Final answer:
If a constraint has a positive slack value in the optimal solution, then it will have a dual price of zero, indicating that the resources are not fully utilized, and increasing the constraint would not improve the objective function.
Step-by-step explanation:
When analyzing the optimal solution in linear programming, if a constraint has a positive slack value, it indicates that the constraint is not binding at the optimal solution. This means that the resources constrained by that particular condition are not fully utilized. In terms of the dual price, also known as the shadow price, it is the value that represents how much the objective function would improve if the constraint's right-hand side is increased by one unit.
In this case, because the constraint is not limiting the solution (it has a positive slack), increasing the right-hand side of the constraint wouldn't improve the objective function since there are already sufficient resources. Therefore, the dual price of a constraint with positive slack is zero. This principle is part of duality in linear programming, which establishes a relationship between the primal problem and the dual problem. A positive slack in the primal problem corresponds to a zero dual price in the dual problem.
Complete question is:
If a constraint has a positive slack value in the optimal solution then it:
will have a negative dual price
will have a positive dual price
will have a dual price of zero.
has no restrictions for its dual price