  This paper explores fast, polynomial time heuristic approximate solutions to
the NP-hard problem of scheduling jobs on N identical machines. The jobs are
independent and are allowed to be stopped and restarted on another machine at a
later time. They have well-defined deadlines, and relative priorities
quantified by non-negative real weights. The objective is to find schedules
which minimize the total weighted tardiness (TWT) of all jobs. We show how this
problem can be mapped into quadratic form and present a polynomial time
heuristic solution based on the Hopfield Neural Network (HNN) approach. It is
demonstrated, through the results of extensive numerical simulations, that this
solution outperforms other popular heuristic methods. The proposed heuristic is
both theoretically and empirically shown to be scalable to large problem sizes
(over 100 jobs to be scheduled), which makes it applicable to grid computing
scheduling, arising in fields such as computational biology, chemistry and
finance.
