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A factory produces two types of drink, an ¶energy·drink and a ¶refresher· drink. The day·s output isto be planned. Each drink requires syrup, vitamin
supplement and concentrated flavouring, as shownin the table.
The last row in the table shows how much of eachingredient is available for the day·s production.
How can the factory manager decide how muchof each drink to make?
THE PROBLEM
Linear Programming : Introductory Example
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SyrupVitamin
supplementConcentratedflavouring
1 litre ofenergy drink
0.25 litres 0.4 units 6 cc
1 litre ofrefresher
drink0.25 litres 0.2 unit 4 cc
Availabilities 250 litres 300 units 4.8 litres
Energy drink sells at £1 per litre
Refresher drink sells at 80 p per litre
THE PROBLEM
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Syrup constraint:
Let x represent number of litres of energy drink
Let y represent number of litres of refresher drink
0.25x + 0.25y e 250
x + y e 1000
FORMULATION
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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 e 300
2x + y e 1500
FORMULATION
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Concentrated flavouring constraint:
Let x represent number of litres of energy drink
Let y represent number of litres of refresher drink
6x + 4y e 4800
3x + 2y e 2400
FORMULATION
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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
FORMULATION
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- 200 200 400
-200
200Empty grid toaccommodatethe 3inequalities
SOLUTION
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- 200 200 400
-200
2001st constraint
Draw boundaryline:
x + y = 1000
x y
0 1000
1
000 0
SOLUTION
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- 200 200 400
-200
2001st constraint
Shade outunwantedregion:
x + ye 1
000
SOLUTION
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- 200 200 400
-200
200Empty grid to
accommodatethe 3inequalities
SOLUTION
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-
-
2nd constraint
Draw boundaryline:
2x + y = 1500
x y
0 1500
750 0
SOLUTION
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-
-
2nd constraint
Shade outunwantedregion:
2x + ye 1
500
SOLUTION
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- 200 200 400
-200
200Empty grid to
accommodatethe 3inequalities
SOLUTION
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-
-
3rd constraint
Draw boundaryline:
3x + 2y = 2400
x y
0 1200
800 0
SOLUTION
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-
3rd constraint
Shade outunwanted region:
3x + 2y e 2400
SOLUTION
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- 200 200 400
-200
200All three
constraints:
First:
x + y e 1000
SOLUTION
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- 200 200 400
-200
200All three
constraints:
First:
x + y e 1000
Second:
2x + y e 1500
SOLUTION
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- 200 200 400
-200
200All three
constraints:
First:
x + y e 1000
Second:
2x + y e 1500
Third:
3x + 2y e 2400
SOLUTION
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- 200 200 400
-200
200All three
constraints:
First:
x + y e 1000
Second:
2x + y e 1500
Third:
3x + 2y e 2400Adding:
x u 0 and y u 0
SOLUTION
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- 200 200 400
-200
200Feasible region
is the unshadedarea andsatisfies:
x + y e 1000
2x + y e 1500
3x + 2y e 2400
x u 0 and y u 0
SOLUTION
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Evaluate the
objective functionx + 0.8y
at vertices of thefeasible region:
O: 0 + 0 =0
A: 0 + 0.8x1000= 800
B: 400 + 0.8x600= 880
C: 600 + 0.8x300= 840
D: 750 + 0 =750
-
-
O
A
B
C
D
Maximum income = £880 at (400, 600)
SOLUTION