Basic Variables

Definition

Basic variables are the variables in a linear programming problem that are part of the solution at any given feasible solution point. These variables correspond to the columns of a tableau that are used to express the constraints and objective function in standard form. Understanding basic variables is crucial as they help define the current solution and contribute to the overall optimization process.

5 Must Know Facts For Your Next Test

1. Basic variables correspond to the pivot columns in a tableau, indicating which variables are active in the current solution.
2. In a basic feasible solution, the number of basic variables is equal to the number of constraints in the problem, leading to a unique solution when all constraints are satisfied.
3. Basic variables can change when moving from one vertex of the feasible region to another, reflecting different possible solutions during optimization.
4. The values of basic variables determine how much of each resource is being used, providing insight into resource allocation within the constraints.
5. Basic variables play a key role in identifying optimal solutions; if an optimal solution exists, it will always occur at one of the vertices defined by basic variables.

Review Questions

• How do basic variables relate to the concept of feasible solutions in linear programming?
  • Basic variables are directly tied to feasible solutions since they represent the active dimensions of those solutions within the constraints. In a basic feasible solution, these variables assume non-zero values while non-basic variables are set to zero. This relationship helps visualize how solutions interact with the constraints, defining potential optimal points within the feasible region.
• Discuss how changes in basic variables affect the overall solution of a linear programming problem.
  • When basic variables change, it typically indicates a movement from one vertex of the feasible region to another. This alteration can lead to different allocations of resources, impacting both feasibility and optimality. By analyzing how basic variables transition during iterations in methods like the Simplex algorithm, one can understand how various potential solutions may arise and converge towards an optimal outcome.
• Evaluate the importance of basic variables in determining optimal solutions and their implications for resource management.
  • Basic variables are crucial for pinpointing optimal solutions in linear programming as they signify which resources are actively utilized under given constraints. Their values inform decision-makers about how effectively resources are allocated and help identify trade-offs. Understanding shifts in basic variable values can reveal insights into optimizing resource management strategies, ensuring efficiency and alignment with organizational goals.

Related terms

Non-basic Variables: Non-basic variables are those variables in a linear programming problem that are not included in the current solution and typically take on a value of zero.

Tableau: A tableau is a matrix representation used in linear programming that organizes the coefficients of the constraints and the objective function to facilitate calculations.

Feasible Region: The feasible region is the set of all possible points that satisfy the constraints of a linear programming problem, where basic variables define the corners or vertices.