Reduced Costs

Definition

Reduced costs refer to the amount by which the objective function coefficient of a variable must improve before that variable can enter the solution of a linear programming problem. In the context of optimization, it helps determine the contribution of each variable to the overall objective, indicating how much more benefit can be gained by increasing that variable's value. This concept is particularly important when analyzing sensitivity and optimality in linear programming and integer programming scenarios.

5 Must Know Facts For Your Next Test

1. Reduced costs are calculated as the difference between the objective function coefficient and the sum of the shadow prices multiplied by the coefficients of the variable in the constraints.
2. If a variable has a reduced cost greater than zero in a maximization problem, it indicates that increasing this variable will not improve the objective value under current conditions.
3. For a basic feasible solution, reduced costs provide insight into which non-basic variables might become beneficial if included in the solution.
4. In integer programming, reduced costs can also indicate whether it's worthwhile to relax integrality constraints for a particular variable.
5. Understanding reduced costs is essential for sensitivity analysis, allowing decision-makers to evaluate how changes in coefficients affect optimal solutions.

Review Questions

• How do reduced costs influence decision-making in linear programming problems?
  • Reduced costs play a crucial role in decision-making as they indicate whether increasing the value of certain variables will lead to an improved objective function. If the reduced cost for a variable is positive in a maximization problem, it suggests that increasing this variable will not add any benefit. This information helps prioritize which variables to adjust to enhance overall performance and guide strategic decisions effectively.
• Discuss how reduced costs relate to slack variables and their impact on feasible solutions.
  • Reduced costs are directly tied to slack variables since both concepts are integral to understanding optimality in linear programming. Slack variables represent unused resources in constraints, and their existence affects the calculation of reduced costs. When evaluating a basic feasible solution, if slack variables exist, it indicates that some resources aren't fully utilized, which may lead to certain non-basic variables having positive reduced costs. Understanding this relationship helps identify potential adjustments for optimizing resource allocation.
• Evaluate how changes in objective function coefficients can impact reduced costs and overall optimal solutions in linear programming.
  • Changes in objective function coefficients significantly affect reduced costs, as these coefficients are fundamental components of their calculation. When a coefficient increases or decreases, it can shift which variables are considered optimal. For example, an increase in a variable's coefficient might reduce its associated reduced cost, making it more attractive to include in the optimal solution. Consequently, fluctuations in these coefficients can alter the landscape of feasible solutions and require re-evaluation of strategy to maintain optimality.

Related terms

Optimal Solution: The best feasible solution that maximizes or minimizes the objective function in a linear programming problem.

Slack Variables: Extra variables added to a linear programming model to convert inequalities into equalities, helping to find feasible solutions.

Shadow Price: The amount by which the objective function value would improve if there were one additional unit of a resource available in a linear programming model.