Optimal analysis of profit maximization production inventory system under an imperfect environment and shortage

The theory of optimal control plays an important role in solving several real-life inventory problems. However, uncertainty is often seen as a major challenge in formulating the appropriate model corresponding to real-world inventory. To overcome this challenge, This study introduces a mathematical model for an imperfect production inventory system, departing from the conventional assumption of perfect quality in all produced items. The model accounts for defective items, classified as scrap, imperfect quality, and re-workable items, with the rework process initiated immediately following regular production. Shortages arising from factors such as increased demand or disruptions in the production process are also considered. The model aims to minimize the impact of shortages on manufacturing and inventory management, offering a comprehensive framework for optimal decision-making in the presence of faulty goods, shortages, price-dependent demand, and the necessity for reworking and scrapping defective items. Overall, this production inventory model provides a framework for making optimal decisions regarding production and inventory management in the presence of faulty goods, shortages, price-dependent demand, and the need for reworking and scrapping of faulty goods.