Business Mathematics
www.uni-corvinus.hu/~u2w6ol
Rétallér Orsi
Graphical solution
The problem
max z = 3x1 + 2x2
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40
x1 ≥ 0
x2 ≥ 0
Graphical solution
Feasible
region
Is there always one solution?
Possible LP solutions
One optimum
 Alternative optimums (Infinite solutions)
 Infeasibility
 Unboundedness

Possible LP solutions
One optimum
 Alternative optimums (Infinite solutions)
 Infeasibility
 Unboundedness

Possible LP solutions
One optimum
 Alternative optimums (Infinite solutions)
 Infeasibility
 Unboundedness

Alternative optimum
max z = 4x1 + x2
8x1 + 2x2 ≤ 16
5x1 + 2x2 ≤ 12
x1 ≥ 0
x2 ≥ 0
Alternative optimum
Possible LP solutions
One optimum
 Alternative optimums (Infinite solutions)
 Infeasibility
 Unboundedness

Infeasibility
max z = x1 + x2
x 1 + x2 ≤ 4
x1 - x2 ≥ 5
x1 ≥ 0
x2 ≥ 0
Infeasibility
Possible LP solutions
One optimum
 Alternative optimums (Infinite solutions)
 Infeasibility
 Unboundedness

Unboundedness
max z = -x1 + 3x2
x1 - x2 ≤ 4
x1 + 2x2 ≥ 4
x1 ≥ 0
x2 ≥ 0
Unboundedness
Sensitivity analysis
Sensitivity analysis
When is the yellow point
the optimal solution?
Sensitivity analysis
The problem
max z = 3x1 + 2x2
2x1 + x2 = 100
x1 + x2 = 80
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40
x1 ≥ 0
x2 ≥ 0
Sensitivity analysis
2x1 + x2 = 100
x1 + x2 = 80
Range of optimality:
[1;2]
Duality theorem
Problem – 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.
Problem – Winston
Resource
Desk
Table
Chair
Lumber
(board ft)
8
6
1
Finishing
(hours)
4
2
1,5
Carpentry
(hours)
2
1,5
0,5
Problem – 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 problem
max z = 60x1 + 30x2 + 20x3
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
The 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
Problem – Winston
Resource
Desk
Table
Chair
Lumber
(board ft)
8
6
1
Finishing
(hours)
4
2
1,5
Carpentry
(hours)
2
1,5
0,5
The primal problem
max z = 60x1 + 30x2 + 20x3
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
The dual problem
min w = 48y1 + 20y2 + 8y3
Problem – Winston
Resource
Desk
Table
Chair
Lumber
(board ft)
8
6
1
Finishing
(hours)
4
2
1,5
Carpentry
(hours)
2
1,5
0,5
The primal problem
max z = 60x1 + 30x2 + 20x3
8x1 + 6x2 + 1x3 ≤ 48
4x1 + 2x2 + 1,5x3 ≤ 20
2x1 + 1,5x2 + 0,5x3 ≤ 8
x1, x2, x3≥ 0
The dual problem
min w = 48y1 + 20y2 + 8y3
8y1 +
4y2 + 2y3 ≥ 60
The dual problem
min w = 48y1 + 20y2 + 8y3
8y1 + 4y2 + 2y3 ≥ 60
6y1 + 2y2 + 1,5y3 ≥ 30
1y1 + 1,5y2 + 0,5y3 ≥ 20
y1, y2, y3 ≥ 0
Traditional minimum task
min z = 5x1 + 2x2
2x1 + 3x2 ≥ 2
2x1 + x2 ≥ 4
max w = 2y1 + 4y2 + 6y3
x1 – x2 ≥ 6
2y1 + 2y2 + y3 ≤ 5
x1, x2 ≥ 0
3y1 + y2 – y3 ≤ 2
y1, y2, y3 ≥ 0
Traditional minimum task
min z = 5x1 + 2x2
2x1 + 3x2 ≥ 2
2x1 + x2 ≥ 4
max w = 2y1 + 4y2 + 6y3
x1 – x2 ≥ 6
2y1 + 2y2 + y3 ≤ 5
x1, x2 ≥ 0
3y1 + y2 – y3 ≤ 2
y1, y2, y3 ≥ 0
A little help for duality
Maximum task Minimum task
≥
yi ≤ 0
≤
yi ur
=
≥
yi ≤ 0
≤
yi ≥ 0
=
yi ur
Boundaries Variables
Variables Boundaries
yi ≥ 0
Nontraditional minimum task
min z = 2x1 + 4x2 + 6x3
x1 + 2x2 + x3 ≥ 2
x1
– x3 ≥ 1
x 2 + x3 = 1
2x1 + x2
≤3
x1 ur, x2, x3 ≥ 0
Nontraditional minimum task
max w = 2y1 + y2 + y3 + 3y4
y1 + y2
+ y4 = 2
2y1
+ y3 + y4 ≤ 4
y1 – y2 + y3
≤6
y1, y2 ≥ 0, y3 ur, y4 ≤ 0
Thank you for your attention!