We study the minimum spanning arborescence problem on the complete digraph K → n , where an edge e has a weight We and a cost Ce, each of which is an independent uniform random variable Us, where 0 < s ≤ 1 and U is uniform [ 0 , 1 ]. There is also a constraint that the spanning arborescence T must satisfy C (T) ≤ c 0 . We establish, for a range of values for c 0 , s , the asymptotic value of the optimum weight via the consideration of a dual problem.