Revision Linear Optimisation. A miller can buy wheat from three suppliers: Airey Farm, Berry Farm...

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Linear Optimisation

A miller can buy wheat from three suppliers: Airey Farm,

Berry Farm and Cherry Farm. In each case the wheat is

contaminated with two things – bran and husks. When combined, the wheat must contain no more than 5% bran and no more than 4% husks. The miller wishes to

make 50 tonnes of wheat in total, by purchasing from each

farm. Each farm’s wheat contains the following amounts

of bran and husks.

Identify the decision variables

• Let x = tonnes of wheat purchased from Airey Farm

• Let y = tonnes of wheat purchased from Berry Farm

• Let z = tonnes of wheat purchased from Cherry Farm

• Write z in terms of x and y

Define the constraint due to bran

• 3% of x + 5% of y + 7% of z is less than or equal to 5% of 50 tonnes.

€

0.03x + 0.05y + 0.07z ≤ 0.05 × 50

€

3x + 5y + 7(50 − x − y) ≤ 5(50)

€

4x + 2y ≥100

Define the constraint due to husks

• 5% of x + 2% of y + 6% of z is less than or equal to 4% of 50 tonnes.

€

0.05x + 0.02y + 0.06z ≤ 0.04 × 50

€

5x + 2y + 6(50 − x − y) ≤ 4(50)

€

x + 4y ≥100

Non-negativity

€

x ≥ 0

€

y ≥ 0

Supply of wheat from Cherry Farm must be non-negative too.

€

50 − x − y ≥ 0

€

x + y ≤ 50

Graph the solution region

100

50

25

50

Find the vertices of the solution region

100

50

25

50

(0, 50)

(14.29, 21.43)

(33.33, 16.67)

Define the objective function

• Cost = $70x + $60y + $40z• Cost = $70x + $60y + $40(50 - x - y)

• Cost = 30x + 20y + 2000

Substitute each vertex’s coordinates into the objective

function• Cost = 30x + 20y + 2000• A(0, 50) Cost = $3000• B(14.29, 21.43) Cost = $2857.30• C(33.33, 16.67) Cost = $3333.30

• B gives the minimum costs so purchase

14.29 from Airey, 21.43 from Berry and 14.28 from Cherry

Question 2Barmy Bicycle Builders makes bicycles at two factories and ships them to two separatedistributors.Factory A produces 100 bicycles; Factory B produces 150.Distributor X wants 70 bicycles;Distributor Y wants 90 bicycles.Furthermore the shipping costs between factories (in dollars per bicycle) and distributorsare shown in the table below.

Factory A Factory BDistributor X $6 $7Distributor Y $3 $5

Define the decision variables

• Let x = number of bicycles shipped to distributor X from factory A

• Therefore 70 - x is the number shipped to X from factory B

• Let y = number of bicycles shipped to distributor Y from factory A

• Therefore 90 - y is the number shipped to y from factory B

Summarise the decision variables, supply and demand

in a tableFactory A Factory B Total

demand from each distribut

or

Distributor X

Distributor Y

Total supply

from each factory

Summarise the decision variables, supply and demand

in a tableFactory A Factory B Total

demand from each distribut

or

Distributor X

x 70 - x 70

Distributor Y

y 90 - y 90

Total supply

from each factory

100 150

Define constraints based on the fact that all shipments must be non-negative amounts

• Factory A:

€

x + y ≤100

Define constraints based on the fact that all shipments must be non-negative amounts

• Factory B:

€

70 − x + 90 − y ≤150

€

x + y ≥10

Define constraints based on the fact that all shipments must be non-negative amounts

• Non-negativity:

x ≥0y≥0

70 −x≥0 ⇒ x≤7090 −y≥0 ⇒ y≤90

Graph the solution region

100

100

Determine the vertices

100

100 (0, 10)

(0, 90)

(10, 90)

(70, 30)

(70, 0)

(10, 0)

Determine the objective function

• Factory A to distributor X: x bicycles at $6

• Factory A to distributor Y: y bicycles at $3

• Factory B to distributor X: 70 - x bicycles at $7

• Factory B to distributor Y: 90 - y bicycles at $5

• Total cost =

€

6x + 3y + 7(70 − x) + 5(90 − y)

€

Cost = − x − 2y + 940

Determine the cost for each vertex

• A(0, 10) Cost = 920• B(0, 90) Cost = 760 • C(10, 90) Cost = 750• D(70, 30) Cost = 810• E(70, 0) Cost = 870• F(10, 0) Cost = 930

• Minimum costs occur when x = 10 and y = 90

Amount to ship from each factory

• Ship 10 from factory A to distributor X

• Ship 90 from factory A to distributor Y

• Ship 60 from factory B to distributor X

• Ship 0 from factory B to distributor Y