Mathematical modeling for solving manufacturing run time problem with defective rate and random machine breakdown Yuan-Shyi Peter Chiu a , Hong-Dar Lin a,⇑ , Huei-Hsin Chang b a Department of Industrial Engineering and Management, Chaoyang University of Technology, Jifong East Road, Wufong, Taichung 413, Taiwan b Department of Finance, Chaoyang University of Technology, Jifong East Road, Wufong, Taichung 413, Taiwan article info Article history: Received 8 September 2010 Received in revised form 21 December 2010 Accepted 22 December 2010 Available online 13 January 2011 Keywords: Manufacturing Replenishment run time Machine failure Defective Production and inventory control abstract This paper employs mathematical modeling for solving manufacturing run time problem with random defective rate and stochastic machine breakdown. In real life manufacturing systems, generation of non- conforming items and unexpected breakdown of production equipment are inevitable. For the purpose of addressing these practical issues, this paper studies a system that may produce defective items randomly and it is also subject to a random equipment failure. A no resumption inventory control policy is adopted when breakdown occurs. Under such a policy, the interrupted lot is aborted and malfunction machine is immediately under repair. A new lot will be started only when all on-hand inventory are depleted. Mod- eling and numerical analyses are used to establish the solution procedure for such a problem. As a result, the optimal manufacturing run time that minimizes the long-run average production–inventory cost is derived. A numerical example is provided to show how the solution procedure works as well as the usages of research results. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction This paper employs mathematical modeling for solving manu- facturing run time problem with random defective rate and sto- chastic machine breakdown. The economic order quantity (EOQ) model (Harris, 1913) was first introduced several decades ago to assist corporations in minimizing total inventory costs. It used mathematical techniques to balance the setup and stock holding costs and derived an optimal order size that minimizes the long- run average cost. In the manufacturing sector, when products are made in-house instead of being purchased from outside suppliers, the economic production quantity (EPQ) model is often used to deal with non-instantaneous inventory replenishment rate in or- der to obtain minimum production–inventory cost per unit time (Silver et al., 1998). Despite the simplicity of EOQ and EPQ models, they are still broadly applied today (Nahmias, 2009; Osteryoung et al., 1986). Many production–inventory models with more com- plicated and practical factors have since been extensively studied (see for example, Jaber, 2007; Wee and Shum, 1999). The classic EPQ model implicitly assumes that all items made are of perfect quality. However, in real world manufacturing sys- tems, due to process deterioration and/or other factors, generation of defective items is inevitable. Practically, these nonconforming items sometimes can be reworked hence overall production costs can be reduced. For example, the manufacturing processes in printed circuit board assembly, or in plastic injection molding, etc., sometimes employs rework as an acceptable process in terms of level of quality. Hence, many studies have been carried out to address the issues of imperfect production and reduction of its cor- responding quality costs (Cheng, 1991; Cheung and Hausman, 1997; Chiu and Chiu, 2006; Chiu et al., 2009a; Jaber, 2006; Jamal et al., 2004; Rahim and Ben-Daya, 2001; Rosenblatt and Lee, 1986; Taleizadeh et al., 2010; Wee, 1993). Articles related to the aforementioned issues are surveyed as follows. Rosenblatt and Lee (1986) proposed an EPQ model that deals with imperfect quality. They assumed that at some random point in time the process might shift from an in-control to an out-of-con- trol state, and a fixed percentage of defective items are produced. Approximate solutions for obtaining the optimal lot size were developed. Cheng (1991) examined an economic order quantity model with demand-dependent unit production cost and imperfect production processes. Wee (1993) developed and formulated an economic production policy for deteriorating items with partial back-ordering. Two numerical examples are used to illustrate the theory and computational results indicated that the proposed pol- icy leads to lower cost. Cheung and Hausman (1997) developed an analytical model of preventive maintenance (PM) and safety stock (SS) strategies in a production environment. They illustrated the trade-off between investing in the two options (PM and SS), and provided optimality conditions under either one or both strategies that minimize the associated costs. Both deterministic and expo- nential repair time distributions are analyzed in detail in their 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2010.12.015 ⇑ Corresponding author. Tel.: +886 4 2332 3000x4252; fax: +886 4 2374 2327. E-mail address: hdlin@cyut.edu.tw (H.-D. Lin). Computers & Industrial Engineering 60 (2011) 576–584 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie