More on Simplex

Abstract

In the previous chapter, the basics of the simplex and revised simplex algorithms were developed. Degeneracy and convergence issues were addressed. In this chapter we continue with several enhancements to the basic algorithm and related issues. In Section 6.2 we continue the development of sensitivity analysis which was first discussed in Chapters 2 and 3. We show that an analysis of the simplex tableau provides dual variable values, reduced costs and range analysis for the right hand side and objective function, but the simplex algorithm does not provide results which are as reliable as those obtained using projection and inverse projection. In Section 6.3 we develop the dual simplex algorithm. This is the simplex algorithm applied to the dual problem but using the primal simplex tableau. In our earlier development of the simplex algorithm we assumed that all variables had a lower bound of zero and an upper bound of oo. In Section 6.4 we show how to incorporate variables with arbitrary lower and upper bounds into the simplex algorithm. The simplex algorithm requires an initial feasible basis. In Section 6.5 we show how to find an initial basic feasible solution. In Step 3 of the simplex algorithm it is necessary to select a variable to enter the basis. In Section 6.6 we describe several methods for selecting the entering variable. In Section 6.7 we discuss other numerical issues such as tolerances, the minimum ratio test and scaling. Concluding remarks are given in Section 6.8. Exercises are provided in Section 6.9. One topic not covered in this chapter is preprocessing. By preprocessing we mean analyzing the problem prior to solution in order to discover obvious redundancies, tighter variable bounds, eliminate variables and constraints, etc. This is very important in reducing the solution time for solving linear programs. Our discussion of preprocessing linear programs is postponed until Chapter 15.