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

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

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 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 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 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 FORMULATION

Empty grid to accommodate the 3 inequalities SOLUTION

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

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

Empty grid to accommodate the 3 inequalities SOLUTION

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

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

Empty grid to accommodate the 3 inequalities SOLUTION

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

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

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

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 SOLUTION

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

Feasible region is the unshaded area and satisfies: x + y  x + y  x + 2y  2400 x  0 and y  0 SOLUTION

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

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