If the scores are equally important then it would seem to me that summing the scores is the best way because that would do exactly what you asked, rank with equal importance.

If you don't wish to do that then perhaps I don't fully understand what you are asking? Would you prefer a formula that would rank the totals within the formula without having an added column or are you trying to rank some other way?

This formula will rank the totals without duplicate ranks, without physically adding another column

=SUMPRODUCT((B2+C2<B$2:B$18+C$2:C$18)+0)+SUMPRODUCT((B2+C2=B$2:B2+C$2:C2)+0)

If that doesn't give the results you want then please suggest expected results