Business Mathematics Rétallér Orsi
Graphical solution
The problem max z = 3x 1 + 2x 2 2x 1 + x 2 ≤ 100 x 1 + x 2 ≤ 80 x 1 ≤ 40 x 1 ≥ 0 x 2 ≥ 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 = 4x 1 + x 2 8x 1 + 2x 2 ≤ 16 5x 1 + 2x 2 ≤ 12 x 1 ≥ 0 x 2 ≥ 0
Alternative optimum
Possible LP solutions One optimum Alternative optimums (Infinite solutions) Infeasibility Unboundedness
Infeasibility max z = x 1 + x 2 x 1 + x 2 ≤ 4 x 1 - x 2 ≥ 5 x 1 ≥ 0 x 2 ≥ 0
Infeasibility
Possible LP solutions One optimum Alternative optimums (Infinite solutions) Infeasibility Unboundedness
Unboundedness max z = -x 1 + 3x 2 x 1 - x 2 ≤ 4 x 1 + 2x 2 ≥ 4 x 1 ≥ 0 x 2 ≥ 0
Unboundedness
Sensitivity analysis
When is the yellow point the optimal solution?
Sensitivity analysis
The problem max z = 3x 1 + 2x 2 2x 1 + x 2 ≤ 100 x 1 + x 2 ≤ 80 x 1 ≤ 40 x 1 ≥ 0 x 2 ≥ 0 2x 1 + x 2 = 100 x 1 + x 2 = 80
Sensitivity analysis 2x 1 + x 2 = 100 x 1 + x 2 = 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.
ResourceDeskTableChair Lumber (board ft) 861 Finishing (hours) 421,5 Carpentry (hours) 21,50,5 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 8x 1 + 6x 2 + 1x 3 ≤ 48 4x 1 + 2x 2 + 1,5x 3 ≤ 20 2x 1 + 1,5x 2 + 0,5x 3 ≤ 8 x 1, x 2, x 3 ≥ 0 max z = 60x x x 3
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 y 1, y 2, y 3
ResourceDeskTableChair Lumber (board ft) 861 Finishing (hours) 421,5 Carpentry (hours) 21,50,5 Problem – Winston
The primal problem 8x 1 + 6x 2 + 1x 3 ≤ 48 4x 1 + 2x 2 + 1,5x 3 ≤ 20 2x 1 + 1,5x 2 + 0,5x 3 ≤ 8 x 1, x 2, x 3 ≥ 0 max z = 60x x x 3
The dual problem min w = 48y y 2 + 8y 3
ResourceDeskTableChair Lumber (board ft) 861 Finishing (hours) 421,5 Carpentry (hours) 21,50,5 Problem – Winston
The primal problem 8x 1 + 6x 2 + 1x 3 ≤ 48 4x 1 + 2x 2 + 1,5x 3 ≤ 20 2x 1 + 1,5x 2 + 0,5x 3 ≤ 8 x 1, x 2, x 3 ≥ 0 max z = 60x x x 3
The dual problem min w = 48y y 2 + 8y 3 8y 1 + 4y 2 + 2y 3 ≥ 60
The dual problem 8y 1 + 4y 2 + 2y 3 ≥ 60 6y 1 + 2y 2 + 1,5y 3 ≥ 30 1y 1 + 1,5y 2 + 0,5y 3 ≥ 20 y 1, y 2, y 3 ≥ 0 min w = 48y y 2 + 8y 3
Traditional minimum task 2x 1 + 3x 2 ≥ 2 2x 1 + x 2 ≥ 4 x 1 – x 2 ≥ 6 x 1, x 2 ≥ 0 min z = 5x 1 + 2x 2 2y 1 + 2y 2 + y 3 ≤ 5 3y 1 + y 2 – y 3 ≤ 2 y 1, y 2, y 3 ≥ 0 max w = 2y 1 + 4y 2 + 6y 3
Traditional minimum task 2x 1 + 3x 2 ≥ 2 2x 1 + x 2 ≥ 4 x 1 – x 2 ≥ 6 x 1, x 2 ≥ 0 min z = 5x 1 + 2x 2 2y 1 + 2y 2 + y 3 ≤ 5 3y 1 + y 2 – y 3 ≤ 2 y 1, y 2, y 3 ≥ 0 max w = 2y 1 + 4y 2 + 6y 3
A little help for duality Maximum taskMinimum task Variables y i ≥ 0≥ Boundaries y i ≤ 0≤ y i ur= Boundaries ≥y i ≤ 0 Variables ≤y i ≥ 0 =y i ur
Nontraditional minimum task x 1 + 2x 2 + x 3 ≥ 2 x 1 – x 3 ≥ 1 x 2 + x 3 = 1 2x 1 + x 2 ≤ 3 x 1 ur, x 2, x 3 ≥ 0 min z = 2x 1 + 4x 2 + 6x 3
Nontraditional minimum task y 1 + y 2 + y 4 = 2 2y 1 + y 3 + y 4 ≤ 4 y 1 – y 2 + y 3 ≤ 6 y 1, y 2 ≥ 0, y 3 ur, y 4 ≤ 0 max w = 2y 1 + y 2 + y 3 + 3y 4
Thank you for your attention!