Mathematical and analytical approach for the management of defective items in a multi-stage production system

Reworking defective items any in multi-stage production systems have become a common issue for production industries. Due to random defective rate of production, it may contain several imperfect items as wastes. The reduction of those wastes through reworking in each stage of a cycle or at the end of stage of a cycle by an identifying opportunity design technique. A design is developed for making such type multi-stage production system. For this, imperfect products may be reworked within each cycle while avoiding any shortages or reworked after end cycle with shortages in each cycle. The aim of the model is to diminish wastes by reworking strategy for minimizing the total cost. For this purpose, the proposed study develops a multi-stage production model for an optimum production batch policy with random production of imperfect products. Two-stage inspection is considered for detecting those faulty products and ensuring that there are no faulty items anymore. As it is a long-run production system, setup cost plays an important role, an attempt at setup cost reduction has been made to reduce it by a discrete investment for each stage along with a budget constraint. The Lagrangian method is employed to solve the model and the global minimum cost for optimal decision variables is obtained. Ten lemmas are established to obtain the global optimum solution. The model shows that a large investment to reduce the setup cost is better than a constant setup cost. It is found that the imperfect items should be reworked after the end cycle. There are eight numerical examples, eight special cases, a sensitivity analysis of the key parameters, and sixteen graphical representations to demonstrate the proposed model. (C) 2019 Elsevier Ltd. All rights reserved.