When is a dual value considered valid in linear programming?

A dual value, also known as a shadow price, represents the change in the objective function of a linear programming problem when the right-hand side of a constraint is increased by one unit. For this concept to hold true, several conditions must be met.

The correct answer indicates that dual values are valid within the range of feasibility. This means that the changes in the right-hand side of a constraint that the dual value refers to must not cause the solution to become infeasible. In the feasible region, the dual values give meaningful insights into how much the objective function would improve or worsen with slight changes to the constraint limits. This allows decision-makers to optimize their choices based on the available resources or constraints.

In contrast, if the solution is infeasible, dual values are not defined since the model does not produce a feasible solution for evaluation. Non-negative variables are necessary for many linear programming problems to remain valid, but this alone does not guarantee that the dual values are applicable. Tightly bound constraints may indicate geometric aspects of the feasible region but do not directly determine the validity of a dual value. Therefore, the crux of the dual value's validity lies in operating within a feasible range where the changes still yield a solution that satisfies all constraints.