revision linear optimisation. a miller can buy wheat from three suppliers: airey farm, berry farm...
TRANSCRIPT
![Page 1: Revision Linear Optimisation. A miller can buy wheat from three suppliers: Airey Farm, Berry Farm and Cherry Farm. In each case the wheat is contaminated](https://reader035.vdocuments.us/reader035/viewer/2022062803/56649efa5503460f94c0c5a7/html5/thumbnails/1.jpg)
Revision
Linear Optimisation
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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.
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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
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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
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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
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Non-negativity
€
x ≥ 0
€
y ≥ 0
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Supply of wheat from Cherry Farm must be non-negative too.
€
50 − x − y ≥ 0
€
x + y ≤ 50
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Graph the solution region
100
50
25
50
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Find the vertices of the solution region
100
50
25
50
(0, 50)
(14.29, 21.43)
(33.33, 16.67)
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Define the objective function
• Cost = $70x + $60y + $40z• Cost = $70x + $60y + $40(50 - x - y)
• Cost = 30x + 20y + 2000
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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
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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
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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
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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
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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
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Define constraints based on the fact that all shipments must be non-negative amounts
• Factory A:
€
x + y ≤100
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Define constraints based on the fact that all shipments must be non-negative amounts
• Factory B:
€
70 − x + 90 − y ≤150
€
x + y ≥10
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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
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Graph the solution region
100
100
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Determine the vertices
100
100 (0, 10)
(0, 90)
(10, 90)
(70, 30)
(70, 0)
(10, 0)
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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
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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
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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
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