| Primal formulation |
|---|
| \[\begin{align}\min& \sum_{i,j} \color{darkblue}{\mathit{cost}}_{i,j}\cdot\color{darkred}x_{i,j} \\ & \sum_j \color{darkred}x_{i,j}\le \color{darkblue}{\mathit{supply}}_{i}\perp \color{darkred}u_i \le 0 && \forall i \\ & \sum_i \color{darkred}x_{i,j}\ge \color{darkblue}{\mathit{demand}}_{j}\perp \color{darkred}v_j \ge 0 && \forall j \\ & \color{darkred}x_{i,j}\ge 0\end{align} \] |
Here \(\perp\) indicates "with dual ...". The duals for this model are \(\color{darkred}u_i\) and \(\color{darkred}v_j\). In the model, we have added the (optimal) signs of the duals. It may come as a surprise that the optimality of a solution can be established by just looking at the signs of the marginals (duals and reduced cost). When we print the results of this model, we can see something like:
