Transportation simplex algorithm

The transportation simplex computes row and column potentials, enters a negative-reduced-cost cell, and pivots around its induced alternating cycle.

The northwest-corner method repeatedly allocates the smaller remaining supply and demand in the current top-left cell to construct an initial transportation basis.

For transportation potentials, the reduced cost is ${\bar{c}}_{ij}=c_{ij}-u_i-v_j$; a negative value identifies an improving entering cell.

Adding a nonbasic edge to a transportation tree creates one cycle; alternating equal additions and subtractions around it preserves every row and column sum.