Number partitioning targeting ratio of subset sums and equal size


I’ve seen a number of questions and answers related to the partitioning problem of dividing a set into 2 subsets of equal size and sum that use greedy or dynamic programming solutions to get approximate answers. However, I am looking to split a set into 2 subsets of minimum difference in size but with a target ratio of sums.

I have tried variations of greedy algorithms that will work to minimize the largest difference of the two metrics at any point in the calculation, but this just results in a result that splits the difference between the two. I’m happy with an approximate solution since it needs to run in a reasonable amount of time.