Try this

=SUMPRODUCT(MIN((Sheet2!A1:A3=Sheet1!A9)*(Sheet2!B1:B3=Sheet1!B9)*(Sheet2!C1:C3=Sheet1!C9)*(Sheet2!D1:D3=Sheet1!D9)*((Sheet2!E1:E3))))

I put 3 rows of the same ABCD values with column E reading 6, 15 and 9 and the above pulled out 6 which is correct.
You'll need to put this formula against each row in Sheet 1