Reduced cost

In linear programming, reduced cost, or opportunity cost, is the cost for increasing a variable by a small amount, i.e., the first derivative from a certain point on the polyhedron that constraints the problem. When the point is a vertex in the polyhedron, the variable with the most extreme cost, negatively for minimisation and positively maximisation, is sometimes referred to as the steepest edge.

Given a system minimize $mathbf\{c\}^Tmathbf\{x\}$ subject to $mathbf\{Ax\}leqmathbf\{b\},\; mathbf\{x\}geq\; 0$, the reduced cost vector can be computed as $mathbf\{c\}\; -\; mathbf\{y\}mathbf\{A\}$, where $mathbf\{y\}$ is the dual cost vector.

It follows directly that for an minimisation problem, any non-basic variables at their lower bounds with strictly negative reduced costs is eligible to enter that basis, while any basic variables must have a reduced cost that is exactly 0. For a maximisation problem, the non-basic variables at their lower bounds that are eligible for entering the basis have a strictly positive reduced cost.

Interpretation

For the case where x and y are optimal, the reduced costs can help explain why variables attain the value they do. For each variables, the corresponding sum that gives the reduced cost show which constraints forces the variable up and down. For non-basic variables, the distance to zero over the gives the minimal change in the object coefficient to change the solution vector x.

Reduced costs in pivot strategy

In principle, a good pivot strategy would be to select whichever variable has the greatest reduced cost. However, the steepest edge might ultimately not be the most attractive, as the edge might be very short, thus affording only a small betterment of the object function value. From a computational view, another problem is that to compute the steepest edge, an inner product much be computed for every variable in the system, making the computational cost too high in many cases. The Devex algorithm attempts to overcome the latter problem by estimating the reduced costs rather than calculating them at every pivot step, exploiting that a pivot step might not alter the reduced costs of all variables dramatically.