Linear Programming inequalities

Feb 27, 2016

I find that the first challenge in solving LP problems is the construction of the inequalities. Often, the question asks you to “show” a particular inequality and to construct a second inequality (assuming two variables).

The setup of the situation is of the following form.

There are several versions or type of object that are to be created. For example making: cakes, radio programs, type of exercise program, type of garden, type of festival etc.

Each of the objects is made from at least two different components or ingredients. For example:

Object Type

Components or ingredients

Cake (sponge, fruit, flap-jack, brownie)

Eggs, flour, milk

Radio program (presenter 1, presenter 2)

News, music, adverts

Exercise programme (weight-loss, endurance, cardio)

Swimming, rowing, running

Type of packing for parcel to be posted (fragile glass, electronics, pens, DVDs)

Shredded paper, polystyrene, air-filled padding

Childrens’ party bag contents (toddlers, teenagers, pre-teens)

Balloons, sweets, toys

Layout of garden (modern, classical, coastal)

Flowers, grass, paving

Music Festival

Rock music, soul music, folk music

Film Festival (new films, classics, foreign language)

Action adventure, animation, Science Fiction

Dance Festival (youth, retirees, solo dance)

Ballroom, Folk, Ballet, Street

Type of badge (advert, club membership, comic)

Print image, fix cover, fix pin

Type of building (apartment, house, office block)

Concrete, bricks, wooden wall.

Also there will be constraints on the amount of components available) e.g.

Object Type

Maximum or minimum quantities of Components or ingredients available

Cake

50 Eggs,

5kg flour,

4000 ml milk

Radio program

At least 5 minutes per hour News,

No more than 20 minutes per hour music, At least 2 minutes per hour adverts

No more than 10 minutes per hour adverts

Exercise programme

At least one length Swimming,

Maximum 5 minutes rowing,

Maximum 10 minutes running

Type of packing for parcel to be posted

10 kg Shredded paper,

3 kg polystyrene,

30 kg air-filled padding

Childrens’ party bag contents

300 Balloons available,

3 kg sweets available,

50 toys available

Layout of garden

400 Flower plants,

30 metres of grass,

15 metre squared paving

Music Festival

6 Rock bands,

10 soul music bands,

12 folk music groups

Film Festival (new films, classics, foreign language)

Action adventure, animation, Science Fiction

Dance Festival (youth, retirees, solo dance)

7 Ballroom couples,

4 Folk groups,

12 Ballet dancers,

3 Street dance troupes

To construct the inequalities you clearly identify how much of each component or ingredient you require for one each of the object e.g.

1 sponge requires 2 Eggs, 50g flour, 125ml milk

1 fruit cake 3 Eggs, 125g flour, 25ml milk

1 flap-jack cake, 1 Eggs, 75g flour, no milk

1 brownie, 2 Eggs, 125 g flour, 35ml milk

you multiply each type of object by the (unknown) variables x, y, z (use more variables if required) e.g.

w sponges,

x fruit cakes,

y flap-jack cakes,

z brownies,

Then carry this multiple through to the components or ingredients e.g.

w sponges requires 2w Eggs, 50w g flour, 125w ml milk

x fruit cakes, 3x Eggs, 125x g flour, 25x ml milk

y flap-jack cakes, 1y Eggs, 75y g flour, no milk

z brownies, 2z Eggs, 125z g flour, 35z ml milk

So we total each ingredient or component

Eggs: 2w + 3x + y + 2z

Flour: 50w + 125x + 75y + 125z

Milk: 125w + 25x + 0 + 35z

Then we apply the limits for each components:

50 Eggs,

5kg flour,

4000 ml milk

2w + 3x + y + 2z ≤ 50

50w + 125x + 75y + 125z ≤ 5000

125w + 25x + 35z ≤ 4000

So finally we have the inequalities. Phew !

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