Lot sizing for an imperfect production process with quality corrective interruptions and improvements, and reduction in setups
Introduction
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, 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, 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, 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, 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, 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.
Section snippets
Mathematical modeling
In this section, a mathematical model to determine the economic order/manufacture quantity (EOQ/EMQ) for a production process that generates defects requiring reworks is developed. The model developed in this section could be considered as an immediate extension to the work of Khouja (2005) by assuming reduction in setups and that the process benefits from any changes for eliminating the defects.
Numerical results
In this section, the model presented in Section 2.2.4 is solved for the same data set used by Khouja (2005) with the model of Khouja being the baseline model since it assumes no reduction in setups (b = 0) and improvement in quality (c = 0). All numerical examples were solved using Excel Solver enhanced by VBA codes.
Consider the numerical example of (Khouja (2005); page 409) in which r = 1000 units/year, p = 3200 units/year, S = $400/major setup, s = $80/minor setup, h = $4/unit/year, w = $6/unit, τ = 0.0015625,
Summary and conclusions
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
Acknowledgments
The author thanks the Natural Sciences and Engineering Research Council of Canada (NSERC) for supporting his research. He also thanks the anonymous referees for their helpful comments and suggestions.
References (47)
- et al.
The impact of defects on a process with rework
European Journal of Operational Research
(1995) - et al.
The economics of yield-driven processes
Journal of Operations Management
(1999) - et al.
A revised EMQ/JIT production-run model: an examination of inventory and production costs
International Journal of Production Economics
(2004) Lot sizes and setup frequency with learning and process quality
European Journal of Operational Research
(1989)- et al.
Comparison between Just-in-Time and EOQ system
Engineering Costs & Production Economics
(1990) - et al.
The value of setup cost reduction and process improvement for the economic production quantity model with defects
European Journal of Operational Research
(2006) - et al.
Expected number analysis of a two-server queuing network subject to inter-stage inspection and rework
Computers & Operations Research
(1995) - et al.
Cost and performance analysis of a robotic rework cell
International Journal of Production Economics
(1999) - et al.
Rework: a symptom of a dysfunctional supply-chain
European Journal of Purchasing & Supply Management
(1999) - et al.
Optimal lot sizing with regular maintenance interruptions
Applied Mathematical Modelling
(1997)