Note on Shim and Kim’s lower bounds for scheduling on identical parallel machines to minimize total tardiness

This note introduces a new lower bound for the problem of scheduling on parallel identical machines to minimize total tardiness that is based on the concepts used in the two lower bounds developed by Shim and Kim [Shim, S.O., Kim, Y.D., 2007. Scheduling on parallel identical machines to minimize total tardiness. European Journal of Operational Research 177, 135–146]. The note shows that the new lower bound dominates the three lower bounds used in Shim and Kim’s branch-and-bound algorithm and can be used in place of these lower bounds to lower the enumeration required.