Pivot once as indicated in each simplex tableau. Read the...

Pivot once as indicated in each simplex tableau. Read the solution from the result. $$ \begin{array}{rrrrrr|r}{x_{1}} & {x_{2}} & {x_{3}} & {s_{1}} & {s_{2}} & {s_{3}} & {z} \\ {4} & {2} & {3} & {1} & {0} & {0} & {0} & {22} \\ {2} & {2} & {5} & {0} & {1} & {0} & {0} & {28} \\ {1} & {3} & {2} & {0} & {0} & {1} & {0} & {45} \\ \hline-3 & {-2} & {-4} & {0} & {0} & {0} & {1} & {0}\end{array} ] $$

Pivot once as indicated in each simplex tableau. Read the solution from the result.
$$
\begin{array}{rrrrrr|r}{x_{1}} & {x_{2}} & {x_{3}} & {s_{1}} & {s_{2}} & {s_{3}} & {z} \\ {4} & {2} & {3} & {1} & {0} & {0} & {0} & {22} \\ {2} & {2} & {5} & {0} & {1} & {0} & {0} & {28} \\ {1} & {3} & {2} & {0} & {0} & {1} & {0} & {45} \\ \hline-3 & {-2} & {-4} & {0} & {0} & {0} & {1} & {0}\end{array} ]
$$

Key Concepts

Pivot Operation

A pivot operation is a fundamental step in the Simplex Method that involves selecting a pivot element in the tableau to exchange a basic variable with a non-basic variable. The aim of the pivot is to improve the value of the objective function while maintaining feasibility, by performing row operations that reconfigure the tableau according to the rules of linear algebra.

Basic Feasible Solution

A basic feasible solution is a solution to a linear programming problem that satisfies all of its constraints and is found by setting the non-basic variables to zero. In the context of the Simplex Method, each pivot operation leads to a new basic feasible solution, and the algorithm iterates through these solutions in search of the one that optimizes the objective function.

Simplex Method

The Simplex Method is an algorithm used in linear programming to optimize a linear objective function, subject to a set of linear equality and inequality constraints. It proceeds iteratively by moving from one vertex (or basic feasible solution) of the feasible region to an adjacent one, in such a way that the objective function is continually improved until an optimum is reached.

Simplex Tableau

A Simplex Tableau is a tabular representation of a linear programming problem that organizes all the coefficients of the objective function and constraints into a matrix form. This structured format facilitates the implementation of the Simplex Method by allowing the systematic application of pivot operations and tracking of the basic and non-basic variables during the solution process.

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