Formula to find the minimum value with constraint

I have a simple table with A, B, and C columns and D, E, F rows and numeric values in each cell of the table. I need to get one value from each column (only one value per column) that will result in the lowest possible number when adding those values up. There is one constraint: I can not get cells that belong to the same row.

On the example I am providing, the sum of the cells highlighted in green totals 1978 which is the lowest possible result on that table provided that I'm selecting one cell per column and each selected cell belongs to an unique row.

Is there an excel formula for this problem? I looking for a formula to provide what the lowest possible number is and also need to identify the cells being selected.

Thanks in advance.