Multi-objective optimization of lithography alignment using fractional dynamics and fuzzy meta-goal programming

Lithography alignment in semiconductor manufacturing demands nanometer-scale precision amidst inherent challenges such as mechanical vibrations, thermal drift, and actuator nonlinearities. Traditional integer-order control strategies often fail to optimally balance competing objectives like positional accuracy, settling time, and energy efficiency. This paper introduces a Fuzzy Meta Goal Programming (FMGP) framework integrated with fractional calculus to address these limitations. The alignment process is modeled using a fractional-order differential equation (FDE) governed by the Caputo derivative, which captures memory-dependent dynamics and viscoelastic behavior. The FMGP approach formulates three meta-goals—positional error minimization, time efficiency, and control effort reduction—as fuzzy membership functions, enabling systematic trade-off resolution under uncertainty. Discretized via the Grünwald–Letnikov method, the FDE is solved iteratively while optimizing piecewise constant control inputs through evolutionary algorithms (NSGA-II) and gradient-based methods. Numerical simulations demonstrate that the proposed framework achieves 23% higher positional accuracy (≤1 nm error) and 15% faster settling time compared to integer-order PID and LQR controllers, with a 20% reduction in energy consumption under vibrational disturbances. Sensitivity analysis confirms robustness to parameter variations, while comparative studies highlight the superiority of fractional-order dynamics in mitigating hysteresis and overshoot. The results underscore the potential of FMGP-based fractional control in advancing lithography systems, with broader applicability to precision manufacturing processes such as atomic force microscopy and laser machining. This work bridges a critical gap between multi-objective optimization and fractional calculus, offering a scalable, data-driven paradigm for high-precision industrial automation.