An improved way to calculate imperfect items during long-run production in an integrated inventory model with backorders

This paper investigates an improved way to calculate imperfect items in an integrated inventory model with distribution free approach for lead time demand to simultaneously optimize lot size, safety factor, number of shipments, and lead time. The backorder rate depends on the reduced lead time. Due to the quality of products, each item is inspected by the buyer and defective items are sent to the supplier during delivery of the next lot. Some investments are used to improve the quality of products and reduce the setup cost. The model is solved by using distribution free approach with known mean and standard deviation. An improved solution methodology is designed and a lemma is constructed to obtain global optimum solution of the integrated model. An improved algorithm is designed to approach the global optimal solution. Five numerical examples and two special cases are provided to validate the suggested model. A case study is conducted for which the global optimum solution is obtained using the proposed solution methodology. Simplified versions of the proposed model, illustrated through numerical examples as special cases, converged over the existing theories. Some managerial insights are provided to help managers implement the suggested model in real time situations.