نگاه فنی بر بهینه سازی تصمیمات موجود در زنجیره تامین چند مشتری چند مرحله ای

We first generalize Khouja [Khouja, M., 2003. Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transportation Research Part E: Logistics and Transportation Review 39 (3), 193–208] integrated model considering the integer multipliers mechanism and next individually derive the optimal solution to the three- and four-stage model using the perfect squares method, which is a simple algebraic approach so that ordinary readers unfamiliar with differential calculus can understand the optimal solution procedure with ease. We subsequently deduce the optimal expressions for Khouja (2003) and Cárdenas-Barrón [Cárdenas-Barrón, L.E., 2007. Optimal inventory decisions in a multi-stage multi-customer supply chain: a note. Transportation Research Part E: Logistics and Transportation Review 43 (5), 647–654] model, and identify the associated errors in Khouja (2003). We present two numerical examples for illustrative purposes. We finally shed light on some future research by extending or modifying the generalized model.

Increasing attention has been given to the management of a multi-stage multi-firm (or multi-customer) supply chain in recent years. This is due to increasing competitiveness, short life cycles of modern electronic products and the quick global changes in today’s businesses. The integration of the supply chain provides a key to successful international business operations. This is because the integrated approach improves the global system performance and cost effectiveness. Besides integrating all members in a supply chain, to improve the traditional method of solving inventory problems is also necessary. Without using derivatives, Grubbström (1995) first derived the optimal expressions for the classical economic order quantity (EOQ) model using the unity decomposition method, which is an algebraic approach. Adopting this method, Grubbström and Erdem, 1999 and Cárdenas-Barrón, 2001, respectively, derived the optimal expressions for an EOQ and economic production quantity (EPQ) model with complete backorders. In this note, a generalized model for a three- or four-stage multi-firm production-inventory integrated system is solved using the revised version of the perfect squares method, which is also an algebraic approach; whereby optimal expressions of decision variables and the objective function are derived. In addition to the papers with regard to solving some inventory models without derivatives surveyed by and classified in Table 1 of Cárdenas-Barrón (2007), we review some recently relevant papers as follows: using the unity decomposition method, Chiu et al. (2006) derived the optimal expressions for an EPQ model with complete backorders, a random proportion of defectives, and an immediate imperfect rework process while Cárdenas-Barrón (2008) derived those for an EPQ model with no shortages, a fixed proportion of defectives, and an immediate or a N-cycle perfect rework process. Using the complete squares method and perfect squares method proposed by Chang et al., 2005, Wee and Chung, 2007 and Chung and Wee, 2007, respectively, derived the optimal expressions for a two- and three-stage single-firm supply chain inventory model with complete backorders, and lot streaming (which means that any shipments can be made from a production batch before the whole batch is finished). Leung (2008a) proposed revised versions of the complete and perfect squares methods to derive the optimal expressions for an EOQ model with partial backorders and Leung (2008b) also adopted them to derive those for an EOQ model when the quantity backordered and the quantity received are both uncertain. Teng (2008) proposed the arithmetic–geometric-mean-inequality method to derive the optimal expressions for the classical EOQ model. Wee et al. (2009) proposed a modified version of the cost-difference comparisons method originated from Minner (2007) to individually derive the optimal expressions for an EOQ and EPQ model with complete backorders.

We present a review of some work relating to algebraic solutions to inventory systems from 2006 to 2009 in Section 1. The main contribution of the note to the literature is twofold: first, we establish a more pragmatic model than that of Khouja (2003) by including the five realistic conditions listed in Section 2. Secondly, we provide a much more simplified optimal solution procedure which is developed using the perfect squares method so that users may easily understand and apply than that provided in Khouja (2003) or Cárdenas-Barrón (2007), and illustrate the procedure through the two numerical examples. The limitation of our model and its optimal solution procedure may be that both cannot be readily modified to adapt the use of the integer powers of two multipliers mechanism in order that the joint total relevant cost can be further reduced. Khouja (2003) shows numerically that saving may be significant when going from equal-cycle-time to integer multipliers mechanism, while the saving is less pronounced when going from integer multipliers to integer powers of two multipliers mechanism. We generally believe that it is not worthwhile to employ the integer powers of two multipliers mechanism with which inherently complicated computations are accompanied. Three ready extensions of our model that constitutes future research endeavors in this field are as follows: first, following the evolutions of a three- and four- stage multi-firm supply chain shown in Sections 3 and 5, we can readily formulate and algebraically analyze the integrated model of a five- or higher-stage multi-firm supply chain. Secondly, modifying designations (3), (4) and (5) and following the evolutions shown in Sections 3 and 5, we can solve the integrated model of a n -stage (n=2,3,…)(n=2,3,…) multi-firm supply chain for an equal-cycle-time or an integer multiplier at each stage with lot streaming, and deduce the optimal solution to Ben-Daya and Al-Nassar’s (2008) model. Thirdly, using not only the perfect squares method but also the complete squares method advocated in Leung, 2008a and Leung, 2008b, we can solve the integrated model of a n-stage multi-firm supply chain for an equal-cycle-time or an integer multiplier at each stage with complete backorders allowed for some/all downstream firms (or retailers), and without or with lot streaming, and individually deduce the optimal solution to Wee and Chung’s (2007) model (where the three inspection costs are assigned to be zero) and Wee and Chung’s (2007) model.