Reduced Cost in Linear Programming
Associated with each variable is a reduced cost value. However, the reduced cost value is only non-zero when the optimal value of a variable is zero. A somewhat intuitive way to think about the reduced cost variable is to think of it as indicating how much the cost of the activity represented by the variable must be reduced before any of that activity will be done. More precisely,
... the reduced cost value indicates how much the objective function coefficient on the corresponding variable must be improved before the value of the variable will be positive in the optimal solution.
In the case of a minimization problem, "improved" means "reduced." So, in the case of a cost-minimization problem, where the objective function coefficients represent the per-unit cost of the activities represented by the variables, the "reduced cost" coefficients indicate how much each cost coefficient would have to be reduced before the activity represented by the corresponding variable would be cost-effective. In the case of a maximization problem, "improved" means "increased." In this case, where, for example, the objective function coefficient might represent the net profit per unit of the activity, the reduced cost value indicates how much the profitability of the activity would have to be increased in order for the activity to occur in the optimal solution. The units of the reduced cost values are the same as the units of the corresponding objective function coefficients.
If the optimal value of a variable is positive (not zero), then the reduced cost is always zero. If the optimal value of a variable is zero and the reduced cost corresponding to the variable is also zero, then there is at least one other corner that is also in the optimal solution. The value of this variable will be positive at one of the other optimal corners.
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