مدل جبری کلی برای بهینه سازی تصمیم گیری های موجودی در زنجیره تأمین چند مرحله ای چند شرکتی متمرکز یا غیرمتمرکز

First of all, a number of integrated models with/without lot streaming under the integer multiplier coordination mechanism is generalized by allowing lot streaming and three types of inspection for some/all upstream firms. Secondly, the optimal solutions to the three- and four-stage models are individually derived, both using the perfect squares method, which is a simple algebraic approach so that ordinary readers unfamiliar with differential calculus can easily understand how to obtain the optimal solution procedures. Thirdly, optimal expressions for some well-known models are deduced. Fourthly, expressions for sharing the coordination benefits based on Goyal’s (1976) scheme are derived, and a further sharing scheme is introduced. Fifthly, two numerical examples for illustrative purposes are presented. Finally, some future research works involving extension or modification of the generalized model are suggested.

Supply chain management has enabled numerous firms to enjoy great advantages by integrating all activities associated with the flow of material, information and capital between suppliers of raw materials and the ultimate customers. The benefits of a property managed supply chain include reduced costs, faster product delivery, greater efficiency, and lower costs for both the business and its customers. These competitive advantages are achieved through improved supply chain relationships and tightened links between chain partners such as suppliers, manufacturing facilities, distribution centers, wholesalers, and end users (Berger et al., 2004). 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 paper, a generalized model for a three- or four-stage multi-firm production-inventory integrated system is solved using the perfect squares method adopted in Leung, 2008a, Leung, 2008b, Leung, 2009a, Leung, 2009b and Leung, 2010, which is also an algebraic approach; whereby optimal expressions of decision variables and the objective function are derived. Assume that there is an uninterrupted production run. In the case of lot streaming in stage i (=1, … , n − 1), shipments can be made from a production batch even before the whole batch is finished. According to Joglekar (1988, pp. 1397–8) the average inventory with lot streaming, for example, in stage 2 of a three-stage supply chain, is View the MathML sourceT3D2j2[φ2j+(K2-1)φ¯2j] units, which is the same as Eq. (7) of Ben-Daya and Al-Nassar (2008). However, some or all suppliers/manufacturers/assemblers cannot accommodate lot streaming because of regulations, material handling equipment, or production restrictions (Silver et al., 1998, p. 657). Without lot streaming, no shipments can be made from a production batch until the whole batch is finished. Sucky (2005) discussed the integrated single-vendor single-buyer system, with and without lot streaming, in detail. The opportunity of lot streaming affects supplier’s average inventory. According to Goysl’s (1988, p. 237), the average inventory without lot streaming, for example, in stage 2 of a 3-stage supply chain, is View the MathML sourceT3D2j2(φ2jK2+K2-1) units, which is the same as term 2 in Eq. (5) of Khouja (2003). In the inventory/production literature, all researchers have constructed their models under the assumption of either allowing lot streaming for all firms involving production (Khouja, 2003) or not (Ben-Daya and Al-Nassar, 2008), or both extremes (Sucky, 2005 and Leung, 2010). The main purpose of the paper is twofold: First, we build a generalized model incorporating a mixture of the two extremes, and solve it algebraically. As a result, we can deduce and solve such special models as Khouja, 2003, Cárdenas-Barrón, 2007, Ben-Daya and Al-Nassar, 2008, Seliaman and Ahmad, 2009 and Leung, 2009a. In addition, with appropriate assignments as in Section 5 of Leung (2010), we can also deduce and solve other special models: Yang and Wee, 2002, Wu and Ouyang, 2003, Wee and Chung, 2007 and Chung and Wee, 2007. Second, we derive expressions for sharing the coordination benefits based on Goyal’s (1976) scheme, and introduce a further sharing scheme. Some good review articles exist that provide an extensive overview of the topic under study and can be helpful as guidance through the literature. We mention surveys by Goyal and Gupta, 1989, Goyal and Deshmukh, 1992, Bhatnagar et al., 1993, Maloni and Benton, 1997, Sarmah et al., 2006 and Ben-Daya et al., 2008. The well-known models of Goyal, 1976, Banerjee, 1986, Lu, 1995 and Hill, 1997 are extended by Ben-Daya et al. (2008) as well. Other recently related articles include Chan and Kingsman, 2007, Chiou et al., 2007, Cha et al., 2008, Leng and Parlar, 2009a, Leng and Parlar, 2009b and Leng and Zhu, 2009

The main contribution of the paper to the literature is threefold: First, we establish the n-stage (n = 2, 3, 4, …) model, which is more pragmatic than that of Leung (2009a), by including Assumptions (6), (7) and (12). Secondly, we derive expressions for sharing the coordination benefits based on Goyal’s (1976) scheme, and on a further sharing scheme. Thirdly, we deduce and solve such special models as Khouja, 2003, Cárdenas-Barrón, 2007, Ben-Daya and Al-Nassar, 2008, Seliaman and Ahmad, 2009 and Leung, 2009a. The limitation of our model manifest in Examples 1 and 2 is that the number of suppliers in Stage 1 is arbitrarily assigned. Concerning the issue of “How many suppliers are best?”, we can refer to Berger et al., 2004, Ruiz-Torres and Mahmoodi, 2006 and Ruiz-Torres and Mahmoodi, 2007 to decide the optimal number of suppliers at the very beginning. Three ready extensions of our model that warrant future research endeavors in this field are: First, following the evolution of three- and four-stage multi-firm supply chains shown in Sections 3 and 4, we can readily formulate and algebraically analyze the integrated model of a five- or higher-stage multi-firm supply chain. In addition, a remark relating to determining optimal integral values of K’s is as follows: To be more specific, letting n = 5, we have at most 24 (=4 × 3 × 2 × 1) options to determine the optimal values of K1, K2, K3 and K4. However, Option (1), evaluating in the order of K1, K2, K3 and K4, might dominate other options when the holding costs decrease from upstream to downstream firms. Although this conjecture is confirmed by our two examples, a formal analysis is still necessary. Secondly, using complete and perfect squares, we can solve the integrated model of a n-stage multi-firm supply chain either for an equal-cycle-time, or an integer multiplier at each stage, with not only a linear (see Leung, 2010) but also a fixed shortage cost for either the complete, or a fixed ratio partial backordering allowed for some/all downstream firms (i.e. retailers), and with lot streaming allowed for some/all upstream firms (i.e. suppliers, manufacturers and assemblers). Thirdly, severity of green issues gives rise to consider integrated deteriorating production-inventory models incorporating the factor of environmental consciousness such as Yu et al., 2008, Chung and Wee, 2008 and Wee and Chung, 2009. Rework, a means to reduce waste disposal, is examined in Chiu et al., 2006 and Leung, 2009b who 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. Reuse, another means to reduce waste disposal, is investigated in El Saadany and Jaber, 2008 and Jaber and Rosen, 2008. Incorporating rework or reuse in our model will be a challenging piece of future research.