business mathematics u2w6ol rétallér orsi
TRANSCRIPT
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Business MathematicsBusiness Mathematicswww.uni-corvinus.hu/~u2w6ol
Rétallér Orsi
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Graphical solutionGraphical solution
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The problemThe problem
max z = 3x1 + 2x2
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40
x1 ≥ 0
x2 ≥ 0
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Graphical solutionGraphical solution
Feasible region
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Is there always one Is there always one solution?solution?
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Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
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Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
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Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
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Alternative optimumAlternative optimum
max z = 4x1 + x2
8x1 + 2x2 ≤ 16
5x1 + 2x2 ≤ 12
x1 ≥ 0
x2 ≥ 0
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Alternative optimumAlternative optimum
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Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
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InfeasibilityInfeasibility
max z = x1 + x2
x1 + x2 ≤ 4
x1 - x2 ≥ 5
x1 ≥ 0
x2 ≥ 0
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InfeasibilityInfeasibility
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Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
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UnboundednessUnboundedness
max z = -x1 + 3x2
x1 - x2 ≤ 4
x1 + 2x2 ≥ 4
x1 ≥ 0
x2 ≥ 0
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UnboundednessUnboundedness
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Sensitivity analysisSensitivity analysis
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Sensitivity analysisSensitivity analysis
When is the yellow point the optimal solution?
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Sensitivity analysisSensitivity analysis
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The problemThe problem
max z = 3x1 + 2x2
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40
x1 ≥ 0
x2 ≥ 0
2x1 + x2 = 100
x1 + x2 = 80
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Sensitivity analysisSensitivity analysis
2x1 + x2 = 100
x1 + x2 = 80Range of optimality:
[1;2]
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Duality theoremDuality theorem
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Problem – WinstonProblem – Winston
The Dakota Furniture Company manufactures desks, tables, and chairs. The manufacture of each type of furniture requires lumber and two types of skilled labor: finishing and carpentry. The amount of each resource needed to make each type of furniture is given in the following table.
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Resource Desk Table Chair
Lumber (board ft)
8 6 1
Finishing(hours)
4 2 1,5
Carpentry(hours)
2 1,5 0,5
Problem – WinstonProblem – Winston
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Problem – WinstonProblem – Winston
At present, 48 board feet of lumber, 20 finishing hours, and 8 carpentry hours are available. A desk sells for $60, a table for $30, and a chair for $20. Since the available resources have already been purchased, Dakota wants to maximize total revenue.
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Formalizing the problemFormalizing the problem
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
max z = 60x1 + 30x2 + 20x3
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The new problemThe new problem
For how much could a company buy all the resources of the Dakota company?
(Dual task)
The prices for the resources are indicated as y1, y2, y3
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Resource Desk Table Chair
Lumber (board ft)
8 6 1
Finishing(hours)
4 2 1,5
Carpentry(hours)
2 1,5 0,5
Problem – WinstonProblem – Winston
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The primal problemThe primal problem
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
max z = 60x1 + 30x2 + 20x3
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The dual problemThe dual problem
min w = 48y1 + 20y2 + 8y3
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Resource Desk Table Chair
Lumber (board ft)
8 6 1
Finishing(hours)
4 2 1,5
Carpentry(hours)
2 1,5 0,5
Problem – WinstonProblem – Winston
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The primal problemThe primal problem
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
max z = 60x1 + 30x2 + 20x3
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The dual problemThe dual problem
min w = 48y1 + 20y2 + 8y3
8y1 + 4y2 + 2y3 ≥ 60
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The dual problemThe dual problem
8y1 + 4y2 + 2y3 ≥ 60
6y1 + 2y2 + 1,5y3 ≥ 30
1y1 + 1,5y2 + 0,5y3 ≥ 20
y1, y2, y3 ≥ 0
min w = 48y1 + 20y2 + 8y3
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Traditional minimum taskTraditional minimum task
2x1 + 3x2 ≥ 2
2x1 + x2 ≥ 4
x1 – x2 ≥ 6
x1, x2 ≥ 0
min z = 5x1 + 2x2
2y1 + 2y2 + y3 ≤ 5
3y1 + y2 – y3 ≤ 2
y1, y2, y3 ≥ 0
max w = 2y1 + 4y2 + 6y3
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Traditional minimum taskTraditional minimum task
2x1 + 3x2 ≥ 2
2x1 + x2 ≥ 4
x1 – x2 ≥ 6
x1, x2 ≥ 0
min z = 5x1 + 2x2
2y1 + 2y2 + y3 ≤ 5
3y1 + y2 – y3 ≤ 2
y1, y2, y3 ≥ 0
max w = 2y1 + 4y2 + 6y3
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A little help for dualityA little help for duality
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Nontraditional minimum Nontraditional minimum tasktask
x1 + 2x2 + x3 ≥ 2
x1 – x3 ≥ 1
x2 + x3 = 1
2x1 + x2 ≤ 3
x1 ur, x2, x3 ≥ 0
min z = 2x1 + 4x2 + 6x3
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Nontraditional minimum Nontraditional minimum tasktask
y1 + y2 + y4 = 2
2y1 + y3 + y4 ≤ 4
y1 – y2 + y3 ≤ 6
y1, y2 ≥ 0, y3 ur, y4 ≤ 0
max w = 2y1 + y2 + y3 + 3y4
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Thank you for your Thank you for your attention!attention!