Simplex Method

 However, we were only able to solve for an optimal solution if the problem had only two decision variables as the life problems containing more than two decision variables cannot be solved by graphical method. The L.P. problems with two or more than two variables can be solved by using a systematic procedure called the simplex Method. The simplex method is the name given to the solution algorithm for solving LP problems developed by George B. Dantzig in 1947. By solution algorithm we refer to an integrative procedure having fixed computation rules that leads to a solution to the problem in a finite number of steps. By using the simplex method one is capable of solving large LP problems efficiently. The simplex method allows us to evaluate the corner points in such a way that each successive corner point gives the same or better solution than the previous one. This search for examining the same or better solution than the previous one. This search for examining successive corner points will continue until no better solution can be found.

        Remark: The simplex method is applicable to any problem that can be formulated in terms of linear objective function subject to a set of linear objective function subject to set linear constraints. There are no theoretical restrictions placed on the number of decision variables or constraints contained in a problem as the simplex method accompanied by computational facility of a large computer can solve linear programming problems with several thousand variables and several thousand constraints.

Development of simplex method

Computational procedure in the simplex method is based on the following key property.

"The optimum solution to a linear programming problem if it exists all ways occurs at one of the corner points of the feasible solution space.

        The simplex algorithm is a systematic and efficient procedure for finding corner point solution and taking them for optimality. The evaluation of corner points always starts at the point of origin (initial basic feasible solution) which is one of the corners of the feasible solution space. This solutions is then tested for possible, i. e., it tests whether an improvement in the objective function is possible by moving to adjacent corner point of he feasible solution space. If the improvement is possible then at a new corner point the solution is against tasted for optimality. This iterative search for a better corner point is repeated until an optimum solution. If it exists is determined.

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