Ch.4Duality and Post Optimal Analysis

Dr. Ayham Jaaron

Introduction

• One of the most important discoveries in the early development of linear programming was the concept of duality and its many important ramifications.

• This discovery revealed that every linear programming problem has associated with it another linear programming problem called the dual. The relationships between the dual problem and the original problem (called the primal) prove to be extremely useful in a variety of ways.

• We shall describe many valuable applications of duality theory in this chapter.

Definition of the Dual problem

• The dual problem is an LP defined directly and systematically from the primal (original) LP model.

• The two problems are so closely related that the primal solution of one problem automatically provides the optimal solution to the other.

• The primal problem represents a resource allocation case where the dual problem represents a resource valuation problem.

• Duality help simplification of the simplex problem.

Rules for constructing the dual problem

Primal Problem Dual Problem

Objective Objective Constraints Type Variable sign

Maximization Minimization ≥ unrestricted

Minimization Maximization ≤ unrestricted

Example (1)

• Write the dual for the following primal problem: • Maximize Z= 5x1 + 12x2 + 4x3

Subject to: x1+2x2+x3 ≤ 10 2x1- x2 + 3x3 = 8 x1,x2,x3 ≥ 0What if you considered artificial variables to change to

standard form rather than equation form???.....Try

Example (2)

• Write the dual for the following primal problem

Minimize Z= 15x1+ 12x2 Subject to x1 + 2x2 ≥ 3 2x1 - 4x2 ≤ 5 x1,x2 ≥ 0

Example (3)

• Maximize Z = 5x1 + 6x2• Subject to: x1 + 2x2 = 5 -x1 + 5x2 ≥ 3 4x1 + 7x2 ≤ 8 x1 unrestricted, x2 ≥ 0

Optimal Dual Solution

• This section provides two methods for solving the optimal of the dual problems.

• However, dual of the dual is itself the primal, which means that the dual solution can also be used to yield the optimal primal solution automatically.

Optimal dual solution.....

Problem

• For the following primal problem, find the optimal dual solution:

• Maximize Z= 5x1 + 12x2 + 4x3 Subject to: x1+2x2+x3 ≤ 10 2x1- x2 + 3x3 = 8 x1,x2,x3 ≥ 0

Problem 3- page 162

Write the dual and determine its optimal solution in two ways

Problem 4- Page 163

Problem6- page 163

Problem 6- Page 163

Verification methods

Examples

The following table represents the optimal primal solution for the above LP model. Using the optimal inverse provided in the table, verify that the given table represents a correct solution for the original LP model?

Example..continued..

Example...continued..solution..

Primal-dual objective values

Primal-dual relationship

Book problems- page 166

Continued....

Book Problems- Page 167

Problem 4- continued.....

Continued...optimal simplex table is