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Chapter Guide: Chapter 3 dealt with the sensitivity of the optimal solution by determining the ranges for the model parameters that will keep the optimum basic solution unchanged.

**CHAPTER** **4**

**Duality ****and**** Post-Optimal Analysis**

*Chapter Guide. *Chapter 3 dealt with the
sensitivity of the optimal solution by determining the ranges for the model
parameters that will keep the optimum basic solution unchanged. A natural
sequel to sensitivity analysis is *post-optimal
analysis,* where the goal is to determine the new optimum that results from
making targeted changes in the model parameters. Although post-optimal analysis
can be carried out using the simplex tableau computations in Section 3.6, this
chapter is based entirely on the dual problem.

At a
minimum, you will need to study the dual problem and its economic
inter-pretation (Sections 4.1,4.2, and 4.3). The mathematical definition of the
dual problem in Section 4.1 is purely abstract. Yet, when you study Section
4.3, you will see that the dual problem leads to intriguing economic
interpretations of the LP model, including *dual*
*prices *and* reduced costs. *It* *also provides the foundation for
the development of the new* dual simplex
algorithm, *a prerequisite for post-optimal analysis. The dual simplex algorithm
is also needed for integer programming in Chapter 9.

The *generalized simplex algorithm* in Section
4.4.2 is intended to show that the simplex method is not rigid, in the sense
that you can modify the rules to handle prob-lems that start both infeasible
and nonoptimal. However, this material may be skipped without loss of
continuity.

You may
use TORA's interactive mode to reinforce your understanding of the
computational details of the dual simplex method.

This
chapter includes 14 solved examples, 56 end-of-section problems, and 2 cases.
The cases are in Appendix E on the CD.

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