A note on the continuous mixing set

The continuous mixing set is S={(s,r,z)∈ℜ×ℜ+n×Zn:s+rj+wjzj≥fj,j=1,…,n}, where w1,…,wn>0 and f1,…,fn∈ℜ. Let m=|{w1,…,wn}|. We show that when w1|⋯|wn, optimization over S can be performed in time O(nm+1), and in time O(nlogn) when w1=⋯=wn=1.