Business Mathematics www uni-corvinus hu u 2 w 6

Business Mathematics www. uni-corvinus. hu/~u 2 w 6 ol Rétallér Orsi

Graphical solution

The problem max z = 3 x 1 + 2 x 2 2 x 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

Alternative optimum max z = 4 x 1 + x 2 8 x 1 + 2 x 2 ≤ 16 5 x 1 + 2 x 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 ≤ 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 + 3 x 2 x 1 - x 2 ≤ 4 x 1 + 2 x 2 ≥ 4 x 1 ≥ 0 x 2 ≥ 0

Unboundedness

Sensitivity analysis

Sensitivity analysis When is the yellow point the optimal solution?

Sensitivity analysis

The problem max z = 3 x 1 + 2 x 2 2 x 1 + x 2 = 100 x 1 + x 2 = 80 2 x 1 + x 2 ≤ 100 x 1 + x 2 ≤ 80 x 1 ≤ 40 x 1 ≥ 0 x 2 ≥ 0

Sensitivity analysis 2 x 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.

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 = 60 x 1 + 30 x 2 + 20 x 3 8 x 1 + 6 x 2 + 1 x 3 ≤ 48 4 x 1 + 2 x 2 + 1, 5 x 3 ≤ 20 2 x 1 + 1, 5 x 2 + 0, 5 x 3 ≤ 8 x 1, x 2, x 3≥ 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 y 1, y 2, y 3

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 = 60 x 1 + 30 x 2 + 20 x 3 8 x 1 + 6 x 2 + 1 x 3 ≤ 48 4 x 1 + 2 x 2 + 1, 5 x 3 ≤ 20 2 x 1 + 1, 5 x 2 + 0, 5 x 3 ≤ 8 x 1, x 2, x 3≥ 0

The dual problem min w = 48 y 1 + 20 y 2 + 8 y 3

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 = 60 x 1 + 30 x 2 + 20 x 3 8 x 1 + 6 x 2 + 1 x 3 ≤ 48 4 x 1 + 2 x 2 + 1, 5 x 3 ≤ 20 2 x 1 + 1, 5 x 2 + 0, 5 x 3 ≤ 8 x 1, x 2, x 3≥ 0

The dual problem min w = 48 y 1 + 20 y 2 + 8 y 3 8 y 1 + 4 y 2 + 2 y 3 ≥ 60

The dual problem min w = 48 y 1 + 20 y 2 + 8 y 3 8 y 1 + 4 y 2 + 2 y 3 ≥ 60 6 y 1 + 2 y 2 + 1, 5 y 3 ≥ 30 1 y 1 + 1, 5 y 2 + 0, 5 y 3 ≥ 20 y 1, y 2, y 3 ≥ 0

Traditional minimum task min z = 5 x 1 + 2 x 2 2 x 1 + 3 x 2 ≥ 2 2 x 1 + x 2 ≥ 4 max w = 2 y 1 + 4 y 2 + 6 y 3 x 1 – x 2 ≥ 6 2 y 1 + 2 y 2 + y 3 ≤ 5 x 1, x 2 ≥ 0 3 y 1 + y 2 – y 3 ≤ 2 y 1, y 2, y 3 ≥ 0

A little help for duality Maximum task Minimum task ≥ yi ≤ 0 ≤ yi ur = ≥ yi ≤ 0 ≤ yi ≥ 0 = yi ur Boundaries Variables Boundaries yi ≥ 0

Nontraditional minimum task min z = 2 x 1 + 4 x 2 + 6 x 3 x 1 + 2 x 2 + x 3 ≥ 2 x 1 – x 3 ≥ 1 x 2 + x 3 = 1 2 x 1 + x 2 ≤ 3 x 1 ur, x 2, x 3 ≥ 0

Nontraditional minimum task max w = 2 y 1 + y 2 + y 3 + 3 y 4 y 1 + y 2 + y 4 = 2 2 y 1 + y 3 + y 4 ≤ 4 y 1 – y 2 + y 3 ≤ 6 y 1, y 2 ≥ 0, y 3 ur, y 4 ≤ 0

Thank you for your attention!