[Paper Review] A novel Hopfield neural network approach for minimizing total weighted tardiness of jobs scheduled on identical machines

Norbert Fogarasi, Kálmán Tornai+1 more|arXiv (Cornell University)|Jul 26, 2012

Neural Networks and Applications28 references3 citations

TL;DR

This paper proposes a novel Hopfield Neural Network (HNN)-based heuristic for minimizing total weighted tardiness (TWT) in scheduling jobs on identical parallel machines with preemption. By formulating the problem as a quadratic optimization task, the HNN approach achieves superior performance—producing solutions 25–56% better than EDD and 25–84% better than WSPT—while remaining scalable to over 100 jobs, making it suitable for real-time grid computing applications in finance and computational biology.

ABSTRACT

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.

Motivation & Objective

• To address the NP-hard problem of minimizing total weighted tardiness (TWT) in scheduling independent jobs with deadlines and non-negative weights on identical parallel machines.
• To develop a fast, scalable heuristic solution that outperforms existing classical scheduling heuristics like EDD and WSPT.
• To enable practical application in real-time, large-scale computing environments such as grid computing in computational finance and biology.
• To demonstrate the HNN-based approach is both theoretically and empirically scalable to problems with over 100 jobs.
• To evaluate the HNN’s performance against established heuristics across diverse problem instances and sizes.

Proposed method

• The TWT scheduling problem is mapped into a quadratic programming formulation suitable for neural network optimization.
• A Hopfield Neural Network (HNN) framework is employed to solve the quadratic optimization problem, leveraging energy minimization to find near-optimal schedules.
• The HNN uses dynamic update rules based on Lagrangian relaxation and penalty parameters (alpha, beta, gamma) to enforce scheduling constraints.
• The algorithm includes iterative parameter adjustment (as per Algorithm 1) to improve convergence and solution quality.
• Solutions are generated through multiple random initializations, with the best solution selected from a small number of runs.
• The method is designed for parallel execution, enabling efficient deployment on multicore or GPU-based architectures to reduce runtime.

Experimental results

Research questions

• RQ1 Can a Hopfield Neural Network be effectively adapted to solve the NP-hard total weighted tardiness (TWT) problem on identical parallel machines?
• RQ2 How does the HNN-based heuristic compare in solution quality to classical heuristics such as EDD, WSPT, and LWPF across varying problem sizes?
• RQ3 To what extent is the HNN approach scalable to large-scale scheduling problems (e.g., 100+ jobs) relevant to real-world grid computing?
• RQ4 What is the empirical runtime performance of the HNN method, and can it be optimized for real-time or production use?
• RQ5 Can intelligent initialization (e.g., using WSPT or LWPF results) improve HNN convergence and solution quality over random starts?

Key findings

• The HNN heuristic outperforms EDD, WSPT, and LWPF by producing total weighted tardiness values that are 25–56% of EDD’s, 25–84% of WSPT’s, and 34–91% of LWPF’s on average.
• In 98.6–100% of 500 randomly generated problems per size, the HNN produced a better solution than the next-best heuristic (LWPF), demonstrating consistent superiority.
• For a 100-job problem, the HNN achieves near-optimal solutions in approximately 6 seconds per iteration using non-optimized MATLAB code on a Core i7@3GHz processor.
• The theoretical convergence complexity of the HNN is O(L²N²), which is higher than EDD/WSPT (O(LN)), but practical performance remains acceptable due to parallelization potential.
• The HNN method scales well to large problems, with runtime remaining within acceptable limits for real-world applications such as overnight financial risk computation.
• The HNN’s performance is robust across diverse problem instances, with no significant degradation in solution quality as problem size increases.