3.2 Basic and non-basic variables
2 min read•july 24, 2024
The is a powerful tool for solving linear programming problems. It uses basic and non-basic variables to find optimal solutions, with basic variables forming the current solution and non-basic variables set to zero.
Understanding how basic solutions are represented in simplex tableaux is crucial. The method involves identifying basic variables, extracting their values, and updating the tableau through to improve the solution iteratively.
Basic and Non-Basic Variables in the Simplex Method
Basic vs non-basic variables
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- Basic variables form current correspond to columns in simplex tableau equal number of constraints (slack variables)
- Non-basic variables excluded from current solution set to zero correspond to columns in simplex tableau (decision variables)
Basic solutions in simplex tableaux
- Locate identity matrix columns representing basic variables
- Extract values from right-hand side column for values
- Set non-basic variables to zero
- Resulting solution represents current basic feasible solution ()
Basic variables and basis matrix
- square matrix of basic variable columns dimensions match constraint count
- Basis matrix properties include invertibility enables basic feasible solution computation
- Each basis matrix column links to specific basic variable determines coefficients for basic variables
Entering and leaving variables
- selected to join basis chosen by most negative reduced cost in objective row (largest improvement potential)
- basic variable exiting basis determined by smallest non-negative ratio (maintains feasibility)
- intersection of entering variable column and leaving variable row
- Update process employs transforms pivot column to unit vector adjusts tableau elements accordingly
Key Terms to Review (18)
Basic Feasible Solution: A basic feasible solution is a point in the feasible region of a linear programming problem that satisfies all constraints and is formed by setting some variables to zero while solving the remaining equations. This solution represents a corner point of the feasible region, where it is possible to evaluate the objective function for optimization. Understanding this concept is crucial as it lays the foundation for identifying optimal solutions and differentiating between basic and non-basic variables.
Basic Variable: A basic variable is a variable in a linear programming problem that has a non-zero value in a basic feasible solution, representing a certain amount of resources or constraints being utilized. These variables are crucial in the Simplex method, as they help define the vertices of the feasible region, where optimal solutions can be found. In contrast, non-basic variables are set to zero in these solutions, indicating that they are not being actively used.
Basis Matrix: A basis matrix is a matrix that is formed by selecting a subset of the columns from the original constraint matrix in a linear programming problem, specifically those that correspond to the basic variables. This matrix is essential for representing a feasible solution within the context of the simplex method, allowing for the identification and manipulation of basic and non-basic variables to optimize the objective function.
Decision Variable: A decision variable is a key element in optimization problems that represents the choices available to the decision-maker. These variables are the unknowns that are determined within the model to optimize an objective function, while also satisfying certain constraints. Decision variables play a critical role in defining the feasible region of solutions and are directly tied to the outcomes of the optimization process.
Entering Variable: An entering variable is a non-basic variable in a linear programming model that is selected to increase its value in order to improve the objective function during optimization. Its introduction into the set of basic variables often leads to an enhancement of the current solution, and understanding how it interacts with basic and non-basic variables is crucial for implementing the Simplex algorithm effectively.
Gaussian elimination: Gaussian elimination is a systematic method for solving systems of linear equations, transforming the system's augmented matrix into row echelon form and ultimately into reduced row echelon form. This process helps identify basic and non-basic variables by simplifying the equations and making it easier to back-substitute for solutions. The method is pivotal in linear algebra and optimization, as it provides a clear path to finding unique solutions or determining the conditions under which solutions exist.
Identity Matrix: An identity matrix is a square matrix with ones on the diagonal and zeros elsewhere. It plays a crucial role in linear algebra, particularly when working with systems of equations and transformations. The identity matrix serves as the multiplicative identity in matrix multiplication, similar to how the number 1 acts in regular arithmetic, meaning that when any matrix is multiplied by the identity matrix, it remains unchanged.
Leaving Variable: A leaving variable is a basic variable that is removed from the solution set during the pivot operation in the simplex algorithm. This occurs when another variable, called the entering variable, replaces it to move towards an optimal solution. The concept of leaving variables is crucial for understanding how the simplex algorithm iteratively improves the objective function while maintaining feasibility within the constraints.
Linear Programming Problem: A linear programming problem is a mathematical optimization problem in which the goal is to maximize or minimize a linear objective function, subject to a set of linear equality and inequality constraints. This type of problem involves decision variables that are continuous, and it can be visually represented in two dimensions, making it possible to use graphical methods for solutions, especially when there are two variables. Understanding the distinction between basic and non-basic variables is crucial in formulating and solving these problems effectively.
Minimum Ratio Test: The Minimum Ratio Test is a technique used in linear programming to determine which basic variable should leave the solution in the simplex algorithm. It compares the ratios of the current solution values to the coefficients of the entering variable and helps identify the limiting factor, ensuring that the solution remains feasible while optimizing the objective function.
Non-basic variable: A non-basic variable is a variable in a linear programming problem that is not part of the solution at a particular vertex of the feasible region. These variables typically take a value of zero in the optimal solution, contrasting with basic variables that are active and determine the solution's coordinates. Understanding the distinction between basic and non-basic variables is essential for interpreting the results of linear programming models and for the simplex algorithm's operations.
Non-identity matrix: A non-identity matrix is a matrix that is not equal to the identity matrix, which has ones on the diagonal and zeros elsewhere. In the context of basic and non-basic variables, non-identity matrices often play a crucial role in optimization problems, particularly when representing coefficients in systems of equations or transformations that do not preserve the original space.
Objective Function: An objective function is a mathematical expression that defines the goal of an optimization problem, typically by representing the quantity to be maximized or minimized. It serves as the foundation for decision-making in various optimization models, guiding the selection of optimal solutions based on specific criteria.
Optimal Solution: An optimal solution is the best possible outcome that satisfies all constraints in a decision-making problem, often maximizing or minimizing a specific objective function. This concept is crucial in determining the most efficient way to allocate resources or make choices within a set of defined parameters.
Pivot Element: A pivot element is a specific non-basic variable in a linear programming tableau that is used to perform row operations during the Simplex algorithm. This element is crucial because it determines which variable will enter the basis and which will leave, ultimately influencing the solution's direction towards optimality. The choice of pivot element affects the feasibility and efficiency of the algorithm, making it a key component in managing basic and non-basic variables.
Pivoting: Pivoting refers to the process of exchanging one basic variable for a non-basic variable in a linear programming solution to improve the overall objective function. This technique is essential in optimizing solutions by systematically moving from one vertex of the feasible region to another until the best possible outcome is achieved. It helps determine the most efficient allocation of resources and is a fundamental concept in the simplex method for solving linear programming problems.
Simplex method: The simplex method is a widely used algorithm for solving linear programming problems, particularly those in standard form. It systematically examines the vertices of the feasible region defined by the constraints to find the optimal solution while maintaining feasibility throughout the process.
Slack Variable: A slack variable is an additional variable introduced in a linear programming problem to transform an inequality constraint into an equality constraint. By adding a slack variable, the unused resources or capacities in a constraint are explicitly accounted for, allowing for a clearer representation of the feasible region. This concept is crucial for identifying optimal solutions as it helps define the boundaries of the solution space.