Transportation problems

The stepping-stone method

Goods move from supply points to demand points at given unit costs. Balance supply and demand (dummy if needed), start with the north-west corner, then compute shadow costs $R_i+K_j=C_{ij}$ on occupied cells and improvement indices $I_{ij}=C_{ij}-R_i-K_j$ on empty ones. A negative index means a cheaper plan: move stock round a stepping-stone loop.