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In this paper, we consider a deficient production system with permissible shortages. The production system consists of a unique machine that manufactures a number of products that a part of them are imperfect in form of rework or scrap. These defective products are identified by 100% inspection during production, then, they are whether reworked or disposed of after normal production process. Like real-world production systems, there are diverse kinds of errors creating dissimilar breakdown severity and rework. Moreover, reworks have non-zero setup times that makes the problem closer to real-world instances where machines require some preparations before starting a new production cycle. Thus, we introduce an economic production quantity (EPQ) problem for an imperfect manufacturing system with non-zero setup times for rework items. The rework items are classified into several categories based on their type of failure and rework rate. The aim of this study is to obtain optimum production time and shortage in each period that minimizes total inventory system costs. Convexity of the objective function and exact solution procedure for the current nonlinear optimization problem are also proposed. Finally, a numerical example is proposed to assess efficiency and validation of proposed algorithm.

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17This study considers a multi-product multi-machine economic production quantity inventory problem in an 18 imperfect production system that produces two types of defective items: items that require rework and scrapped 19items. The shortage is allowed and fully backordered. The scrapped items are disposed with a disposal cost and the 20 rework is done at the end of the normal production period. Moreover, a potential set of available machines for 21 utilization is considered such that each has a specific production rate per item. Each machine has its own utilization 2 allocation of items on each machine are obtained thru a genetic algorithm. Then, using the convexity attribute of the 1 second level problem the optimum cycle length per machine is determined. The proposed hybrid genetic algorithm 2 outperformed conventional genetic algorithm and a GAMS solver, considering solution quality and solving time. 3Finally, a sensitivity analysis is also given.4