Sorting Based On Matching Characteristics?

The scenario is as follows:

Each row describes a physical object. Each object has several columns for various physical characteristics. Each object is in one of several numbered bins (1, 2, 3, ...). One column on the end combines the rest of the characteristics into one line {=B2&C2&D2...}

Here's what I would like to happen:

I would like one last column to be a "new bin number" for each object. If the combined characteristics column of a given object is unique in the list, I want "bin new" to not change. If the combined characteristics column of a given object is not unique, but the matching object has the same value for the original bin number as the object in question, then "bin new" again does not change. If, however, there are matching objects in other bins, I need "bin new" for that object to return the lowest "old bin" value of all of the matching objects. Consider the simple example:

mVBbUpP.png
Link: [URL] .....

From there, I can sort the list by the "new" bin column, and that would preserve the existing list, except matching objects would get bumped up to the earliest instance of that object occurring in the list. In essence, my final sorted list needs to keep the original bin groups together, except in the case of matches where the matched objects in higher bin groups get placed with their lower bin group counterparts. This "new bin" scheme is the clearest way that I see that can accomplish this.

My attempt at a solution:

xls file: binsort.xls

I thought of using arrays. For the example above, I would highlight G2 through G10 and input the following formula:

{=Min(if($E$2:$E$10=E2,$B$2:$B$10))}

and then hit ctrl+shift+enter. The issue is that the =E2 part does not iterate throughout, so it is always comparing the combined values in the 2nd row with itself, and thus returning "1" for every value in the "bin new" column. You can see this in the .xls file provided: column F is what I am trying to reproduce via the function, but column G is what I end up with instead.