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?

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

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 + 0.25y  250  x + y  1000

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)