Reduced cost

Definition

Reduced cost is a concept in linear programming that indicates how much the objective function coefficient of a non-basic variable must improve before that variable can enter the solution basis. It highlights the opportunity cost of including a non-basic variable in the optimal solution, which is crucial for determining feasibility and optimality during optimization processes.

5 Must Know Facts For Your Next Test

1. Reduced cost is only relevant for non-basic variables, as basic variables are already part of the current solution.
2. A reduced cost of zero indicates that a non-basic variable is already at its optimal level and can be considered for inclusion without altering the objective function value.
3. If the reduced cost is positive in a minimization problem, it implies that increasing that variable will increase the overall cost, making it unattractive for inclusion in the solution.
4. In maximization problems, a negative reduced cost indicates that increasing the non-basic variable would improve the objective function value.
5. Understanding reduced costs helps in sensitivity analysis, providing insight into how changes in coefficients affect the optimal solution.

Review Questions

• How does reduced cost influence the decision-making process in linear programming?
  • Reduced cost plays a significant role in decision-making during linear programming because it determines whether a non-basic variable should be included in the optimal solution. If the reduced cost is negative in maximization or positive in minimization, it signals that including this variable would not be beneficial. Thus, recognizing and analyzing reduced costs helps decision-makers identify potential improvements to their current solutions and optimize resource allocation effectively.
• Discuss how understanding reduced costs can aid in sensitivity analysis for a linear programming model.
  • Understanding reduced costs is essential for conducting sensitivity analysis as it provides insight into how changes in the objective function coefficients impact optimal solutions. By evaluating how reduced costs shift with different coefficient values, analysts can predict how sensitive their solutions are to variations in input data. This helps identify which variables have significant impacts on overall outcomes and guides adjustments to maximize efficiency and profitability.
• Evaluate the implications of having a zero reduced cost for a non-basic variable in a linear programming scenario.
  • A zero reduced cost for a non-basic variable implies that changing its level will not affect the overall value of the objective function; hence, it is at an optimal point within its constraints. This situation allows for flexibility since it can be included or excluded without affecting the current optimal solution. It indicates that this variable holds potential for future adjustments depending on changes in other factors, emphasizing its importance in strategic decision-making and resource management.

Related terms

objective function: A mathematical expression that defines the goal of a linear programming problem, usually aimed at maximizing profit or minimizing costs.

basic feasible solution: A solution to a linear programming problem that satisfies all constraints and has a number of basic variables equal to the number of dimensions of the decision space.

shadow price: The amount by which the objective function value would improve if there were one additional unit of a resource available, reflecting the value of constrained resources.