MS&E111
Lecture 7
Introduction to Optimization
April 26, 2006
Prof. Amin Saberi
Scribed by: Yusuf Ozuysal
1
Duality Theory and Sensitivity Analysis
In the previous lecture, we introduced the dual of a linear program. Given a linear program
maximize
c
T
x
subject to
Ax
≤
b
x
≥
0
(1)
Its dual program will be
minimize
b
T
y
subject to
A
T
y
≥
c
y
≥
0
(2)
In this lecture, we will see that we can gain a lot of intuition from the dual of a linear
program and its optimum solution. In particular, we will see that the optimum solution of
the dual program can be used for sensitivity analysis.
1.1
Complementary slackness
We have already seen that by strong duality theorem, the optimum solution of the dual LP
can be used as a certificate of optimality for the solution of the primal program. But this is
not the only use of a dual solution. The values of the variables in the optimum dual solution
can tell us about the “importance” of their corresponding constraints in the primal program.
The next example will make this more clear:
Example: Production Management
Let us consider a simplified model of an automobile manufacturer that produces cars and
trucks.
Manufacturing is organized into four departments: sheet metal stamping, engine
assembly, automobile assembly, and truck assembly.
The capacity of each department is
limited. The following table provides the percentages of each department’s monthly capacity
that is consumed by constructing a thousand cars or a thousand trucks, respectively:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '06
 UNKNOWN
 Linear Programming, Optimization, Sensitivity analysis, objective function, Dual problem, optimum solution

Click to edit the document details