ЛЕК ФОРМЫ D1 - LP Example

8/7/2019 ЛЕК ФОРМЫ D1 - LP Example

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A factory produces two types of drink, an ¶energy·drink and a ¶refresher· drink. The day·s output isto be planned. Each drink requires syrup, vitamin

supplement and concentrated flavouring, as shownin the table.

The last row in the table shows how much of eachingredient is available for the day·s production.

How can the factory manager decide how muchof each drink to make?

THE PROBLEM

Linear Programming : Introductory Example



SyrupVitamin

supplementConcentratedflavouring

1 litre ofenergy drink

0.25 litres 0.4 units 6 cc

1 litre ofrefresher

drink0.25 litres 0.2 unit 4 cc

Availabilities 250 litres 300 units 4.8 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 + 0.25y e 250

x + y e 1000

FORMULATION



Vitamin supplement constraint:



0.4x + 0.2y e 300

2x + y e 1500

FORMULATION



Concentrated flavouring constraint:



6x + 4y e 4800

3x + 2y e 2400

FORMULATION



Objective function:


� Energy drink sells for £1 per litre


� Refresher drink sells for 80 pence per litre

Maximise x+ 0.8y

FORMULATION



- 200 200 400

-200

200Empty grid toaccommodatethe 3inequalities

SOLUTION



- 200 200 400

-200

2001st constraint

Draw boundaryline:

x + y = 1000

x y

0 1000

1

000 0

SOLUTION



- 200 200 400

-200

2001st constraint

Shade outunwantedregion:

x + ye 1

000

SOLUTION



- 200 200 400

-200

200Empty grid to

accommodatethe 3inequalities

SOLUTION



-

-

2nd constraint

Draw boundaryline:

2x + y = 1500

x y

0 1500

750 0

SOLUTION



-

-

2nd constraint

Shade outunwantedregion:

2x + ye 1

500

SOLUTION



- 200 200 400

-200

200Empty grid to

accommodatethe 3inequalities

SOLUTION



-

-

3rd constraint

Draw boundaryline:

3x + 2y = 2400

x y

0 1200

800 0

SOLUTION



-

-

3rd constraint

Shade outunwanted region:

3x + 2y e 2400

SOLUTION



- 200 200 400

-200

200All three

constraints:

First:

x + y e 1000

SOLUTION



- 200 200 400

-200

200All three

constraints:

First:

x + y e 1000

Second:

2x + y e 1500

SOLUTION



- 200 200 400

-200

200All three

constraints:

First:

x + y e 1000

Second:

2x + y e 1500

Third:

3x + 2y e 2400

SOLUTION



- 200 200 400

-200

200All three

constraints:

First:

x + y e 1000

Second:

2x + y e 1500

Third:

3x + 2y e 2400Adding:

x u 0 and y u 0

SOLUTION



- 200 200 400

-200

200Feasible region

is the unshadedarea andsatisfies:

x + y e 1000

2x + y e 1500

3x + 2y e 2400

x u 0 and y u 0

SOLUTION



Evaluate the

objective functionx + 0.8y

at vertices of thefeasible region:

O: 0 + 0 =0

A: 0 + 0.8x1000= 800

B: 400 + 0.8x600= 880

C: 600 + 0.8x300= 840

D: 750 + 0 =750

-

-

O

A

B

C

D

Maximum income = £880 at (400, 600)

SOLUTION

ЛЕК ФОРМЫ D1 - LP Example

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