Optimizing an imperfect production system with varying production cost under selling price and advertising-driven demand

This study develops a production inventory model that accounts for imperfect production processes, acknowledging that a certain percentage of items produced are defective, reason being labor inefficiencies and machinery malfunctions. These defective items are identified and are segregated during the quality check and inspection phase and sold at a discounted price. The model incorporates economies of scale, whereby production costs decreases as productivity increases, and represents production costs as a diminishing function of productivity. Furthermore, the demand rate is modeled as dependent on advertising frequency and selling price, impacting significantly the product demand. The primary objective of this research is to determine the optimal advertising frequency, selling price, and the production cycle that maximize overall system profit. A numerical illustration is provided to validate the model, and a sensitivity analysis is conducted to assess the stability of the system. The findings reveal that increased demand reduces production time and raises pricing potential that leads to boosting of profits. Higher production rates improve efficiency and lower unit costs, while a higher defect rate and rising holding costs adversely impact profitability, which illustrates the importance of quality control and efficient inventory management. These insights highlight the model's practicality and innovation, offering valuable guidance for optimizing operations, managing costs, and achieving sustainable growth.