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

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

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

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

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

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

Empty grid to accommodate the 3 inequalitiesSOLUTION

1st constraintDraw boundary line:x + y = 1000SOLUTION

xy0100010000

1st constraintShade out unwanted region:x + y 1000SOLUTION

Empty grid to accommodate the 3 inequalitiesSOLUTION

2nd constraintDraw boundary line:2x + y = 1500SOLUTION

xy015007500

2nd constraintShade out unwanted region:2x + y 1500SOLUTION

Empty grid to accommodate the 3 inequalitiesSOLUTION

3rd constraintDraw boundary line:3x + 2y = 2400SOLUTION

xy012008000

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

All three constraints:First:x + y 1000SOLUTION

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

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

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

Feasible region is the unshaded area and satisfies:x + y 10002x + y 15003x + 2y 2400x 0 and y 0 SOLUTION

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