تعیین اندازه دسته تولید برای یک فرایند تولید ناقص با وقفه های اصلاحی کیفیت و بهبود و کاهش در تنظیمات

The just-in-time (JIT) management philosophy advocates the elimination of waste or activities that add cost and not value to the product. Eliminating waste in the production process could be attained through smaller batch (lot) sizes and reduction of in-process inventory, where concepts such as setup reduction and increased quality are fundamental. In a JIT environment workers are authorized to stop production if a quality or a production problem arises, e.g., the production process going out-of-control. In such a case, the production process is interrupted for quality maintenance to bring the process in control again. This paper investigates the lot sizing problem for reduction in setups, with reworks, and interruptions to restore process quality. This paper assumes the rate of generating defects to benefit from any changes for eliminating the defects, and thus reduces with each quality restoration action. A mathematical model is developed and numerical examples are provided with results discussed.

The classical lot sizing problem, also known as the economic order/manufacture (EOQ/EMQ) quantity model, has captivated the attention of researchers since the earliest decades of the past century. The simplistic assumptions of the EOQ model that make its mathematics easy to use and understand is probably why the EOQ problem has been widely accepted and used by researchers and practitioners alike. Harris (1915) is assumed to be the first to provide a scientific approach to inventory management by developing the EOQ square root formulae. Since, there has been a plethora of work that extended upon the work of Harris with a reasonably good survey of these works provided in Silver, Pyke, and Peterson (1998). These extensions relaxed one or more assumptions inherent in the EOQ model to develop mathematical models that more closely conform to real-world inventory systems. Among these assumptions is that items produced and stocked are of perfect quality. This is an unrealistic assumption since the product quality is directly affected by the reliability of the production process (Cheng, 1991). Readers may refer to the work of Wright and Mehrez (1998) who provided a taxonomy of the research that includes the relationship between quality and inventory. In spite of the emphasis in quality control, some manufacturing processes today are imperfect and result in defective items that require reworking (Buzacott, 1999). Electronics manufacturing (e.g., printed circuit boards assembly, semiconductor wafers fabrication, etc.) is an example of such imperfect processes (Agnihothri and Kenett, 1995, Gopalan and Kannan, 1995 and Geren and Redford, 1999), and of batch (lot) manufacturing (Zargar, 1995). Besides electronics manufacturing, rework is an important issue in many process industries, such as the glass, steel, and pharmaceutical (Flapper, Fransoo, Broekmeulen, & Inderfurth, 2002), which are batch (lot) manufacturing too. Reworking defective items requires additional effort that adds cost and not value to the product, which the just-in-time (JIT) philosophy considers as waste to be eliminated (e.g., Waters, 2003). The JIT advocates that inventory is a blanket that covers problems is production and quality. Reducing inventory uncovers these problems making it easier for management to solve. These problems could be deracinated through the implementation of continuous improvement programs. Ideas such as reduction of lot sizes and setups, shorter lead-times, zero defects, preventive maintenance, and flexible workforce are among many concepts inherent in continuous improvement. These concepts enticed researchers in inventory management to put the classical EOQ/EMQ model in context with JIT to better understand the latter (e.g., Cao and Schniederjans, 2004, Chyr et al., 1990 and Jones, 1991). Porteus (1986) is among the earliest researchers who investigated the EOQ/EMQ model in conjunction with setup reduction and quality improvements. He showed that reducing the setup cost (time) and subsequently the lot size results in less reworks. The work of Porteus (1986) has been the cornerstone for many models. Some of these models are surveyed here. Chand (1989) studied the effect of learning in setups and process quality on the optimal lot sizes and the setup frequency. Khouja and Mehrez (1994) formulated an EMQ model with production rates as decision variables and assumed the percentage of good quality items in a lot decreases as the production rate increases resulting in more reworks. Urban (1998) investigated a production lot-size model that explicitly incorporates the effect of learning on the relationship (positive or negative) between the run length and defect rate. Khouja (1999) considered the economic lot scheduling problem with controllable production rates and imperfect quality. In a subsequent paper, Khouja (2000) extended the economic lot scheduling and delivery problem to the case of imperfect quality. Recently, Khouja (2003) formulated and solved two-stage supply chain inventory models in which the proportion of defective products increases with increased production lot sizes. Most recently, Freimer, Thomas, and Tyworth (2006) considered the EMQ model with defects produced according to some time-varying function. A primary assumption to the work of Porteus (1986) is that a process could go out-of-control with a given probability each time an item is produced, where the process produces defective items until the entire lot is produced. After which the process is corrected and resumed in control at the beginning of the subsequent lot. Conversely, in a JIT manufacturing environment line workers have the authority to stop the line if a quality or a production problem arises (Inman & Brandon, 1992). This perhaps what enticed Khouja (2005) to reformulate the model of Porteus (1986) in which adjustment to the process can be made within a production cycle to restore it to an “in-control” state. Prior to the work of Khouja, 2005 and Salameh and Jaber, 1997 investigated the EOQ/EMQ model with regular maintenance interruptions as preventive action. They have not attributed this interruption to restore the quality of the production process, but to avoid a major machine breakdown. Independently, the works of Khouja, 2005 and Salameh and Jaber, 1997 assumed inventory to behave in exactly the same manner, and that the setup cost function, S(n) = S + ns, consist of a fixed component, S, and a variable component, s, where n is the number of minor setups within a cycle. The cost of a major setup, S, involves all tasks required for preparing and adjusting the production process, while s is the cost of a minor setup which involves only the tasks required to restore the process to an “in-control” state. This paper integrates the works of Chand (1989) and that of Khouja (2005) by assuming that the major setup cost reduces with every setup (e.g., because of learning effects), and that the rate of generating defects reduces because the production process benefits from any changes for eliminating the defects (e.g., because of learning effects). It also corrects some of the assumptions in the works of Khouja, 2005 and Chand, 1989, which are addressed in further details later in this paper. The remainder of this paper is organized as follows. Section 2 addresses the above limitations by developing a new mathematical model. Section 3 is for numerical examples and discussion of results. Finally, this paper concludes in Section 4.

Porteus (1986) extended the economic lot sizing problem for a process producing defective items which require reworking. Porteus (1986) assumed that the process is not interrupted for corrective action before the entire lot size is produced. Khouja (2005) extended the work of Porteus (1986) by assuming single or multiple system stoppage to restore the state of the production process, an action that incurs additional costs (minor setups). However, the work of Khouja (2005) has limitations, which are the assumptions that the major setup component and the rate of generating defects are constant. This paper addressed these limitations by assuming that the setup cost reduces because of learning effects, and that the rate of generating defects reduces because the production process benefits from any changes for eliminating the defects, and thus reduces with every minor setup. This is done by integrating the works of Khouja, 2005 and Chand, 1989. Chand (1989) extended the work of Porteus (1986) by assuming learning to occur in setup and process quality. Unlike the work of Khouja, this paper assumes that holding cost accounts for the cost of producing a single unit (good or defective) and the additional cost of reworking a defective unit. This paper also corrects the work of Chand (1989) by assuming a realistic learning curve. The results indicate that learning in setups and improvement in quality reduces the annual costs significantly, with possible savings of up to 40%. The results also show that accounting for the cost of reworking defective units when calculating the unit holding cost may not be unrealistic given that some researchers suggested using higher holding cost than the cost of money. This paper assumed that all defective units could be reworked; i.e., no scrap. Relaxing this assumption may require accounting for recovery, recycling and disposal costs. This may be an immediate extension of the work presented herein. Another extension is to investigate the developed model in a manufacturer-buyer supply chain context.