Maximum stable matching/allocation
Your problem is equivalent to MAX SMTI (Stable Marriage with Ties and Incomplete lists).  You can find the current best approximation algorithm for MAX SMTI in the following paper:
Z. Kiraly, Linear Time Local Approximation Algorithm for Maximum
Stable Marriage, Algorithms 2013, 6, 471-484.

There are many papers on MAX SMTI, but unfortunately I am not aware of any paper that handles the weighted version of MAX SMTI.