Optimizing Inventory Management in Manufacturing Using Variational Inequalities

Thesis Abstract

Effective inventory management remains a critical challenge for manufacturing organizations seeking to minimize costs while maximizing operational efficiency. Fluctuations in demand, supply chain disruptions, and the complexity of production processes necessitate robust optimization frameworks that can adapt to dynamic conditions. This study aims to develop and validate a comprehensive optimization model for inventory management in manufacturing settings by applying variational inequality (VI) theory, which provides a versatile mathematical framework for modeling equilibrium conditions in complex systems. The specific objectives are to formulate the inventory management problem as a variational inequality, analyze the impact of production and demand variability on inventory levels, and propose solution algorithms that enable decision-makers to efficiently determine optimal reorder points and safety stock levels. The research adopts a quantitative, case study methodology centered on a medium-sized manufacturing firm in the automotive parts industry, employing a population of inventory data spanning three years (2019–2021). A stratified random sampling technique was used to select a sample of 150 inventory items representing different categories of components with varying turnover rates. Data collection involved extracting quantitative data from the organization’s Enterprise Resource Planning (ERP) system, complemented by structured interviews with inventory managers to corroborate quantitative findings. The primary analytical approach involves formulating the inventory control problem as a variational inequality model based on the Bellman and Complementarity theories, which are aligned with the theoretical framework of supply chain equilibrium and stochastic demand modeling. Numerical solutions are obtained through iterative algorithms such as the projected gradient method and the extragradient algorithm, implemented using MATLAB. The anticipated findings of the study include the derivation of a set of equilibrium conditions for inventory levels that account for stochastic demand, lead times, and supply variability. The model is expected to demonstrate superior accuracy in determining reorder points compared to traditional economic order quantity (EOQ) models, especially under conditions of demand uncertainty. The solution algorithms are projected to exhibit computational efficiency and convergence properties suitable for practical implementation in manufacturing environments. These results aim to inform inventory decision-making processes, facilitating more responsive and cost-effective inventory policies that align with the stochastic nature of real-world manufacturing operations. This research contributes to the existing body of knowledge by extending the application of variational inequality theory to inventory management, a domain traditionally dominated by deterministic models. It provides a novel analytical framework that integrates demand uncertainty and supply variability within an equilibrium modeling context, offering manufacturing firms a sophisticated tool for optimizing inventory levels dynamically. Furthermore, the study enhances the theoretical understanding of supply chain equilibrium under stochastic conditions, bridging the gap between mathematical optimization theory and practical inventory control. The main conclusion underscores the potential of variational inequality-based models to revolutionize inventory management practices in manufacturing, highlighting their capacity to improve responsiveness, reduce holding and stockout costs, and better accommodate uncertainties. Recommendations include adopting the developed model within enterprise resource planning systems, training inventory managers in variational inequality approaches, and extending the methodology to multi-echelon supply chains. Future research avenues involve integrating real-time data analytics and machine learning techniques with the VI framework for adaptive inventory policies, as well as exploring multi-objective optimization to balance cost, service levels, and sustainability considerations. Overall, this study aims to provide a rigorous mathematical foundation and practical insights that can significantly enhance inventory management strategies in the manufacturing industry.