I recently read a question from someone who had solved a linear program (minimization) using a reliable solver, extracted the primal and dual solutions, recomputed reduced costs manually and encountered a negative reduced cost in a supposed optimal solution. That appeared to be a bit paradoxical. I don't have enough information to be sure, but I suspect variable bounds are involved.
Fourteen years ago (!) I wrote a post about how to find the dual value of a bound on a variable when the bound is entered as a bound rather than a functional constraint. It belatedly occurred to me that an example might be helpful (and might shed some light on the paradoxical reduced cost), so here goes. Let's start with a simple LP.\begin{alignat*}{1} \max\ 5x+y\\ \textrm{s.t. }x+2y & \le 8\\ x & \le 2\\ y & \le 10 \end{alignat*} where $x\ge 0$ and $y\ge 0.$ The problem is easy to solve graphically, as shown below.
The optimal solution is $(x,y) = (2,3)$ with objective value 13. Let's assume we feed it to an LP solver as written (three constraints). The dual values of the constraints are $(0.5, 4.5, 0).$ In particular, note the dual value of 4.5 for the upper bound on $x$ (which is binding). If we increase the bound from $2$ to $2 + \epsilon,$ the optimal corner shifts from $(2,3)$ to $(2 + \epsilon, 3- \frac{1}{2}\epsilon),$ with a resulting change in objective value from 13 to $13+4.5\epsilon.$ Also note that the reduced cost of $x$ is $5 - 1 \times 0.5 - 1 \times 4.5 - 0 \times 0 = 0$ and the reduced cost of $y$ is $1 - 2\times 0.5 - 0 \times 4.5 -1 \times 0 = 0,$ which conforms to our expectation that basic variables have zero reduced costs.
Now suppose that I enter the variable bounds directly as bounds, rather than as constraints, so that the LP has a single inequality constraint. Most (all?) contemporary solvers allow this. The optimal solution is unchanged, as is the dual value (0.5) for the first (and now sole) constraint. What follows was confirmed using CPLEX, but I suspect the experience with other solvers will be similar. The only dual value available is the dual for the first constraint. CPLEX, as best I can tell, does not allow you to ask for a dual value associated with a variable bound. So does that mean that the reduced cost of $x$ is just $5 - 1 \times 0.5 = 4.5?$ Yes, that is what CPLEX reports as the reduced cost of $x,$ and correctly so.
The fact that at optimum a basic variable has zero reduced cost applies to the original simplex method. There is a variant of the simplex method for bounded variables, and in that version a variable can be nonbasic at its upper bound, with a nonzero (possibly favorable) reduced cost. Moreover, as I mentioned in that earlier post, in this approach the reduced cost of a variable at its bound is the dual value of the bound constraint. So it is not coincidence that the reduced cost of $x$ when entering its bound as a bound (4.5) matches the dual value of the bound when entering it as a constraint.
