Question

If a constraint has a shadow price of zero, what does this mathematically indicate about the current usage of that specific resource?

Accepted Answer

A shadow price of zero indicates that the resource associated with the constraint is currently slack, meaning there is unused capacity of that resource. In linear programming, the shadow price represents the amount the objective function value would increase if the constraint limit were relaxed by one unit. Because the resource is already in surplus, adding one more unit of it provides no additional benefit to the objective function, such as increasing total profit or further reducing costs. This mathematical state occurs when the constraint is non-binding, meaning the optimal solution does not fully consume the available resource. For example, if a factory has 100 labor hours available but only uses 80 to reach the optimal production level, the remaining 20 hours represent slack. Adding another hour of labor to this already underutilized pool does nothing to improve the output, resulting in a shadow price of zero.

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If a constraint has a shadow price of zero, what does this mathematically indicate about the current usage of that specific resource?

A shadow price of zero indicates that the resource associated with the constraint is currently slack, meaning there is unused capacity of that resource. In linear programming, the shadow price represents the amount the objective function value would increase if the constraint l....