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A factory produces two types of drink, an energy drink and a
refresher drink. The days 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 days production.How can the factory
manager decide how much of each drink to make?THE PROBLEMLinear
Programming : Introductory Example
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Energy drink sells at 1 per litreRefresher drink sells at 80 p
per litreTHE PROBLEM
SyrupVitamin supplementConcentrated flavouring5 litres of energy
drink1.25 litres2 units30 cc5 litres of refresher drink1.25 litres1
unit20 ccAvailabilities250 litres300 units4.8 litres
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Syrup constraint:Let x represent number of litres of energy
drinkLet y represent number of litres of refresher drink
0.25x + 0.25y 250 x + y 1000FORMULATION
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Vitamin supplement constraint:Let x represent number of litres
of energy drinkLet y represent number of litres of refresher
drink
0.4x + 0.2y 300 2x + y 1500FORMULATION
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Concentrated flavouring constraint:Let x represent number of
litres of energy drinkLet y represent number of litres of refresher
drink
6x + 4y 4800 3x + 2y 2400FORMULATION
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Objective function:Let x represent number of litres of energy
drinkEnergy drink sells for 1 per litre
Let y represent number of litres of refresher drinkRefresher
drink sells for 80 pence per litre
Maximise x + 0.8yFORMULATION
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Empty grid to accommodate the 3 inequalitiesSOLUTION
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1st constraintDraw boundary line:x + y = 1000SOLUTION
xy0100010000
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1st constraintShade out unwanted region:x + y 1000SOLUTION
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Empty grid to accommodate the 3 inequalitiesSOLUTION
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2nd constraintDraw boundary line:2x + y = 1500SOLUTION
xy015007500
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2nd constraintShade out unwanted region:2x + y 1500SOLUTION
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Empty grid to accommodate the 3 inequalitiesSOLUTION
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3rd constraintDraw boundary line:3x + 2y = 2400SOLUTION
xy012008000
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3rd constraintShade out unwanted region:3x + 2y 2400SOLUTION
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All three constraints:First:x + y 1000SOLUTION
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All three constraints:First:x + y 1000Second:2x + y
1500SOLUTION
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All three constraints:First:x + y 1000Second:2x + y 1500Third:3x
+ 2y 2400SOLUTION
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All three constraints:First:x + y 1000Second:2x + y 1500Third:3x
+ 2y 2400Adding:x 0 and y 0 SOLUTION
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Feasible region is the unshaded area and satisfies:x + y 10002x
+ y 15003x + 2y 2400x 0 and y 0 SOLUTION
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Evaluate the objective function x + 0.8yat vertices of the
feasible region:O: 0 + 0 = 0A: 0 + 0.8x1000 = 800B: 400 + 0.8x600 =
880C: 600 + 0.8x300= 840D: 750 + 0 = 750
OABCDMaximum income = 800 at (400, 600)SOLUTION