Question

Find a basic feasible solution to: $$ \begin{aligned} &2 \mathrm{x}_{1}+\mathrm{x}_{2}+2 \mathrm{x}_{3}=4 \\ &3 \mathrm{x}_{1}+3 \mathrm{x}_{2}+\mathrm{x}_{3}=3 \\ &\mathrm{x}_{1} \geq 0, \mathrm{x}_{2} \geq 0, \mathrm{x}_{2} \geq 0 \end{aligned} $$

Short answer

The basic feasible solution to the given system is \((x_1, x_2, x_3, a_1, a_2) = (0, 0, 0, 4, 3)\), where \(a_1\) and \(a_2\) are artificial variables introduced to create an identity matrix in the augmented coefficient matrix. Note that this solution is not optimal and may need further refining using the Simplex Method.

Step-by-step solution

Write the given system in its standard form

The standard form of a linear system includes slack variables for each inequality (equal to zero, in our case) that represent the difference between the left and right sides of the inequality. However, all our equations are given in the equality form, so we don't need to introduce slack variables. The given system can be considered already in its standard form.

Identify the basic variables

Basic variables are the ones that correspond to the identity matrix in the coefficient matrix of the system. In this case, none of the variables x₁, x₂, and x₃ form an identity matrix (notice the coefficients 2, 3, and 1 in the first column). Therefore, we don't have a basic feasible solution for the system in its current state.

Introduce artificial variables

To find a basic feasible solution, we will introduce artificial variables to create an identity matrix in the augmented coefficient matrix. Add the variables a₁ and a₂ so that the matrix forms an identity matrix. The new system becomes: $$ \begin{aligned} & 2x_1 + x_2 + 2x_3 + a_1 = 4 \\ & 3x_1 + 3x_2 + x_3 + a_2 = 3 \\ & x_1 \geq 0, x_2 \geq 0, x_3 \geq 0, a_1 \geq 0, a_2 \geq 0 \end{aligned} $$ Now the basic variables are a₁ and a₂.

Calculate the basic feasible solution

The values of the basic variables a₁ and a₂ are equal to the corresponding right-hand-side values. So we have: \(a_1 = 4\) \(a_2 = 3\) And the remaining variables \(x_1, x_2\), and \(x_3\) are equal to 0 at this stage. The basic feasible solution is therefore: \((x_1, x_2, x_3, a_1, a_2) = (0, 0, 0, 4, 3)\) Note: This basic feasible solution does not describe an optimal solution to the given system, since artificial variables are used only to facilitate finding the feasible starting point. By further refining this solution iteratively through the Simplex Method, we could potentially find an optimal solution without involving artificial variables.

Key concepts

Basic Feasible Solution

In linear programming, a basic feasible solution is a foundational concept that provides an initial solution to a system of linear equations or inequalities. It's where some variables, called basic variables, take non-zero values while others, the non-basic variables, are zero.
To find a basic feasible solution, we usually aim to set up a system where the number of equations equals the number of variables that are allowed to be non-zero. These non-zero variables form what is called the 'basis', and they correspond with the columns forming an identity matrix within our system's augmented matrix.
This solution is feasible if it satisfies all the constraints of the problem, including the non-negativity restrictions. However, it's important to note that not every basic feasible solution is optimal.
When we apply this to our exercise, the original set of equations didn't provide such a perfect match because of their coefficients. However, by introducing artificial variables, we were able to manipulate this system to find a basic feasible solution.

Artificial Variables

Artificial variables are a technique used in linear programming to assist in finding a basic feasible solution. When your system of equations doesn't naturally lend itself to forming a basic feasible solution, you might introduce artificial variables to create an identity matrix in the augmented system.
These variables are temporarily added to convert the set of equations into a form that allows you to solve for an initial basic feasible solution easily. However, they do not have any actual physical or practical significance in the problem itself.
In the given exercise, artificial variables \(a_1\) and \(a_2\) were added to the equations to help achieve a basic feasible solution. This means that our system of equations could be manipulated to make a starting point from which an actual solution could be derived, although these artificial variables are not part of the final or true solution as they represent auxiliary components introduced solely for the purpose of solving.

Simplex Method

The Simplex Method is a well-known iterative process used in linear programming to find the optimal solution of an objective function, given a set of linear constraints. It's particularly handy when dealing with multi-variable systems with many constraints.
The method takes the basic feasible solution as a starting point and systematically improves upon it. In essence, it finds a path through vertices of the feasible region, defined by the constraints, towards the optimal solution.
While the initial step often involves introducing artificial variables to obtain a starting solution, the Simplex Method iteratively refines this result by pivoting through basic feasible solutions. The aim is to drive these artificial variables out of the solution and find the real-valued solutions that satisfy the original constraints.
In our context, after identifying an initial basic feasible solution using artificial variables, the Simplex Method would be the tool used to iteratively refine this solution, ultimately seeking an optimal configuration where all the artificial variables are zero and only the realistic variables remain as determined by the original problem statement.