The Simplex Algorithm: Use a Simplex Tableau

Basic Concept

The Simplex Algorithm is a method used for solving optimization problems, particularly those related to linear programming.
The decisions are made systematically by the algorithm to find the most profitable solution, commonly tracked using a simplex tableau.
The Simplex Tableau is a tabular representation of the problem constraints, objective function, and decision variables.

The simplex tableau is set up with a column for each variable (decision variables and slack variables) and a row for each constraint, plus an additional row to represent the objective function.
The right-hand side entries correspond to the constraint values, and the bottom row corresponds to the coefficients of the decision variables in the objective function.
The final column on the right and the final row at the bottom are known as the RHS column and the z-row, respectively.
With an ample understanding of the problem’s constraints and objectives, the initial or starting simplex tableau can be formed.

A pivot operation is used to switch a basic variable with a non-basic variable for optimising the solution.
The pivot row is chosen by dividing each entry in the RHS column by the corresponding entry in the column of the entering variable. The smallest non-negative ratio indicates the pivot row.
The pivot column is determined based on the most negative entry in the z-row, indicating the entering variable.
The number where the pivot row and pivot column intersect is known as the pivot element.

After a series of pivoting, when all of the coefficients of the z-row are non-negative, an optimal solution is achieved.
The optimal solution can be read off the tableau — the variables corresponding to the columns with a single ‘1’ and all other elements ‘0’ in the final tableau correspond to the basic variables and these basic-variables’ coefficients in the RHS column represent their respective optimal values.
Problems could also result in no feasible solutions or have no optimal finite solution (unbounded), and these scenarios can also be identified with the Simplex Algorithm.

Developed by George Dantzig in the mid-20th century, the Simplex Algorithm revolutionised the field of operations research and remains a standard method for solving linear programming problems.
It’s utilized in fields like economics, manufacturing, transportation, and military operations for tasks including resource allocation, production scheduling, and logistics.

Despite its usefulness, the Simplex Algorithm can be computationally demanding for large-scale problems. This drawback is known as the curse of dimensionality.
Creating tableau and performing pivot operations may be prone to human error. Hence, it is crucial to carefully perform each step.
There might also be situations where the simplex algorithm can cycle indefinitely without finding an optimal solution, although these are considered exceptional cases.
To handle these complexities, various forms of the Simplex Algorithm have been developed, each with its strategies to handle different problem types.