 Transportation dual certificate with capacity inequalities

For shipments $x_{ij}\ge 0$ with factory row sums at most capacities $s_i$ and shop column sums exactly demands $d_j$, the displayed dual potentials give $\sum c_{ij}{x}_{ij}\ge {\sum }_i{u}_i{s}_i+{\sum }_j{v}_j{d}_j$. The sign $u_i\le 0$ ensures $u_i{\sum }_j{x}_{ij}\ge u_i{s}_i$. Equality proves optimality. It occurs when every used cell is tight and every row with spare capacity has $u_i=0$, the relevant complementary slackness conditions. A dummy demand column with zero costs converts unused capacity into a balanced transportation problem.