Combinatorial Optimization for Disrupted Logistics and Battery Thermal Dynamics

Abstract

This paper introduces the Laplace-Weierstrass (LW) transform, which fuses the classical Laplace transform with Weierstrass Gaussian smoothing to embed built-in regularization for dynamic systems. We present a corrected existence theorem, an explicit inversion formula, a convolution theorem, fractional-order operational calculus, and stable numerical methods. These advances enable efficient, robust solutions to linear and integro-differential equations even with noisy or incomplete data. The framework is applied to electric vehicle battery systems for lithium-ion modeling, parameter estimation, SoC/SoH analysis, and hybrid optimization under thermal constraints. It is further applied to resilient supply chains to model inventory dynamics with lead-time delays, disruption recovery, and bullwhip-effect mitigation. The approach yields computationally tractable tools for system modeling and digital twins. Future work focuses on hybrid quantum-classical methods that leverage quantum annealing for large-scale logistics optimization.

Introduction

This work presents an advanced Laplace–Weierstrass (LW) transform, a mathematical framework that combines the Laplace transform (for analyzing time-dependent systems) and the Weierstrass transform (Gaussian smoothing for noise reduction). The combined transform is designed to solve complex differential, delay, and fractional-order equations while simultaneously filtering noisy data, making it particularly useful for modern engineering applications.

The motivation for this research comes from two rapidly growing fields:

• Electric vehicle (EV) battery systems, where accurate battery modeling, state-of-charge (SoC), state-of-health (SoH), and thermal management require reliable analysis of noisy sensor data.
• Resilient supply chain management, where inventory dynamics, transportation delays, uncertain demand, and disruption recovery require robust mathematical models.

Previous research introduced the LW transform but lacked several important mathematical foundations, including a rigorous inversion method, convolution theorem, fractional calculus extensions, and practical engineering applications. This work addresses these gaps by developing:

• A corrected existence theorem.
• An explicit inversion formula.
• A convolution theorem.
• Fractional-order operational calculus.
• Stable numerical algorithms for practical computation.

The study also extends the LW transform to distribution theory, allowing it to process noisy, discontinuous, or impulsive signals commonly encountered in battery monitoring systems and supply chain data. Operational properties of the transform convert differentiation, integration, convolution, delays, and fractional derivatives into simpler algebraic operations while preserving Gaussian smoothing. This greatly simplifies solving differential and delay equations.

A major contribution is the integration of the LW transform with hybrid quantum–classical optimization. In this framework:

• The LW transform converts high-dimensional dynamic systems into lower-dimensional algebraic models.
• Gaussian smoothing reduces noise and dimensionality.
• The resulting optimization problem is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) problem.
• A quantum annealer or quantum-inspired solver efficiently solves the discrete optimization component.

The hybrid framework is applied to two major engineering domains:

1. Resilient Supply Chains
   • Models inventory dynamics and lead-time delays.
   • Reduces the bullwhip effect.
   • Optimizes logistics, routing, warehouse allocation, and disruption recovery.
   • Provides scalable solutions for uncertain and dynamic supply networks.
2. Advanced Battery Systems
   • Models battery thermal behavior and delayed feedback.
   • Improves SoC and SoH estimation.
   • Supports battery cooling, cell balancing, predictive maintenance, and module reconfiguration.
   • Reduces the impact of noisy Battery Management System (BMS) sensor data.

The paper also presents efficient numerical implementation methods using:

• FFT-based Gaussian convolution and Gauss–Hermite quadrature for smoothing.
• Talbot contour Laplace inversion for recovering time-domain solutions.
• Adaptive selection of the Gaussian smoothing parameter to balance noise suppression and model accuracy.

The proposed numerical algorithms have computational complexity of approximately O(N log N), making the framework suitable for real-time estimation, optimization, digital twins, and embedded engineering applications.

Conclusion

The hybrid quantum-classical Laplace–Weierstrass framework presented in this work provides a powerful and practical bridge between classical operational calculus for delay differential equations and near term quantum optimization solvers. By converting constant and distributed delays into simple multiplications in the transform domain and employing Gaussian smoothing for regularization and dimensionality reduction, the LW transform supplies clean, low-dimensional inputs to a QUBO layer that is efficiently solved by quantum annealing. The numerical implementation, based on FFT convolution and robust Laplace inversion techniques (Talbot contour and fixed Talbot methods), ensures stability and real time capability even under noisy sensor data typical of battery management systems and supply chain information feeds. Automatic regularization parameter selection via the discrepancy principle or cross validation further enhances robustness without manual intervention. This architecture has been shown to be particularly effective for two strategically important domains: resilient supply chain optimization under disruption (including ULD configuration, dynamic routing, and inventory reallocation subject to lead time delays and the bullwhip effect) and advanced battery thermal and electrochemical management with delayed feedback in cooling circuits and state observers. The clean separation of continuous delay dynamics (handled classically via the LW transform) from discrete combinatorial decisions (handled by quantum annealing) offers a scalable and noise-resilient pathway toward practical quantum advantage on industrially relevant problem sizes. Future research directions include extending the framework to stochastic delay systems, fractional order models with memory effects, and tighter co-design with quantum hardware error mitigation and pulse level control. The theoretical clarity, numerical robustness, and demonstrated applicability of the LW–quantum hybrid approach positions it as a valuable methodological contribution for both academic research in operational calculus and industrial deployment in quantum enhanced logistics and intelligent energy systems. By unifying rigorous transform domain analysis of delay operators with the combinatorial power of quantum annealing, this framework contributes to the growing toolkit for solving large scale, memory rich optimization problems that lie at the intersection of advanced battery systems and resilient, quantum enhanced supply chains.

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Copyright © 2026 Padmaja ., Abhilasha G. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.