The multidimensional knapsack problem (MKP) is a classic problem in combinatorial optimisation. Several authors have proposed to use surrogate relaxation to compute upper bounds for the MKP. In their papers, the surrogate dual is solved heuristically. We show that, using a modern dual simplex solver as a subroutine, one can solve the surrogate dual exactly in reasonable computing times. On the other hand, the resulting upper bound tends to be strong only for relatively small MKP instances.