= Production costs in a transportation problem
{title2=$c'_{ij}=c_{ij}+p_i$}
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.
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