Post on 02-Jan-2016
Business MathematicsBusiness Mathematicswww.uni-corvinus.hu/~u2w6ol
Rétallér Orsi
Graphical solutionGraphical solution
The problemThe problem
max z = 3x1 + 2x2
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40
x1 ≥ 0
x2 ≥ 0
Graphical solutionGraphical solution
Feasible region
Is there always one Is there always one solution?solution?
Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
Alternative optimumAlternative optimum
max z = 4x1 + x2
8x1 + 2x2 ≤ 16
5x1 + 2x2 ≤ 12
x1 ≥ 0
x2 ≥ 0
Alternative optimumAlternative optimum
Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
InfeasibilityInfeasibility
max z = x1 + x2
x1 + x2 ≤ 4
x1 - x2 ≥ 5
x1 ≥ 0
x2 ≥ 0
InfeasibilityInfeasibility
Possible LP solutionsPossible LP solutions
One optimumAlternative optimums (Infinite
solutions)InfeasibilityUnboundedness
UnboundednessUnboundedness
max z = -x1 + 3x2
x1 - x2 ≤ 4
x1 + 2x2 ≥ 4
x1 ≥ 0
x2 ≥ 0
UnboundednessUnboundedness
Sensitivity analysisSensitivity analysis
Sensitivity analysisSensitivity analysis
When is the yellow point the optimal solution?
Sensitivity analysisSensitivity analysis
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
Sensitivity analysisSensitivity analysis
2x1 + x2 = 100
x1 + x2 = 80Range of optimality:
[1;2]
Duality theoremDuality theorem
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.
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
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.
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
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
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
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
The dual problemThe dual problem
min w = 48y1 + 20y2 + 8y3
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
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
The dual problemThe dual problem
min w = 48y1 + 20y2 + 8y3
8y1 + 4y2 + 2y3 ≥ 60
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
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
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
A little help for dualityA little help for duality
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
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
Thank you for your Thank you for your attention!attention!