1. Linear Programming : Introductory Example
THE PROBLEM
A factory produces two types of drink, an ‘energy’ drink and a ‘refresher’ drink. The day’s output is to be planned. Each drink requires syrup, vitamin supplement and concentrated flavouring, as shown in the table.
The last row in the table shows how much of each ingredient is available for the day’s production.
How can the factory manager decide how much of each drink to make?

2. Concentrated flavouring
THE PROBLEM
Syrup
Vitamin supplement
Concentrated flavouring
5 litres of energy drink
1.25 litres
2 units
30 cc
5 litres of refresher drink
1 unit
20 cc
Availabilities
250 litres
300 units
4.8 litres
Energy drink sells at £1 per litre
Refresher drink sells at 80 p per litre

3. Let x represent number of litres of energy drink
FORMULATION
Syrup constraint:
Let x represent number of litres of energy drink
Let y represent number of litres of refresher drink
0.25x y  250
 x + y  1000

4. Vitamin supplement constraint:
FORMULATION
Vitamin supplement constraint:
Let x represent number of litres of energy drink
Let y represent number of litres of refresher drink
0.4x + 0.2y  300
 2x + y  1500

5. Concentrated flavouring constraint:
FORMULATION
Concentrated flavouring constraint:
Let x represent number of litres of energy drink
Let y represent number of litres of refresher drink
6x + 4y  4800
 3x + 2y  2400

6. Let x represent number of litres of energy drink
FORMULATION
Objective function:
Let x represent number of litres of energy drink
Energy drink sells for £1 per litre
Let y represent number of litres of refresher drink
Refresher drink sells for 80 pence per litre
Maximise x + 0.8y

7. Empty grid to accommodate the 3 inequalities
SOLUTION
Empty grid to accommodate the 3 inequalities

8. SOLUTION
1st constraint
Draw boundary line:
x + y = 1000 x y
1000

9. Shade out unwanted region:
SOLUTION
1st constraint
Shade out unwanted region:
x + y  1000

10. Empty grid to accommodate the 3 inequalities
SOLUTION
Empty grid to accommodate the 3 inequalities

11. SOLUTION
2nd constraint
Draw boundary line:
2x + y = 1500 x y
1500
750

12. Shade out unwanted region:
SOLUTION
2nd constraint
Shade out unwanted region:
2x + y  1500

13. Empty grid to accommodate the 3 inequalities
SOLUTION
Empty grid to accommodate the 3 inequalities

14. x y 1200 800 3rd constraint Draw boundary line: 3x + 2y = 2400
SOLUTION
3rd constraint
Draw boundary line:
3x + 2y = 2400 x y
1200
800

15. Shade out unwanted region:
SOLUTION
3rd constraint
Shade out unwanted region:
3x + 2y  2400

16. All three constraints: First: x + y  1000
SOLUTION
All three constraints:
First:
x + y  1000

17. All three constraints: First: x + y  1000 Second: 2x + y  1500
SOLUTION
All three constraints:
First:
x + y  1000
Second:
2x + y  1500

18. All three constraints: First: x + y  1000 Second: 2x + y  1500
SOLUTION
All three constraints:
First:
x + y  1000
Second:
2x + y  1500
Third:
3x + 2y  2400

19. All three constraints: First: x + y  1000 Second: 2x + y  1500
SOLUTION
All three constraints:
First:
x + y  1000
Second:
2x + y  1500
Third:
3x + 2y  2400
Adding:
x  0 and y  0

20. Feasible region is the unshaded area and satisfies:
SOLUTION
Feasible region is the unshaded area and satisfies:
x + y  1000
2x + y  1500
3x + 2y  2400
x  0 and y  0

21. Evaluate the objective function x + 0.8y
SOLUTION
Evaluate the objective function
x + 0.8y
at vertices of the feasible region:
O: = 0
A: x1000
= 800
B: x600
= 880
C: x300
= 840
D: = 750 A B C D O
Maximum income = £880 at (400, 600)

22. Alternatively, draw a straight line x + 0.8y = k.
SOLUTION
Alternatively, draw a straight line x + 0.8y = k. A
Move a ruler parallel to this line until it reaches the edge of the feasible region. B
The furthest point you can move it to is point B. C D O
At B (400, 600) the value of the objective function is 880.
Maximum income = £880 at (400, 600)