Controlling defective items in a complex multi-phase manufacturing system

In manufacturing systems, defective items are produced for machine drift and error. Usually, an imperfect production rate is random, and if the items are not reworked, these are considered trash and harm the environment. The proposed model aims to reduce waste by reworking defective products and maximizing profit. For profit maximization or overall cost minimization of the manufacturing system, setup cost has significant. A discrete investment for each phase is introduced with an inequality investment constraint for reducing the setup cost. Selling price-dependent demand is trained for more generalized applications for various industries. The proposed model is a multi-phase manufacturing system with optimum batch size, selling price, and investment with an irregular, imperfect production rate. Defects are detected at the first inspection, and the reworked items are checked if the reworked items are all non-defective in the second inspection. The model conducts a two-stage inspection. One is for detecting defective items, and another is for checking if all items are not defective after reworking. The model is solved with the Karush-Kuhn-Tucker (KKT) method, and the global maximum profit is obtained. The model shows that all investments should be assigned to maximize the profit and the optimal solution. Reducing setup cost with the investment is better than a constant setup cost.