Production costs in a transportation problem ($c_{ij}^{\prime }=c_{ij}+p_i$)

When factory $i$ incurs unit production cost $p_i$ only on goods actually shipped, add that cost to every real shipping destination in its row. Unused capacity incurs no production cost; a dummy unused-capacity column therefore keeps zero cost rather than receiving $p_i$. This changes the optimal factory utilization when capacity exceeds total demand. Transportation dual certificate with capacity inequalities proves optimality without assuming every factory must produce at full capacity.