You won't be able to do this. That's because your trying to use the 'Project' fact table incorrectly. This table can give you some cross-project type data, but if you want to count the number of projects of a particular type, you will need the associated fact table, e.g. Contract Workspace (Procurement).
You could do this per project type fairly easily. Run the Supplier Detail report, add in Contract Workspace (Procurement) in the secondary fact, add Project Count as a measure, remove all non-required columns, view in aggregate view. Save as a pivot or chart on your dashboard as something like '# of CW's by Supplier'. Repeat the process for SP's and SPM's.
You could then add these individual reports to a compound report so with 1 click you can see them all.
There may be another way, but this is the best option I can think of.