Reduced Cost

Definition

Reduced cost is a concept in linear programming that indicates how much the objective function coefficient of a variable would need to improve before that variable can enter the solution with a positive value. This term helps in determining the potential profitability of including a new variable in an optimization problem, especially during processes like column generation where new variables or 'columns' are introduced iteratively to improve the solution.

5 Must Know Facts For Your Next Test

1. Reduced cost is calculated as the difference between the current objective function coefficient and the shadow price associated with that variable.
2. A reduced cost greater than zero for a maximization problem suggests that introducing that variable will not improve the current solution.
3. In column generation, reduced costs help identify which new variables could lead to an improved solution when they are added to the model.
4. For a minimization problem, a reduced cost less than zero indicates that adding that variable would decrease the objective function value.
5. Reduced costs play a critical role in determining whether a variable should enter or leave the basis in the context of optimization algorithms.

Review Questions

• How does reduced cost impact decision-making when considering new variables during optimization?
  • Reduced cost provides critical information about whether introducing a new variable will be beneficial for improving the overall objective function. If the reduced cost is positive in a maximization problem, it indicates that including the variable won’t enhance the solution and may be discarded. Conversely, if it’s negative, it suggests that including the variable could lead to a better outcome, guiding decision-makers on which columns to consider for inclusion.
• Discuss how reduced costs are computed and their significance in the column generation process.
  • Reduced costs are computed by subtracting the product of dual variables and constraint coefficients from the objective function coefficients. In column generation, this computation is essential because it identifies whether new columns (variables) should be added to improve the current solution. A negative reduced cost for a column signifies potential profitability and justifies its inclusion in the model, thus refining the overall solution progressively.
• Evaluate the relationship between reduced costs and dual variables in linear programming optimization.
  • Reduced costs and dual variables are intricately linked within linear programming optimization, as they both provide insights into resource allocation and profit maximization. The value of reduced costs reflects how much the objective function would improve if a new variable were introduced, while dual variables indicate how much an increase in resource availability affects optimal outcomes. Understanding this relationship helps optimize resource allocation more effectively, facilitating smarter decisions during processes such as column generation.

Related terms

Dual Variables: Values associated with the constraints of a linear programming problem, representing the marginal worth of resources within the optimal solution.

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

Simplex Method: An algorithm used for solving linear programming problems by iterating through feasible solutions to find the optimal one.