4 The Simplex Method
As we have seen, a linear programming problem forms a convex polygon in the best possible scenario. It is imperative to obtain a process that would assist in determining the optimal solution without the need to examine the graphical representation. The need for an algorithm that would perform such process was essential in the early days of the formulations of linear programming problems. Although challenging, the task was accomplished by a mathematician of the twentieth century. In 1947, George Dantzig developed a process that assisted in computing optimal solutions for minimization and maximization linear programming problems, this method is known as the simplex method [6]. Regardless of his great discovery, the linear programming problem needed to be set up in canonical form, so that the process could be utilized. Dantzig’s discovery could be applied to optimize any given objective function, given that the structure was in canonical form. In addition to the proper set up, the number of variables needs to be larger than the number of constraints. Assume that there are m equations and n unknowns. If the number of equations is equal to the number of constraints, then the solution could be derived without considering the objective function. Thus n>m is a prerequisite for a …show more content…
The simplex method does not check for all possible solutions, it iterates once a feasible solution is found, and continues to find the next feasible solution until the optimal solution is found. For problems with small number of variables and linear constraints, the process can be represented in a tableau table. The tableau table assists in comprehending the process that a linear programming problem undergoes while utilizing the simplex method. Let’s consider the following example
As we have seen, a linear programming problem forms a convex polygon in the best possible scenario. It is imperative to obtain a process that would assist in determining the optimal solution without the need to examine the graphical representation. The need for an algorithm that would perform such process was essential in the early days of the formulations of linear programming problems. Although challenging, the task was accomplished by a mathematician of the twentieth century. In 1947, George Dantzig developed a process that assisted in computing optimal solutions for minimization and maximization linear programming problems, this method is known as the simplex method [6]. Regardless of his great discovery, the linear programming problem needed to be set up in canonical form, so that the process could be utilized. Dantzig’s discovery could be applied to optimize any given objective function, given that the structure was in canonical form. In addition to the proper set up, the number of variables needs to be larger than the number of constraints. Assume that there are m equations and n unknowns. If the number of equations is equal to the number of constraints, then the solution could be derived without considering the objective function. Thus n>m is a prerequisite for a …show more content…
The simplex method does not check for all possible solutions, it iterates once a feasible solution is found, and continues to find the next feasible solution until the optimal solution is found. For problems with small number of variables and linear constraints, the process can be represented in a tableau table. The tableau table assists in comprehending the process that a linear programming problem undergoes while utilizing the simplex method. Let’s consider the following example