<p>Inventory management in modern supply chains faces significant challenges such as carbon emissions, fuzzy shortages, order quantity-dependent lead time, price-dependent demand, and lost sales. To address these complexities and promote sustainable practices, this study presents a comprehensive model that integrates these factors into a unified framework for optimizing inventory decisions. The model emphasizes carbon emissions as a vital consideration in inventory management, aiming to minimize environmental impact while meeting customer demand. It also accounts for uncertain demand forecasts through the incorporation of fuzzy shortage concepts. Additionally, the model adapts to order quantity-dependent lead times and price-sensitive demands, allowing for flexible adjustments in response to market conditions. To further enhance performance, a lost sale reduction factor is included to account for stockout related costs. In addition, three different derivative-free optimization techniques, such as pattern search, fimincon, and simulated annealing, are utilized to assess the numerical data, and which approach produces the best results is investigated. Numerical experiments are conducted to evaluate the model’s effectiveness, demonstrating its ability to reduce the carbon footprint, ensure customer satisfaction, and maximize supply chain profits. This research significantly contributes to the literature on sustainable supply chain management by offering a comprehensive inventory management model that holistically addresses multiple crucial factors.</p>