
2×𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐ℎ𝑎𝑖𝑟 𝐵
=4𝑥+2𝑦
The total craftsman’s time available is 32 hours though. Therefore:
𝑥≥2
2𝑥+3𝑦≤30
4𝑥+2𝑦≤32
𝑦≥6
2𝑥+𝑦≤16
Simplify
2𝑥+𝑦≤16

9. Ignore the inequality signs, and work out coordinates where needed…
Linear Programming
x = 2 y
Using these inequalities, we can draw graphs to represent the situation…
Ignore the inequality signs, and work out coordinates where needed… 18 16
𝑥≥2
2𝑥+3𝑦≤30 14
𝑦≥6
2𝑥+𝑦≤16 12 10 8 6
y = 6
2𝑥+3𝑦=30 4
𝐼𝑓 𝑥=0, 𝑦=10
𝐼𝑓 𝑦=0, 𝑥=15 2 x 2 4 6 8 10 12 14 16 18
2𝑥+𝑦=16
2x + y = 16
2x + 3y = 30
𝐼𝑓 𝑥=0, 𝑦=16
𝐼𝑓 𝑦=0, 𝑥=8

10. Linear Programming 𝑥≥2 2𝑥+3𝑦≤30 𝑦≥6 2𝑥+𝑦≤16
x = 2 y
Using these inequalities, we can draw graphs to represent the situation…
Now consider the region on the graph which satisfies all the inequalities. You are usually asked on these questions to shade the unwanted region… (for some reason…)
To the right of the red line
Above the blue line
Below/left of the green line
Below/left of the purple line
The unshaded region represents all combinations of chairs that can be made! 18 16
𝑥≥2
2𝑥+3𝑦≤30 14
𝑦≥6
2𝑥+𝑦≤16 12 10 8 6
y = 6 4 2 x 2 4 6 8 10 12 14 16 18
2x + y = 16
2x + 3y = 30

11. The optimal value will always occur at a corner of the wanted region!
Linear Programming
x = 2 y
Using these inequalities, we can draw graphs to represent the situation…
Remember that in this region we can have any combination of chair A and B as long as they are integer values… 18 16
𝑥≥2
2𝑥+3𝑦≤30 14
𝑦≥6
2𝑥+𝑦≤16 12 10 8 6
y = 6
𝐶ℎ𝑎𝑖𝑟 𝐴 (𝑥)
𝐶ℎ𝑎𝑖𝑟 𝐵 (𝑦) 4 2 6 2 7 2 2 8 x 3 6 2 4 6 8 10 12 14 16 18 3 7
2x + y = 16
2x + 3y = 30 3 8 4 6
The optimal value will always occur at a corner of the wanted region! 4 7 5 6

12. Write down 2 more inequalities to represent this information:
Plenary
A company has a vehicle parking area of 1200m2 with space for x cars and y trucks. Each car requires 20m2 of space, and a truck requires 100m2 of space.
a) Show that: 𝑥+5𝑦≤60
b) There must also be space for at least 40 vehicles and at least 2 trucks
Write down 2 more inequalities to represent this information:
________

13. Summary We have learnt about Linear Programming
We have seen how it can be used to help businesses make decisions
We have also recapped some graph plotting and Inequalities!

14. Starter
Plot the following inequalities on a graph and shade the region that satisfies all 3… y
𝑥>4
𝑦≤8
2𝑥+𝑦≤18 18 16 14 12 10 8 6 4 2 x 2 4 6 8 10 12 14 16 18

15. Write down 2 more inequalities to represent this information:
Plenary
A company has a vehicle parking area of 1200m2 with space for x cars and y trucks. Each car requires 20m2 of space, and a truck requires 100m2 of space.
a) Show that: 𝑥+5𝑦≤60
b) There must also be space for at least 40 vehicles and at least 2 trucks
Write down 2 more inequalities to represent this information:
________