In hybrid flow shop scheduling problem (HFS) with unrelated parallel machines, a set of n jobs are processed on k machines. A mixed integer linear programming (MILP) model for the HFS scheduling problems with unrelated parallel machines has been proposed to minimize the maximum completion time (makespan). Since the problem is shown to be NP-complete, it is necessary to use heuristic methods to ...

We prove an (lg 2 n= lg lg n) lower bound for the depth of n-input sorting networks based on the shuue permutation. The best previously known lower bound was the trivial (lg n) bound, while the best upper bound is given by Batcher's (lg 2 n)-depth bitonic sorting network. The proof technique employed in the lower bound argument may be of independent interest.

This paper provides upper and lower bounds on the information rates achievable with a minimum mean square error (MMSE) Tomlinson-Harashima precoder (THP) assuming ideal interleaving. We also give an exact formula for zero-forcing (ZF) THP achievable information rates. At high SNR, the performance of the MMSE-THP and the ZF-THP identically suuer only the 2.55 bit or 1.53 dB \shaping" loss from c...