Reduced Cost

Definition

Reduced cost is the amount by which the objective function coefficient of a decision variable must increase before that variable can enter the basis of an optimal solution in linear programming. This concept highlights how much more profit is needed to justify including a non-basic variable in the solution, helping to understand the trade-offs involved in optimization.

5 Must Know Facts For Your Next Test

1. A reduced cost for a non-basic variable that is less than zero indicates that increasing this variable could improve the objective function value.
2. If a reduced cost is greater than zero, it suggests that increasing that variable would worsen the objective, making it non-beneficial to include it in the solution.
3. In a maximization problem, a non-basic variable with a negative reduced cost represents a potential increase in profit if allowed to enter the basis.
4. The reduced cost can also be interpreted in economic terms, providing insights into how much more of a resource or product would need to be worth before it's rational to produce more of it.
5. Understanding reduced costs aids in sensitivity analysis, allowing decision-makers to evaluate how changes in costs or resources impact optimal solutions.

Review Questions

• How does reduced cost influence the decision-making process in linear programming?
  • Reduced cost plays a critical role in determining which variables can enter an optimal solution. When analyzing reduced costs, decision-makers can assess whether increasing certain variables would lead to better outcomes or whether they should remain fixed. If the reduced cost is negative for a non-basic variable, it indicates that including it could enhance profitability, prompting further investigation into resource allocation.
• What is the relationship between reduced cost and shadow prices in optimization problems?
  • Reduced cost and shadow prices are interconnected concepts within linear programming. While reduced cost indicates how much an objective coefficient must change for a non-basic variable to enter the solution, shadow prices show how much the objective function value would change with a one-unit increase in resource availability. Together, they provide insights into resource management and pricing strategies in optimization problems.
• Evaluate the implications of reduced costs on strategic planning and resource allocation within an organization.
  • Evaluating reduced costs allows organizations to make informed strategic decisions regarding resource allocation and production levels. By understanding which variables can contribute positively to profits when adjusted, managers can prioritize investments and streamline operations. This evaluation not only optimizes current resources but also aids long-term planning by indicating areas where increases in input costs or resources could yield beneficial returns.

Related terms

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

Dual Problem: A linear programming problem derived from the primal problem, where the objective coefficients of the primal become the right-hand side constraints of the dual and vice versa.

Shadow Price: The change in the objective function value of a linear programming problem if there is a one-unit increase in the right-hand side of a constraint.