Optimising inventory policies for new and take-back used products with concept of complementary products under shortages

Any business needs complementary items since they increase sales, profits, and customer attraction. In this study, retailers sell the new products with complementary products, take back the used products, and retailers sell them again with complementary products. Each product's price-sensitive demands relate to one another and enhance its value. The objective of this research is to optimise the selling price, cycle time, inventory duration, and shortage period to maximise retail profits from new and take-back products, along with complementary items, under conditions of partial backlog shortages. A mathematical inventory model incorporating complementary items was formulated for new and take-back products. Using the classical optimisation method, we obtained the optimum value of the decision variables. The concavity of the objective function was verified using the Hessian matrix method and graphically. The study's numerical findings provide optimum profitability and suggest further research directions. Sensitivity analysis is performed to examine the proposed model's stability.