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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? THE PROBLEM Linear Programming : Introductory Example

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Syrup Vitamin supplement Concentrated flavouring 5 litres of energy drink 1.25 litres2 units30 cc 5 litres of refresher drink 1.25 litres1 unit20 cc Availabilities250 litres300 units4.8 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 250 x + y 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 300 2x + y 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 4800 3x + 2y 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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Empty grid to accommodate the 3 inequalities SOLUTION

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1 st constraint Draw boundary line: x + y = 1000 xy 01000 0 SOLUTION

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

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Empty grid to accommodate the 3 inequalities SOLUTION

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2 nd constraint Draw boundary line: 2x + y = 1500 xy 01500 7500 SOLUTION

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

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Empty grid to accommodate the 3 inequalities SOLUTION

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3 rd constraint Draw boundary line: 3x + 2y = 2400 xy 01200 8000 SOLUTION

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

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All three constraints: First: x + y 1000 SOLUTION

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

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

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

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

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Evaluate the objective function x + 0.8y at vertices of the feasible region: O: 0 + 0 = 0 A: 0 + 0.8 x 1000 = 800 B: 400 + 0.8 x 600 = 880 C: 600 + 0.8 x 300 = 840 D: 750 + 0 = 750 O A B C D Maximum income = £800 at (400, 600) SOLUTION

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