This paper is based on the analytic and computational solution procedures of Gröbner basis and its applications. We show the behavior of the ideals generated by polynomials from a polynomial ring. We also present the idea of a zero dimensional ideal and use of this ideal to solve system of polynomial equations. We then introduce an algorithmic procedure for solving a system of polynomial equations (linear and nonlinear) with a finite number of solutions extending the idea of Gröbner basis. Finally we explore the idea of Gröbner basis for coloring the vertices of a given graph. We illustrate the stated results through a number of examples. Moreover, as for auxilary and making comparison with the analytic results, we use Mathematica 9.0.1 to develop some computer algebra.

This paper studies some decomposition methods, including Dantzig-Wolfe decomposition (DWD), decomposition-based pricing (DBP), Benders decomposition (BD), and a recently proposed improved decomposition (ID) method for solving linear programs (LPs). The authors then develop a new decomposition algorithm for solving LPs in a general form, allowing authors to combine the concept of Benders decomposition and decomposition-based pricing methods. The authors generate conditions for solving problems that have either infeasible or unbounded solutions. As an illustration, the authors give the corresponding models and numerical results for two standard mathematical programs: the two-person zero-sum game and the transportation problem. The authors compare several procedures and identify which one produces the best solution by giving the authors the smallest iteration number. This study reveals that the algorithm along with Benders decomposition produce the most efficient computational solutions of LPs.