Set packing with maximum coverage objective
Optimization version of 3D matching is a special case of your problem. Approximating Maximum 3D Matching is $APX$-complete. However, it is approximable within 3/2$+\epsilon $ for any  $\epsilon &gt;0$

References:

Hurkens, C. A. J., and Schrijver, A. (1989), On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, 
SIAM J. Disc. Math. 2, 68-72. 

Kann, V. (1991), Maximum bounded 3-dimensional matching is MAX SNP-complete, Inform. Process. Lett. 37, 27-35.