Ch.4Duality and Post Optimal Analysis

Introduction

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

• 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

Definition of the Dual problem

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

• The two problems are so closely related – the primal optimum solution automatically

provides the optimal solution to the dual.

• The primal problem represents – a resource allocation case

• 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

First modify the give PLP in the standard equation form and use the following table.

Figure 4.1 Schematic representation of the starting and general simplex tableaus

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 ≥ 0

What 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

Maximize Z= 5x1 + 12x

2 + 4x

3

Subject to: x1+2x

2+x

3 ≤ 10

2x1- x

2 + 3x

3 = 8

x1,x

2,x

3 ≥ 0

Minimize w = 10y1 + 8y

2

Subject to: y1 + 2y

2 ≥5

2y1 - y

2 ≥ 12

y1 + 3y

2 ≥ 4

y1 ≥ 0

y2 unrestricted

Maximize Z= 5x1 + 12x

2 + 4x

3

Subject to: x1+2x

2+x

3 ≤ 10

2x1- x

2 + 3x

3 = 8

x1,x

2,x

3 ≥ 0

Optimal Dual Solution

• The two methods for finding the optimal value 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.....

Maximize Z= 5x1 + 12x

2 + 4x

3

Subject to: x1+2x

2+x

3 ≤ 10

2x1- x

2 + 3x

3 = 8

x1,x

2,x

3 ≥ 0

Verification methods

Maximize Z= 5x1 + 12x

2 + 4x

3

Subject to: x1+2x

2+x

3 ≤ 10

2x1- x

2 + 3x

3 = 8

x1,x

2,x

3 ≥ 0

Minimize w = 10y1 + 8y

2

Subject to: y1 + 2y

2 ≥5

2y1 - y

2 ≥ 12

y1 + 3y

2 ≥ 4

y1 ≥ 0

y2 unrestricted

Example...continued..solution..

Primal-dual relationship

Primal-dual objective values

Economic Interpretations

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