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This paper explores novel, polynomial time, heuristic, approximate solutions to the NP-hard problem of finding the optimal job schedule on identical machines which minimizes total weighted tardiness (TWT). We map the TWT problem to quadratic optimization and demonstrate that the Hopfield Neural Network (HNN) can successfully solve it. Furthermore, the solution can be significantly sped up by choosing the initial state of the HNN as the result of a known simple heuristic, we call this Smart Hopfield Neural Network (SHNN). We also demonstrate, through extensive simulations, that by considering random perturbations to the Largest Weighted Process First (LWPF) and SHNN methods, we can introduce further improvements to the quality of the solution, we call the latter Perturbed Smart Hopfield Neural Network (PSHNN). Finally, we argue that due to parallelization techniques, such as the use of GPGPU, the additional cost of these improvements is small. Numerical simulations demonstrate that PSHNN outperforms HNN in over 99% of all randomly generated cases by an average of 3-7%, depending on the problem size. On a specific, large scale scheduling problem arising in computational finance at Morgan Stanley, one of the largest financial institutions in the world, PSHNN produced a 5% improvement over the next best heuristic.

Improvements to the Hopfield Neural Network Solution of the TWT Scheduling Problem