Reading a Solver Sensitivity Report

When you solve a linear program with a software solver (Excel Solver, LINDO, Gurobi, etc.), you can request a sensitivity report that summarizes how the optimal solution would respond to small changes in the model's coefficients.

The report has two parts.

The Variable Cells section has one row per decision variable, with columns

$$\text{Final Value},\;\text{Reduced Cost},\;\text{Objective Coefficient},\;\text{Allowable Increase},\;\text{Allowable Decrease}.$$

The Constraints section has one row per functional constraint, with columns

$$\text{Final Value},\;\text{Shadow Price},\;\text{Constraint RHS},\;\text{Allowable Increase},\;\text{Allowable Decrease}.$$

We begin with the Variable Cells.

The reduced cost of a decision variable is the rate of change in the optimal objective per unit increase of that variable away from its current optimal value (with the remaining basic variables adjusting to preserve feasibility).

• A basic variable (one taking a positive value in the optimum) has reduced cost $0$.
• A nonbasic variable sits at its lower bound — typically $0$ — with reduced cost $\le 0$ for a maximization and $\ge 0$ for a minimization.

Forcing a nonbasic variable $x_j$ from $0$ up to some level $\Delta$ (along the local parametric direction) changes the optimal objective by

$$\Delta z = (\text{reduced cost})\cdot \Delta.$$

For example, suppose a max LP with $z^* = 50$ returns the row

$$\begin{array}{|c|c|c|c|c|c|}\hline \text{Var}&\text{Final}&\text{Red. Cost}&\text{Coef}&\text{Allow Inc}&\text{Allow Dec}\\\hline x_2 & 0 & -2 & 6 & 2 & 10^{30}\\\hline\end{array}$$

Forcing $x_2$ from $0$ up to $3$ changes the objective by $(-2)(3) = -6$, giving a new objective of $50 - 6 = 44$. Equivalently, $x_2$'s coefficient would have to rise by more than $2$ — above $8$ — before $x_2$ might enter the optimal basis.