Lower bounds on network flow (${\ell }_e\le f_e\le c_e$)

A prescribed minimum amount on each directed edge can be removed by substituting $f_e={\ell }_e+x_e$. The new capacity constraints are $0\le x_e\le c_e-{\ell }_e$. For a circulation with vertex demands using net inflow $d_v$, the adjusted demand is $d_v^{\prime }=d_v-({\sum }_u{\ell }_{uv}-{\sum }_w{\ell }_{vw})$. Solving the adjusted problem and adding $\ell$ back gives a one-to-one correspondence of feasible solutions. In particular, removing a lower bound on $u\to v$ increases $u$'s residual demand and decreases $v$'s by that amount.