Question

Consider the system in standard form: $$ \begin{aligned} &\mathrm{x}_{1}+\mathrm{x}_{4}+\mathrm{x}_{5}-\mathrm{x}_{6}=5 \\ &\mathrm{x}_{2}+2 \mathrm{x}_{4}-3 \mathrm{x}_{5}+\mathrm{x}_{6}=3 \\ &\mathrm{x}_{3}-\mathrm{x}_{4}+2 \mathrm{x}_{5}-\mathrm{x}_{6}=-1 \end{aligned} $$ Set up the system of equations in tableau form, obtain a basic feasible solution having \(\mathrm{x}_{4} \mathrm{x}_{5}, \mathrm{x}_{6}\) as basic variables.

Short answer

The given system of linear equations can be represented in tableau form as: $$ \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & -1 & 5 \\ 0 & 1 & 0 & 2 & -3 & 1 & 3 \\ 0 & 0 & 1 & -1 & 2 & -1 & -1 \end{bmatrix} $$ As \(x_4, x_5\), and \(x_6\) are already in the pivot positions, we don't need to perform any row operations. The basic feasible solution is directly read from the tableau as: \((x_1, x_2, x_3, x_4, x_5, x_6) = (0, 0, 0, 5, 3, -1)\).

Step-by-step solution

Set up the system in tableau form

First, we need to set up the given system of linear equations in tableau form. Our tableau will be a matrix that contains the coefficients of the variables and the constants. Our given system is: $\begin{aligned} & x_1 + x_4 + x_5 - x_6 &= 5 \\ & x_2 + 2x_4 - 3x_5 + x_6 &= 3 \\ & x_3 - x_4 + 2x_5 - x_6 &= -1 \end{aligned}$ The tableau will look like this: $$ \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & -1 & 5 \\ 0 & 1 & 0 & 2 & -3 & 1 & 3 \\ 0 & 0 & 1 & -1 & 2 & -1 & -1 \end{bmatrix} $$

Perform matrix row operations

We want to obtain a feasible solution with x4, x5, and x6 as basic variables. Since x4, x5, and x6 are already in our pivot positions, we don't need to perform any row operations in this case. We have the following simplex tableau: $$ \begin{bmatrix} 1 & 0 & 0 & 1 & 1 & -1 & 5 \\ 0 & 1 & 0 & 2 & -3 & 1 & 3 \\ 0 & 0 & 1 & -1 & 2 & -1 & -1 \end{bmatrix} $$

Read the basic feasible solution

Now, we can directly read the basic feasible solution from the tableau. We have: \(x_1 = 0, x_4 = 5\) \(x_2 = 0, x_5 = 3\) \(x_3 = 0, x_6 = -1\) Thus, the basic feasible solution having \(x_4, x_5\), and \(x_6\) as basic variables is: \((x_1, x_2, x_3, x_4, x_5, x_6) = (0, 0, 0, 5, 3, -1)\)

Key concepts

Simplex Method

The Simplex Method is an algorithm used in linear programming for optimizing a linear objective function, subject to a set of linear inequalities. It is famous for solving linear programming problems efficiently by moving along the edges of the feasible region to find the optimal solution. This method systematically checks vertices of the feasible region (or the boundaries in higher dimensions) to locate the maximum or minimum value of the objective function.
A critical aspect of the Simplex Method is the iterative process where a series of pivot operations are performed on the tableau, transitioning from one vertex to the next until the optimal solution is reached. Each pivot involves selecting an entering and leaving variable based on specific criteria related to improving the objective function, often involving the largest coefficient in the objective function row (entering) and the smallest ratio criterion (leaving).
The beauty of the Simplex Method is its efficiency, which stems from skipping the interior points and just evaluating corner solutions, significantly reducing computation in comparison to other methods, particularly in problems with a large number of constraints and variables.

Tableau Form

The Tableau Form is a structured tableau—a sort of visual matrix—that is fundamental in implementing the Simplex Method. This tableau includes rows that represent each constraint of the linear programming problem and columns for each variable involved, including slack or artificial variables transformed from inequalities into equalities.
In mathematical terms, the tableau consists of the coefficients for the linear equations derived from the constraints of the problem, along with additional columns for slack variables, which help convert inequalities into equalities. It also typically includes a rightmost column showing the constant terms (representing resource availability or limits) and a bottom row for the objective function coefficients.
For example, the given problem's tableau form is structured to easily implement row operations to transition between feasible solutions. It stands as an essential step since any operations we do involving pivoting or swapping base variables happen in this tableau, guiding us towards the Simplex Method's ultimate goal—reaching an optimal feasible solution efficiently.

Basic Feasible Solution

A Basic Feasible Solution (BFS) represents a feasible solution to the linear programming problem that meets all constraints and maximizes or minimizes the objective function. This solution is named 'basic' because it is formed by a subset of the variables equaling zero, and is viewed as a 'vertex' in the geometric sense of the problem's feasibility set.
In the context of the Simplex Method, a BFS is often an intermediate point that the algorithm checks and pivots from. It consists of setting a number of decision variables to zero (up to the constraints' degree), allowing the remaining variables, known as basic variables, to be solved directly from the system of equations.
Obtaining a BFS is crucial, as it acts as the starting solution for the Simplex Method algorithm. The initial BFS in the current exercise involves choosing the variables that naturally appear as pivots without any further reduction, aligning with the given system in standard form. This BFS gives a practical starting point for the iterative optimization process by providing a feasible solution that can be systematically improved using pivot operations.

Pivot Positions

Pivot Positions are key spots in the tableau and pivotal points in the Simplex Method process. Pivots determine which basic and non-basic variables need to be exchanged to move from one basic feasible solution to another in the search for the optimal solution. Specifically, a pivot is a matrix element used to perform Gaussian elimination to change base variables while keeping the remaining system consistent and feasible.
In practical terms, the pivot column is identified first—often the one whose objective function coefficient is most negative (in a minimization problem). The pivot row is then chosen using the smallest non-negative ratio test from available row's rightmost constants relative to the pivot column's positive elements.
For this exercise, the requirement was to ensure that certain variables (namely, the variables indexed 4, 5, and 6) are left as the basic variables—this means they were already in the right pivot positions. By analyzing the tableau, it’s seen that no further pivoting actions were necessary since these variables were already aligned properly according to their initial setup, leaving a clear path to determine the BFS indicated above.