Sum of data across multiple columns & rows based on Criteria

I have a data set that ranges across multiple rows and columns.  I am building a nested formula and looking to sum QTY based on multiple criteria in one part of it.  I tried sumifs, however, that does not work.  Below is my example using a hypothetical sumifs (keeping in mind, this does not work).  Any suggestions on how to return my expected result would be greatly appreciated.