Securing a Dimension with Members Having the Same Name

Here's an SSAS security issue that doesn't come up very much at all. In fact, when this recently came up, I had forgotten about a solution I provided way back in 2005 (which was the only other time I've encountered this). The problem is that when securing an attribute member (a role's Dimension Data tab) using the member's name and not key (using the "Allowed Member Set" text box of the "Advanced" tab), and there is more than one member with that same name, only the first one will be secured.