Ch.4
Duality 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 ≥ 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
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